Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces

<!DOCTYPE html> <html lang="en" xmlns:og="http://ogp.me/ns#" xmlns:fb="https://www.facebook.com/2008/fbml"> <head> <meta charset="utf-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <meta content="mdpi" name="sso-service" /> <meta content="width=device-width, initial-scale=1.0" name="viewport" /> <title>Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces</title><link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/font-awesome.min.css?eb190a3a77e5e1ee?1732286508"> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/jquery.multiselect.css?f56c135cbf4d1483?1732286508"> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/chosen.min.css?d7ca5ca9441ef9e1?1732286508"> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/main2.css?69b39374e6b554b7?1732286508"> <link rel="mask-icon" href="https://pub.mdpi-res.com/img/mask-icon-128.svg?c1c7eca266cd7013?1732286508" color="#4f5671"> <link rel="apple-touch-icon" sizes="180x180" href="https://pub.mdpi-res.com/icon/apple-touch-icon-180x180.png?1732286508"> <link rel="apple-touch-icon" sizes="152x152" href="https://pub.mdpi-res.com/icon/apple-touch-icon-152x152.png?1732286508"> <link rel="apple-touch-icon" sizes="144x144" href="https://pub.mdpi-res.com/icon/apple-touch-icon-144x144.png?1732286508"> <link rel="apple-touch-icon" sizes="120x120" href="https://pub.mdpi-res.com/icon/apple-touch-icon-120x120.png?1732286508"> <link rel="apple-touch-icon" sizes="114x114" href="https://pub.mdpi-res.com/icon/apple-touch-icon-114x114.png?1732286508"> <link rel="apple-touch-icon" sizes="76x76" href="https://pub.mdpi-res.com/icon/apple-touch-icon-76x76.png?1732286508"> <link rel="apple-touch-icon" sizes="72x72" href="https://pub.mdpi-res.com/icon/apple-touch-icon-72x72.png?1732286508"> <link rel="apple-touch-icon" sizes="57x57" href="https://pub.mdpi-res.com/icon/apple-touch-icon-57x57.png?1732286508"> <link rel="apple-touch-icon" href="https://pub.mdpi-res.com/icon/apple-touch-icon-57x57.png?1732286508"> <link rel="apple-touch-icon-precomposed" href="https://pub.mdpi-res.com/icon/apple-touch-icon-57x57.png?1732286508"> <link rel="manifest" href="/manifest.json"> <meta name="theme-color" content="#ffffff"> <meta name="application-name" content=" "/> <link rel="apple-touch-startup-image" href="https://pub.mdpi-res.com/img/journals/universe-logo-sq.png?8600e93ff98dbf14"> <link rel="apple-touch-icon" href="https://pub.mdpi-res.com/img/journals/universe-logo-sq.png?8600e93ff98dbf14"> <meta name="msapplication-TileImage" content="https://pub.mdpi-res.com/img/journals/universe-logo-sq.png?8600e93ff98dbf14"> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/jquery-ui-1.10.4.custom.min.css?80647d88647bf347?1732286508"> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/magnific-popup.min.css?04d343e036f8eecd?1732286508"> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/xml2html/article-html.css?230b005b39af4260?1732286508"> <style> h2, #abstract .related_suggestion_title { } .batch_articles a { color: #000; } a, .batch_articles .authors a, a:focus, a:hover, a:active, .batch_articles a:focus, .batch_articles a:hover, li.side-menu-li a { } span.label a { color: #fff; } #main-content a.title-link:hover, #main-content a.title-link:focus, #main-content div.generic-item a.title-link:hover, #main-content div.generic-item a.title-link:focus { } #main-content #middle-column .generic-item.article-item a.title-link:hover, #main-content #middle-column .generic-item.article-item a.title-link:focus { } .art-authors a.toEncode { color: #333; font-weight: 700; } #main-content #middle-column ul li::before { } .accordion-navigation.active a.accordion__title, .accordion-navigation.active a.accordion__title::after { } .accordion-navigation li:hover::before, .accordion-navigation li:hover a, .accordion-navigation li:focus a { } .relative-size-container .relative-size-image .relative-size { } .middle-column__help__fixed a:hover i, } input[type="checkbox"]:checked:after { } input[type="checkbox"]:not(:disabled):hover:before { } #main-content .bolded-text { } #main-content .hypothesis-count-container { } #main-content .hypothesis-count-container:before { } .full-size-menu ul li.menu-item .dropdown-wrapper { } .full-size-menu ul li.menu-item > a.open::after { } #title-story .title-story-orbit .orbit-caption { #background: url('/img/design/000000_background.png') !important; background: url('/img/design/ffffff_background.png') !important; color: rgb(51, 51, 51) !important; } #main-content .content__container__orbit { background-color: #000 !important; } #main-content .content__container__journal { color: #fff; } .html-article-menu .row span { } .html-article-menu .row span.active { } .accordion-navigation__journal .side-menu-li.active::before, .accordion-navigation__journal .side-menu-li.active a { color: rgba(13,52,73,0.75) !important; font-weight: 700; } .accordion-navigation__journal .side-menu-li:hover::before , .accordion-navigation__journal .side-menu-li:hover a { color: rgba(13,52,73,0.75) !important; } .side-menu-ul li.active a, .side-menu-ul li.active, .side-menu-ul li.active::before { color: rgba(13,52,73,0.75) !important; } .side-menu-ul li.active a { } .result-selected, .active-result.highlighted, .active-result:hover, .result-selected, .active-result.highlighted, .active-result:focus { } .search-container.search-container__default-scheme { } nav.tab-bar .open-small-search.active:after { } .search-container.search-container__default-scheme .custom-accordion-for-small-screen-link::after { color: #fff; } @media only screen and (max-width: 50em) { #main-content .content__container.journal-info { color: #fff; } #main-content .content__container.journal-info a { color: #fff; } } .button.button--color { } .button.button--color:hover, .button.button--color:focus { } .button.button--color-journal { position: relative; background-color: rgba(13,52,73,0.75); border-color: #fff; color: #fff !important; } .button.button--color-journal:hover::before { content: ''; position: absolute; top: 0; left: 0; height: 100%; width: 100%; background-color: #ffffff; opacity: 0.2; } .button.button--color-journal:visited, .button.button--color-journal:hover, .button.button--color-journal:focus { background-color: rgba(13,52,73,0.75); border-color: #fff; color: #fff !important; } .button.button--color path { } .button.button--color:hover path { fill: #fff; } #main-content #search-refinements .ui-slider-horizontal .ui-slider-range { } .breadcrumb__element:last-of-type a { } #main-header { } #full-size-menu .top-bar, #full-size-menu li.menu-item span.user-email { } .top-bar-section li:not(.has-form) a:not(.button) { } #full-size-menu li.menu-item .dropdown-wrapper li a:hover { } #full-size-menu li.menu-item a:hover, #full-size-menu li.menu.item a:focus, nav.tab-bar a:hover { } #full-size-menu li.menu.item a:active, #full-size-menu li.menu.item a.active { } #full-size-menu li.menu-item a.open-mega-menu.active, #full-size-menu li.menu-item div.mega-menu, a.open-mega-menu.active { } #full-size-menu li.menu-item div.mega-menu li, #full-size-menu li.menu-item div.mega-menu a { border-color: #9a9a9a; } div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732286508"> <meta name="title" content="Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces"> <meta name="description" content="DUNE is a next-generation long-baseline neutrino oscillation experiment. It is expected to measure, with unprecedented precision, the atmospheric oscillation parameters, including the CP-violating phase δCP. Moreover, several studies have suggested that its unique features should allow DUNE to probe several new physics scenarios. In this work, we explore the performances of the DUNE far detector in constraining new physics if a high-energy neutrino flux is employed (HE-DUNE). We take into account three different scenarios: Lorentz Invariance Violation (LIV), Long-Range Forces (LRFs) and Large Extra Dimensions (LEDs). Our results show that HE-DUNE should be able to set bounds competitive to the current ones and, in particular, it can outperform the standard DUNE capabilities in constraining CPT-even LIV parameters and the compactification radius RED of the LED model." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces"> <meta name="dc.creator" content="Alessio Giarnetti"> <meta name="dc.creator" content="Simone Marciano"> <meta name="dc.creator" content="Davide Meloni"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Universe 2024, Vol. 10, Page 357"> <meta name="dc.date" content="2024-09-05"> <meta name ="dc.identifier" content="10.3390/universe10090357"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="DUNE is a next-generation long-baseline neutrino oscillation experiment. It is expected to measure, with unprecedented precision, the atmospheric oscillation parameters, including the CP-violating phase δCP. Moreover, several studies have suggested that its unique features should allow DUNE to probe several new physics scenarios. In this work, we explore the performances of the DUNE far detector in constraining new physics if a high-energy neutrino flux is employed (HE-DUNE). We take into account three different scenarios: Lorentz Invariance Violation (LIV), Long-Range Forces (LRFs) and Large Extra Dimensions (LEDs). Our results show that HE-DUNE should be able to set bounds competitive to the current ones and, in particular, it can outperform the standard DUNE capabilities in constraining CPT-even LIV parameters and the compactification radius RED of the LED model." > <meta name="dc.subject" content="neutrino mixing" > <meta name="dc.subject" content="DUNE" > <meta name="dc.subject" content="BSM" > <meta name ="prism.issn" content="2218-1997"> <meta name ="prism.publicationName" content="Universe"> <meta name ="prism.publicationDate" content="2024-09-05"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="9"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="357" > <meta name="citation_issn" content="2218-1997"> <meta name="citation_journal_title" content="Universe"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces"> <meta name="citation_publication_date" content="2024/9"> <meta name="citation_online_date" content="2024/09/05"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="9"> <meta name="citation_firstpage" content="357"> <meta name="citation_author" content="Giarnetti, Alessio"> <meta name="citation_author" content="Marciano, Simone"> <meta name="citation_author" content="Meloni, Davide"> <meta name="citation_doi" content="10.3390/universe10090357"> <meta name="citation_id" content="mdpi-universe10090357"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2218-1997/10/9/357"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2218-1997/10/9/357/pdf?version=1725541553"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2218-1997/10/9/357/pdf?version=1725541553"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2218-1997/10/9/357/pdf?version=1725541553"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2218-1997/10/9/357/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2218-1997/10/9/357/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2218-1997/10/9/357/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2218-1997/10/9/357/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2218-1997/10/9/357/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2218-1997/10/9/357/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/universe-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2218-1997/10/9/357" /> <meta property="og:title" content="Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces" /> <meta property="og:description" content="DUNE is a next-generation long-baseline neutrino oscillation experiment. It is expected to measure, with unprecedented precision, the atmospheric oscillation parameters, including the CP-violating phase δCP. Moreover, several studies have suggested that its unique features should allow DUNE to probe several new physics scenarios. In this work, we explore the performances of the DUNE far detector in constraining new physics if a high-energy neutrino flux is employed (HE-DUNE). We take into account three different scenarios: Lorentz Invariance Violation (LIV), Long-Range Forces (LRFs) and Large Extra Dimensions (LEDs). Our results show that HE-DUNE should be able to set bounds competitive to the current ones and, in particular, it can outperform the standard DUNE capabilities in constraining CPT-even LIV parameters and the compactification radius RED of the LED model." /> <meta property="og:image" content="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001-550.jpg?1725541642" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-WPK7SW5'); </script> <script type="text/plain" data-cookieconsent="statistics"> _linkedin_partner_id = "2846186"; window._linkedin_data_partner_ids = window._linkedin_data_partner_ids || []; window._linkedin_data_partner_ids.push(_linkedin_partner_id); </script><script type="text/javascript"> (function(){var s = document.getElementsByTagName("script")[0]; var b = document.createElement("script"); b.type = "text/javascript";b.async = true; b.src = "https://snap.licdn.com/li.lms-analytics/insight.min.js"; s.parentNode.insertBefore(b, s);})(); </script> <script type="text/plain" data-cookieconsent="statistics" data-cfasync="false" src="//script.crazyegg.com/pages/scripts/0116/4951.js" async="async" ></script> </head> <body> <div class="direction direction_right" id="small_right" style="border-right-width: 0px; padding:0;"> <i class="fa fa-caret-right fa-2x"></i> </div> <div class="big_direction direction_right" id="big_right" style="border-right-width: 0px;"> <div style="text-align: right;"> Next Article in Journal<br> <div><a href="/2218-1997/10/9/358">Three-Dimensional Quantum Black Holes: A Primer</a></div> </div> </div> <div class="direction" id="small_left" style="border-left-width: 0px"> <i class="fa fa-caret-left fa-2x"></i> </div> <div class="big_direction" id="big_left" style="border-left-width: 0px;"> <div> Previous Article in Journal<br> <div><a href="/2218-1997/10/9/356">Non-Canonical Dark Energy Parameter Evolution in a Canonical Quintessence Cosmology</a></div> </div> </div> <div style="clear: both;"></div> <div id="menuModal" class="reveal-modal reveal-modal-new reveal-modal-menu" aria-hidden="true" data-reveal role="dialog"> <div class="menu-container"> <div class="UI_NavMenu"> <div class="content__container " > <div class="custom-accordion-for-small-screen-link " > <h2>Journals</h2> </div> <div class="target-item custom-accordion-for-small-screen-content show-for-medium-up"> <div class="menu-container__links"> <div style="width: 100%; float: left;"> <a href="/about/journals">Active Journals</a> <a href="/about/journalfinder">Find a Journal</a> <a href="/about/journals/proposal">Journal Proposal</a> <a href="/about/proceedings">Proceedings Series</a> </div> </div> </div> </div> <a href="/topics"> <h2>Topics</h2> </a> <div class="content__container " > <div class="custom-accordion-for-small-screen-link " > <h2>Information</h2> </div> <div class="target-item custom-accordion-for-small-screen-content show-for-medium-up"> <div class="menu-container__links"> <div style="width: 100%; max-width: 200px; float: left;"> <a href="/authors">For Authors</a> <a href="/reviewers">For Reviewers</a> <a href="/editors">For Editors</a> <a href="/librarians">For Librarians</a> <a href="/publishing_services">For Publishers</a> <a href="/societies">For Societies</a> <a href="/conference_organizers">For Conference Organizers</a> </div> <div style="width: 100%; max-width: 250px; float: left;"> <a href="/openaccess">Open Access Policy</a> <a href="/ioap">Institutional Open Access Program</a> <a href="/special_issues_guidelines">Special Issues Guidelines</a> <a href="/editorial_process">Editorial Process</a> <a href="/ethics">Research and Publication Ethics</a> <a href="/apc">Article Processing Charges</a> <a href="/awards">Awards</a> <a href="/testimonials">Testimonials</a> </div> </div> </div> </div> <a href="/authors/english"> <h2>Editing Services</h2> </a> <div class="content__container " > <div class="custom-accordion-for-small-screen-link " > <h2>Initiatives</h2> </div> <div class="target-item custom-accordion-for-small-screen-content show-for-medium-up"> <div class="menu-container__links"> <div style="width: 100%; float: left;"> <a href="https://sciforum.net" target="_blank" rel="noopener noreferrer">Sciforum</a> <a href="https://www.mdpi.com/books" target="_blank" rel="noopener noreferrer">MDPI Books</a> <a href="https://www.preprints.org" target="_blank" rel="noopener noreferrer">Preprints.org</a> <a href="https://www.scilit.net" target="_blank" rel="noopener noreferrer">Scilit</a> <a href="https://sciprofiles.com" target="_blank" rel="noopener noreferrer">SciProfiles</a> <a href="https://encyclopedia.pub" target="_blank" rel="noopener noreferrer">Encyclopedia</a> <a href="https://jams.pub" target="_blank" rel="noopener noreferrer">JAMS</a> <a href="/about/proceedings">Proceedings Series</a> </div> </div> </div> </div> <div class="content__container " > <div class="custom-accordion-for-small-screen-link " > <h2>About</h2> </div> <div class="target-item custom-accordion-for-small-screen-content show-for-medium-up"> <div class="menu-container__links"> <div style="width: 100%; float: left;"> <a href="/about">Overview</a> <a href="/about/contact">Contact</a> <a href="https://careers.mdpi.com" target="_blank" rel="noopener noreferrer">Careers</a> <a href="/about/announcements">News</a> <a href="/about/press">Press</a> <a href="http://blog.mdpi.com/" target="_blank" rel="noopener noreferrer">Blog</a> </div> </div> </div> </div> </div> <div class="menu-container__buttons"> <a class="button UA_SignInUpButton" href="/user/login">Sign In / Sign Up</a> </div> </div> </div> <div id="captchaModal" class="reveal-modal reveal-modal-new reveal-modal-new--small" data-reveal aria-label="Captcha" aria-hidden="true" role="dialog"></div> <div id="actionDisabledModal" class="reveal-modal" data-reveal aria-labelledby="actionDisableModalTitle" aria-hidden="true" role="dialog" style="width: 300px;"> <h2 id="actionDisableModalTitle">Notice</h2> <form action="/email/captcha" method="post" id="emailCaptchaForm"> <div class="row"> <div id="js-action-disabled-modal-text" class="small-12 columns"> </div> <div id="js-action-disabled-modal-submit" class="small-12 columns" style="margin-top: 10px; display: none;"> You can make submissions to other journals <a href="https://susy.mdpi.com/user/manuscripts/upload">here</a>. </div> </div> </form> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> <div id="rssNotificationModal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="rssNotificationModalTitle" aria-hidden="true" role="dialog"> <div class="row"> <div class="small-12 columns"> <h2 id="rssNotificationModalTitle">Notice</h2> <p> You are accessing a machine-readable page. In order to be human-readable, please install an RSS reader. </p> </div> </div> <div class="row"> <div class="small-12 columns"> <a class="button button--color js-rss-notification-confirm">Continue</a> <a class="button button--grey" onclick="$(this).closest('.reveal-modal').find('.close-reveal-modal').click(); return false;">Cancel</a> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> <div id="drop-article-label-openaccess" class="f-dropdown medium" data-dropdown-content aria-hidden="true" tabindex="-1"> <p> All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to <a href="https://www.mdpi.com/openaccess">https://www.mdpi.com/openaccess</a>. </p> </div> <div id="drop-article-label-feature" class="f-dropdown medium" data-dropdown-content aria-hidden="true" tabindex="-1"> <p> Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications. </p> <p> Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewers. </p> </div> <div id="drop-article-label-choice" class="f-dropdown medium" data-dropdown-content aria-hidden="true" tabindex="-1"> <p> Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal. <div style="margin-top: -10px;"> <div id="drop-article-label-choice-journal-link" style="display: none; margin-top: -10px; padding-top: 10px;"> </div> </div> </p> </div> <div id="drop-article-label-resubmission" class="f-dropdown medium" data-dropdown-content aria-hidden="true" tabindex="-1"> <p> Original Submission Date Received: <span id="drop-article-label-resubmission-date"></span>. </p> </div> <div id="container"> <noscript> <div id="no-javascript"> You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled. </div> </noscript> <div class="fixed"> <nav class="tab-bar show-for-medium-down"> <div class="row full-width collapse"> <div class="medium-3 small-4 columns"> <a href="/"> <img class="full-size-menu__mdpi-logo" src="https://pub.mdpi-res.com/img/design/mdpi-pub-logo-black-small1.svg?da3a8dcae975a41c?1732286508" style="width: 64px;" title="MDPI Open Access Journals"> </a> </div> <div class="medium-3 small-4 columns right-aligned"> <div class="show-for-medium-down"> <a href="#" style="display: none;"> <i class="material-icons" onclick="$('#menuModal').foundation('reveal', 'close'); return false;">clear</i> </a> <a class="js-toggle-desktop-layout-link" title="Toggle desktop layout" style="display: none;" href="/toggle_desktop_layout_cookie"> <i class="material-icons">zoom_out_map</i> </a> <a href="#" class="js-open-small-search open-small-search"> <i class="material-icons show-for-small only">search</i> </a> <a title="MDPI main page" class="js-open-menu" data-reveal-id="menuModal" href="#"> <i class="material-icons">menu</i> </a> </div> </div> </div> </nav> </div> <section class="main-section"> <header> <div class="full-size-menu show-for-large-up"> <div class="row full-width"> <div class="large-1 columns"> <a href="/"> <img class="full-size-menu__mdpi-logo" src="https://pub.mdpi-res.com/img/design/mdpi-pub-logo-black-small1.svg?da3a8dcae975a41c?1732286508" title="MDPI Open Access Journals"> </a> </div> <div class="large-8 columns text-right UI_NavMenu"> <ul> <li class="menu-item"> <a href="/about/journals" data-dropdown="journals-dropdown" aria-controls="journals-dropdown" aria-expanded="false" data-options="is_hover: true; hover_timeout: 200">Journals</a> <ul id="journals-dropdown" class="f-dropdown dropdown-wrapper dropdown-wrapper__small" data-dropdown-content aria-hidden="true" tabindex="-1"> <li> <div class="row"> <div class="small-12 columns"> <ul> <li> <a href="/about/journals"> Active Journals </a> </li> <li> <a href="/about/journalfinder"> Find a Journal </a> </li> <li> <a href="/about/journals/proposal"> Journal Proposal </a> </li> <li> <a href="/about/proceedings"> Proceedings Series </a> </li> </ul> </div> </div> </li> </ul> </li> <li class="menu-item"> <a href="/topics">Topics</a> </li> <li class="menu-item"> <a href="/authors" data-dropdown="information-dropdown" aria-controls="information-dropdown" aria-expanded="false" data-options="is_hover:true; hover_timeout:200">Information</a> <ul id="information-dropdown" class="f-dropdown dropdown-wrapper" data-dropdown-content aria-hidden="true" tabindex="-1"> <li> <div class="row"> <div class="small-5 columns right-border"> <ul> <li> <a href="/authors">For Authors</a> </li> <li> <a href="/reviewers">For Reviewers</a> </li> <li> <a href="/editors">For Editors</a> </li> <li> <a href="/librarians">For Librarians</a> </li> <li> <a href="/publishing_services">For Publishers</a> </li> <li> <a href="/societies">For Societies</a> </li> <li> <a href="/conference_organizers">For Conference Organizers</a> </li> </ul> </div> <div class="small-7 columns"> <ul> <li> <a href="/openaccess">Open Access Policy</a> </li> <li> <a href="/ioap">Institutional Open Access Program</a> </li> <li> <a href="/special_issues_guidelines">Special Issues Guidelines</a> </li> <li> <a href="/editorial_process">Editorial Process</a> </li> <li> <a href="/ethics">Research and Publication Ethics</a> </li> <li> <a href="/apc">Article Processing Charges</a> </li> <li> <a href="/awards">Awards</a> </li> <li> <a href="/testimonials">Testimonials</a> </li> </ul> </div> </div> </li> </ul> </li> <li class="menu-item"> <a href="/authors/english">Editing Services</a> </li> <li class="menu-item"> <a href="/about/initiatives" data-dropdown="initiatives-dropdown" aria-controls="initiatives-dropdown" aria-expanded="false" data-options="is_hover: true; hover_timeout: 200">Initiatives</a> <ul id="initiatives-dropdown" class="f-dropdown dropdown-wrapper dropdown-wrapper__small" data-dropdown-content aria-hidden="true" tabindex="-1"> <li> <div class="row"> <div class="small-12 columns"> <ul> <li> <a href="https://sciforum.net" target="_blank" rel="noopener noreferrer"> Sciforum </a> </li> <li> <a href="https://www.mdpi.com/books" target="_blank" rel="noopener noreferrer"> MDPI Books </a> </li> <li> <a href="https://www.preprints.org" target="_blank" rel="noopener noreferrer"> Preprints.org </a> </li> <li> <a href="https://www.scilit.net" target="_blank" rel="noopener noreferrer"> Scilit </a> </li> <li> <a href="https://sciprofiles.com" target="_blank" rel="noopener noreferrer"> SciProfiles </a> </li> <li> <a href="https://encyclopedia.pub" target="_blank" rel="noopener noreferrer"> Encyclopedia </a> </li> <li> <a href="https://jams.pub" target="_blank" rel="noopener noreferrer"> JAMS </a> </li> <li> <a href="/about/proceedings"> Proceedings Series </a> </li> </ul> </div> </div> </li> </ul> </li> <li class="menu-item"> <a href="/about" data-dropdown="about-dropdown" aria-controls="about-dropdown" aria-expanded="false" data-options="is_hover: true; hover_timeout: 200">About</a> <ul id="about-dropdown" class="f-dropdown dropdown-wrapper dropdown-wrapper__small" data-dropdown-content aria-hidden="true" tabindex="-1"> <li> <div class="row"> <div class="small-12 columns"> <ul> <li> <a href="/about"> Overview </a> </li> <li> <a href="/about/contact"> Contact </a> </li> <li> <a href="https://careers.mdpi.com" target="_blank" rel="noopener noreferrer"> Careers </a> </li> <li> <a href="/about/announcements"> News </a> </li> <li> <a href="/about/press"> Press </a> </li> <li> <a href="http://blog.mdpi.com/" target="_blank" rel="noopener noreferrer"> Blog </a> </li> </ul> </div> </div> </li> </ul> </li> </ul> </div> <div class="large-3 columns text-right full-size-menu__buttons"> <div> <a class="button button--default-inversed UA_SignInUpButton" href="/user/login">Sign In / Sign Up</a> <a class="button button--default js-journal-active-only-link js-journal-active-only-submit-link UC_NavSubmitButton" href=" https://susy.mdpi.com/user/manuscripts/upload?journal=universe " data-disabledmessage="new submissions are not possible.">Submit</a> </div> </div> </div> </div> <div class="header-divider"> </div> <div class="search-container hide-for-small-down row search-container__homepage-scheme"> <form id="basic_search" style="background-color: inherit !important;" class="large-12 medium-12 columns " action="/search" method="get"> <div class="row search-container__main-elements"> <div class="large-2 medium-2 small-12 columns text-right1 small-only-text-left"> <div class="show-for-medium-up"> <div class="search-input-label"> </div> </div> <span class="search-container__title">Search<span class="hide-for-medium"> for Articles</span><span class="hide-for-small">:</span></span> </div> <div class="custom-accordion-for-small-screen-content"> <div class="large-2 medium-2 small-6 columns "> <div class=""> <div class="search-input-label">Title / Keyword</div> </div> <input type="text" placeholder="Title / Keyword" id="q" tabindex="1" name="q" value="" /> </div> <div class="large-2 medium-2 small-6 columns "> <div class=""> <div class="search-input-label">Author / Affiliation / Email</div> </div> <input type="text" id="authors" placeholder="Author / Affiliation / Email" tabindex="2" name="authors" value="" /> </div> <div class="large-2 medium-2 small-6 columns "> <div class=""> <div class="search-input-label">Journal</div> </div> <select id="journal" tabindex="3" name="journal" class="chosen-select"> <option value="">All Journals</option> <option value="acoustics" > Acoustics </option> <option value="amh" > Acta Microbiologica Hellenica (AMH) </option> <option value="actuators" > Actuators </option> <option value="admsci" > Administrative Sciences </option> <option value="adolescents" > Adolescents </option> <option value="arm" > Advances in Respiratory Medicine (ARM) </option> <option value="aerobiology" > Aerobiology </option> <option value="aerospace" > Aerospace </option> <option value="agriculture" > Agriculture </option> <option value="agriengineering" > AgriEngineering </option> <option value="agrochemicals" > Agrochemicals </option> <option value="agronomy" > Agronomy </option> <option value="ai" > AI </option> <option value="air" > Air </option> <option value="algorithms" > Algorithms </option> <option value="allergies" > Allergies </option> <option value="alloys" > Alloys </option> <option value="analytica" > Analytica </option> <option value="analytics" > Analytics </option> <option value="anatomia" > Anatomia </option> <option value="anesthres" > Anesthesia Research </option> <option value="animals" > Animals </option> <option value="antibiotics" > Antibiotics </option> <option value="antibodies" > Antibodies </option> <option value="antioxidants" > Antioxidants </option> <option value="applbiosci" > Applied Biosciences </option> <option value="applmech" > Applied Mechanics </option> <option value="applmicrobiol" > Applied Microbiology </option> <option value="applnano" > Applied Nano </option> <option value="applsci" > Applied Sciences </option> <option value="asi" > Applied System Innovation (ASI) </option> <option value="appliedchem" > AppliedChem </option> <option value="appliedmath" > AppliedMath </option> <option value="aquacj" > Aquaculture Journal </option> <option value="architecture" > Architecture </option> <option value="arthropoda" > Arthropoda </option> <option value="arts" > Arts </option> <option value="astronomy" > Astronomy </option> <option value="atmosphere" > Atmosphere </option> <option value="atoms" > Atoms </option> <option value="audiolres" > Audiology Research </option> <option value="automation" > Automation </option> <option value="axioms" > Axioms </option> <option value="bacteria" > Bacteria </option> <option value="batteries" > Batteries </option> <option value="behavsci" > Behavioral Sciences </option> <option value="beverages" > Beverages </option> <option value="BDCC" > Big Data and Cognitive Computing (BDCC) </option> <option value="biochem" > BioChem </option> <option value="bioengineering" > Bioengineering </option> <option value="biologics" > Biologics </option> <option value="biology" > Biology </option> <option value="blsf" > Biology and Life Sciences Forum </option> <option value="biomass" > Biomass </option> <option value="biomechanics" > Biomechanics </option> <option value="biomed" > BioMed </option> <option value="biomedicines" > Biomedicines </option> <option value="biomedinformatics" > BioMedInformatics </option> <option value="biomimetics" > Biomimetics </option> <option value="biomolecules" > Biomolecules </option> <option value="biophysica" > Biophysica </option> <option value="biosensors" > Biosensors </option> <option value="biotech" > BioTech </option> <option value="birds" > Birds </option> <option value="blockchains" > Blockchains </option> <option value="brainsci" > Brain Sciences </option> <option value="buildings" > Buildings </option> <option value="businesses" > Businesses </option> <option value="carbon" > C </option> <option value="cancers" > Cancers </option> <option value="cardiogenetics" > Cardiogenetics </option> <option value="catalysts" > Catalysts </option> <option value="cells" > Cells </option> <option value="ceramics" > Ceramics </option> <option value="challenges" > Challenges </option> <option value="ChemEngineering" > ChemEngineering </option> <option value="chemistry" > Chemistry </option> <option value="chemproc" > Chemistry Proceedings </option> <option value="chemosensors" > Chemosensors </option> <option value="children" > Children </option> <option value="chips" > Chips </option> <option value="civileng" > CivilEng </option> <option value="cleantechnol" > Clean Technologies (Clean Technol.) </option> <option value="climate" > Climate </option> <option value="ctn" > Clinical and Translational Neuroscience (CTN) </option> <option value="clinbioenerg" > Clinical Bioenergetics </option> <option value="clinpract" > Clinics and Practice </option> <option value="clockssleep" > Clocks & Sleep </option> <option value="coasts" > Coasts </option> <option value="coatings" > Coatings </option> <option value="colloids" > Colloids and Interfaces </option> <option value="colorants" > Colorants </option> <option value="commodities" > Commodities </option> <option value="complications" > Complications </option> <option value="compounds" > Compounds </option> <option value="computation" > Computation </option> <option value="csmf" > Computer Sciences & Mathematics Forum </option> <option value="computers" > Computers </option> <option value="condensedmatter" > Condensed Matter </option> <option value="conservation" > Conservation </option> <option value="constrmater" > Construction Materials </option> <option value="cmd" > Corrosion and Materials Degradation (CMD) </option> <option value="cosmetics" > Cosmetics </option> <option value="covid" > COVID </option> <option value="crops" > Crops </option> <option value="cryo" > Cryo </option> <option value="cryptography" > Cryptography </option> <option value="crystals" > Crystals </option> <option value="cimb" > Current Issues in Molecular Biology (CIMB) </option> <option value="curroncol" > Current Oncology </option> <option value="dairy" > Dairy </option> <option value="data" > Data </option> <option value="dentistry" > Dentistry Journal </option> <option value="dermato" > Dermato </option> <option value="dermatopathology" > Dermatopathology </option> <option value="designs" > Designs </option> <option value="diabetology" > Diabetology </option> <option value="diagnostics" > Diagnostics </option> <option value="dietetics" > Dietetics </option> <option value="digital" > Digital </option> <option value="disabilities" > Disabilities </option> <option value="diseases" > Diseases </option> <option value="diversity" > Diversity </option> <option value="dna" > DNA </option> <option value="drones" > Drones </option> <option value="ddc" > Drugs and Drug Candidates (DDC) </option> <option value="dynamics" > Dynamics </option> <option value="earth" > Earth </option> <option value="ecologies" > Ecologies </option> <option value="econometrics" > Econometrics </option> <option value="economies" > Economies </option> <option value="education" > Education Sciences </option> <option value="electricity" > Electricity </option> <option value="electrochem" > Electrochem </option> <option value="electronicmat" > Electronic Materials </option> <option value="electronics" > Electronics </option> <option value="ecm" > Emergency Care and Medicine </option> <option value="encyclopedia" > Encyclopedia </option> <option value="endocrines" > Endocrines </option> <option value="energies" > Energies </option> <option value="esa" > Energy Storage and Applications (ESA) </option> <option value="eng" > Eng </option> <option value="engproc" > Engineering Proceedings </option> <option value="entropy" > Entropy </option> <option value="environsciproc" > Environmental Sciences Proceedings </option> <option value="environments" > Environments </option> <option value="epidemiologia" > Epidemiologia </option> <option value="epigenomes" > Epigenomes </option> <option value="ebj" > European Burn Journal (EBJ) </option> <option value="ejihpe" > European Journal of Investigation in Health, Psychology and Education (EJIHPE) </option> <option value="fermentation" > Fermentation </option> <option value="fibers" > Fibers </option> <option value="fintech" > FinTech </option> <option value="fire" > Fire </option> <option value="fishes" > Fishes </option> <option value="fluids" > Fluids </option> <option value="foods" > Foods </option> <option value="forecasting" > Forecasting </option> <option value="forensicsci" > Forensic Sciences </option> <option value="forests" > Forests </option> <option value="fossstud" > Fossil Studies </option> <option value="foundations" > Foundations </option> <option value="fractalfract" > Fractal and Fractional (Fractal Fract) </option> <option value="fuels" > Fuels </option> <option value="future" > Future </option> <option value="futureinternet" > Future Internet </option> <option value="futurepharmacol" > Future Pharmacology </option> <option value="futuretransp" > Future Transportation </option> <option value="galaxies" > Galaxies </option> <option value="games" > Games </option> <option value="gases" > Gases </option> <option value="gastroent" > Gastroenterology Insights </option> <option value="gastrointestdisord" > Gastrointestinal Disorders </option> <option value="gastronomy" > Gastronomy </option> <option value="gels" > Gels </option> <option value="genealogy" > Genealogy </option> <option value="genes" > Genes </option> <option value="geographies" > Geographies </option> <option value="geohazards" > GeoHazards </option> <option value="geomatics" > Geomatics </option> <option value="geometry" > Geometry </option> <option value="geosciences" > Geosciences </option> <option value="geotechnics" > Geotechnics </option> <option value="geriatrics" > Geriatrics </option> <option value="glacies" > Glacies </option> <option value="gucdd" > Gout, Urate, and Crystal Deposition Disease (GUCDD) </option> <option value="grasses" > Grasses </option> <option value="hardware" > Hardware </option> <option value="healthcare" > Healthcare </option> <option value="hearts" > Hearts </option> <option value="hemato" > Hemato </option> <option value="hematolrep" > Hematology Reports </option> <option value="heritage" > Heritage </option> <option value="histories" > Histories </option> <option value="horticulturae" > Horticulturae </option> <option value="hospitals" > Hospitals </option> <option value="humanities" > Humanities </option> <option value="humans" > Humans </option> <option value="hydrobiology" > Hydrobiology </option> <option value="hydrogen" > Hydrogen </option> <option value="hydrology" > Hydrology </option> <option value="hygiene" > Hygiene </option> <option value="immuno" > Immuno </option> <option value="idr" > Infectious Disease Reports </option> <option value="informatics" > Informatics </option> <option value="information" > Information </option> <option value="infrastructures" > Infrastructures </option> <option value="inorganics" > Inorganics </option> <option value="insects" > Insects </option> <option value="instruments" > Instruments </option> <option value="iic" > Intelligent Infrastructure and Construction </option> <option value="ijerph" > International Journal of Environmental Research and Public Health (IJERPH) </option> <option value="ijfs" > International Journal of Financial Studies (IJFS) </option> <option value="ijms" > International Journal of Molecular Sciences (IJMS) </option> <option value="IJNS" > International Journal of Neonatal Screening (IJNS) </option> <option value="ijpb" > International Journal of Plant Biology (IJPB) </option> <option value="ijt" > International Journal of Topology </option> <option value="ijtm" > International Journal of Translational Medicine (IJTM) </option> <option value="ijtpp" > International Journal of Turbomachinery, Propulsion and Power (IJTPP) </option> <option value="ime" > International Medical Education (IME) </option> <option value="inventions" > Inventions </option> <option value="IoT" > IoT </option> <option value="ijgi" > ISPRS International Journal of Geo-Information (IJGI) </option> <option value="J" > J </option> <option value="jal" > Journal of Ageing and Longevity (JAL) </option> <option value="jcdd" > Journal of Cardiovascular Development and Disease (JCDD) </option> <option value="jcto" > Journal of Clinical & Translational Ophthalmology (JCTO) </option> <option value="jcm" > Journal of Clinical Medicine (JCM) </option> <option value="jcs" > Journal of Composites Science (J. Compos. Sci.) </option> <option value="jcp" > Journal of Cybersecurity and Privacy (JCP) </option> <option value="jdad" > Journal of Dementia and Alzheimer's Disease (JDAD) </option> <option value="jdb" > Journal of Developmental Biology (JDB) </option> <option value="jeta" > Journal of Experimental and Theoretical Analyses (JETA) </option> <option value="jfb" > Journal of Functional Biomaterials (JFB) </option> <option value="jfmk" > Journal of Functional Morphology and Kinesiology (JFMK) </option> <option value="jof" > Journal of Fungi (JoF) </option> <option value="jimaging" > Journal of Imaging (J. Imaging) </option> <option value="jintelligence" > Journal of Intelligence (J. Intell.) </option> <option value="jlpea" > Journal of Low Power Electronics and Applications (JLPEA) </option> <option value="jmmp" > Journal of Manufacturing and Materials Processing (JMMP) </option> <option value="jmse" > Journal of Marine Science and Engineering (JMSE) </option> <option value="jmahp" > Journal of Market Access & Health Policy (JMAHP) </option> <option value="jmp" > Journal of Molecular Pathology (JMP) </option> <option value="jnt" > Journal of Nanotheranostics (JNT) </option> <option value="jne" > Journal of Nuclear Engineering (JNE) </option> <option value="ohbm" > Journal of Otorhinolaryngology, Hearing and Balance Medicine (JOHBM) </option> <option value="jop" > Journal of Parks </option> <option value="jpm" > Journal of Personalized Medicine (JPM) </option> <option value="jpbi" > Journal of Pharmaceutical and BioTech Industry (JPBI) </option> <option value="jor" > Journal of Respiration (JoR) </option> <option value="jrfm" > Journal of Risk and Financial Management (JRFM) </option> <option value="jsan" > Journal of Sensor and Actuator Networks (JSAN) </option> <option value="joma" > Journal of the Oman Medical Association (JOMA) </option> <option value="jtaer" > Journal of Theoretical and Applied Electronic Commerce Research (JTAER) </option> <option value="jvd" > Journal of Vascular Diseases (JVD) </option> <option value="jox" > Journal of Xenobiotics (JoX) </option> <option value="jzbg" > Journal of Zoological and Botanical Gardens (JZBG) </option> <option value="journalmedia" > Journalism and Media </option> <option value="kidneydial" > Kidney and Dialysis </option> <option value="kinasesphosphatases" > Kinases and Phosphatases </option> <option value="knowledge" > Knowledge </option> <option value="labmed" > LabMed </option> <option value="laboratories" > Laboratories </option> <option value="land" > Land </option> <option value="languages" > Languages </option> <option value="laws" > Laws </option> <option value="life" > Life </option> <option value="limnolrev" > Limnological Review </option> <option value="lipidology" > Lipidology </option> <option value="liquids" > Liquids </option> <option value="literature" > Literature </option> <option value="livers" > Livers </option> <option value="logics" > Logics </option> <option value="logistics" > Logistics </option> <option value="lubricants" > Lubricants </option> <option value="lymphatics" > Lymphatics </option> <option value="make" > Machine Learning and Knowledge Extraction (MAKE) </option> <option value="machines" > Machines </option> <option value="macromol" > Macromol </option> <option value="magnetism" > Magnetism </option> <option value="magnetochemistry" > Magnetochemistry </option> <option value="marinedrugs" > Marine Drugs </option> <option value="materials" > Materials </option> <option value="materproc" > Materials Proceedings </option> <option value="mca" > Mathematical and Computational Applications (MCA) </option> <option value="mathematics" > Mathematics </option> <option value="medsci" > Medical Sciences </option> <option value="msf" > Medical Sciences Forum </option> <option value="medicina" > Medicina </option> <option value="medicines" > Medicines </option> <option value="membranes" > Membranes </option> <option value="merits" > Merits </option> <option value="metabolites" > Metabolites </option> <option value="metals" > Metals </option> <option value="meteorology" > Meteorology </option> <option value="methane" > Methane </option> <option value="mps" > Methods and Protocols (MPs) </option> <option value="metrics" > Metrics </option> <option value="metrology" > Metrology </option> <option value="micro" > Micro </option> <option value="microbiolres" > Microbiology Research </option> <option value="micromachines" > Micromachines </option> <option value="microorganisms" > Microorganisms </option> <option value="microplastics" > Microplastics </option> <option value="minerals" > Minerals </option> <option value="mining" > Mining </option> <option value="modelling" > Modelling </option> <option value="mmphys" > Modern Mathematical Physics </option> <option value="molbank" > Molbank </option> <option value="molecules" > Molecules </option> <option value="mti" > Multimodal Technologies and Interaction (MTI) </option> <option value="muscles" > Muscles </option> <option value="nanoenergyadv" > Nanoenergy Advances </option> <option value="nanomanufacturing" > Nanomanufacturing </option> <option value="nanomaterials" > Nanomaterials </option> <option value="ndt" > NDT </option> <option value="network" > Network </option> <option value="neuroglia" > Neuroglia </option> <option value="neurolint" > Neurology International </option> <option value="neurosci" > NeuroSci </option> <option value="nitrogen" > Nitrogen </option> <option value="ncrna" > Non-Coding RNA (ncRNA) </option> <option value="nursrep" > Nursing Reports </option> <option value="nutraceuticals" > Nutraceuticals </option> <option value="nutrients" > Nutrients </option> <option value="obesities" > Obesities </option> <option value="oceans" > Oceans </option> <option value="onco" > Onco </option> <option value="optics" > Optics </option> <option value="oral" > Oral </option> <option value="organics" > Organics </option> <option value="organoids" > Organoids </option> <option value="osteology" > Osteology </option> <option value="oxygen" > Oxygen </option> <option value="parasitologia" > Parasitologia </option> <option value="particles" > Particles </option> <option value="pathogens" > Pathogens </option> <option value="pathophysiology" > Pathophysiology </option> <option value="pediatrrep" > Pediatric Reports </option> <option value="pets" > Pets </option> <option value="pharmaceuticals" > Pharmaceuticals </option> <option value="pharmaceutics" > Pharmaceutics </option> <option value="pharmacoepidemiology" > Pharmacoepidemiology </option> <option value="pharmacy" > Pharmacy </option> <option value="philosophies" > Philosophies </option> <option value="photochem" > Photochem </option> <option value="photonics" > Photonics </option> <option value="phycology" > Phycology </option> <option value="physchem" > Physchem </option> <option value="psf" > Physical Sciences Forum </option> <option value="physics" > Physics </option> <option value="physiologia" > Physiologia </option> <option value="plants" > Plants </option> <option value="plasma" > Plasma </option> <option value="platforms" > Platforms </option> <option value="pollutants" > Pollutants </option> <option value="polymers" > Polymers </option> <option value="polysaccharides" > Polysaccharides </option> <option value="populations" > Populations </option> <option value="poultry" > Poultry </option> <option value="powders" > Powders </option> <option value="proceedings" > Proceedings </option> <option value="processes" > Processes </option> <option value="prosthesis" > Prosthesis </option> <option value="proteomes" > Proteomes </option> <option value="psychiatryint" > Psychiatry International </option> <option value="psychoactives" > Psychoactives </option> <option value="psycholint" > Psychology International </option> <option value="publications" > Publications </option> <option value="qubs" > Quantum Beam Science (QuBS) </option> <option value="quantumrep" > Quantum Reports </option> <option value="quaternary" > Quaternary </option> <option value="radiation" > Radiation </option> <option value="reactions" > Reactions </option> <option value="realestate" > Real Estate </option> <option value="receptors" > Receptors </option> <option value="recycling" > Recycling </option> <option value="rsee" > Regional Science and Environmental Economics (RSEE) </option> <option value="religions" > Religions </option> <option value="remotesensing" > Remote Sensing </option> <option value="reports" > Reports </option> <option value="reprodmed" > Reproductive Medicine (Reprod. Med.) </option> <option value="resources" > Resources </option> <option value="rheumato" > Rheumato </option> <option value="risks" > Risks </option> <option value="robotics" > Robotics </option> <option value="ruminants" > Ruminants </option> <option value="safety" > Safety </option> <option value="sci" > Sci </option> <option value="scipharm" > Scientia Pharmaceutica (Sci. Pharm.) </option> <option value="sclerosis" > Sclerosis </option> <option value="seeds" > Seeds </option> <option value="sensors" > Sensors </option> <option value="separations" > Separations </option> <option value="sexes" > Sexes </option> <option value="signals" > Signals </option> <option value="sinusitis" > Sinusitis </option> <option value="smartcities" > Smart Cities </option> <option value="socsci" > Social Sciences </option> <option value="siuj" > Société Internationale d’Urologie Journal (SIUJ) </option> <option value="societies" > Societies </option> <option value="software" > Software </option> <option value="soilsystems" > Soil Systems </option> <option value="solar" > Solar </option> <option value="solids" > Solids </option> <option value="spectroscj" > Spectroscopy Journal </option> <option value="sports" > Sports </option> <option value="standards" > Standards </option> <option value="stats" > Stats </option> <option value="stresses" > Stresses </option> <option value="surfaces" > Surfaces </option> <option value="surgeries" > Surgeries </option> <option value="std" > Surgical Techniques Development </option> <option value="sustainability" > Sustainability </option> <option value="suschem" > Sustainable Chemistry </option> <option value="symmetry" > Symmetry </option> <option value="synbio" > SynBio </option> <option value="systems" > Systems </option> <option value="targets" > Targets </option> <option value="taxonomy" > Taxonomy </option> <option value="technologies" > Technologies </option> <option value="telecom" > Telecom </option> <option value="textiles" > Textiles </option> <option value="thalassrep" > Thalassemia Reports </option> <option value="therapeutics" > Therapeutics </option> <option value="thermo" > Thermo </option> <option value="timespace" > Time and Space </option> <option value="tomography" > Tomography </option> <option value="tourismhosp" > Tourism and Hospitality </option> <option value="toxics" > Toxics </option> <option value="toxins" > Toxins </option> <option value="transplantology" > Transplantology </option> <option value="traumacare" > Trauma Care </option> <option value="higheredu" > Trends in Higher Education </option> <option value="tropicalmed" > Tropical Medicine and Infectious Disease (TropicalMed) </option> <option value="universe" selected='selected'> Universe </option> <option value="urbansci" > Urban Science </option> <option value="uro" > Uro </option> <option value="vaccines" > Vaccines </option> <option value="vehicles" > Vehicles </option> <option value="venereology" > Venereology </option> <option value="vetsci" > Veterinary Sciences </option> <option value="vibration" > Vibration </option> <option value="virtualworlds" > Virtual Worlds </option> <option value="viruses" > Viruses </option> <option value="vision" > Vision </option> <option value="waste" > Waste </option> <option value="water" > Water </option> <option value="wild" > Wild </option> <option value="wind" > Wind </option> <option value="women" > Women </option> <option value="world" > World </option> <option value="wevj" > World Electric Vehicle Journal (WEVJ) </option> <option value="youth" > Youth </option> <option value="zoonoticdis" > Zoonotic Diseases </option> </select> </div> <div class="large-2 medium-2 small-6 columns "> <div class=""> <div class="search-input-label">Article Type</div> </div> <select id="article_type" tabindex="4" name="article_type" class="chosen-select"> <option value="">All Article Types</option> <option value="research-article">Article</option> <option value="review-article">Review</option> <option value="rapid-communication">Communication</option> <option value="editorial">Editorial</option> <option value="abstract">Abstract</option> <option value="book-review">Book Review</option> <option value="brief-communication">Brief Communication</option> <option value="brief-report">Brief Report</option> <option value="case-report">Case Report</option> <option value="clinicopathological-challenge">Clinicopathological Challenge</option> <option value="article-commentary">Comment</option> <option value="commentary">Commentary</option> <option value="concept-paper">Concept Paper</option> <option value="conference-report">Conference Report</option> <option value="correction">Correction</option> <option value="creative">Creative</option> <option value="data-descriptor">Data Descriptor</option> <option value="discussion">Discussion</option> <option value="Entry">Entry</option> <option value="essay">Essay</option> <option value="expression-of-concern">Expression of Concern</option> <option value="extended-abstract">Extended Abstract</option> <option value="field-guide">Field Guide</option> <option value="guidelines">Guidelines</option> <option value="hypothesis">Hypothesis</option> <option value="interesting-image">Interesting Images</option> <option value="letter">Letter</option> <option value="books-received">New Book Received</option> <option value="obituary">Obituary</option> <option value="opinion">Opinion</option> <option value="perspective">Perspective</option> <option value="proceedings">Proceeding Paper</option> <option value="project-report">Project Report</option> <option value="protocol">Protocol</option> <option value="registered-report">Registered Report</option> <option value="reply">Reply</option> <option value="retraction">Retraction</option> <option value="note">Short Note</option> <option value="study-protocol">Study Protocol</option> <option value="systematic_review">Systematic Review</option> <option value="technical-note">Technical Note</option> <option value="tutorial">Tutorial</option> <option value="viewpoint">Viewpoint</option> </select> </div> <div class="large-1 medium-1 small-6 end columns small-push-6 medium-reset-order large-reset-order js-search-collapsed-button-container"> <div class="search-input-label"> </div> <input type="submit" id="search" value="Search" class="button button--dark button--full-width searchButton1 US_SearchButton" tabindex="12"> </div> <div class="large-1 medium-1 small-6 end columns large-text-left small-only-text-center small-pull-6 medium-reset-order large-reset-order js-search-collapsed-link-container"> <div class="search-input-label"> </div> <a class="main-search-clear search-container__link" href="#" onclick="openAdvanced(''); return false;">Advanced<span class="show-for-small-only"> Search</span></a> </div> </div> </div> <div class="search-container__advanced" style="margin-top: 0; padding-top: 0px; background-color: inherit; color: inherit;"> <div class="row"> <div class="large-2 medium-2 columns show-for-medium-up"> </div> <div class="large-2 medium-2 small-6 columns "> <div class=""> <div class="search-input-label">Section</div> </div> <select id="section" tabindex="5" name="section" class="chosen-select"> <option value=""></option> </select> </div> <div class="large-2 medium-2 small-6 columns "> <div class=""> <div class="search-input-label">Special Issue</div> </div> <select id="special_issue" tabindex="6" name="special_issue" class="chosen-select"> <option value=""></option> </select> </div> <div class="large-1 medium-1 small-6 end columns "> <div class="search-input-label">Volume</div> <input type="text" id="volume" tabindex="7" name="volume" placeholder="..." value="10" /> </div> <div class="large-1 medium-1 small-6 end columns "> <div class="search-input-label">Issue</div> <input type="text" id="issue" tabindex="8" name="issue" placeholder="..." value="9" /> </div> <div class="large-1 medium-1 small-6 end columns "> <div class="search-input-label">Number</div> <input type="text" id="number" tabindex="9" name="number" placeholder="..." value="" /> </div> <div class="large-1 medium-1 small-6 end columns "> <div class="search-input-label">Page</div> <input type="text" id="page" tabindex="10" name="page" placeholder="..." value="" /> </div> <div class="large-1 medium-1 small-6 columns small-push-6 medium-reset order large-reset-order medium-reset-order js-search-expanded-button-container"></div> <div class="large-1 medium-1 small-6 columns large-text-left small-only-text-center small-pull-6 medium-reset-order large-reset-order js-search-expanded-link-container"></div> </div> </div> </form> <form id="advanced-search" class="large-12 medium-12 columns"> <div class="search-container__advanced"> <div id="advanced-search-template" class="row advanced-search-row"> <div class="large-2 medium-2 small-12 columns show-for-medium-up"> </div> <div class="large-2 medium-2 small-3 columns connector-div"> <div class="search-input-label"><span class="show-for-medium-up">Logical Operator</span><span class="show-for-small">Operator</span></div> <select class="connector"> <option value="and">AND</option> <option value="or">OR</option> </select> </div> <div class="large-3 medium-3 small-6 columns search-text-div"> <div class="search-input-label">Search Text</div> <input type="text" class="search-text" placeholder="Search text"> </div> <div class="large-2 medium-2 small-6 large-offset-0 medium-offset-0 small-offset-3 columns search-field-div"> <div class="search-input-label">Search Type</div> <select class="search-field"> <option value="all">All fields</option> <option value="title">Title</option> <option value="abstract">Abstract</option> <option value="keywords">Keywords</option> <option value="authors">Authors</option> <option value="affiliations">Affiliations</option> <option value="doi">Doi</option> <option value="full_text">Full Text</option> <option value="references">References</option> </select> </div> <div class="large-1 medium-1 small-3 columns"> <div class="search-input-label"> </div> <div class="search-action-div"> <div class="search-plus"> <i class="material-icons">add_circle_outline</i> </div> </div> <div class="search-action-div"> <div class="search-minus"> <i class="material-icons">remove_circle_outline</i> </div> </div> </div> <div class="large-1 medium-1 small-6 large-offset-0 medium-offset-0 small-offset-3 end columns"> <div class="search-input-label"> </div> <input class="advanced-search-button button button--dark search-submit" type="submit" value="Search"> </div> <div class="large-1 medium-1 small-6 end columns show-for-medium-up"></div> </div> </div> </form> </div> <div class="header-divider"> </div> <div class="breadcrumb row full-row"> <div class="breadcrumb__element"> <a href="/about/journals">Journals</a> </div> <div class="breadcrumb__element"> <a href="/journal/universe">Universe</a> </div> <div class="breadcrumb__element"> <a href="/2218-1997/10">Volume 10</a> </div> <div class="breadcrumb__element"> <a href="/2218-1997/10/9">Issue 9</a> </div> <div class="breadcrumb__element"> <a href="#">10.3390/universe10090357</a> </div> </div> </header> <div id="main-content" class=""> <div class="row full-width row-fixed-left-column"> <div id="left-column" class="content__column large-3 medium-3 small-12 columns"> <div class="content__container"> <a href="/journal/universe"> <img src="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" alt="universe-logo" title="Universe" style="max-height: 60px; margin: 0 0 0 0;"> </a> <div class="generic-item no-border"> <a class="button button--color button--full-width js-journal-active-only-link js-journal-active-only-submit-link UC_ArticleSubmitButton" href="https://susy.mdpi.com/user/manuscripts/upload?form%5Bjournal_id%5D%3D133" data-disabledmessage="creating new submissions is not possible."> Submit to this Journal </a> <a class="button button--color button--full-width js-journal-active-only-link UC_ArticleReviewButton" href="https://susy.mdpi.com/volunteer/journals/review" data-disabledmessage="volunteering as journal reviewer is not possible."> Review for this Journal </a> <a class="button button--color-inversed button--color-journal button--full-width js-journal-active-only-link UC_ArticleEditIssueButton" href="/journalproposal/sendproposalspecialissue/universe" data-path="/2218-1997/10/9/357" data-disabledmessage="proposing new special issue is not possible."> Propose a Special Issue </a> </div> <div class="generic-item link-article-menu show-for-small"> <a href="#" class="link-article-menu show-for-small"> <span class="closed">►</span> <span class="open" style="display: none;">▼</span> Article Menu </a> </div> <div class="hide-small-down-initially UI_ArticleMenu"> <div class="generic-item"> <h2>Article Menu</h2> </div> <ul class="accordion accordion__menu" data-accordion data-options="multi_expand:true;toggleable: true"> <li class="accordion-navigation"> <a href="#academic_editors" class="accordion__title">Academic Editor</a> <div id="academic_editors" class="content active"> <div class="academic-editor-container " title="Department of Astronomy and Theoretical Physics, Lund University, 221 00 Lund, Sweden"> <div class="sciprofiles-link" style="display: inline-block"><a class="sciprofiles-link__link" href="https://sciprofiles.com/profile/150564?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" target="_blank" rel="noopener noreferrer"><img class="sciprofiles-link__image" src="/profiles/150564/thumb/Roman_Pasechnik.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Roman Pasechnik</span></a></div> </div> </div> </li> <li class="accordion-direct-link"> <a href="/2218-1997/10/9/357/scifeed_display" data-reveal-id="scifeed-modal" data-reveal-ajax="true">Subscribe SciFeed</a> </li> <li class="accordion-direct-link js-article-similarity-container" style="display: none"> <a href="#" class="js-similarity-related-articles">Recommended Articles</a> </li> <li class="accordion-navigation"> <a href="#related" class="accordion__title">Related Info Link</a> <div id="related" class="content UI_ArticleMenu_RelatedLinks"> <ul> <li class="li-link"> <a href="https://scholar.google.com/scholar?q=Exploring%20New%20Physics%20with%20Deep%20Underground%20Neutrino%20Experiment%20High-Energy%20Flux%3A%20The%20Case%20of%20Lorentz%20Invariance%20Violation%2C%20Large%20Extra%20Dimensions%20and%20Long-Range%20Forces" target="_blank" rel="noopener noreferrer">Google Scholar</a> </li> </ul> </div> </li> <li class="accordion-navigation"> <a href="#authors" class="accordion__title">More by Authors Links</a> <div id="authors" class="content UI_ArticleMenu_AuthorsLinks"> <ul class="side-menu-ul"> <li> <a class="expand" onclick='$(this).closest("li").next("div").toggle(); return false;'>on DOAJ</a> </li> <div id="AuthorDOAJExpand" style="display:none;"> <ul class="submenu"> <li class="li-link"> <a href='http://doaj.org/search/articles?source=%7B%22query%22%3A%7B%22query_string%22%3A%7B%22query%22%3A%22%5C%22Alessio%20Giarnetti%5C%22%22%2C%22default_operator%22%3A%22AND%22%2C%22default_field%22%3A%22bibjson.author.name%22%7D%7D%7D' target="_blank" rel="noopener noreferrer">Giarnetti, A.</a> <li> </li> <li class="li-link"> <a href='http://doaj.org/search/articles?source=%7B%22query%22%3A%7B%22query_string%22%3A%7B%22query%22%3A%22%5C%22Simone%20Marciano%5C%22%22%2C%22default_operator%22%3A%22AND%22%2C%22default_field%22%3A%22bibjson.author.name%22%7D%7D%7D' target="_blank" rel="noopener noreferrer">Marciano, S.</a> <li> </li> <li class="li-link"> <a href='http://doaj.org/search/articles?source=%7B%22query%22%3A%7B%22query_string%22%3A%7B%22query%22%3A%22%5C%22Davide%20Meloni%5C%22%22%2C%22default_operator%22%3A%22AND%22%2C%22default_field%22%3A%22bibjson.author.name%22%7D%7D%7D' target="_blank" rel="noopener noreferrer">Meloni, D.</a> <li> </li> </ul> </div> <li> <a class="expand" onclick='$(this).closest("li").next("div").toggle(); return false;'>on Google Scholar</a> </li> <div id="AuthorGoogleExpand" style="display:none;"> <ul class="submenu"> <li class="li-link"> <a href="https://scholar.google.com/scholar?q=Alessio%20Giarnetti" target="_blank" rel="noopener noreferrer">Giarnetti, A.</a> <li> </li> <li class="li-link"> <a href="https://scholar.google.com/scholar?q=Simone%20Marciano" target="_blank" rel="noopener noreferrer">Marciano, S.</a> <li> </li> <li class="li-link"> <a href="https://scholar.google.com/scholar?q=Davide%20Meloni" target="_blank" rel="noopener noreferrer">Meloni, D.</a> <li> </li> </ul> </div> <li> <a class="expand" onclick='$(this).closest("li").next("div").toggle(); return false;'>on PubMed</a> </li> <div id="AuthorPubMedExpand" style="display:none;"> <ul class="submenu"> <li class="li-link"> <a href="http://www.pubmed.gov/?cmd=Search&term=Alessio%20Giarnetti" target="_blank" rel="noopener noreferrer">Giarnetti, A.</a> <li> </li> <li class="li-link"> <a href="http://www.pubmed.gov/?cmd=Search&term=Simone%20Marciano" target="_blank" rel="noopener noreferrer">Marciano, S.</a> <li> </li> <li class="li-link"> <a href="http://www.pubmed.gov/?cmd=Search&term=Davide%20Meloni" target="_blank" rel="noopener noreferrer">Meloni, D.</a> <li> </li> </ul> </div> </ul> </div> </li> </ul> <span style="display:none" id="scifeed_hidden_flag"></span> <span style="display:none" id="scifeed_subscribe_url">/ajax/scifeed/subscribe</span> </div> </div> <div class="content__container responsive-moving-container large medium active hidden" data-id="article-counters"> <div id="counts-wrapper" class="row generic-item no-border" data-equalizer> <div id="js-counts-wrapper__views" class="small-12 hide columns count-div-container"> <a href="#metrics" > <div class="count-div" data-equalizer-watch> <span class="name">Article Views</span> <span class="count view-number"></span> </div> </a> </div> <div id="js-counts-wrapper__citations" class="small-12 columns hide count-div-container"> <a href="#metrics" > <div class="count-div" data-equalizer-watch> <span class="name">Citations</span> <span class="count citations-number Var_ArticleMaxCitations">-</span> </div> </a> </div> </div> </div> <div class="content__container"> <div class="hide-small-down-initially"> <ul class="accordion accordion__menu" data-accordion data-options="multi_expand:true;toggleable: true"> <li class="accordion-navigation"> <a href="#table_of_contents" class="accordion__title">Table of Contents</a> <div id="table_of_contents" class="content active"> <div class="menu-caption" id="html-quick-links-title"></div> </div> </li> </ul> </div> </div>  <div id="pbgrd-sky"></div> <script src="https://cdn.pbgrd.com/core-mdpi.js"></script> <style>.content__container { min-width: 300px; }</style>  </div> <div id="middle-column" class="content__column large-9 medium-9 small-12 columns end middle-bordered"> <div class="middle-column__help"> <div class="middle-column__help__fixed show-for-medium-up"> <span id="js-altmetrics-donut" href="#" target="_blank" rel="noopener noreferrer" style="display: none;"> <span data-badge-type='donut' class='altmetric-embed' data-doi='10.3390/universe10090357'></span> <span>Altmetric</span> </span> <a href="#" class="UA_ShareButton" data-reveal-id="main-share-modal" title="Share"> <i class="material-icons">share</i> <span>Share</span> </a> <a href="#" data-reveal-id="main-help-modal" title="Help"> <i class="material-icons">announcement</i> <span>Help</span> </a> <a href="javascript:void(0);" data-reveal-id="cite-modal" data-counterslink = "https://www.mdpi.com/2218-1997/10/9/357/cite" > <i class="material-icons">format_quote</i> <span>Cite</span> </a> <a href="https://sciprofiles.com/discussion-groups/public/10.3390/universe10090357?utm_source=mpdi.com&utm_medium=publication&utm_campaign=discuss_in_sciprofiles" target="_blank" rel="noopener noreferrer" title="Discuss in Sciprofiles"> <i class="material-icons">question_answer</i> <span>Discuss in SciProfiles</span> </a> <a href="#" class="" data-hypothesis-trigger-endorses-tab title="Endorse"> <i data-hypothesis-endorse-trigger class="material-icons" >thumb_up</i> <div data-hypothesis-endorsement-count data-hypothesis-trigger-endorses-tab class="hypothesis-count-container"> ... </div> <span>Endorse</span> </a> <a href="#" data-hypothesis-trigger class="js-hypothesis-open UI_ArticleAnnotationsButton" title="Comment"> <i class="material-icons">textsms</i> <div data-hypothesis-annotation-count class="hypothesis-count-container"> ... </div> <span>Comment</span> </a> </div> <div id="main-help-modal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="modalTitle" aria-hidden="true" role="dialog"> <div class="row"> <div class="small-12 columns"> <h2 style="margin: 0;">Need Help?</h2> </div> <div class="small-6 columns"> <h3>Support</h3> <p> Find support for a specific problem in the support section of our website. </p> <a target="_blank" href="/about/contactform" class="button button--color button--full-width"> Get Support </a> </div> <div class="small-6 columns"> <h3>Feedback</h3> <p> Please let us know what you think of our products and services. </p> <a target="_blank" href="/feedback/send" class="button button--color button--full-width"> Give Feedback </a> </div> <div class="small-6 columns end"> <h3>Information</h3> <p> Visit our dedicated information section to learn more about MDPI. </p> <a target="_blank" href="/authors" class="button button--color button--full-width"> Get Information </a> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> </div> <div class="middle-column__main "> <div class="page-highlight"> <style type="text/css"> img.review-status { width: 30px; } </style> <div id="jmolModal" class="reveal-modal" data-reveal aria-labelledby="Captcha" aria-hidden="true" role="dialog"> <h2>JSmol Viewer</h2> <div class="row"> <div class="small-12 columns text-center"> <iframe style="width: 520px; height: 520px;" frameborder="0" id="jsmol-content"></iframe> <div class="content"></div> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> <div itemscope itemtype="http://schema.org/ScholarlyArticle" id="abstract" class="abstract_div"> <div class="js-check-update-container"></div> <div class="html-content__container content__container content__container__combined-for-large__first" style="overflow: auto; position: inherit;"> <div class='html-profile-nav'> <div class='top-bar'> <div class='nav-sidebar-btn show-for-large-up' data-status='opened' > <i class='material-icons'>first_page</i> </div> <a id="js-button-download" class="button button--color-inversed" style="display: none;" href="/2218-1997/10/9/357/pdf?version=1725541553" data-name="Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces" data-journal="universe"> <i class="material-icons custom-download"></i> Download PDF </a> <div class='nav-btn'> <i class='material-icons'>settings</i> </div> <a href="/2218-1997/10/9/357/reprints" id="js-button-reprints" class="button button--color-inversed"> Order Article Reprints </a> </div> <div class='html-article-menu'> <div class='html-first-step row'> <div class='html-font-family large-6 medium-6 small-12 columns'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns'> Font Type: </div> <div class='large-8 medium-8 small-12 columns'> <span class="html-article-menu-option"><i style='font-family:Arial, Arial, Helvetica, sans-serif;' data-fontfamily='Arial, Arial, Helvetica, sans-serif'>Arial</i></span> <span class="html-article-menu-option"><i style='font-family:Georgia1, Georgia, serif;' data-fontfamily='Georgia1, Georgia, serif'>Georgia</i></span> <span class="html-article-menu-option"><i style='font-family:Verdana, Verdana, Geneva, sans-serif;' data-fontfamily='Verdana, Verdana, Geneva, sans-serif' >Verdana</i></span> </div> </div> </div> <div class='html-font-resize large-6 medium-6 small-12 columns'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns'>Font Size:</div> <div class='large-8 medium-8 small-12 columns'> <span class="html-article-menu-option a1" data-percent="100">Aa</span> <span class="html-article-menu-option a2" data-percent="120">Aa</span> <span class="html-article-menu-option a3" data-percent="160">Aa</span> </div> </div> </div> </div> <div class='row'> <div class='html-line-space large-6 medium-6 small-12 columns'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns' >Line Spacing:</div> <div class='large-8 medium-8 small-12 columns'> <span class="html-article-menu-option a1" data-line-height="1.5em"> <i class="fa"></i> </span> <span class="html-article-menu-option a2" data-line-height="1.8em"> <i class="fa"></i> </span> <span class="html-article-menu-option a3" data-line-height="2.1em"> <i class="fa"></i> </span> </div> </div> </div> <div class='html-column-width large-6 medium-6 small-12 columns'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns' >Column Width:</div> <div class='large-8 medium-8 small-12 columns'> <span class="html-article-menu-option a1" data-column-width="20%"> <i class="fa"></i> </span> <span class="html-article-menu-option a2" data-column-width="10%"> <i class="fa"></i> </span> <span class="html-article-menu-option a3" data-column-width="0%"> <i class="fa"></i> </span> </div> </div> </div> </div> <div class='row'> <div class='html-font-bg large-6 medium-6 small-12 columns end'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns'>Background:</div> <div class='large-8 medium-8 small-12 columns'> <div class="html-article-menu-option html-nav-bg html-nav-bright" data-bg="bright"> <i class="fa fa-file-text"></i> </div> <div class="html-article-menu-option html-nav-bg html-nav-dark" data-bg="dark"> <i class="fa fa-file-text-o"></i> </div> <div class="html-article-menu-option html-nav-bg html-nav-creme" data-bg="creme"> <i class="fa fa-file-text"></i> </div> </div> </div> </div> </div> </div> </div> <article ><div class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2218-1997/10/9/357"></span> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <h1 class="title hypothesis_container" itemprop="name"> Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13117502' data-options='is_hover:true, hover_timeout:5000'> Alessio Giarnetti</div><div id="profile-card-drop13117502" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Alessio Giarnetti</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/3578960?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Alessio%20Giarnetti" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Alessio%20Giarnetti&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Alessio%20Giarnetti" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> *</sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13117502" href="/cdn-cgi/l/email-protection#aa85c9c4ce87c9cdc385c685cfc7cbc3c687dad8c5decfc9dec3c5c4899a9a9ace9c9b9a9e9b989b989a929acf9ecc9a9c9a929a9a9b999acc9a9e9b9f9b9f9a92989b9b9e9acc9a929b999acf9ac99a9a9f989ecc9a929b9f"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0001-8487-8045" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732286508" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13117503' data-options='is_hover:true, hover_timeout:5000'> Simone Marciano</div><div id="profile-card-drop13117503" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Simone Marciano</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/3718904?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Simone%20Marciano" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Simone%20Marciano&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Simone%20Marciano" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup></sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13117503" href="/cdn-cgi/l/email-protection#1a3579747e37797d733576357f777b7376376a68756e7f796e737574392a2a2b7b2d292b7f2b792b7e2b2c2f7e2b7f2b282a2b2b2a2b7b2b282b7e2b7929292a2c2b7e2b7b2a2b2b792b7f2b282e2a2f7e2b7b2a2d"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0002-2243-7495" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732286508" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a> and </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13117504' data-options='is_hover:true, hover_timeout:5000'> Davide Meloni</div><div id="profile-card-drop13117504" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Davide Meloni</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/1659267?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Davide%20Meloni" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Davide%20Meloni&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Davide%20Meloni" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup></sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13117504" href="/cdn-cgi/l/email-protection#b49bd7dad099d7d3dd9bd89bd1d9d5ddd899c4c6dbc0d1d7c0dddbda97848484818280858684d08484848580d5848d8485848c84d684d584d08680858584d584d0858284d6848d8481818380d584d08584"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0001-7680-6957" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1732286508" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-name ">Dipartimento di Matematica e Fisica, Università di Roma Tre, INFN Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Universe</em> <b>2024</b>, <em>10</em>(9), 357; <a href="https://doi.org/10.3390/universe10090357">https://doi.org/10.3390/universe10090357</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 29 July 2024</span> / <span style="display: inline-block">Revised: 23 August 2024</span> / <span style="display: inline-block">Accepted: 28 August 2024</span> / <span style="display: inline-block">Published: 5 September 2024</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/universe/sections/high_energy_nuclear_and_particle_physics">High Energy Nuclear and Particle Physics</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1472235" aria-controls="drop-supplementary-1472235" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1472235" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2218-1997/10/9/357/pdf?version=1725541553" data-name="Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces" data-journal="universe">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2218-1997/10/9/357/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2218-1997/10/9/357/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2218-1997/10/9/357/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2218-1997/10/9/357/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001.png?1725541638" title=" <strong>Figure 1</strong><br/> <p><math display="inline"><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance (left panels) and <math display="inline"><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance (right panels) probabilities in the presence of off-diagonal CPT violating and conserving LIV parameters. In particular, the top, middle and bottom panels show the effect of <math display="inline"><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> (<math display="inline"><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math>), <math display="inline"><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display="inline"><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>) and <math display="inline"><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display="inline"><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>), respectively. Black lines correspond to the standard oscillation case and red (orange) lines to the probabilities obtained for <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> </mrow> </semantics></math> GeV (<math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> </mrow> </semantics></math>). Solid and dashed curves depict the effects of LIV phases (generically indicated <math display="inline"><semantics> <mo>Φ</mo> </semantics></math>) when <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, respectively. The grey and green shadowed regions illustrate the standard and the high-energy DUNE flux.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002.png?1725541643" title=" <strong>Figure 2</strong><br/> <p><math display="inline"><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> (dotted), <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> (solid) and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> (dashed) allowed contours in the <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>−</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> (left panels) and <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>−</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> planes (right panels) for HE-DUNE. The red curves in the bottom panels depict the effect of the addition of the <math display="inline"><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance channel in the analysis.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003.png?1725541647" title=" <strong>Figure 3</strong><br/> <p>Same as <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a>, but for the Long-Range Force case. Left (right) plot shows the <math display="inline"><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance (<math display="inline"><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance) probability. The red, magenta and orange curves refer to the <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> cases, respectively. The potentials <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> have been fixed to <math display="inline"><semantics> <mrow> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math> eV.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004.png?1725541648" title=" <strong>Figure 4</strong><br/> <p>HE-DUNE sensitivity to the LRF potentials. Red, magenta and orange lines correspond to the <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> cases, respectively.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005.png?1725541650" title=" <strong>Figure 5</strong><br/> <p>The 95% CL excluded regions in the <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mo>−</mo> <msub> <mi>G</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> plane, fixing the LRF potentials to the 95% CL HE-DUNE limits showed in <a href="#universe-10-00357-t003" class="html-table">Table 3</a>. See text for details.</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006.png?1725541652" title=" <strong>Figure 6</strong><br/> <p>Same as <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a> but in the Large Extra Dimension case. Red (orange) curves have been obtained fixing <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math> eV (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> eV).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007.png?1725541653" title=" <strong>Figure 7</strong><br/> <p><math display="inline"><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> (dotted) <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> (solid) and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> (dashed) allowed regions in the <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </semantics></math> plane for HE-DUNE.</p> "> </a> </div> <a class="button button--color-inversed" href="/2218-1997/10/9/357/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">DUNE is a next-generation long-baseline neutrino oscillation experiment. It is expected to measure, with unprecedented precision, the atmospheric oscillation parameters, including the CP-violating phase <math display="inline"><semantics> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> </semantics></math>. Moreover, several studies have suggested that its unique features should allow DUNE to probe several new physics scenarios. In this work, we explore the performances of the DUNE far detector in constraining new physics if a high-energy neutrino flux is employed (HE-DUNE). We take into account three different scenarios: Lorentz Invariance Violation (LIV), Long-Range Forces (LRFs) and Large Extra Dimensions (LEDs). Our results show that HE-DUNE should be able to set bounds competitive to the current ones and, in particular, it can outperform the standard DUNE capabilities in constraining CPT-even LIV parameters and the compactification radius <math display="inline"><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> of the LED model.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=neutrino+mixing">neutrino mixing</a>; <a href="/search?q=DUNE">DUNE</a>; <a href="/search?q=BSM">BSM</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-universe-10-00357' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>Neutrino oscillation discovery [<a href="#B1-universe-10-00357" class="html-bibr">1</a>] represented a milestone in the history of particle physics. Indeed, the observation of this phenomenon unveiled that neutrinos have (tiny) masses compared to the other fermions of the Standard Model (SM) of particle physics. In addition, the small uncertainties achieved in the measurements of mixing angles carry us into a precision era in the neutrino sector, thanks to an effort that lasted 25 years and that has involved different particle sources: the Sun [<a href="#B2-universe-10-00357" class="html-bibr">2</a>], the Earth atmosphere [<a href="#B3-universe-10-00357" class="html-bibr">3</a>], nuclear reactors [<a href="#B4-universe-10-00357" class="html-bibr">4</a>] and accelerator facilities [<a href="#B5-universe-10-00357" class="html-bibr">5</a>]. The oscillation parameters involved in solar oscillation, namely the solar mixing angle <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>12</mn> </msub> </semantics></math> and the solar mass splitting <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>21</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>, have been precisely measured by several solar neutrino experiments and by a peculiar reactor experiment: KamLAND [<a href="#B6-universe-10-00357" class="html-bibr">6</a>,<a href="#B7-universe-10-00357" class="html-bibr">7</a>,<a href="#B8-universe-10-00357" class="html-bibr">8</a>]. The reactor mixing angle <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>13</mn> </msub> </semantics></math>, instead, was discovered and measured with an astonishing precision in 2012 [<a href="#B9-universe-10-00357" class="html-bibr">9</a>], leaving the atmospheric oscillation sector less constrained. Indeed, the mixing angle <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math> is almost maximal (<math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>23</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>40</mn> <mo>°</mo> <mo>,</mo> <mn>50</mn> <mo>°</mo> <mo>]</mo> </mrow> </mrow> </semantics></math> [<a href="#B10-universe-10-00357" class="html-bibr">10</a>]) and suffers from the so-called <span class='html-italic'>octant degeneracy,</span> which makes the determination of the octant in which <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math> lies (Higher Octant, HO, <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>23</mn> </msub> <mo>></mo> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math> or Lower Octant, LO, <math display='inline'><semantics> <mrow> <msub> <mi>θ</mi> <mn>23</mn> </msub> <mo><</mo> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math>) very difficult to be determined. Moreover, according to the current neutrino oscillation data, the atmospheric mass splitting <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> has an absolute value 30 times larger than the solar one and can still assume both positive and negative values. This is the so-called <span class='html-italic'>mass hierarchy problem</span>.</div><div class='html-p'>The solution to these long-lasting problems might be given by the next generation of long-baseline (LBL) accelerator experiments. Such experiments are mainly sensitive to the atmospheric neutrino frequency and employ a well-known, focused, artificial muon neutrino flux coming from an accelerator facility. The two experiments that are expected to start their data-taking in the current decade are T2HK in Japan [<a href="#B11-universe-10-00357" class="html-bibr">11</a>] and DUNE [<a href="#B12-universe-10-00357" class="html-bibr">12</a>] in the USA. The importance of such experiments lies not only in the solution of the octant and hierarchy problems but also in their unprecedented capabilities to measure the CP-violating phase <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>CP</mi> </msub> </semantics></math>, for which T2K and NO<math display='inline'><semantics> <mi>ν</mi> </semantics></math>A [<a href="#B13-universe-10-00357" class="html-bibr">13</a>,<a href="#B14-universe-10-00357" class="html-bibr">14</a>,<a href="#B15-universe-10-00357" class="html-bibr">15</a>] have provided a first (weak) signal. Even though both experiments will be sensitive to the same neutrino oscillation regime, DUNE will have the advantage of running with a broad-band beam, allowing observation of neutrinos whose energy extends beyond the first oscillation maximum. To go even higher in energy, the possibility of using a high-energy (HE) neutrino beam in DUNE has been widely discussed. This would allow, for instance, collection of the largest <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> event sample ever observed [<a href="#B16-universe-10-00357" class="html-bibr">16</a>]. In recent years, a large number of studies have shown that the employment of this particular flux might be extremely useful in exploring tiny new physics effects in neutrino oscillation in the presence, for instance, of sterile neutrinos, non-unitarity of the PMNS matrix, Non-Standard Interactions or quantum decoherence [<a href="#B16-universe-10-00357" class="html-bibr">16</a>,<a href="#B17-universe-10-00357" class="html-bibr">17</a>,<a href="#B18-universe-10-00357" class="html-bibr">18</a>,<a href="#B19-universe-10-00357" class="html-bibr">19</a>,<a href="#B20-universe-10-00357" class="html-bibr">20</a>,<a href="#B21-universe-10-00357" class="html-bibr">21</a>]. In this context, the advantage of using the HE flux lies in the fact that some models predicts new effects enhanced by neutrino energy. The extremely large collectable sample of events in all neutrino flavors provides a unique tool to study these energy-enhanced new physics effects.</div><div class='html-p'>In this paper, we explore and discuss the DUNE capabilities in its HE configuration to measure the new physics parameters involved in neutrino oscillation in three particular models: the Lorentz Invariance Violation (LIV), the Long-Range Forces (LRFs) and the Large Extra Dimensions (LEDs) models. In the first two cases, the Lagrangian interaction is supplemented with additional operators whose coefficients must be bounded from above, while in the last case, the space-time framework is enlarged by at least one spatial dimension (experienced by right-handed neutrinos only) whose radius belongs to the sub-millimeter range. Even though such new physics models have been investigated by other authors in the literature, our results show that the bounds that HE-DUNE can set on the model parameters are comparable with the existing ones and, for the CPT-even LIV parameters and for the radius of the LED model, can outperform the capabilities of the standard DUNE setup.</div><div class='html-p'>The paper will be organized as follows: in <a href="#sec2-universe-10-00357" class="html-sec">Section 2</a>, we will describe the DUNE experiment and its HE configuration; in <a href="#sec3-universe-10-00357" class="html-sec">Section 3</a>, <a href="#sec4-universe-10-00357" class="html-sec">Section 4</a> and <a href="#sec5-universe-10-00357" class="html-sec">Section 5</a>, we discuss the LIV, LRF and LED models, respectively. Finally, in <a href="#sec6-universe-10-00357" class="html-sec">Section 6</a>, we draw our conclusions.</div></section><section id='sec2-universe-10-00357' type=''><h2 data-nested='1'> 2. The DUNE Experiment and the High-Energy Flux</h2><div class='html-p'>The DUNE (Deep Underground Neutrino Experiment) experiment is a proposed long-baseline experiment in the USA [<a href="#B12-universe-10-00357" class="html-bibr">12</a>,<a href="#B22-universe-10-00357" class="html-bibr">22</a>,<a href="#B23-universe-10-00357" class="html-bibr">23</a>,<a href="#B24-universe-10-00357" class="html-bibr">24</a>]. The near detector complex, composed of different multi-purpose near detectors [<a href="#B25-universe-10-00357" class="html-bibr">25</a>] as well as the accelerator facilities, is being built at Fermilab; on the other hand, the 40kt LAr-TPC detector will be located in South Dakota, roughly 1300 km away from the neutrino beam source. The on-axis neutrino beams will be mainly composed of <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> or <math display='inline'><semantics> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>μ</mi> </msub> </semantics></math> depending on the current circulating in the focusing horns; this will allow the experiment to run in both <span class='html-italic'>neutrino</span> and <span class='html-italic'>antineutrino</span> modes. The main purpose of the experiment is to precisely measure the oscillation parameters in the atmospheric sector. In particular, DUNE is expected to reach a great sensitivity to the mass hierarchy and an unprecedented sensitivity to the <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math> octant. Moreover, the experiment is also designed to maximize the sensitivity to the PMNS matrix phase <math display='inline'><semantics> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> </semantics></math>. In order to perform such a measurement, the proposed neutrino flux is a broad-band beam that peaks at around 2.5 GeV so sit at the first atmospheric oscillation maximum. This should allow observation of not only a huge sample of <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> events (<math display='inline'><semantics> <mrow> <mi mathvariant="script">O</mi> <mo>(</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> <mo>)</mo> </mrow> </semantics></math> per year) but also several <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance events (<math display='inline'><semantics> <mrow> <mi mathvariant="script">O</mi> <mo>(</mo> <msup> <mn>10</mn> <mn>2</mn> </msup> <mo>)</mo> </mrow> </semantics></math> per year) [<a href="#B12-universe-10-00357" class="html-bibr">12</a>,<a href="#B22-universe-10-00357" class="html-bibr">22</a>,<a href="#B23-universe-10-00357" class="html-bibr">23</a>,<a href="#B24-universe-10-00357" class="html-bibr">24</a>]. A very intense flux of <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> will also arrive at the far detector; however, given the Charged Current (CC) <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> interaction energy threshold (3.1 GeV), only a minor fraction of these events might be observed. To overcome this problem and observe a larger number of <math display='inline'><semantics> <mi>τ</mi> </semantics></math> neutrinos, a broader, high-energetic flux peaked at around 5 GeV has also been considered by the DUNE collaboration [<a href="#B26-universe-10-00357" class="html-bibr">26</a>,<a href="#B27-universe-10-00357" class="html-bibr">27</a>,<a href="#B28-universe-10-00357" class="html-bibr">28</a>]. The main disadvantage of using this flux is that the performances in measuring standard oscillation parameters are poorer [<a href="#B16-universe-10-00357" class="html-bibr">16</a>]. However, other than collecting the largest <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> sample ever obtained (<math display='inline'><semantics> <mrow> <mi mathvariant="script">O</mi> <mo>(</mo> <msup> <mn>10</mn> <mn>2</mn> </msup> <mo>)</mo> </mrow> </semantics></math> in a year [<a href="#B16-universe-10-00357" class="html-bibr">16</a>]), as already mentioned, the employment of such a flux has been demonstrated to be very promising in constraining new physics scenarios [<a href="#B16-universe-10-00357" class="html-bibr">16</a>,<a href="#B17-universe-10-00357" class="html-bibr">17</a>,<a href="#B18-universe-10-00357" class="html-bibr">18</a>,<a href="#B19-universe-10-00357" class="html-bibr">19</a>]. Thus, a high-energy flux run, at least in addition to the standard DUNE one [<a href="#B18-universe-10-00357" class="html-bibr">18</a>], might be extremely useful to provide unique information in the context of neutrino oscillation. In this work, we focus on the capabilities of the DUNE experiment to probe some beyond-the-Standard Model (BSM) theories that make an imprint on the neutrino oscillation probabilities, taking full advantage of the high-energy flux. From now on, we will refer to this DUNE configuration as HE-DUNE. In order to make a comparison with the standard DUNE results, for the HE-DUNE, we will use the same efficiencies, energy resolutions and systematic uncertainties provided by the collaboration for the standard DUNE [<a href="#B29-universe-10-00357" class="html-bibr">29</a>,<a href="#B30-universe-10-00357" class="html-bibr">30</a>]. In order to include the possible effects of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance in constraining the new physics parameters space, we follow [<a href="#B16-universe-10-00357" class="html-bibr">16</a>,<a href="#B18-universe-10-00357" class="html-bibr">18</a>]. In particular, we make the hypothesis that the HE-DUNE might be able to observe 30% of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> events with subsequent <math display='inline'><semantics> <mrow> <mi>τ</mi> <mo>→</mo> <mi>e</mi> </mrow> </semantics></math> decays and 30% of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> events with subsequent hadronic <math display='inline'><semantics> <mi>τ</mi> </semantics></math> decays. The systematic uncertainty for this channel has been set to a conservative 25% normalization error<a href="#fn001-universe-10-00357" class="html-fn">1</a>. Misidentified <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> and NC events have been considered as a background to the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> channel, according to [<a href="#B16-universe-10-00357" class="html-bibr">16</a>,<a href="#B18-universe-10-00357" class="html-bibr">18</a>].</div><div class='html-p'>The total running time for HE-DUNE has been fixed to 5 years in neutrino mode and 5 years in antineutrino mode. Finally, the whole analysis has been carried out neglecting the events observed at the near detectors and considering the final 40kt far detector for all the 10 years of data taking.</div></section><section id='sec3-universe-10-00357' type=''><h2 data-nested='1'> 3. The Lorentz Invariance Violation Case</h2><div class='html-p'>The Lorentz invariance is one of the fundamental symmetries of the SM and it is related to the space-time structure. The other essential symmetry of the quantum field theory is the CPT symmetry<a href="#fn002-universe-10-00357" class="html-fn">2</a>. Since the SM fails to unify all the forces governing the Universe, it has been taken into account the possibility that the SM is an effective theory of a wider framework that unifies not only electromagnetic and weak interactions, but also strong interactions and gravity. The energy scale of such a general theory should be the Planck mass (<math display='inline'><semantics> <msub> <mi>M</mi> <mi>P</mi> </msub> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>19</mn> </msup> </semantics></math> GeV). In these SM extensions, CPT and Lorentz symmetries might be violated [<a href="#B31-universe-10-00357" class="html-bibr">31</a>,<a href="#B32-universe-10-00357" class="html-bibr">32</a>,<a href="#B33-universe-10-00357" class="html-bibr">33</a>,<a href="#B34-universe-10-00357" class="html-bibr">34</a>,<a href="#B35-universe-10-00357" class="html-bibr">35</a>,<a href="#B36-universe-10-00357" class="html-bibr">36</a>,<a href="#B37-universe-10-00357" class="html-bibr">37</a>,<a href="#B38-universe-10-00357" class="html-bibr">38</a>,<a href="#B39-universe-10-00357" class="html-bibr">39</a>,<a href="#B40-universe-10-00357" class="html-bibr">40</a>]; in particular, it has been shown that CPT breaking always leads to Lorentz Invariance Violation (LIV) as well [<a href="#B34-universe-10-00357" class="html-bibr">34</a>]. Neutrino experiments could be able to probe LIV through suitable modification of the oscillation probabilities induced by the presence of new terms in the full theory. In the next subsection, we will show how the neutrino probabilities can be affected by LIV.</div><section id='sec3dot1-universe-10-00357' type=''><h4 class='html-italic' data-nested='2'> 3.1. Theoretical Framework</h4><div class='html-p'>In the presence of LIV, the neutrino Lagrangian density term can be written as [<a href="#B38-universe-10-00357" class="html-bibr">38</a>,<a href="#B41-universe-10-00357" class="html-bibr">41</a>,<a href="#B42-universe-10-00357" class="html-bibr">42</a>] <div class='html-disp-formula-info' id='FD1-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mover accent="true"> <mi>ψ</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>i</mi> <menclose notation="updiagonalstrike"> <mo>∂</mo> </menclose> <mo>−</mo> <mi>M</mi> <mo>+</mo> <mi>Q</mi> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>+</mo> <mi>h</mi> <mo>.</mo> <mi>c</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where <math display='inline'><semantics> <mi>ψ</mi> </semantics></math> is the neutrino fermionic field and the effect of the LIV is encoded in the generic operator Q. If we restrict ourselves to renormalizable Dirac couplings, the Lorentz Invariance violating Lagrangian terms can be written as [<a href="#B42-universe-10-00357" class="html-bibr">42</a>,<a href="#B43-universe-10-00357" class="html-bibr">43</a>]:<div class='html-disp-formula-info' id='FD2-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi mathvariant="script">L</mi> <mi>LIV</mi> </msup> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mfenced separators="" open="(" close=")"> <msubsup> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>μ</mi> </msubsup> <msub> <mover accent="true"> <mi>ψ</mi> <mo>¯</mo> </mover> <mi>α</mi> </msub> <msub> <mi>γ</mi> <mi>μ</mi> </msub> <msub> <mi>ψ</mi> <mi>β</mi> </msub> <mo>+</mo> <msubsup> <mi>b</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>μ</mi> </msubsup> <msub> <mover accent="true"> <mi>ψ</mi> <mo>¯</mo> </mover> <mi>α</mi> </msub> <msub> <mi>γ</mi> <mn>5</mn> </msub> <msub> <mi>γ</mi> <mi>μ</mi> </msub> <msub> <mi>ψ</mi> <mi>β</mi> </msub> <mo>−</mo> <mi>i</mi> <msubsup> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> <msub> <mover accent="true"> <mi>ψ</mi> <mo>¯</mo> </mover> <mi>α</mi> </msub> <msub> <mi>γ</mi> <mi>μ</mi> </msub> <msub> <mo>∂</mo> <mi>ν</mi> </msub> <msub> <mi>ψ</mi> <mi>β</mi> </msub> <mo>−</mo> <mi>i</mi> <msubsup> <mi>d</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> <msub> <mover accent="true"> <mi>ψ</mi> <mo>¯</mo> </mover> <mi>α</mi> </msub> <msub> <mi>γ</mi> <mn>5</mn> </msub> <msub> <mi>γ</mi> <mi>μ</mi> </msub> <msub> <mo>∂</mo> <mi>ν</mi> </msub> <msub> <mi>ψ</mi> <mi>β</mi> </msub> </mfenced> <mo>+</mo> <mi>h</mi> <mo>.</mo> <mi>c</mi> <mo>.</mo> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> The first two terms are CPT-odd, while the third and the fourth terms are CPT-even. Thus, the LIV effect in the interaction Hamiltonian can be encoded in two Hermitian matrices:<div class='html-disp-formula-info' id='FD3-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>a</mi> <mo>˜</mo> </mover> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>μ</mi> </msubsup> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>)</mo> </mrow> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>μ</mi> </msubsup> <mspace width="1.em"/> <mspace width="1.em"/> <mi>and</mi> <mspace width="1.em"/> <mspace width="1.em"/> <msubsup> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mo>)</mo> </mrow> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> These matrices modify the standard neutrino oscillation Hamiltonian by adding a new term <math display='inline'><semantics> <msub> <mi>H</mi> <mi>LIV</mi> </msub> </semantics></math> to the vacuum and matter contributions:<div class='html-disp-formula-info' id='FD4-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>H</mi> <mi>ν</mi> </msub> <mo>=</mo> <msub> <mi>H</mi> <mi mathvariant="normal">m</mi> </msub> <mo>+</mo> <msub> <mi>H</mi> <mi>æ</mi> </msub> <mo>+</mo> <msub> <mi>H</mi> <mi>LIV</mi> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where, as usual, <div class='html-disp-formula-info' id='FD5-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>H</mi> <mi mathvariant="normal">m</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>E</mi> </mrow> </mfrac> </mstyle> <mi>U</mi> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>21</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <msup> <mi>U</mi> <mo>†</mo> </msup> <mspace width="1.em"/> <mi>and</mi> <mspace width="1.em"/> <msub> <mi>H</mi> <mi>æ</mi> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>G</mi> <mi>F</mi> </msub> <msub> <mi>N</mi> <mi>e</mi> </msub> <mi>diag</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> Here, <span class='html-italic'>E</span> is the neutrino energy, <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msubsup> <mi>m</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>−</mo> <msubsup> <mi>m</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mrow> </semantics></math> are the neutrino mass splittings, <span class='html-italic'>U</span> is the neutrino mixing matrix, which depends on three mixing angles (<math display='inline'><semantics> <msub> <mi>θ</mi> <mn>12</mn> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>13</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math>) and one complex phase <math display='inline'><semantics> <msub> <mi>δ</mi> <mi>CP</mi> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>G</mi> <mi>F</mi> </msub> </semantics></math> is the Fermi constant and <math display='inline'><semantics> <msub> <mi>N</mi> <mi>e</mi> </msub> </semantics></math> is the number density of electrons in the medium traversed by neutrinos. The term <math display='inline'><semantics> <msub> <mi>H</mi> <mi>æ</mi> </msub> </semantics></math> corresponds to the well-known Mikheev–Smirnov–Wolfenstein (MSW) mechanism [<a href="#B44-universe-10-00357" class="html-bibr">44</a>].</div><div class='html-p'>The last term <math display='inline'><semantics> <msub> <mi>H</mi> <mi>LIV</mi> </msub> </semantics></math> is the one induced by the LIV. On a general ground, it reads <math display='inline'><semantics> <mrow> <msub> <mi>H</mi> <mi>LIV</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mi>E</mi> <mrow> <mo>(</mo> <msup> <mover accent="true"> <mi>a</mi> <mo>˜</mo> </mover> <mi>μ</mi> </msup> <msub> <mi>p</mi> <mi>μ</mi> </msub> <mo>−</mo> <msup> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>p</mi> <mi>μ</mi> </msub> <msub> <mi>p</mi> <mi>ν</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, where <math display='inline'><semantics> <msub> <mi>p</mi> <mi>μ</mi> </msub> </semantics></math> is the neutrino four-momentum. However, focusing only on time-like LIV matrix components (<math display='inline'><semantics> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>) and considering a Sun-centered isotropic inertial frame (see [<a href="#B43-universe-10-00357" class="html-bibr">43</a>] for details), the Lorentz Invariance Violation effects are governed by the parameters <math display='inline'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>a</mi> <mo>˜</mo> </mover> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mn>0</mn> </msubsup> <mo>≡</mo> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mn>00</mn> </msubsup> <mo>≡</mo> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math>. Being elements of Hermitian matrices, diagonal <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>)</mo> </mrow> <mrow> <mi>α</mi> <mi>α</mi> </mrow> </msub> </semantics></math> are real, while off-diagonal <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>)</mo> </mrow> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> are complex parameters uniquely determined by their moduli, which we denote as <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>)</mo> </mrow> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math>, and their phases <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math>. Thus, the LIV Hamiltonian reads:<div class='html-disp-formula-info' id='FD6-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>H</mi> <mi>LIV</mi> </msub> <mo>=</mo> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> <mo>*</mo> </msubsup> </mtd> <mtd> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>μ</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> <mo>*</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> <mo>*</mo> </msubsup> </mtd> <mtd> <msub> <mi>a</mi> <mrow> <mi>τ</mi> <mi>τ</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> <mi>E</mi> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> <mo>*</mo> </msubsup> </mtd> <mtd> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>μ</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> <mo>*</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> <mo>*</mo> </msubsup> </mtd> <mtd> <mrow> <msub> <mi>c</mi> <mrow> <mi>τ</mi> <mi>τ</mi> </mrow> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div> where, as already mentioned, <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math>) are CPT-odd (CPT-even) LIV parameters. Notice that the factor <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math> comes from the fact that the trace of <math display='inline'><semantics> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> </semantics></math> is not observable and its diagonal space components must be related to its 00 component [<a href="#B43-universe-10-00357" class="html-bibr">43</a>]. It is worth to mention that <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> matrix has a similar structure to the propagation Non Standard Interaction (NSI) matrix. Even though there exists a direct correspondence between NSI and LIV parameters, it is important to notice that the two models affect neutrino oscillation in a different way: while NSIs necessitate neutrinos to travel through a matter medium, LIV modifies the oscillation probabilities also in vacuum [<a href="#B41-universe-10-00357" class="html-bibr">41</a>,<a href="#B42-universe-10-00357" class="html-bibr">42</a>,<a href="#B45-universe-10-00357" class="html-bibr">45</a>,<a href="#B46-universe-10-00357" class="html-bibr">46</a>].</div><div class='html-p'>Another important aspect of LIV effects is that the CPT-even ones are amplified by neutrino energy. For this reason, we expect a high-energy flux for the DUNE experiment to be more efficient in constraining them. From now on, we will only focus on the off-diagonal LIV parameters since they affect the most oscillations of neutrinos in a long-baseline experiment, as we will discuss below.</div><div class='html-p'>In order to have a feeling of the effect of the LIV on the oscillation parameters, we can write the correction to the <math display='inline'><semantics> <mrow> <msub> <mi>ν</mi> <mi>μ</mi> </msub> <mo>→</mo> <msub> <mi>ν</mi> <mi>e</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>ν</mi> <mi>μ</mi> </msub> <mo>→</mo> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> SM probabilities at the first order in <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> as [<a href="#B41-universe-10-00357" class="html-bibr">41</a>].<a href="#fn003-universe-10-00357" class="html-fn">3</a><div class='html-disp-formula-info' id='FD7-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mi>P</mi> <mrow> <mi>μ</mi> <mi>e</mi> </mrow> <mi>LIV</mi> </msubsup> <mo>∼</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mn>2</mn> <mi>L</mi> <msub> <mi>s</mi> <mn>13</mn> </msub> <mo form="prefix">sin</mo> <mn>2</mn> <msub> <mi>θ</mi> <mn>23</mn> </msub> <mo form="prefix">sin</mo> <mo>Δ</mo> <mo>{</mo> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mo>[</mo> <mo>−</mo> <msub> <mi>c</mi> <mn>23</mn> </msub> <mo form="prefix">sin</mo> <mo>Δ</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mo>+</mo> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>c</mi> <mn>23</mn> </msub> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msubsup> <mi>s</mi> <mrow> <mn>23</mn> </mrow> <mn>2</mn> </msubsup> <mo form="prefix">sin</mo> <mo>Δ</mo> </mrow> <mrow> <msubsup> <mi>c</mi> <mrow> <mn>23</mn> </mrow> <mn>2</mn> </msubsup> <mo>Δ</mo> </mrow> </mfrac> </mstyle> <mo>+</mo> <mo form="prefix">cos</mo> <mo>Δ</mo> </mfenced> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>]</mo> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mo>+</mo> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mfenced separators="" open="[" close="]"> <msub> <mi>s</mi> <mn>23</mn> </msub> <mo form="prefix">sin</mo> <mo>Δ</mo> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>s</mi> <mn>23</mn> </msub> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo form="prefix">sin</mo> <mo>Δ</mo> </mrow> <mo>Δ</mo> </mfrac> </mstyle> <mo>−</mo> <mo form="prefix">cos</mo> <mo>Δ</mo> </mfenced> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> <mo>+</mo> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mo>)</mo> </mrow> </mfenced> <mo>}</mo> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo form="prefix">sin</mo> <msub> <mi>θ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo form="prefix">cos</mo> <msub> <mi>θ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> <mi>L</mi> <mo>/</mo> <mn>4</mn> <mi>E</mi> </mrow> </semantics></math> with <span class='html-italic'>L</span> is the distance travelled by the neutrino and finally <math display='inline'><semantics> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> is either <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> or <math display='inline'><semantics> <mrow> <mrow> <mo>−</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mi>E</mi> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> (the related phases are indicated with <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math>). It is clear that the leading corrections to the standard <math display='inline'><semantics> <msub> <mi>P</mi> <mrow> <mi>μ</mi> <mi>e</mi> </mrow> </msub> </semantics></math> are driven by the <math display='inline'><semantics> <mrow> <mi>e</mi> <mo>−</mo> <mi>μ</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>e</mi> <mo>−</mo> <mi>τ</mi> </mrow> </semantics></math> LIV parameters. For the muon neutrino disappearance, instead, we obtain [<a href="#B41-universe-10-00357" class="html-bibr">41</a>]:<div class='html-disp-formula-info' id='FD8-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mi>P</mi> <mrow> <mi>μ</mi> <mi>μ</mi> </mrow> <mi>LIV</mi> </msubsup> <mo>∼</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <mn>2</mn> <msub> <mi>θ</mi> <mn>23</mn> </msub> <mfenced separators="" open="[" close="]"> <mn>2</mn> <mo>Δ</mo> <msup> <mo form="prefix">sin</mo> <mn>2</mn> </msup> <msub> <mi>θ</mi> <mn>13</mn> </msub> <mo>−</mo> <mn>2</mn> <mi>L</mi> <mo form="prefix">sin</mo> <mn>2</mn> <msub> <mi>θ</mi> <mn>23</mn> </msub> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mo form="prefix">cos</mo> <msub> <mo>Φ</mo> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mfenced> <mspace width="0.166667em"/> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div> the main dependence on the LIV parameters is given by <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>)<a href="#fn004-universe-10-00357" class="html-fn">4</a>.</div><div class='html-p'>Several bounds on the LIV parameter have been obtained using long-baseline accelerator neutrinos [<a href="#B41-universe-10-00357" class="html-bibr">41</a>,<a href="#B42-universe-10-00357" class="html-bibr">42</a>,<a href="#B46-universe-10-00357" class="html-bibr">46</a>,<a href="#B47-universe-10-00357" class="html-bibr">47</a>,<a href="#B48-universe-10-00357" class="html-bibr">48</a>,<a href="#B49-universe-10-00357" class="html-bibr">49</a>,<a href="#B50-universe-10-00357" class="html-bibr">50</a>,<a href="#B51-universe-10-00357" class="html-bibr">51</a>,<a href="#B52-universe-10-00357" class="html-bibr">52</a>,<a href="#B53-universe-10-00357" class="html-bibr">53</a>,<a href="#B54-universe-10-00357" class="html-bibr">54</a>,<a href="#B55-universe-10-00357" class="html-bibr">55</a>,<a href="#B56-universe-10-00357" class="html-bibr">56</a>,<a href="#B57-universe-10-00357" class="html-bibr">57</a>,<a href="#B58-universe-10-00357" class="html-bibr">58</a>], short baseline accelerator neutrinos [<a href="#B59-universe-10-00357" class="html-bibr">59</a>,<a href="#B60-universe-10-00357" class="html-bibr">60</a>], reactor neutrinos [<a href="#B61-universe-10-00357" class="html-bibr">61</a>,<a href="#B62-universe-10-00357" class="html-bibr">62</a>], solar neutrinos [<a href="#B63-universe-10-00357" class="html-bibr">63</a>], high-energy astrophysical neutrinos [<a href="#B64-universe-10-00357" class="html-bibr">64</a>,<a href="#B65-universe-10-00357" class="html-bibr">65</a>,<a href="#B66-universe-10-00357" class="html-bibr">66</a>,<a href="#B67-universe-10-00357" class="html-bibr">67</a>] and atmospheric neutrinos [<a href="#B68-universe-10-00357" class="html-bibr">68</a>,<a href="#B69-universe-10-00357" class="html-bibr">69</a>,<a href="#B70-universe-10-00357" class="html-bibr">70</a>,<a href="#B71-universe-10-00357" class="html-bibr">71</a>] (see also [<a href="#B72-universe-10-00357" class="html-bibr">72</a>,<a href="#B73-universe-10-00357" class="html-bibr">73</a>] for reviews on this, and other, new physics models). Since we are interested in the DUNE experiment performances with a high-energy flux, in the following, we summarize the bounds that DUNE in its standard configuration [<a href="#B23-universe-10-00357" class="html-bibr">23</a>,<a href="#B24-universe-10-00357" class="html-bibr">24</a>,<a href="#B30-universe-10-00357" class="html-bibr">30</a>] might set on off-diagonal LIV parameters. In [<a href="#B41-universe-10-00357" class="html-bibr">41</a>], the authors found, at 95% CL:<div class='html-disp-formula-info' id='FD9-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>1.00</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> <mspace width="1.em"/> <mi>GeV</mi> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="1.em"/> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo><</mo> <mn>0.66</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>1.05</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> <mspace width="1.em"/> <mi>GeV</mi> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="1.em"/> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo><</mo> <mn>1.65</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>1.26</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> <mspace width="1.em"/> <mi>GeV</mi> <mspace width="0.166667em"/> <mo>,</mo> <mspace width="1.em"/> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo><</mo> <mn>0.97</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div> which are one order (two orders) of magnitude stronger than the bounds predicted from the current LBL experiments NO<math display='inline'><semantics> <mi>ν</mi> </semantics></math>A and T2K on CPT-violating (CPT-conserving) LIV parameters in Ref. [<a href="#B41-universe-10-00357" class="html-bibr">41</a>]<a href="#fn005-universe-10-00357" class="html-fn">5</a>. For <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>, the authors of [<a href="#B41-universe-10-00357" class="html-bibr">41</a>] observed that there exists a strong correlation among them and the standard <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> </semantics></math>. This allows for a second minimum in the sensitivity analysis and thus weaker bounds on the parameters; however, given that we fix the atmospheric mixing angle to the lower octant, we will not be able to observe such a behavior. Neglecting this degeneracy, the standard DUNE limits obtained in [<a href="#B41-universe-10-00357" class="html-bibr">41</a>] for <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> are stronger than the ones in Equation (<a href="#FD9-universe-10-00357" class="html-disp-formula">9</a>) and are as follows:<div class='html-disp-formula-info' id='FD10-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>3.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> <mspace width="1.em"/> <mi>GeV</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>4.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> <mspace width="1.em"/> <mi>GeV</mi> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div></div></section><section id='sec3dot2-universe-10-00357' type=''><h4 class='html-italic' data-nested='2'> 3.2. HE-DUNE Results</h4><div class='html-p'>The HE configuration of the DUNE experiment, as already discussed in <a href="#sec2-universe-10-00357" class="html-sec">Section 2</a>, might allow accelerator neutrino energies from roughly 1 to 15 GeV. In <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a>, we show the electron appearance (left) and muon disappearance (right) oscillation probabilities with and without LIV. Black lines depict the SM oscillation probabilities computed using the best fits summarized in <a href="#universe-10-00357-t001" class="html-table">Table 1</a> [<a href="#B10-universe-10-00357" class="html-bibr">10</a>,<a href="#B74-universe-10-00357" class="html-bibr">74</a>]. Red lines have been obtained setting CP-odd LIV parameters <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> to <math display='inline'><semantics> <mrow> <mn>2.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> <mi>GeV</mi> </mrow> </semantics></math> in top, middle and bottom panels, respectively. Orange lines show the effect of <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> with a magnitude of <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> </semantics></math>. We also considered two extreme values of the corresponding LIV phase <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math>, namely <math display='inline'><semantics> <mn>0</mn> <mo>°</mo> </semantics></math> (dashed lines) and <math display='inline'><semantics> <mn>90</mn> <mo>°</mo> </semantics></math> (solid lines). The benchmark values of the LIV parameters have been chosen of the same order of magnitude as the DUNE limits obtained with the standard neutrino flux. The shaded regions correspond to the unoscillated standard (grey) and HE (green) DUNE fluxes in arbitrary units. It is clear that in the appearance case, the most important CPT-odd LIV parameters are <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>, as clearly visible in Equation (<a href="#FD7-universe-10-00357" class="html-disp-formula">7</a>). From the same equation, one can observe that, at the first oscillation maximum, the correction proportional to <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> is positive for both considered values of the LIV phase; on the other hand, it has a plus (minus) sign for <math display='inline'><semantics> <mrow> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>) when the <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> parameter is taken into account. Regarding CPT-even LIV parameters, their effects on <math display='inline'><semantics> <msub> <mi>P</mi> <mrow> <mi>μ</mi> <mi>e</mi> </mrow> </msub> </semantics></math> become more important for higher neutrino energies, above <math display='inline'><semantics> <mrow> <mo>∼</mo> <mn>4</mn> </mrow> </semantics></math> GeV. Such an energy is located at the high-energy tail of the standard flux and at the peak of the HE flux. For this reason, we expect HE-DUNE to be more sensitive to energy-enhanced CPT-even LIV parameters than DUNE in its standard configuration. We also observe that the most relevant parameter in this case is <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math>.</div><div class='html-p'>For the disappearance probability, as already mentioned and explicitly shown in Equation (<a href="#FD8-universe-10-00357" class="html-disp-formula">8</a>), the most important LIV parameters are <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>. In <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a>, we can also observe a slight sensitivity of this channel to the <math display='inline'><semantics> <mrow> <mi>e</mi> <mo>−</mo> <mi>μ</mi> </mrow> </semantics></math> parameters, especially in the high-energy region mostly covered by the HE-DUNE.</div><div class='html-p'>We now study the performances of DUNE in its high-energy configuration in constraining the LIV parameters. In order to perform our numerical analysis, we used the GLoBES software v3.2.18 [<a href="#B75-universe-10-00357" class="html-bibr">75</a>,<a href="#B76-universe-10-00357" class="html-bibr">76</a>] and its new physics tool [<a href="#B77-universe-10-00357" class="html-bibr">77</a>]. All the results have been obtained using a Poissonian <math display='inline'><semantics> <msup> <mi>χ</mi> <mn>2</mn> </msup> </semantics></math> defined as:<div class='html-disp-formula-info' id='FD11-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mover accent="true"> <mi>λ</mi> <mo>→</mo> </mover> <mo>,</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mn>2</mn> <mfenced separators="" open="(" close=")"> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>−</mo> <msub> <mi>O</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>O</mi> <mi>i</mi> </msub> <mo form="prefix">log</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>O</mi> <mi>i</mi> </msub> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> </mrow> </mfrac> </mstyle> </mfenced> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>a</mi> <mn>2</mn> </msup> <msubsup> <mi>σ</mi> <mi>a</mi> <mn>2</mn> </msubsup> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div> where <math display='inline'><semantics> <mover accent="true"> <mi>λ</mi> <mo>→</mo> </mover> </semantics></math> is the set of input oscillation parameters, <math display='inline'><semantics> <msub> <mi>σ</mi> <mi>a</mi> </msub> </semantics></math> is the normalization error, <span class='html-italic'>n</span> is the number of energy bins that we fixed to 80 [<a href="#B30-universe-10-00357" class="html-bibr">30</a>], <math display='inline'><semantics> <msub> <mi>O</mi> <mi>i</mi> </msub> </semantics></math> are the observed rates and <math display='inline'><semantics> <msub> <mi>T</mi> <mi>i</mi> </msub> </semantics></math> are the theoretical rates employed in the fit. The systematic uncertainties are included using the <span class='html-italic'>pull method</span> described in [<a href="#B78-universe-10-00357" class="html-bibr">78</a>,<a href="#B79-universe-10-00357" class="html-bibr">79</a>]. The validity of our analysis has been tested by reproducing the standard DUNE results on standard [<a href="#B30-universe-10-00357" class="html-bibr">30</a>] and non-standard oscillation physics [<a href="#B41-universe-10-00357" class="html-bibr">41</a>,<a href="#B80-universe-10-00357" class="html-bibr">80</a>,<a href="#B81-universe-10-00357" class="html-bibr">81</a>,<a href="#B82-universe-10-00357" class="html-bibr">82</a>]. Moreover, we checked that the number of events obtained with the DUNE high-energy flux was consistent with the one cited in [<a href="#B16-universe-10-00357" class="html-bibr">16</a>].</div><div class='html-p'>In <a href="#universe-10-00357-f002" class="html-fig">Figure 2</a>, we show the allowed <math display='inline'><semantics> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </semantics></math><math display='inline'><semantics> <mi>σ</mi> </semantics></math> contours in the <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>−</mo> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> (top) and <math display='inline'><semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>−</mo> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> planes (bottom). All the analyses have been performed fitting the data obtained using the SM parameters scanning over the LIV parameters one at a time. When computing the <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </semantics></math>, defined as:<div class='html-disp-formula-info' id='FD12-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <mo>Δ</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div> we fix all the not-showed LIV parameters to zero and we marginalize in the fit on all the oscillation parameters but the solar ones; we use Gaussian priors with the <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> uncertainties<a href="#fn006-universe-10-00357" class="html-fn">6</a> as summarized in <a href="#universe-10-00357-t001" class="html-table">Table 1</a> for <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>13</mn> </msub> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> while leaving <math display='inline'><semantics> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> </semantics></math> free to vary in all its <math display='inline'><semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo>)</mo> </mrow> <mo>°</mo> </semantics></math> range. The red lines in the bottom plots corresponding to <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> depict the effect of the inclusion in the analysis of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance channel. We have verified that, as expected, in all other LIV parameters, the effect of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance sample is marginal, a situation very similar to that discussed in [<a href="#B16-universe-10-00357" class="html-bibr">16</a>,<a href="#B18-universe-10-00357" class="html-bibr">18</a>], where a non-negligible impact of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> events was observed in the <math display='inline'><semantics> <mrow> <mi>μ</mi> <mo>−</mo> <mi>τ</mi> </mrow> </semantics></math> sector only in the context of propagation NSI.</div><div class='html-p'>The 95% limits on the moduli of the LIV parameters, obtained after marginalizing over the corresponding phases, are summarized in <a href="#universe-10-00357-t002" class="html-table">Table 2</a>. We observe that by fixing the octant and thus neglecting the degenerate LIV solutions found in [<a href="#B41-universe-10-00357" class="html-bibr">41</a>], the HE-DUNE limits on the <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> parameters are weaker than the standard DUNE ones. This mainly comes from the fact that the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance probability (strongly affected by <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>) and consequently the number of <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> events is larger at the first oscillation maximum than at the HE-DUNE energies. On the other hand, the limit on <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> is one order of magnitude more stringent in the HE-DUNE case due to the larger number of <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance events. However, notice that this oscillation channel is strongly affected by the atmospheric mixing angle and by matter effects [<a href="#B83-universe-10-00357" class="html-bibr">83</a>]; thus, the different procedure used in the LIV analysis in Ref. [<a href="#B41-universe-10-00357" class="html-bibr">41</a>], where <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math> has been left free to vary in its 3<math display='inline'><semantics> <mi>σ</mi> </semantics></math> allowed range, might have amplified the differences between their standard DUNE and our HE-DUNE results. In fact, we checked that using our same minimization procedure, the standard DUNE bound on <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> would only be worse than the HE-DUNE one by a factor of 2. The interplay between the magnitude of the LIV parameters and their phases is mostly visible in the case of <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> since the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance probability is directly proportional to <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo form="prefix">cos</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mrow> </semantics></math> at the leading order. For this reason, the sensitivity is substantially worse when <math display='inline'><semantics> <mrow> <mo form="prefix">cos</mo> <msub> <mo>Φ</mo> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mrow> </semantics></math>∼0. When considering the other two parameters <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> </semantics></math>, the strong correlations between the new physics phase and the standard phase <math display='inline'><semantics> <msub> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> </msub> </semantics></math> make the interpretation of the results as a function of <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> less clear. However, it can be seen that the limits on <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> are only marginally impacted by the value of the LIV phase.</div><div class='html-p'>As for the HE-DUNE limits on the CPT-even <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> LIV parameters, as expected, they are better than the standard DUNE ones since their effects are amplified by the neutrino energy; the only exception is given by <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>, which, for <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>∼<math display='inline'><semantics> <mn>90</mn> <mo>°</mo> </semantics></math>, experiences a worsening in the sensitivity; indeed, as can be seen in Equation (<a href="#FD7-universe-10-00357" class="html-disp-formula">7</a>) and <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a>, the probabilities when <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>∼<math display='inline'><semantics> <mn>90</mn> <mo>°</mo> </semantics></math> are very close to standard ones. Also in this case, except for <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (see Equation (<a href="#FD8-universe-10-00357" class="html-disp-formula">8</a>)), the role of the LIV phase on the <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> limits cannot be easily understood from analytical formulae, but our numerical results show that the limits on <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> do not drastically depend on the new <math display='inline'><semantics> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> phase value.</div></section></section><section id='sec4-universe-10-00357' type=''><h2 data-nested='1'> 4. The Long-Range Forces Case</h2><div class='html-p'>As is well known, the neutrino flavor transition can be strongly affected by the presence of a matter medium, which can induce an effective potential modifying the interaction Hamiltonian. However, the presence of BSM interactions between neutrinos and ordinary matter particles can in principle modify the matter potential term in the neutrino Hamiltonian. This is the case, for instance, of the widely studied vector and scalar Non-Standard Interactions (NSIs) [<a href="#B21-universe-10-00357" class="html-bibr">21</a>,<a href="#B84-universe-10-00357" class="html-bibr">84</a>,<a href="#B85-universe-10-00357" class="html-bibr">85</a>,<a href="#B86-universe-10-00357" class="html-bibr">86</a>,<a href="#B87-universe-10-00357" class="html-bibr">87</a>,<a href="#B88-universe-10-00357" class="html-bibr">88</a>,<a href="#B89-universe-10-00357" class="html-bibr">89</a>,<a href="#B90-universe-10-00357" class="html-bibr">90</a>,<a href="#B91-universe-10-00357" class="html-bibr">91</a>,<a href="#B92-universe-10-00357" class="html-bibr">92</a>]. Another interesting and less studied example is provided by the Long-Range Forces (LRFs) [<a href="#B93-universe-10-00357" class="html-bibr">93</a>,<a href="#B94-universe-10-00357" class="html-bibr">94</a>,<a href="#B95-universe-10-00357" class="html-bibr">95</a>,<a href="#B96-universe-10-00357" class="html-bibr">96</a>,<a href="#B97-universe-10-00357" class="html-bibr">97</a>,<a href="#B98-universe-10-00357" class="html-bibr">98</a>,<a href="#B99-universe-10-00357" class="html-bibr">99</a>,<a href="#B100-universe-10-00357" class="html-bibr">100</a>,<a href="#B101-universe-10-00357" class="html-bibr">101</a>,<a href="#B102-universe-10-00357" class="html-bibr">102</a>], which will be described in detail in this section.</div><section id='sec4dot1-universe-10-00357' type=''><h4 class='html-italic' data-nested='2'> 4.1. Theoretical Framework</h4><div class='html-p'>The SM gauge group can be extended by the anomaly-free combination of the <math display='inline'><semantics> <mrow> <mi>U</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> symmetries <math display='inline'><semantics> <msub> <mi>L</mi> <mi>e</mi> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>L</mi> <mi>μ</mi> </msub> </semantics></math>, <math display='inline'><semantics> <msub> <mi>L</mi> <mi>τ</mi> </msub> </semantics></math> and <span class='html-italic'>B</span> associated with lepton family number and baryon number. These combinations can be, for instance, <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> [<a href="#B103-universe-10-00357" class="html-bibr">103</a>,<a href="#B104-universe-10-00357" class="html-bibr">104</a>,<a href="#B105-universe-10-00357" class="html-bibr">105</a>]<a href="#fn007-universe-10-00357" class="html-fn">7</a>. Other combinations have been discussed in the context of neutrino oscillation in [<a href="#B97-universe-10-00357" class="html-bibr">97</a>]. The gauge boson of these symmetries is a massive neutral vector <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math> that can mediate new physics interactions between neutrinos and matter. In the case of a large mediator mass, it is possible to study its effect in an Effective Field Theory (EFT) approach, which results in neutrino propagation affected in a vector NSI fashion. If, on the other hand, the mediator is extremely light, the flavor-dependent interaction forces between neutrinos and matter might become important over large distances. Given the huge interaction distance, proportional to <math display='inline'><semantics> <mi>λ</mi> </semantics></math>∼<math display='inline'><semantics> <msubsup> <mi>m</mi> <mrow> <msup> <mi>Z</mi> <mo>′</mo> </msup> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </semantics></math>, the matter potential term affecting neutrino oscillation will depend on the matter contained within a radius <math display='inline'><semantics> <mi>λ</mi> </semantics></math>.</div><div class='html-p'>Let us first discuss the new interactions arising from these additional symmetries. In addition to the interactions mediated by the SM Z boson, for an <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>α</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>β</mi> </msub> </mrow> </semantics></math> symmetry, new Lagrangian terms for the <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math>-induced interactions can be written as:<div class='html-disp-formula-info' id='FD13-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi mathvariant="script">L</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mo>=</mo> <msubsup> <mi>g</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>Z</mi> <mi>μ</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>l</mi> <mo>¯</mo> </mover> <mi>α</mi> </msub> <msup> <mi>γ</mi> <mi>μ</mi> </msup> <msub> <mi>l</mi> <mi>α</mi> </msub> <mo>−</mo> <msub> <mover accent="true"> <mi>l</mi> <mo>¯</mo> </mover> <mi>β</mi> </msub> <msup> <mi>γ</mi> <mi>μ</mi> </msup> <msub> <mi>l</mi> <mi>β</mi> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>α</mi> </msub> <msup> <mi>γ</mi> <mi>μ</mi> </msup> <msub> <mi>P</mi> <mi>L</mi> </msub> <msub> <mi>ν</mi> <mi>α</mi> </msub> <mo>−</mo> <msub> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>β</mi> </msub> <msup> <mi>γ</mi> <mi>μ</mi> </msup> <msub> <mi>P</mi> <mi>L</mi> </msub> <msub> <mi>ν</mi> <mi>β</mi> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>α</mi> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>l</mi> <mi>α</mi> </msub> </semantics></math> are the neutrino and the charged lepton fields and <math display='inline'><semantics> <msub> <mi>P</mi> <mi>L</mi> </msub> </semantics></math> is the left-handed projection operator. There also exists a <math display='inline'><semantics> <mrow> <mi>Z</mi> <mo>−</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> </mrow> </semantics></math> mixed term <math display='inline'><semantics> <msub> <mi mathvariant="script">L</mi> <mi>mix</mi> </msub> </semantics></math> that can introduce new four-fermion interactions proportional to the coupling <math display='inline'><semantics> <mrow> <msubsup> <mi>g</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>ξ</mi> <mo>−</mo> <mo form="prefix">sin</mo> <msub> <mi>θ</mi> <mi>W</mi> </msub> <mi>χ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, where <math display='inline'><semantics> <mi>ξ</mi> </semantics></math> is the rotation angle between the gauge bosons eigenstates, <math display='inline'><semantics> <mi>χ</mi> </semantics></math> is their kinetic mixing angle and <math display='inline'><semantics> <msub> <mi>θ</mi> <mi>W</mi> </msub> </semantics></math> is the usual Weinberg angle [<a href="#B98-universe-10-00357" class="html-bibr">98</a>]. This term allows not only neutrino–lepton new interactions but also new contributions to the neutrino–nucleon scattering. However, since, at large distances, <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="script">L</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mo>≫</mo> <msub> <mi mathvariant="script">L</mi> <mi>mix</mi> </msub> </mrow> </semantics></math>, the mixed term is important only in the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> case, for which neutrino–electron scattering mediated by <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math> is prohibited.</div><div class='html-p'>All these new interactions involving <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math> induce a Yukawa-like potential coming from electrons and neutrons in the Universe which can affect neutrino oscillations [<a href="#B109-universe-10-00357" class="html-bibr">109</a>,<a href="#B110-universe-10-00357" class="html-bibr">110</a>]. For a neutrino at a distance <span class='html-italic'>d</span> from a source of a number <math display='inline'><semantics> <msub> <mi>N</mi> <mi>e</mi> </msub> </semantics></math> of electrons and for an <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>β</mi> </msub> </mrow> </semantics></math> symmetry, it can be written as [<a href="#B81-universe-10-00357" class="html-bibr">81</a>,<a href="#B96-universe-10-00357" class="html-bibr">96</a>,<a href="#B98-universe-10-00357" class="html-bibr">98</a>]:<div class='html-disp-formula-info' id='FD14-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>G</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> <mn>2</mn> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>N</mi> <mi>e</mi> </msub> <mrow> <mn>4</mn> <mi>π</mi> <mi>d</mi> </mrow> </mfrac> </mstyle> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mi>d</mi> </mrow> </msup> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div> Under the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> symmetry, instead, the LRF effect comes from the mixing between <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math> and the SM <span class='html-italic'>Z</span> boson. Considering the Universe to be neutral, the net potential, in this case, is only due to a <math display='inline'><semantics> <msub> <mi>N</mi> <mi>n</mi> </msub> </semantics></math> number of neutrons, which generate a potential of the form:<div class='html-disp-formula-info' id='FD15-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>G</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> <mn>2</mn> </msubsup> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>e</mi> <mrow> <mo form="prefix">sin</mo> <msub> <mi>θ</mi> <mi>W</mi> </msub> <mo form="prefix">cos</mo> <msub> <mi>θ</mi> <mi>W</mi> </msub> </mrow> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>N</mi> <mi>n</mi> </msub> <mrow> <mn>4</mn> <mi>π</mi> <mi>d</mi> </mrow> </mfrac> </mstyle> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mi>d</mi> </mrow> </msup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div> where <span class='html-italic'>e</span> is the electric charge. The effective coupling <math display='inline'><semantics> <msub> <mi>G</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> is the equivalent of the coupling <math display='inline'><semantics> <msubsup> <mi>g</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mo>′</mo> </msubsup> </semantics></math> in the case of an <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>β</mi> </msub> </mrow> </semantics></math> symmetry while, for the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, it is related to the Lagrangian parameters through the relation <math display='inline'><semantics> <mrow> <msub> <mi>G</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>g</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>ξ</mi> <mo>−</mo> <mo form="prefix">sin</mo> <msub> <mi>θ</mi> <mi>W</mi> </msub> <mi>χ</mi> <mo>)</mo> </mrow> </mrow> </msqrt> </mrow> </semantics></math>.</div><div class='html-p'>Let us now consider the effect of LRF on the neutrino oscillations. In general, as previously mentioned, the neutrino propagation Hamiltonian always contains the vacuum and the standard matter potential terms shown in Equation (<a href="#FD5-universe-10-00357" class="html-disp-formula">5</a>). The effect of the potential <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> is to add a new contribution of different structures depending on the considered symmetry:<div class='html-disp-formula-info' id='FD16-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mrow> <mi>L</mi> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>−</mo> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mspace width="1.em"/> <mi>for</mi> <mspace width="4.pt"/> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> <mo>,</mo> <mspace width="1.em"/> <msub> <mi>H</mi> <mrow> <mi>L</mi> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>−</mo> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mspace width="1.em"/> <mi>for</mi> <mspace width="4.pt"/> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mrow> <mi>L</mi> <mi>R</mi> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>−</mo> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mspace width="1.em"/> <mi>for</mi> <mspace width="4.pt"/> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div> Notice that the matrices in Equation (<a href="#FD16-universe-10-00357" class="html-disp-formula">16</a>) are similar to the standard matter potential; the main difference between the usual MSW contribution and the LRF one is that the former is a contact potential due to a very massive mediator (the SM <span class='html-italic'>Z</span> boson) while the latter encodes the effect of distant electron and neutrons sources on neutrino propagation due to an extremely light mediator (the <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math>) with a very large interaction length.</div><div class='html-p'>The computation of the oscillation probabilities is very cumbersome. In fact, in order to have observable effects of the long-range potentials, we need the quantity <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>E</mi> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> to be comparable to the vacuum oscillation frequency (<math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> <mo>/</mo> <mn>2</mn> <mi>E</mi> </mrow> </semantics></math> for LBL experiments); this implies that <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math>∼<math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>C</mi> <mi>C</mi> </mrow> </msub> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </semantics></math> eV, where <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>C</mi> <mi>C</mi> </mrow> </msub> </semantics></math> is the usual matter potential term. Since both potentials must be included in the evaluation, this produces very lengthy expressions for the transition probabilities. An example of analytical treatment of LRF has been carried out in [<a href="#B96-universe-10-00357" class="html-bibr">96</a>,<a href="#B99-universe-10-00357" class="html-bibr">99</a>,<a href="#B111-universe-10-00357" class="html-bibr">111</a>,<a href="#B112-universe-10-00357" class="html-bibr">112</a>]. Notice that, in principle, LRF probabilities can also be deduced from those computed in the presence of diagonal NSI parameters [<a href="#B96-universe-10-00357" class="html-bibr">96</a>,<a href="#B110-universe-10-00357" class="html-bibr">110</a>,<a href="#B112-universe-10-00357" class="html-bibr">112</a>]. From them, one can recognize that there exists a particular value of the neutrino energy for which a resonance occurs; in the case of <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>β</mi> </msub> </mrow> </semantics></math> symmetries, and neglecting the solar mass difference contribution, this condition reads [<a href="#B99-universe-10-00357" class="html-bibr">99</a>,<a href="#B111-universe-10-00357" class="html-bibr">111</a>]:<div class='html-disp-formula-info' id='FD17-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>E</mi> <mi>res</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> <mo form="prefix">cos</mo> <mn>2</mn> <msub> <mi>θ</mi> <mn>13</mn> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>V</mi> <mrow> <mi>C</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div> This means that the matter resonance occurs in the presence of LRF at lower energies with respect to the standard MSW case. As in the LIV case, LRF has been widely studied in the literature in the context of neutrino oscillation. Limits on the LRF potentials and on the effective coupling appearing in <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> were obtained, for example, in [<a href="#B41-universe-10-00357" class="html-bibr">41</a>,<a href="#B81-universe-10-00357" class="html-bibr">81</a>,<a href="#B96-universe-10-00357" class="html-bibr">96</a>,<a href="#B97-universe-10-00357" class="html-bibr">97</a>,<a href="#B100-universe-10-00357" class="html-bibr">100</a>,<a href="#B113-universe-10-00357" class="html-bibr">113</a>,<a href="#B114-universe-10-00357" class="html-bibr">114</a>,<a href="#B115-universe-10-00357" class="html-bibr">115</a>,<a href="#B116-universe-10-00357" class="html-bibr">116</a>,<a href="#B117-universe-10-00357" class="html-bibr">117</a>,<a href="#B118-universe-10-00357" class="html-bibr">118</a>,<a href="#B119-universe-10-00357" class="html-bibr">119</a>]. Focusing on the DUNE performances, the Fermilab-based experiment might set the following 95% CL limits on the LRF potentials [<a href="#B81-universe-10-00357" class="html-bibr">81</a>]:<div class='html-disp-formula-info' id='FD18-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>1.9</mn> <mphantom> <mn>0</mn> </mphantom> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi>eV</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>1.3</mn> <mphantom> <mn>0</mn> </mphantom> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi>eV</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>0.82</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi>eV</mi> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div> The strongest limit can be put on <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> because it strongly affects the disappearance channel, which has huge statistics in DUNE.</div></section><section id='sec4dot2-universe-10-00357' type=''><h4 class='html-italic' data-nested='2'> 4.2. HE-DUNE Results</h4><div class='html-p'>In this section, we will explore the capabilities of HE-DUNE to constrain the LRF potential and the limits it might set on the strength of the new forces as well as on the new mediator mass. First, in <a href="#universe-10-00357-f003" class="html-fig">Figure 3</a>, we plot the effect of long-range potential on the appearance and disappearance probabilities at the DUNE baseline.</div><div class='html-p'>The LRF potentials <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> have been set to <math display='inline'><semantics> <mrow> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math> eV to show their effects when the LRF is of the same order of magnitude as the standard matter potential. It is clear that the appearance probability is enhanced at the first oscillation maximum for all three cases due to the LRF-potential-induced resonances. In particular, <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> has the strongest effect while <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> has the mildest one. At higher energies, important for HE-DUNE, we observe that the <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> decreases the appearance probability while <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> increase it. On the other hand, the disappearance probability is enhanced at its first minimum, with <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> having the biggest impact for energies above 2.5 GeV. Notice that <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> has only a negligible effect on the disappearance probability.</div><div class='html-p'>Using the procedure described in <a href="#sec3dot2-universe-10-00357" class="html-sec">Section 3.2</a>, we estimated the bounds that HE-DUNE might be able to set on the LRF potentials. In <a href="#universe-10-00357-f004" class="html-fig">Figure 4</a>, we show the sensitivity to the three <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> as obtained from <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </semantics></math> as:<div class='html-disp-formula-info' id='FD19-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <mo>Δ</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div> We summarize the HE-DUNE 95% CL limits in <a href="#universe-10-00357-t003" class="html-table">Table 3</a>. Comparing them with those in Equation (<a href="#FD18-universe-10-00357" class="html-disp-formula">18</a>), we observe that the HE-DUNE could set bounds on <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>β</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> which are 20% and 35% weaker than the DUNE ones, respectively. The reason is that, even though the effect of matter potentials is in general increased for large neutrino energies, long-range potentials cause low-energy resonances in neutrino oscillation probabilities, which can be probed better at the standard DUNE energies.</div><div class='html-p'>One might use Equations (<a href="#FD14-universe-10-00357" class="html-disp-formula">14</a>) and (<a href="#FD15-universe-10-00357" class="html-disp-formula">15</a>) to find constraints to both the effective <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math> coupling and its mass. Following Refs. [<a href="#B81-universe-10-00357" class="html-bibr">81</a>,<a href="#B109-universe-10-00357" class="html-bibr">109</a>], if we want to consider all the matter content of the Universe, we need to take into account neutrinos from matter sources away up to <math display='inline'><semantics> <msup> <mn>10</mn> <mn>3</mn> </msup> </semantics></math> Gp, which corresponds to a <math display='inline'><semantics> <msup> <mi>Z</mi> <mo>′</mo> </msup> </semantics></math> mediator mass in the range <math display='inline'><semantics> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mrow> <mo>-</mo> <mo>-</mo> </mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>35</mn> </mrow> </msup> </mrow> </semantics></math> eV. Thus, we are dealing with an effective potential whose most important contributions are:<div class='html-disp-formula-info' id='FD20-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>Earth</mi> </msub> <mo>+</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>Moon</mi> </msub> <mo>+</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>Sun</mi> </msub> <mo>+</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mi>Milky</mi> <mspace width="0.166667em"/> <mi>Way</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>Cosmol</mi> </msub> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'>In Equation (<a href="#FD20-universe-10-00357" class="html-disp-formula">20</a>), we consider the Moon and the Sun as point-like electron and neutron sources, with <math display='inline'><semantics> <mrow> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mi>Moon</mi> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>Moon</mi> </msub> </mrow> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>49</mn> </msup> </mrow> </semantics></math> and <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mi>Sun</mi> </msub> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>4</mn> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>Sun</mi> </msub> </mrow> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>57</mn> </msup> </semantics></math> [<a href="#B81-universe-10-00357" class="html-bibr">81</a>]. On the other hand, we modeled the Earth as a continuous distribution with the same average density such as <math display='inline'><semantics> <mrow> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mi>Earth</mi> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>Earth</mi> </msub> </mrow> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>4</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>51</mn> </msup> </mrow> </semantics></math>; for the Milky Way, we divided the matter content in a thin disk, a thick disk, a central bulge and a diffuse gas, following the reasonings on Refs. [<a href="#B109-universe-10-00357" class="html-bibr">109</a>,<a href="#B120-universe-10-00357" class="html-bibr">120</a>,<a href="#B121-universe-10-00357" class="html-bibr">121</a>], with <math display='inline'><semantics> <mrow> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>Milky</mi> <mspace width="0.166667em"/> <mi>Way</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>Milky</mi> <mspace width="0.166667em"/> <mi>Way</mi> </mrow> </msub> </mrow> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>67</mn> </msup> </semantics></math>. Finally, the cosmological matter has been included in computing the whole potential described in [<a href="#B109-universe-10-00357" class="html-bibr">109</a>] at redshift z = 0 as suggested in [<a href="#B81-universe-10-00357" class="html-bibr">81</a>]; the total number of electrons and neutrons in this case has been fixed to <math display='inline'><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mi>Cosmol</mi> </msub> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>10</mn> <msub> <mrow> <mo>(</mo> <msub> <mi>N</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>Cosmol</mi> </msub> </mrow> </semantics></math>∼<math display='inline'><semantics> <msup> <mn>10</mn> <mn>79</mn> </msup> </semantics></math>. Once all the contributions to the long-range potential are estimated, one can constrain both <math display='inline'><semantics> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>G</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> using the limits on <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> in <a href="#universe-10-00357-t003" class="html-table">Table 3</a>. Our results are shown in <a href="#universe-10-00357-f005" class="html-fig">Figure 5</a> for the three cases <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> (green line), <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> (blue line) and <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> (red line). In the upper part of the plot, we show the interaction length <math display='inline'><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> </mrow> </semantics></math> corresponding to the given mediator mass. The grey vertical bands show the parameter space excluded by two phenomena [<a href="#B81-universe-10-00357" class="html-bibr">81</a>]: black-hole superradiance and weak gravity conjecture. The former is related to the superradiant growth of an accumulation of very light vector bosons around extremely massive and gravitational bounded objects like supermassive black holes [<a href="#B122-universe-10-00357" class="html-bibr">122</a>]. The latter is related to a lower limit which might be set on the coupling in theories containing both gravity as the weakest force and a <math display='inline'><semantics> <mrow> <mi>U</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> gauge interaction [<a href="#B123-universe-10-00357" class="html-bibr">123</a>].</div><div class='html-p'>We can observe that, at the specific distance at which the electron and neutron biggest sources are located, the limits on the effective couplings become stronger. In particular, for mediator masses lower than <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>33</mn> </mrow> </msup> </semantics></math> eV, which correspond to roughly 10 Gpc (where the causal horizon is located), the bounds on <math display='inline'><semantics> <msub> <mi>G</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> become as low as <math display='inline'><semantics> <mrow> <mn>2.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>29</mn> </mrow> </msup> </mrow> </semantics></math> for <math display='inline'><semantics> <msub> <mi>G</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>, <math display='inline'><semantics> <mrow> <mn>2.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>29</mn> </mrow> </msup> </mrow> </semantics></math> for <math display='inline'><semantics> <msub> <mi>G</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <mrow> <mn>7.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>29</mn> </mrow> </msup> </mrow> </semantics></math> for <math display='inline'><semantics> <msub> <mi>G</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>. The latter is the weakest limit since it depends on <math display='inline'><semantics> <msub> <mi>N</mi> <mi>n</mi> </msub> </semantics></math>, which is smaller than <math display='inline'><semantics> <msub> <mi>N</mi> <mi>e</mi> </msub> </semantics></math> for cosmological and solar matter, despite corresponding to <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> which is the potential bounded the most by HE-DUNE.</div></section></section><section id='sec5-universe-10-00357' type=''><h2 data-nested='1'> 5. The Large Extra Dimensions Case</h2><div class='html-p'>Neutrino oscillation is not predicted in the original version of the SM since the Higgs mechanism is not capable of providing non-zero neutrino masses and their smallness, compared to the other fermions, is difficult to contemplate in a theoretical general framework. These difficulties can be overcome by several BSM models [<a href="#B124-universe-10-00357" class="html-bibr">124</a>,<a href="#B125-universe-10-00357" class="html-bibr">125</a>]; among them, the Large Extra Dimensions (LEDs) theory [<a href="#B126-universe-10-00357" class="html-bibr">126</a>,<a href="#B127-universe-10-00357" class="html-bibr">127</a>,<a href="#B128-universe-10-00357" class="html-bibr">128</a>,<a href="#B129-universe-10-00357" class="html-bibr">129</a>,<a href="#B130-universe-10-00357" class="html-bibr">130</a>,<a href="#B131-universe-10-00357" class="html-bibr">131</a>,<a href="#B132-universe-10-00357" class="html-bibr">132</a>,<a href="#B133-universe-10-00357" class="html-bibr">133</a>] not only provides a viable framework but also supplies an explanation for the enormous difference between the Electroweak and the Planck scale [<a href="#B134-universe-10-00357" class="html-bibr">134</a>,<a href="#B135-universe-10-00357" class="html-bibr">135</a>]. The main idea is to introduce sterile right-handed neutrino fields, which are singlet under the SM group but propagate in a <math display='inline'><semantics> <mrow> <mn>4</mn> <mo>+</mo> <msub> <mi>N</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math> dimensional space-time, where <math display='inline'><semantics> <msub> <mi>N</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> is the number of space-like extra dimensions. Usual Yukawa Lagrangian terms can then be built using left-handed neutrinos and Higgs fields which live in the usual 4-dimensional space-time. However, the masses coming from such terms are heavily suppressed with respect to other fermion masses due to the much smaller wave function normalizations in the large volume of the extra dimensions. In the following, we will explore the phenomenological implications of the existence of LED on neutrino oscillation.</div><section id='sec5dot1-universe-10-00357' type=''><h4 class='html-italic' data-nested='2'> 5.1. Theoretical Framework</h4><div class='html-p'>Following the approach proposed for neutrino oscillation studies in the context of LED [<a href="#B82-universe-10-00357" class="html-bibr">82</a>,<a href="#B136-universe-10-00357" class="html-bibr">136</a>,<a href="#B137-universe-10-00357" class="html-bibr">137</a>,<a href="#B138-universe-10-00357" class="html-bibr">138</a>,<a href="#B139-universe-10-00357" class="html-bibr">139</a>,<a href="#B140-universe-10-00357" class="html-bibr">140</a>,<a href="#B141-universe-10-00357" class="html-bibr">141</a>,<a href="#B142-universe-10-00357" class="html-bibr">142</a>,<a href="#B143-universe-10-00357" class="html-bibr">143</a>,<a href="#B144-universe-10-00357" class="html-bibr">144</a>,<a href="#B145-universe-10-00357" class="html-bibr">145</a>,<a href="#B146-universe-10-00357" class="html-bibr">146</a>], we will focus on <math display='inline'><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; this single LED is compactified on a circle of radius <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> and, for the sterile neutrinos, this gives rise to Kaluza–Klein (KK) modes in the 4-dimensional space-time. It is also possible to consider the presence of more extra dimensions, whose compactification radius is much smaller than <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> without changing the effect of LED on the neutrino oscillation. The model is built by adding three massless five-dimensional fermion fields <math display='inline'><semantics> <mrow> <msup> <mi>Ψ</mi> <mi>α</mi> </msup> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>ψ</mi> <mi>L</mi> <mi>α</mi> </msubsup> <mo>,</mo> <msubsup> <mi>ψ</mi> <mi>R</mi> <mi>α</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics></math> to the SM. After the compactification of the fifth dimension with the proper boundary conditions, the <math display='inline'><semantics> <msup> <mi>Ψ</mi> <mi>α</mi> </msup> </semantics></math> fields appear in the 4-dimensional space-time as an infinite tower of KK states <math display='inline'><semantics> <msup> <mi>ψ</mi> <mi>n</mi> </msup> </semantics></math>, where <span class='html-italic'>n</span> is any integer number. Identifying the zero mode as the right-handed neutrinos <math display='inline'><semantics> <mrow> <msubsup> <mi>ν</mi> <mi>R</mi> <mi>α</mi> </msubsup> <mo>=</mo> <msubsup> <mi>ψ</mi> <mi>R</mi> <mrow> <mi>α</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> </mrow> </semantics></math> and writing <math display='inline'><semantics> <mrow> <msubsup> <mi>ν</mi> <mrow> <mi>L</mi> <mo>,</mo> <mi>R</mi> </mrow> <mrow> <mi>α</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>ψ</mi> <mrow> <mi>L</mi> <mo>,</mo> <mi>R</mi> </mrow> <mrow> <mi>α</mi> <mo>(</mo> <mo>−</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow> <mi>L</mi> <mo>,</mo> <mi>R</mi> </mrow> <mrow> <mi>α</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </semantics></math> [<a href="#B82-universe-10-00357" class="html-bibr">82</a>], the neutrino mass term is [<a href="#B82-universe-10-00357" class="html-bibr">82</a>,<a href="#B127-universe-10-00357" class="html-bibr">127</a>,<a href="#B130-universe-10-00357" class="html-bibr">130</a>,<a href="#B134-universe-10-00357" class="html-bibr">134</a>]:<div class='html-disp-formula-info' id='FD21-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi mathvariant="script">L</mi> <mi>LED</mi> </msub> <mo>=</mo> <msubsup> <mi>m</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>D</mi> </msubsup> <mfenced separators="" open="(" close=")"> <msubsup> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mi>R</mi> <mi>α</mi> </msubsup> <msubsup> <mi>ν</mi> <mi>L</mi> <mi>β</mi> </msubsup> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> <msubsup> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mrow> <mi>R</mi> </mrow> <mrow> <mi>α</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <msubsup> <mi>ν</mi> <mi>L</mi> <mi>β</mi> </msubsup> </mfenced> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>n</mi> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mfrac> </mstyle> <msubsup> <mover accent="true"> <mi>ν</mi> <mo>¯</mo> </mover> <mrow> <mi>R</mi> </mrow> <mrow> <mi>α</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <msubsup> <mi>ν</mi> <mi>L</mi> <mrow> <mi>α</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <mi>h</mi> <mo>.</mo> <mi>c</mi> <mo>.</mo> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(21)</label> </div> </div> where <math display='inline'><semantics> <msubsup> <mi>m</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>D</mi> </msubsup> </semantics></math> is the Dirac mass matrix. Rewriting the mass eigenstates as <math display='inline'><semantics> <mrow> <msup> <mi>N</mi> <mi>i</mi> </msup> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <msup> <mi>ν</mi> <mrow> <mi>i</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>ν</mi> <mrow> <mi>i</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math>, then <math display='inline'><semantics> <msub> <mi mathvariant="script">L</mi> <mi>LED</mi> </msub> </semantics></math> can be written as <math display='inline'><semantics> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>3</mn> </msubsup> <msubsup> <mover accent="true"> <mi>N</mi> <mo>¯</mo> </mover> <mi>R</mi> <mi>i</mi> </msubsup> <msup> <mi>M</mi> <mi>i</mi> </msup> <msubsup> <mi>N</mi> <mi>L</mi> <mi>i</mi> </msubsup> </mrow> </semantics></math>, where the infinite mass matrices is:<div class='html-disp-formula-info' id='FD22-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>=</mo> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <msub> <mi>m</mi> <mi>i</mi> </msub> </mtd> <mtd> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>m</mi> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>m</mi> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>m</mi> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mn>1</mn> <mo>/</mo> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mn>2</mn> <mo>/</mo> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow> <mi>n</mi> <mo>/</mo> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> </mtr> </mtable> </mfenced> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(22)</label> </div> </div> in which the <math display='inline'><semantics> <msub> <mi>m</mi> <mi>i</mi> </msub> </semantics></math> are the eigenvalues of the neutrino Dirac mass matrix <math display='inline'><semantics> <msubsup> <mi>m</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> <mi>D</mi> </msubsup> </semantics></math> in Equation (<a href="#FD21-universe-10-00357" class="html-disp-formula">21</a>). The neutrino mixing among the active states is then defined by Equation [<a href="#B82-universe-10-00357" class="html-bibr">82</a>,<a href="#B145-universe-10-00357" class="html-bibr">145</a>] <div class='html-disp-formula-info' id='FD23-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ν</mi> <mi>α</mi> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>3</mn> </munderover> <msub> <mi>U</mi> <mrow> <mi>α</mi> <mi>i</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </munderover> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msubsup> <mi>ν</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(23)</label> </div> </div> where <span class='html-italic'>U</span> is the PMNS mixing matrix and <span class='html-italic'>V</span> is the <span class='html-italic'>effective mixing matrix</span> among the KK excitations. Its elements can be written as:<div class='html-disp-formula-info' id='FD24-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>π</mi> <mn>2</mn> </msup> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>λ</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>/</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(24)</label> </div> </div> with <math display='inline'><semantics> <msubsup> <mi>λ</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> being the eigenvalues of the matrices <math display='inline'><semantics> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>M</mi> <mi>i</mi> <mo>†</mo> </msubsup> <msub> <mi>M</mi> <mi>i</mi> </msub> </mrow> </semantics></math> that can be found as solutions of Equation [<a href="#B82-universe-10-00357" class="html-bibr">82</a>,<a href="#B127-universe-10-00357" class="html-bibr">127</a>,<a href="#B130-universe-10-00357" class="html-bibr">130</a>,<a href="#B137-universe-10-00357" class="html-bibr">137</a>,<a href="#B145-universe-10-00357" class="html-bibr">145</a>] <div class='html-disp-formula-info' id='FD25-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>λ</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>−</mo> <mi>π</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo form="prefix">cot</mo> <mfenced separators="" open="(" close=")"> <mi>π</mi> <msubsup> <mi>λ</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> </mfenced> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(25)</label> </div> </div> Notice that the masses of the KK states are in this case <math display='inline'><semantics> <mrow> <msubsup> <mi>m</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>λ</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>/</mo> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math>; since the solutions of Equation (<a href="#FD25-universe-10-00357" class="html-disp-formula">25</a>) satisfy the relation <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>≤</mo> <msubsup> <mi>λ</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo><</mo> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, we can roughly say that <math display='inline'><semantics> <msubsup> <mi>m</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math>∼<math display='inline'><semantics> <mrow> <mi>n</mi> <mo>/</mo> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math> [<a href="#B82-universe-10-00357" class="html-bibr">82</a>]. Once we fix the experimental observation of the mass splittings to be equal to the differences <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>21</mn> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>λ</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>λ</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> </mrow> <mo>/</mo> <msubsup> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>λ</mi> <mn>3</mn> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>λ</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>]</mo> </mrow> <mo>/</mo> <msubsup> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>, we are left with the standard three neutrinos mixing modified by the effect of the mixing between the active neutrinos and an infinite number of sterile neutrinos. The only non-standard parameters of the model are then the compactification radius <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> and the smallest Dirac mass <math display='inline'><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math>. The oscillation probabilities can be obtained, in a vacuum, as <div class='html-disp-formula-info' id='FD26-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="" open="|" close="|"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>3</mn> </munderover> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mo>∞</mo> </munderover> <msubsup> <mi>U</mi> <mrow> <mi>α</mi> <mi>i</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mi>U</mi> <mrow> <mi>β</mi> <mi>i</mi> </mrow> </msub> <msubsup> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mspace width="0.166667em"/> <mi>exp</mi> <mfenced separators="" open="(" close=")"> <mo>−</mo> <mi>i</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>m</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>L</mi> </mrow> <mrow> <mn>2</mn> <mi>E</mi> </mrow> </mfrac> </mstyle> </mfenced> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(26)</label> </div> </div> where <span class='html-italic'>L</span> is the baseline. In the limit <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math>, we observe that <math display='inline'><semantics> <mrow> <msubsup> <mi>m</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math> for <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>→</mo> <msub> <mi>δ</mi> <mrow> <mn>0</mn> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, making the oscillation phenomenology identical to the standard one.</div><div class='html-p'>It has been shown that, apart from the appearance of new matter resonances at high neutrino energies [<a href="#B136-universe-10-00357" class="html-bibr">136</a>] (<math display='inline'><semantics> <mrow> <mi>E</mi> <mo>≫</mo> <mn>1</mn> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi>TeV</mi> </mrow> </semantics></math>), the LED phenomenology does not change significantly if we include more than two KK modes. Indeed, higher modes would imply larger masses <math display='inline'><semantics> <msubsup> <mi>m</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> and smaller matrix elements <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> [<a href="#B145-universe-10-00357" class="html-bibr">145</a>]. Thus, one might study neutrino oscillation in the presence of LED including in the model only a limited number of KK modes. In this context, it is possible to treat the LED case with a number <math display='inline'><semantics> <msub> <mi>n</mi> <mi>KK</mi> </msub> </semantics></math> of KK modes as a <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>+</mo> <mn>3</mn> <msub> <mi>n</mi> <mi>KK</mi> </msub> </mrow> </semantics></math> sterile neutrino model [<a href="#B82-universe-10-00357" class="html-bibr">82</a>,<a href="#B136-universe-10-00357" class="html-bibr">136</a>], where all the non-standard mixing angles and mass splittings can be written in terms of the two LED parameters<a href="#fn008-universe-10-00357" class="html-fn">8</a>.</div><div class='html-p'>The expression of the oscillation probabilities in the LED case is very cumbersome; however, in the limit <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math>, some expansions for the mass eigenstates and mixing matrix elements have been obtained in Refs. [<a href="#B131-universe-10-00357" class="html-bibr">131</a>,<a href="#B145-universe-10-00357" class="html-bibr">145</a>]:<div class='html-disp-formula-info' id='FD27-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <msubsup> <mi>m</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <mfenced separators="" open="[" close="]"> <mn>1</mn> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>π</mi> <mn>2</mn> </msup> <mn>6</mn> </mfrac> </mstyle> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mo>⋯</mo> </mfenced> <mo>∼</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <msubsup> <mi>m</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>n</mi> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <mn>1</mn> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>n</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mo>+</mo> <mo>⋯</mo> </mfenced> <mo>∼</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>n</mi> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>π</mi> <mn>2</mn> </msup> <mn>6</mn> </mfrac> </mstyle> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mo>⋯</mo> <mo>∼</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mstyle> <mfenced separators="" open="[" close="]"> <mn>1</mn> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>n</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mo>+</mo> <mo>⋯</mo> </mfenced> <mo>∼</mo> <msqrt> <mn>2</mn> </msqrt> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(27)</label> </div> </div> from which it is clear that, as already mentioned, the corrections to the standard oscillation case become negligible as <span class='html-italic'>n</span> increases.</div><div class='html-p'>The complete picture of neutrino oscillation in the presence of LED is obtained when matter effects are added. Then, the oscillation probabilities can be obtained by solving the Schroedinger-like evolution Equation [<a href="#B82-universe-10-00357" class="html-bibr">82</a>] <div class='html-disp-formula-info' id='FD28-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <mi>i</mi> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mstyle> <msub> <mi>N</mi> <mi>i</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>E</mi> </mrow> </mfrac> </mstyle> <msubsup> <mi>M</mi> <mi>i</mi> <mo>†</mo> </msubsup> <msub> <mi>M</mi> <mi>i</mi> </msub> <msub> <mi>N</mi> <mi>i</mi> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>3</mn> </munderover> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow> <mi>n</mi> <mo>→</mo> <mo>∞</mo> </mrow> </munder> <mfenced open="(" close=")"> <mtable> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <msub> <mn>0</mn> <mrow> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow> <msub> <mn>0</mn> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>N</mi> <mi>j</mi> </msub> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(28)</label> </div> </div> where the <math display='inline'><semantics> <msub> <mi>N</mi> <mi>i</mi> </msub> </semantics></math> infinite vector of neutrino states and <math display='inline'><semantics> <msub> <mi>M</mi> <mi>i</mi> </msub> </semantics></math> matrices have already been defined in Equation (<a href="#FD22-universe-10-00357" class="html-disp-formula">22</a>) and above, while the quantity <math display='inline'><semantics> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </semantics></math> is defined as <div class='html-disp-formula-info' id='FD29-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mi>α</mi> </munder> <msubsup> <mi>U</mi> <mrow> <mi>α</mi> <mi>i</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mi>U</mi> <mrow> <mi>α</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>δ</mi> <mrow> <mi>α</mi> <mi>e</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow> <mi>C</mi> <mi>C</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>C</mi> </mrow> </msub> <mo>)</mo> </mrow> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(29)</label> </div> </div> with <span class='html-italic'>U</span> being the standard <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </semantics></math> PMNS matrix, <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>C</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>G</mi> <mi>F</mi> </msub> <msub> <mi>n</mi> <mi>e</mi> </msub> </mrow> </semantics></math> the usual matter potential and <math display='inline'><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>N</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>G</mi> <mi>F</mi> </msub> <msub> <mi>n</mi> <mi>n</mi> </msub> </mrow> </semantics></math> the neutral current matter potential that can no longer be neglected due to the presence of sterile states.</div><div class='html-p'>The limits that current experiments could set on the parameter space in the LED case have been discussed, for instance, in Ref. [<a href="#B145-universe-10-00357" class="html-bibr">145</a>]. Notice also that the short baseline, reactor and gallium anomalies, which have been explained with the presence of light sterile neutrinos, could be explained in the presence of LEDs [<a href="#B82-universe-10-00357" class="html-bibr">82</a>,<a href="#B142-universe-10-00357" class="html-bibr">142</a>,<a href="#B145-universe-10-00357" class="html-bibr">145</a>]. In the context of the future DUNE experiment, the expected performances of the experiment have been explored in detail in Ref. [<a href="#B82-universe-10-00357" class="html-bibr">82</a>]. The upper limits on the <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> parameter depend on the lightest neutrino mass value; in particular, in the scenario when <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> eV and the probabilities only depend on <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math>, we expect for DUNE at 95% CL:<div class='html-disp-formula-info' id='FD30-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>0.32</mn> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> <mspace width="0.166667em"/> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(30)</label> </div> </div> while for <math display='inline'><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math>∼<math display='inline'><semantics> <mrow> <mn>0.05</mn> </mrow> </semantics></math> eV, we expect <div class='html-disp-formula-info' id='FD31-universe-10-00357'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </mtd> <mtd columnalign="left"> <mrow> <mo><</mo> <mn>0.22</mn> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> <mspace width="0.166667em"/> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(31)</label> </div> </div> On the other hand, it is impossible to set an absolute limit on <math display='inline'><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> unless data are generated considering <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math>; in that case, DUNE might set a lower limit on the absolute neutrino mass [<a href="#B82-universe-10-00357" class="html-bibr">82</a>]<a href="#fn009-universe-10-00357" class="html-fn">9</a>.</div></section><section id='sec5dot2-universe-10-00357' type=''><h4 class='html-italic' data-nested='2'> 5.2. HE-DUNE Results</h4><div class='html-p'>In this section, we discuss the effects of LED at HE-DUNE. For the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance and <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance probabilities (see <a href="#universe-10-00357-f006" class="html-fig">Figure 6</a>), we consider as a reference two possible lightest neutrino masses <math display='inline'><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math>, namely <math display='inline'><semantics> <mrow> <mn>0.0</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mn>0.05</mn> </mrow> </semantics></math> eV. For the compactification radius, we choose <math display='inline'><semantics> <mrow> <mn>0.5</mn> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, which corresponds to <math display='inline'><semantics> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mn>0.38</mn> <mspace width="0.166667em"/> <msup> <mi>eV</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>. In our computations, we included three KK modes, even though we checked that our results were only negligibly affected by the inclusion of the second and third KK modes. From <a href="#universe-10-00357-f006" class="html-fig">Figure 6</a>, we see that the main effect of LED is the occurrence of new fast oscillations driven by the large mass splittings between the active states and the heavy KK excitations. The amplitude of the oscillations depends on the values of the <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> matrix elements. In addition, the presence of LED decreases both the appearance and disappearance probabilities at the first oscillation maximum since the value of <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> </semantics></math> is always less than 1; see Equation (<a href="#FD27-universe-10-00357" class="html-disp-formula">27</a>). Moreover, at a fixed <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math>, the probabilities in general decrease as <math display='inline'><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> increases. The differences between probabilities around the first oscillation maximum (DUNE) and the high-energy region (HE-DUNE) are mainly due to the fact that fast oscillation driven by KK becomes slower and with a larger amplitude when the neutrino energy increases. For this reason, with high-energy neutrinos, it might be possible to resolve these oscillations allowing for better constraints on the LED model [<a href="#B73-universe-10-00357" class="html-bibr">73</a>].</div><div class='html-p'>In <a href="#universe-10-00357-f007" class="html-fig">Figure 7</a>, we show the allowed parameters space in the <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </semantics></math> plane in the HE-DUNE case. The analysis has been performed using the procedure described in the previous two sections. Here, we can see that at <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> the weakest limit for <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> reached for <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> is <math display='inline'><semantics> <mrow> <mn>0.258</mn> </mrow> </semantics></math> <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>. This constraint is better than the standard DUNE one. As noted above, one of the main reasons for that is the possibility of recognizing fast oscillations at higher energies, where they occur with a smaller frequency with respect to the lower energy region probed by standard DUNE. Another interesting feature is that for <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo><</mo> <mn>0.04</mn> </mrow> </semantics></math> eV, the HE-DUNE constraint on <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> becomes independent of the lightest neutrino mass. This means that if <math display='inline'><semantics> <msub> <mi>m</mi> <mn>1</mn> </msub> </semantics></math> is, for instance, <math display='inline'><semantics> <mrow> <mn>0.05</mn> </mrow> </semantics></math> eV, then the standard DUNE experiment outperforms HE-DUNE (see Equation (<a href="#FD31-universe-10-00357" class="html-disp-formula">31</a>)).</div></section></section><section id='sec6-universe-10-00357' type='conclusions'><h2 data-nested='1'> 6. Conclusions</h2><div class='html-p'>Neutrino oscillation is the best-established phenomenon beyond the Standard Model of particle physics. Despite several experiments that have been able to measure the oscillation parameters with a few percent uncertainty, there are still some unknowns like the neutrino mass hierarchy and the amount of leptonic CP violation (if any). Moreover, oscillation searches are of great interest since several new physics models can affect neutrino propagation and thus modify the related probabilities. Future oscillation facilities are expected to reach great precision in the measurements of the mixing parameters; at the same time, they could provide a great probe for new physics models involving neutrinos. In this work, we considered the capabilities of the high-energy flux configuration of the future DUNE experiment (HE-DUNE). The possibility of employing this broad neutrino flux, which might reach more than 15 GeV in energy, has been envisaged in order to have access to <math display='inline'><semantics> <mrow> <msub> <mi>ν</mi> <mi>μ</mi> </msub> <mo>→</mo> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> oscillations, which could not be easily observed with the standard DUNE configuration due to the energy threshold of CC <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> interactions. In addition, HE-DUNE could in principle also be very useful to constrain new physics scenarios where the non-standard oscillation effects are more pronounced at high energies. Among them, Non-Standard Interactions, sterile neutrinos, non-unitarity of the PMNS matrix and quantum decoherence have already been studied in the literature [<a href="#B16-universe-10-00357" class="html-bibr">16</a>,<a href="#B17-universe-10-00357" class="html-bibr">17</a>,<a href="#B18-universe-10-00357" class="html-bibr">18</a>,<a href="#B19-universe-10-00357" class="html-bibr">19</a>,<a href="#B20-universe-10-00357" class="html-bibr">20</a>,<a href="#B21-universe-10-00357" class="html-bibr">21</a>].</div><div class='html-p'>In this work, we considered three different new physics models:</div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'><b>Lorentz Invariance Violation (LIV)</b>: in this model, the neutrino Lagrangian density is modified through several Lorentz-violating operators, both CPT-even and CPT-odd. The presence of such operators modifies the neutrino propagation Hamiltonian with the addition of two Hermitian matrices <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> (CPT-violating) and <math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> (CPT-conserving). The effects of the second matrix increase linearly with the neutrino energy. We studied in <a href="#sec3dot2-universe-10-00357" class="html-sec">Section 3.2</a> the sensitivity of HE-DUNE to the off-diagonal LIV parameters. We found that the limits on the moduli of CPT-violating parameters <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> are worse than the ones that the standard DUNE is expected to set. On the other hand, HE-DUNE capabilities should exceed the standard DUNE ones in constraining energy-enhanced effects of CPT-conserving LIV parameters <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>.</div></li><li><div class='html-p'><b>Long-Range Forces (LRFs)</b>: in this model, we expect that new interactions with an ultra-light mediator, with a very long interaction length, arise from a gauge <math display='inline'><semantics> <mrow> <mi>U</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> symmetry of the form <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>α</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>β</mi> </msub> </mrow> </semantics></math>. These interactions can modify the matter potential term in the neutrino oscillation Hamiltonian. We showed that the limits from HE-DUNE on this new potential are rather stringent but not enough to overcome the standard DUNE ones. We also computed the limits on the coupling of the new interaction as well as on the mass of the new mediator. These are correlated with the interaction length since, depending on that, neutrinos might experience the potential generated from various astrophysical matter densities.</div></li><li><div class='html-p'><b>Large Extra Dimensions (LEDs)</b>: if right-handed neutrinos are singlets under the SM group, but they can propagate in space-time with more than four dimensions, the smallness of neutrino masses can be naturally explained. In the case in which one of the new dimensions is compactified in a sphere with a relatively large radius, the Kaluza–Klein excitations of the neutrino states can be treated as sterile neutrinos involved in the oscillation. In this approach, the transition probabilities depend not only on the standard mixing parameters but also on the smallest Dirac neutrino mass and on the compactification radius <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math> of the large extra dimension. We showed that the limit that HE-DUNE might set on <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math>, for small enough lightest neutrino mass, is better than the standard DUNE one. This is because the fast active–sterile oscillations coming from the Kaluza–Klein states might be resolved better at high energies than at lower ones.</div></li></ul></div><div class='html-p'>In conclusion, the DUNE high-energy flux might be useful not only to collect a large sample of <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> events but also to set stringent limits on new physics parameters. This suggests that an HE-DUNE run could provide much information on the BSM neutrino physics, which are complementary to the standard DUNE ones.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization: A.G., S.M. and D.M.; Methodology: A.G., S.M. and D.M.; Formal analysis: A.G., S.M. and D.M.; Writing: A.G., S.M. and D.M. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>This research received no external funding.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>Data are contained within the article.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflicts of interest.</div></section><section class='html-new-fn_group'><table><h2 >Notes</h2><tr id='fn001-universe-10-00357'><td><span class='html-label'>1</span></td><td><div class='html-p'>For the other two oscillation channels, namely <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance and <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance, the collaboration suggested 2% and 5%, respectively.</div></td></tr><tr id='fn002-universe-10-00357'><td><span class='html-label'>2</span></td><td><div class='html-p'>In Quantum Field Theory the <span class='html-italic'>CPT theorem</span> states that the combination of the discrete transformations “Chargeconjugation” (C), “Parity” (P) and “Timereversal” (T) must be a symmetry of the theory.</div></td></tr><tr id='fn003-universe-10-00357'><td><span class='html-label'>3</span></td><td><div class='html-p'>The LIV probabilities shown here have been obtained in [<a href="#B41-universe-10-00357" class="html-bibr">41</a>] neglecting also terms proportional to <math display='inline'><semantics> <msup> <mi>α</mi> <mn>2</mn> </msup> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>α</mi> <mo form="prefix">sin</mo> <msub> <mi>θ</mi> <mn>13</mn> </msub> </mrow> </semantics></math> where <math display='inline'><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>21</mn> </mrow> <mn>2</mn> </msubsup> <mo>/</mo> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math>.</div></td></tr><tr id='fn004-universe-10-00357'><td><span class='html-label'>4</span></td><td><div class='html-p'>The <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance probability can be obtained from unitarity and the leading term will be the one depending again on <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>).</div></td></tr><tr id='fn005-universe-10-00357'><td><span class='html-label'>5</span></td><td><div class='html-p'>Notice that limits from the more energetic atmospheric and astrophysical neutrinos on CPT-even LIV parameters are more stringent than the DUNE ones due to the dependence on the neutrino energy of their effect on the oscillation probabilities.</div></td></tr><tr id='fn006-universe-10-00357'><td><span class='html-label'>6</span></td><td><div class='html-p'>In our analysis, we assume that the octant of <math display='inline'><semantics> <msub> <mi>θ</mi> <mn>23</mn> </msub> </semantics></math> is known; in particular, as suggested by global fits in [<a href="#B10-universe-10-00357" class="html-bibr">10</a>,<a href="#B74-universe-10-00357" class="html-bibr">74</a>], to lie in the lower octant.</div></td></tr><tr id='fn007-universe-10-00357'><td><span class='html-label'>7</span></td><td><div class='html-p'>In [<a href="#B106-universe-10-00357" class="html-bibr">106</a>,<a href="#B107-universe-10-00357" class="html-bibr">107</a>,<a href="#B108-universe-10-00357" class="html-bibr">108</a>], it has been shown that <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>α</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>β</mi> </msub> </mrow> </semantics></math> gauge symmetries can predict viable neutrino masses and mixing with the addition of Higgs-like particles charged under the new symmetries.</div></td></tr><tr id='fn008-universe-10-00357'><td><span class='html-label'>8</span></td><td><div class='html-p'>Notice that a general model with sterile neutrinos cannot be mimicked by a LED model in general; for instance, the presence of Large Extra Dimensions does not generate new sources of CP violation, unlike the sterile neutrinos hypothesis [<a href="#B21-universe-10-00357" class="html-bibr">21</a>,<a href="#B82-universe-10-00357" class="html-bibr">82</a>].</div></td></tr><tr id='fn009-universe-10-00357'><td><span class='html-label'>9</span></td><td><div class='html-p'>In [<a href="#B73-universe-10-00357" class="html-bibr">73</a>,<a href="#B146-universe-10-00357" class="html-bibr">146</a>], DUNE analyses have been repeated using a different experimental configuration; they found weaker limits on <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> </semantics></math>.</div></td></tr></table></section><section id='html-references_list'><h2>References</h2><ol class='html-xxx'><li id='B1-universe-10-00357' class='html-x' data-content='1.'>Fukuda, Y.; Hayakawa, T.; Ichihara, E.; Inoue, K.; Ishihara, K.; Ishino, H.; Itow, Y.; Kajita, T.; 554 Kameda, J.; Kasuga, S.; et al. Evidence for oscillation of atmospheric neutrinos. <span class='html-italic'>Phys. Rev. Lett.</span> <b>1998</b>, <span class='html-italic'>81</span>, 1562–1567. [<a href="https://scholar.google.com/scholar_lookup?title=Evidence+for+oscillation+of+atmospheric+neutrinos&author=Fukuda,+Y.&author=Hayakawa,+T.&author=Ichihara,+E.&author=Inoue,+K.&author=Ishihara,+K.&author=Ishino,+H.&author=Itow,+Y.&author=Kajita,+T.&author=554+Kameda,+J.&author=Kasuga,+S.&publication_year=1998&journal=Phys.+Rev.+Lett.&volume=81&pages=1562%E2%80%931567&doi=10.1103/PhysRevLett.81.1562" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.81.1562" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B2-universe-10-00357' class='html-x' data-content='2.'>Xu, X.J.; Wang, Z.; Chen, S. Solar neutrino physics. <span class='html-italic'>Prog. Part. Nucl. Phys.</span> <b>2023</b>, <span class='html-italic'>131</span>, 104043. [<a href="https://scholar.google.com/scholar_lookup?title=Solar+neutrino+physics&author=Xu,+X.J.&author=Wang,+Z.&author=Chen,+S.&publication_year=2023&journal=Prog.+Part.+Nucl.+Phys.&volume=131&pages=104043&doi=10.1016/j.ppnp.2023.104043" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.ppnp.2023.104043" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B3-universe-10-00357' class='html-x' data-content='3.'>Choubey, S. Atmospheric neutrinos: Status and prospects. <span class='html-italic'>Nucl. Phys. B</span> <b>2016</b>, <span class='html-italic'>908</span>, 235–249. [<a href="https://scholar.google.com/scholar_lookup?title=Atmospheric+neutrinos:+Status+and+prospects&author=Choubey,+S.&publication_year=2016&journal=Nucl.+Phys.+B&volume=908&pages=235%E2%80%93249&doi=10.1016/j.nuclphysb.2016.03.026" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.nuclphysb.2016.03.026" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B4-universe-10-00357' class='html-x' data-content='4.'>Wen, L.J.; Cao, J.; Wang, Y.F. Reactor Neutrino Experiments: Present and Future. <span class='html-italic'>Ann. Rev. Nucl. Part. Sci.</span> <b>2017</b>, <span class='html-italic'>67</span>, 183–211. [<a href="https://scholar.google.com/scholar_lookup?title=Reactor+Neutrino+Experiments:+Present+and+Future&author=Wen,+L.J.&author=Cao,+J.&author=Wang,+Y.F.&publication_year=2017&journal=Ann.+Rev.+Nucl.+Part.+Sci.&volume=67&pages=183%E2%80%93211&doi=10.1146/annurev-nucl-101916-123318" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1146/annurev-nucl-101916-123318" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B5-universe-10-00357' class='html-x' data-content='5.'>Mezzetto, M.; Terranova, F. Three-flavour oscillations with accelerator neutrino beams. <span class='html-italic'>Universe</span> <b>2020</b>, <span class='html-italic'>6</span>, 32. [<a href="https://scholar.google.com/scholar_lookup?title=Three-flavour+oscillations+with+accelerator+neutrino+beams&author=Mezzetto,+M.&author=Terranova,+F.&publication_year=2020&journal=Universe&volume=6&pages=32&doi=10.3390/universe6020032" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3390/universe6020032" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B6-universe-10-00357' class='html-x' data-content='6.'>Gando, A.; Gando, Y.; Ichimura, K.; Ikeda, H.; Inoue, K.; Kibe, Y.; Kishimoto, Y.; Koga, M.; Minekawa, Y.; Mitsui, T.; et al. Precision Measurement of Neutrino Oscillation Parameters with KamLAND. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2008</b>, <span class='html-italic'>100</span>, 221803. [<a href="https://scholar.google.com/scholar_lookup?title=Precision+Measurement+of+Neutrino+Oscillation+Parameters+with+KamLAND&author=Gando,+A.&author=Gando,+Y.&author=Ichimura,+K.&author=Ikeda,+H.&author=Inoue,+K.&author=Kibe,+Y.&author=Kishimoto,+Y.&author=Koga,+M.&author=Minekawa,+Y.&author=Mitsui,+T.&publication_year=2008&journal=Phys.+Rev.+Lett.&volume=100&pages=221803&doi=10.1103/PhysRevLett.100.221803" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.100.221803" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B7-universe-10-00357' class='html-x' data-content='7.'>Gando, A.; Gando, Y.; Ichimura, K.; Ikeda, H.; Inoue, K.; Kibe, Y.; Kishimoto, Y.; Koga, M.; Minekawa, Y.; Mitsui, T.; et al. Constraints on <span class='html-italic'>θ</span><sub>13</sub> from A Three-Flavor Oscillation Analysis of Reactor Antineutrinos at KamLAND. <span class='html-italic'>Phys. Rev. D</span> <b>2011</b>, <span class='html-italic'>83</span>, 052002. [<a href="https://scholar.google.com/scholar_lookup?title=Constraints+on+%CE%B813+from+A+Three-Flavor+Oscillation+Analysis+of+Reactor+Antineutrinos+at+KamLAND&author=Gando,+A.&author=Gando,+Y.&author=Ichimura,+K.&author=Ikeda,+H.&author=Inoue,+K.&author=Kibe,+Y.&author=Kishimoto,+Y.&author=Koga,+M.&author=Minekawa,+Y.&author=Mitsui,+T.&publication_year=2011&journal=Phys.+Rev.+D&volume=83&pages=052002&doi=10.1103/PhysRevD.83.052002" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.83.052002" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B8-universe-10-00357' class='html-x' data-content='8.'>Gando, A.; Gando, Y.; Hanakago, H.; Ikeda, H.; Inoue, K.; Ishidoshiro, K.; Ishikawa, H.; Koga, M.; Matsuda, R.; Matsuda, S.; et al. Reactor On-Off Antineutrino Measurement with KamLAND. <span class='html-italic'>Phys. Rev. D</span> <b>2013</b>, <span class='html-italic'>88</span>, 033001. [<a href="https://scholar.google.com/scholar_lookup?title=Reactor+On-Off+Antineutrino+Measurement+with+KamLAND&author=Gando,+A.&author=Gando,+Y.&author=Hanakago,+H.&author=Ikeda,+H.&author=Inoue,+K.&author=Ishidoshiro,+K.&author=Ishikawa,+H.&author=Koga,+M.&author=Matsuda,+R.&author=Matsuda,+S.&publication_year=2013&journal=Phys.+Rev.+D&volume=88&pages=033001&doi=10.1103/PhysRevD.88.033001" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.88.033001" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B9-universe-10-00357' class='html-x' data-content='9.'>An, F.P.; Bai, J.Z.; Balantekin, A.B.; Band, H.R.; Beavis, D.; Beriguete, W.; Bishai, M.; Blyth, S.; Boddy, K.; Brown, R.L.; et al. Observation of electron-antineutrino disappearance at Daya Bay. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2012</b>, <span class='html-italic'>108</span>, 171803. [<a href="https://scholar.google.com/scholar_lookup?title=Observation+of+electron-antineutrino+disappearance+at+Daya+Bay&author=An,+F.P.&author=Bai,+J.Z.&author=Balantekin,+A.B.&author=Band,+H.R.&author=Beavis,+D.&author=Beriguete,+W.&author=Bishai,+M.&author=Blyth,+S.&author=Boddy,+K.&author=Brown,+R.L.&publication_year=2012&journal=Phys.+Rev.+Lett.&volume=108&pages=171803&doi=10.1103/PhysRevLett.108.171803" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.108.171803" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B10-universe-10-00357' class='html-xx' data-content='10.'>Esteban, I.; Gonzalez-Garcia, M.C.; Maltoni, M.; Schwetz, T.; Zhou, A. The fate of hints: Updated global analysis of three-flavor neutrino oscillations. <span class='html-italic'>J. High Energy Phys.</span> <b>2020</b>, <span class='html-italic'>9</span>, 178. [<a href="https://scholar.google.com/scholar_lookup?title=The+fate+of+hints:+Updated+global+analysis+of+three-flavor+neutrino+oscillations&author=Esteban,+I.&author=Gonzalez-Garcia,+M.C.&author=Maltoni,+M.&author=Schwetz,+T.&author=Zhou,+A.&publication_year=2020&journal=J.+High+Energy+Phys.&volume=9&pages=178&doi=10.1007/JHEP09(2020)178" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP09(2020)178" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B11-universe-10-00357' class='html-xx' data-content='11.'>Abe, K.; Abe, K.; Aihara, H.; Aimi, A.; Akutsu, R.; Andreopoulos, C.; Anghel, I.; Anthony, L.H.V.; Antonova, M.; Ashida, Y.; et al. Hyper-Kamiokande Design Report. <span class='html-italic'>arXiv</span> <b>2018</b>, arXiv:1805.04163. [<a href="https://scholar.google.com/scholar_lookup?title=Hyper-Kamiokande+Design+Report&author=Abe,+K.&author=Abe,+K.&author=Aihara,+H.&author=Aimi,+A.&author=Akutsu,+R.&author=Andreopoulos,+C.&author=Anghel,+I.&author=Anthony,+L.H.V.&author=Antonova,+M.&author=Ashida,+Y.&publication_year=2018&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B12-universe-10-00357' class='html-xx' data-content='12.'>Acciarri, R.; Acero, M.A.; Adamowski, M.; Adams, C.; Adamson, P.; Adhikari, S.; Ahmad, Z.; Albright, C.H.; Alion, T.; Amador, E.; et al. Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF. <span class='html-italic'>arXiv</span> <b>2015</b>, arXiv:1512.06148. [<a href="https://scholar.google.com/scholar_lookup?title=Long-Baseline+Neutrino+Facility+(LBNF)+and+Deep+Underground+Neutrino+Experiment+(DUNE):+Conceptual+Design+Report,+Volume+2:+The+Physics+Program+for+DUNE+at+LBNF&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamowski,+M.&author=Adams,+C.&author=Adamson,+P.&author=Adhikari,+S.&author=Ahmad,+Z.&author=Albright,+C.H.&author=Alion,+T.&author=Amador,+E.&publication_year=2015&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B13-universe-10-00357' class='html-xx' data-content='13.'>Himmel, A. New Oscillation Results from the NOvA Experiment. 2020. Available online: <a href="https://www.osti.gov/biblio/1640225/" target="_blank" rel="noopener noreferrer">https://www.osti.gov/biblio/1640225/</a> (accessed on 15 March 2024).</li><li id='B14-universe-10-00357' class='html-xx' data-content='14.'>Batkiewicz-Kwasniak, M. The Latest T2K Neutrino Oscillation Results and the Future of the T2K and Hyper-Kamiokande Experiments. <span class='html-italic'>Acta Phys. Polon. Supp.</span> <b>2022</b>, <span class='html-italic'>15</span>, 23. [<a href="https://scholar.google.com/scholar_lookup?title=The+Latest+T2K+Neutrino+Oscillation+Results+and+the+Future+of+the+T2K+and+Hyper-Kamiokande+Experiments&author=Batkiewicz-Kwasniak,+M.&publication_year=2022&journal=Acta+Phys.+Polon.+Supp.&volume=15&pages=23&doi=10.5506/APhysPolBSupp.15.3-A23" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.5506/APhysPolBSupp.15.3-A23" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B15-universe-10-00357' class='html-xx' data-content='15.'>Rahaman, U.; Raut, S. On the tension between the latest NOvA and T2K data. <span class='html-italic'>arXiv</span> <b>2021</b>, arXiv:2112.13186. [<a href="https://scholar.google.com/scholar_lookup?title=On+the+tension+between+the+latest+NOvA+and+T2K+data&author=Rahaman,+U.&author=Raut,+S.&publication_year=2021&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B16-universe-10-00357' class='html-xx' data-content='16.'>Ghoshal, A.; Giarnetti, A.; Meloni, D. On the role of the <span class='html-italic'>ν</span><sub>τ</sub> appearance in DUNE in constraining standard neutrino physics and beyond. <span class='html-italic'>J. High Energy Phys.</span> <b>2019</b>, <span class='html-italic'>12</span>, 126. [<a href="https://scholar.google.com/scholar_lookup?title=On+the+role+of+the+%CE%BD%CF%84+appearance+in+DUNE+in+constraining+standard+neutrino+physics+and+beyond&author=Ghoshal,+A.&author=Giarnetti,+A.&author=Meloni,+D.&publication_year=2019&journal=J.+High+Energy+Phys.&volume=12&pages=126&doi=10.1007/JHEP12(2019)126" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP12(2019)126" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B17-universe-10-00357' class='html-xx' data-content='17.'>Masud, M.; Bishai, M.; Mehta, P. Extricating New Physics Scenarios at DUNE with Higher Energy Beams. <span class='html-italic'>Sci. Rep.</span> <b>2019</b>, <span class='html-italic'>9</span>, 352. [<a href="https://scholar.google.com/scholar_lookup?title=Extricating+New+Physics+Scenarios+at+DUNE+with+Higher+Energy+Beams&author=Masud,+M.&author=Bishai,+M.&author=Mehta,+P.&publication_year=2019&journal=Sci.+Rep.&volume=9&pages=352&doi=10.1038/s41598-018-36790-6" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1038/s41598-018-36790-6" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B18-universe-10-00357' class='html-xx' data-content='18.'>De Gouvêa, A.; Kelly, K.J.; Stenico, G.V.; Pasquini, P. Physics with Beam Tau-Neutrino Appearance at DUNE. <span class='html-italic'>Phys. Rev. D</span> <b>2019</b>, <span class='html-italic'>100</span>, 016004. [<a href="https://scholar.google.com/scholar_lookup?title=Physics+with+Beam+Tau-Neutrino+Appearance+at+DUNE&author=De+Gouv%C3%AAa,+A.&author=Kelly,+K.J.&author=Stenico,+G.V.&author=Pasquini,+P.&publication_year=2019&journal=Phys.+Rev.+D&volume=100&pages=016004&doi=10.1103/PhysRevD.100.016004" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.100.016004" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B19-universe-10-00357' class='html-xx' data-content='19.'>De Romeri, V.; Giunti, C.; Stuttard, T.; Ternes, C.A. Neutrino oscillation bounds on quantum decoherence. <span class='html-italic'>J. High Energy Phys.</span> <b>2023</b>, <span class='html-italic'>9</span>, 97. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillation+bounds+on+quantum+decoherence&author=De+Romeri,+V.&author=Giunti,+C.&author=Stuttard,+T.&author=Ternes,+C.A.&publication_year=2023&journal=J.+High+Energy+Phys.&volume=9&pages=97&doi=10.1007/JHEP09(2023)097" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP09(2023)097" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B20-universe-10-00357' class='html-xx' data-content='20.'>Mammen Abraham, R.; Alvarez-Muñiz, J.; Argüelles, C.A.; Ariga, A.; Ariga, T.; Aurisano, A.; Autiero, D.; Bishai, M.; Bostan, N.; Bustamante, M.; et al. Tau neutrinos in the next decade: From GeV to EeV. <span class='html-italic'>J. Phys. G</span> <b>2022</b>, <span class='html-italic'>49</span>, 110501. [<a href="https://scholar.google.com/scholar_lookup?title=Tau+neutrinos+in+the+next+decade:+From+GeV+to+EeV&author=Mammen+Abraham,+R.&author=Alvarez-Mu%C3%B1iz,+J.&author=Arg%C3%BCelles,+C.A.&author=Ariga,+A.&author=Ariga,+T.&author=Aurisano,+A.&author=Autiero,+D.&author=Bishai,+M.&author=Bostan,+N.&author=Bustamante,+M.&publication_year=2022&journal=J.+Phys.+G&volume=49&pages=110501&doi=10.1088/1361-6471/ac89d2" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1361-6471/ac89d2" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B21-universe-10-00357' class='html-xx' data-content='21.'>Giarnetti, A.; Meloni, D. New Sources of Leptonic CP Violation at the DUNE Neutrino Experiment. <span class='html-italic'>Universe</span> <b>2021</b>, <span class='html-italic'>7</span>, 240. [<a href="https://scholar.google.com/scholar_lookup?title=New+Sources+of+Leptonic+CP+Violation+at+the+DUNE+Neutrino+Experiment&author=Giarnetti,+A.&author=Meloni,+D.&publication_year=2021&journal=Universe&volume=7&pages=240&doi=10.3390/universe7070240" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3390/universe7070240" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B22-universe-10-00357' class='html-xx' data-content='22.'>Acciarri, R.; Acero, M.A.; Adamowski, M.; Adams, C.; Adamson, P.; Adhikari, S.; Ahmad, Z.; Albright, C.H.; Alion, T.; Amador, E.; et al. Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 1: The LBNF and DUNE Projects. <span class='html-italic'>arXiv</span> <b>2016</b>, arXiv:1601.05471. [<a href="https://scholar.google.com/scholar_lookup?title=Long-Baseline+Neutrino+Facility+(LBNF)+and+Deep+Underground+Neutrino+Experiment+(DUNE):+Conceptual+Design+Report,+Volume+1:+The+LBNF+and+DUNE+Projects&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamowski,+M.&author=Adams,+C.&author=Adamson,+P.&author=Adhikari,+S.&author=Ahmad,+Z.&author=Albright,+C.H.&author=Alion,+T.&author=Amador,+E.&publication_year=2016&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B23-universe-10-00357' class='html-xx' data-content='23.'>Abi, B.; Acciarri, R.; Acero, M.A.; Adamov, G.; Adams, D.; Adinolfi, M.; Ahmad, Z.; Ahmed, J.; Alion, T.; Monsalve, S.A.; et al. Long-baseline neutrino oscillation physics potential of the DUNE experiment. <span class='html-italic'>Eur. Phys. J. C</span> <b>2020</b>, <span class='html-italic'>80</span>, 978. [<a href="https://scholar.google.com/scholar_lookup?title=Long-baseline+neutrino+oscillation+physics+potential+of+the+DUNE+experiment&author=Abi,+B.&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamov,+G.&author=Adams,+D.&author=Adinolfi,+M.&author=Ahmad,+Z.&author=Ahmed,+J.&author=Alion,+T.&author=Monsalve,+S.A.&publication_year=2020&journal=Eur.+Phys.+J.+C&volume=80&pages=978&doi=10.1140/epjc/s10052-020-08456-z" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-020-08456-z" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B24-universe-10-00357' class='html-xx' data-content='24.'>Abi, B.; Acciarri, R.; Acero, M.A.; Adamov, G.; Adams, D.; Adinolfi, M.; Ahmad, Z.; Ahmed, J.; Alion, T.; Monsalve, S.A.; et al. Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume II: DUNE Physics. <span class='html-italic'>arXiv</span> <b>2020</b>, arXiv:2002.03005. [<a href="https://scholar.google.com/scholar_lookup?title=Deep+Underground+Neutrino+Experiment+(DUNE),+Far+Detector+Technical+Design+Report,+Volume+II:+DUNE+Physics&author=Abi,+B.&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamov,+G.&author=Adams,+D.&author=Adinolfi,+M.&author=Ahmad,+Z.&author=Ahmed,+J.&author=Alion,+T.&author=Monsalve,+S.A.&publication_year=2020&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B25-universe-10-00357' class='html-xx' data-content='25.'>Abud, A.A.; Abi, B.; Acciarri, R.; Acero, M.A.; Adamov, G.; Adams, D.; Adinolfi, M.; Aduszkiewicz, A.; Ahmad, Z.; Ahmed, J.; et al. Deep Underground Neutrino Experiment (DUNE) Near Detector Conceptual Design Report. <span class='html-italic'>Instruments</span> <b>2021</b>, <span class='html-italic'>5</span>, 31. [<a href="https://scholar.google.com/scholar_lookup?title=Deep+Underground+Neutrino+Experiment+(DUNE)+Near+Detector+Conceptual+Design+Report&author=Abud,+A.A.&author=Abi,+B.&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamov,+G.&author=Adams,+D.&author=Adinolfi,+M.&author=Aduszkiewicz,+A.&author=Ahmad,+Z.&author=Ahmed,+J.&publication_year=2021&journal=Instruments&volume=5&pages=31&doi=10.3390/instruments5040031" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.3390/instruments5040031" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B26-universe-10-00357' class='html-xx' data-content='26.'>Fermilab. Available online: <a href="http://home.fnal.gov/~ljf26/DUNEFluxes/" target="_blank" rel="noopener noreferrer">http://home.fnal.gov/~ljf26/DUNEFluxes/</a> (accessed on 15 March 2024).</li><li id='B27-universe-10-00357' class='html-xx' data-content='27.'>Bishai, M.; Dolce, M. Optimization of the LBNF/DUNE Beamline for Tau Neutrinos. Available online: <a href="http://docs.dunescience.org/cgi-bin/RetrieveFile?docid=2013&filename=DOLCE_M_report.pdf&version=1" target="_blank" rel="noopener noreferrer">http://docs.dunescience.org/cgi-bin/RetrieveFile?docid=2013&filename=DOLCE_M_report.pdf&version=1</a> (accessed on 15 March 2024).</li><li id='B28-universe-10-00357' class='html-xx' data-content='28.'>DUNE Neutrino Flux Files Generated with G4LBNF. Available online: <a href="https://glaucus.crc.nd.edu/DUNEFluxes/" target="_blank" rel="noopener noreferrer">https://glaucus.crc.nd.edu/DUNEFluxes/</a> (accessed on 15 March 2024).</li><li id='B29-universe-10-00357' class='html-xx' data-content='29.'>Alion, T.; Back, J.J.; Bashyal, A.; Bass, M.; Bishai, M.; Cherdack, D.; Diwan, M.; Djurcic, Z.; Evans, J.; Fernandez-Martinez, E.; et al. Experiment Simulation Configurations Used in DUNE CDR. <span class='html-italic'>arXiv</span> <b>2016</b>, arXiv:1606.09550. [<a href="https://scholar.google.com/scholar_lookup?title=Experiment+Simulation+Configurations+Used+in+DUNE+CDR&author=Alion,+T.&author=Back,+J.J.&author=Bashyal,+A.&author=Bass,+M.&author=Bishai,+M.&author=Cherdack,+D.&author=Diwan,+M.&author=Djurcic,+Z.&author=Evans,+J.&author=Fernandez-Martinez,+E.&publication_year=2016&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B30-universe-10-00357' class='html-xx' data-content='30.'>Abi, B.; Acciarri, R.; Acero, M.A.; Adamov, G.; Adams, D.; Adinolfi, M.; Ahmad, Z.; Ahmed, J.; Alion, T.; Monsalve, S.A.; et al. Experiment Simulation Configurations Approximating DUNE TDR. <span class='html-italic'>arXiv</span> <b>2021</b>, arXiv:2103.04797. [<a href="https://scholar.google.com/scholar_lookup?title=Experiment+Simulation+Configurations+Approximating+DUNE+TDR&author=Abi,+B.&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamov,+G.&author=Adams,+D.&author=Adinolfi,+M.&author=Ahmad,+Z.&author=Ahmed,+J.&author=Alion,+T.&author=Monsalve,+S.A.&publication_year=2021&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B31-universe-10-00357' class='html-xx' data-content='31.'>Colladay, D.; Kostelecky, V.A. CPT violation and the standard model. <span class='html-italic'>Phys. Rev. D</span> <b>1997</b>, <span class='html-italic'>55</span>, 6760–6774. [<a href="https://scholar.google.com/scholar_lookup?title=CPT+violation+and+the+standard+model&author=Colladay,+D.&author=Kostelecky,+V.A.&publication_year=1997&journal=Phys.+Rev.+D&volume=55&pages=6760%E2%80%936774&doi=10.1103/PhysRevD.55.6760" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.55.6760" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B32-universe-10-00357' class='html-xx' data-content='32.'>Colladay, D.; Kostelecky, V.A. Lorentz violating extension of the standard model. <span class='html-italic'>Phys. Rev. D</span> <b>1998</b>, <span class='html-italic'>58</span>, 116002. [<a href="https://scholar.google.com/scholar_lookup?title=Lorentz+violating+extension+of+the+standard+model&author=Colladay,+D.&author=Kostelecky,+V.A.&publication_year=1998&journal=Phys.+Rev.+D&volume=58&pages=116002&doi=10.1103/PhysRevD.58.116002" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.58.116002" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B33-universe-10-00357' class='html-xx' data-content='33.'>Diaz, J.S.; Kostelecky, A. Lorentz- and CPT-violating models for neutrino oscillations. <span class='html-italic'>Phys. Rev. D</span> <b>2012</b>, <span class='html-italic'>85</span>, 016013. [<a href="https://scholar.google.com/scholar_lookup?title=Lorentz-+and+CPT-violating+models+for+neutrino+oscillations&author=Diaz,+J.S.&author=Kostelecky,+A.&publication_year=2012&journal=Phys.+Rev.+D&volume=85&pages=016013&doi=10.1103/PhysRevD.85.016013" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.85.016013" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B34-universe-10-00357' class='html-xx' data-content='34.'>Greenberg, O.W. CPT violation implies violation of Lorentz invariance. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2002</b>, <span class='html-italic'>89</span>, 231602. [<a href="https://scholar.google.com/scholar_lookup?title=CPT+violation+implies+violation+of+Lorentz+invariance&author=Greenberg,+O.W.&publication_year=2002&journal=Phys.+Rev.+Lett.&volume=89&pages=231602&doi=10.1103/PhysRevLett.89.231602" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.89.231602" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B35-universe-10-00357' class='html-xx' data-content='35.'>Kostelecky, V.A.; Potting, R. CPT and strings. <span class='html-italic'>Nucl. Phys. B</span> <b>1991</b>, <span class='html-italic'>359</span>, 545–570. [<a href="https://scholar.google.com/scholar_lookup?title=CPT+and+strings&author=Kostelecky,+V.A.&author=Potting,+R.&publication_year=1991&journal=Nucl.+Phys.+B&volume=359&pages=545%E2%80%93570&doi=10.1016/0550-3213(91)90071-5" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/0550-3213(91)90071-5" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B36-universe-10-00357' class='html-xx' data-content='36.'>Kostelecky, V.A.; Potting, R. CPT, strings, and meson factories. <span class='html-italic'>Phys. Rev. D</span> <b>1995</b>, <span class='html-italic'>51</span>, 3923–3935. [<a href="https://scholar.google.com/scholar_lookup?title=CPT,+strings,+and+meson+factories&author=Kostelecky,+V.A.&author=Potting,+R.&publication_year=1995&journal=Phys.+Rev.+D&volume=51&pages=3923%E2%80%933935&doi=10.1103/PhysRevD.51.3923" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.51.3923" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B37-universe-10-00357' class='html-xx' data-content='37.'>Kostelecky, V.A.; Potting, R. Expectation values, Lorentz invariance, and CPT in the open bosonic string. <span class='html-italic'>Phys. Lett. B</span> <b>1996</b>, <span class='html-italic'>381</span>, 89–96. [<a href="https://scholar.google.com/scholar_lookup?title=Expectation+values,+Lorentz+invariance,+and+CPT+in+the+open+bosonic+string&author=Kostelecky,+V.A.&author=Potting,+R.&publication_year=1996&journal=Phys.+Lett.+B&volume=381&pages=89%E2%80%9396&doi=10.1016/0370-2693(96)00589-8" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/0370-2693(96)00589-8" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B38-universe-10-00357' class='html-xx' data-content='38.'>Kostelecky, V.A.; Mewes, M. Lorentz and CPT violation in neutrinos. <span class='html-italic'>Phys. Rev. D</span> <b>2004</b>, <span class='html-italic'>69</span>, 016005. [<a href="https://scholar.google.com/scholar_lookup?title=Lorentz+and+CPT+violation+in+neutrinos&author=Kostelecky,+V.A.&author=Mewes,+M.&publication_year=2004&journal=Phys.+Rev.+D&volume=69&pages=016005&doi=10.1103/PhysRevD.69.016005" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.69.016005" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B39-universe-10-00357' class='html-xx' data-content='39.'>Kostelecký, V.A.; Samuel, S. Spontaneous breaking of Lorentz symmetry in string theory. <span class='html-italic'>Phys. Rev. D</span> <b>1989</b>, <span class='html-italic'>39</span>, 683–685. [<a href="https://scholar.google.com/scholar_lookup?title=Spontaneous+breaking+of+Lorentz+symmetry+in+string+theory&author=Kosteleck%C3%BD,+V.A.&author=Samuel,+S.&publication_year=1989&journal=Phys.+Rev.+D&volume=39&pages=683%E2%80%93685&doi=10.1103/PhysRevD.39.683&pmid=9959689" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.39.683" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/9959689" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B40-universe-10-00357' class='html-xx' data-content='40.'>Kostelecký, V.A.; Samuel, S. Phenomenological gravitational constraints on strings and higher-dimensional theories. <span class='html-italic'>Phys. Rev. Lett.</span> <b>1989</b>, <span class='html-italic'>63</span>, 224–227. [<a href="https://scholar.google.com/scholar_lookup?title=Phenomenological+gravitational+constraints+on+strings+and+higher-dimensional+theories&author=Kosteleck%C3%BD,+V.A.&author=Samuel,+S.&publication_year=1989&journal=Phys.+Rev.+Lett.&volume=63&pages=224%E2%80%93227&doi=10.1103/PhysRevLett.63.224" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.63.224" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B41-universe-10-00357' class='html-xx' data-content='41.'>Agarwalla, S.K.; Das, S.; Sahoo, S.; Swain, P. Constraining Lorentz invariance violation with next-generation long-baseline experiments. <span class='html-italic'>J. High Energy Phys.</span> <b>2023</b>, <span class='html-italic'>7</span>, 216. [<a href="https://scholar.google.com/scholar_lookup?title=Constraining+Lorentz+invariance+violation+with+next-generation+long-baseline+experiments&author=Agarwalla,+S.K.&author=Das,+S.&author=Sahoo,+S.&author=Swain,+P.&publication_year=2023&journal=J.+High+Energy+Phys.&volume=7&pages=216&doi=10.1007/JHEP07(2023)216" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP07(2023)216" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B42-universe-10-00357' class='html-xx' data-content='42.'>Kumar Agarwalla, S.; Masud, M. Can Lorentz invariance violation affect the sensitivity of deep underground neutrino experiment? <span class='html-italic'>Eur. Phys. J. C</span> <b>2020</b>, <span class='html-italic'>80</span>, 716. [<a href="https://scholar.google.com/scholar_lookup?title=Can+Lorentz+invariance+violation+affect+the+sensitivity+of+deep+underground+neutrino+experiment?&author=Kumar+Agarwalla,+S.&author=Masud,+M.&publication_year=2020&journal=Eur.+Phys.+J.+C&volume=80&pages=716&doi=10.1140/epjc/s10052-020-8303-1" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-020-8303-1" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B43-universe-10-00357' class='html-xx' data-content='43.'>Kostelecky, A.; Mewes, M. Neutrinos with Lorentz-violating operators of arbitrary dimension. <span class='html-italic'>Phys. Rev. D</span> <b>2012</b>, <span class='html-italic'>85</span>, 096005. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrinos+with+Lorentz-violating+operators+of+arbitrary+dimension&author=Kostelecky,+A.&author=Mewes,+M.&publication_year=2012&journal=Phys.+Rev.+D&volume=85&pages=096005&doi=10.1103/PhysRevD.85.096005" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.85.096005" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B44-universe-10-00357' class='html-xx' data-content='44.'>Wolfenstein, L. Neutrino oscillations in matter. <span class='html-italic'>Phys. Rev. D</span> <b>1978</b>, <span class='html-italic'>17</span>, 2369–2374. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillations+in+matter&author=Wolfenstein,+L.&publication_year=1978&journal=Phys.+Rev.+D&volume=17&pages=2369%E2%80%932374&doi=10.1103/PhysRevD.17.2369" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.17.2369" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B45-universe-10-00357' class='html-xx' data-content='45.'>Barenboim, G.; Ternes, C.A.; Tórtola, M. New physics vs. new paradigms: Distinguishing CPT violation from NSI. <span class='html-italic'>Eur. Phys. J. C</span> <b>2019</b>, <span class='html-italic'>79</span>, 390. [<a href="https://scholar.google.com/scholar_lookup?title=New+physics+vs.+new+paradigms:+Distinguishing+CPT+violation+from+NSI&author=Barenboim,+G.&author=Ternes,+C.A.&author=T%C3%B3rtola,+M.&publication_year=2019&journal=Eur.+Phys.+J.+C&volume=79&pages=390&doi=10.1140/epjc/s10052-019-6900-7" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-019-6900-7" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B46-universe-10-00357' class='html-xx' data-content='46.'>Diaz, J.S. Correspondence between nonstandard interactions and CPT violation in neutrino oscillations. <span class='html-italic'>arXiv</span> <b>2015</b>, arXiv:1506.01936. [<a href="https://scholar.google.com/scholar_lookup?title=Correspondence+between+nonstandard+interactions+and+CPT+violation+in+neutrino+oscillations&author=Diaz,+J.S.&publication_year=2015&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B47-universe-10-00357' class='html-xx' data-content='47.'>Barenboim, G.; Masud, M.; Ternes, C.A.; Tórtola, M. Exploring the intrinsic Lorentz-violating parameters at DUNE. <span class='html-italic'>Phys. Lett. B</span> <b>2019</b>, <span class='html-italic'>788</span>, 308–315. [<a href="https://scholar.google.com/scholar_lookup?title=Exploring+the+intrinsic+Lorentz-violating+parameters+at+DUNE&author=Barenboim,+G.&author=Masud,+M.&author=Ternes,+C.A.&author=T%C3%B3rtola,+M.&publication_year=2019&journal=Phys.+Lett.+B&volume=788&pages=308%E2%80%93315&doi=10.1016/j.physletb.2018.11.040" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2018.11.040" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B48-universe-10-00357' class='html-xx' data-content='48.'>Fiza, N.; Khan Chowdhury, N.R.; Masud, M. Investigating Lorentz Invariance Violation with the long baseline experiment P2O. <span class='html-italic'>J. High Energy Phys.</span> <b>2023</b>, <span class='html-italic'>1</span>, 76. [<a href="https://scholar.google.com/scholar_lookup?title=Investigating+Lorentz+Invariance+Violation+with+the+long+baseline+experiment+P2O&author=Fiza,+N.&author=Khan+Chowdhury,+N.R.&author=Masud,+M.&publication_year=2023&journal=J.+High+Energy+Phys.&volume=1&pages=76&doi=10.1007/JHEP01(2023)076" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP01(2023)076" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B49-universe-10-00357' class='html-xx' data-content='49.'>Adamson, P.; Andreopoulos, C.; Arms, K.E.; Armstrong, R.; Auty, D.J.; Ayres, D.S.; Baller, B.; Barr, G.; Barrett, W.L.; Becker, B.R.; et al. Testing Lorentz Invariance and CPT Conservation with NuMI Neutrinos in the MINOS Near Detector. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2008</b>, <span class='html-italic'>101</span>, 151601. [<a href="https://scholar.google.com/scholar_lookup?title=Testing+Lorentz+Invariance+and+CPT+Conservation+with+NuMI+Neutrinos+in+the+MINOS+Near+Detector&author=Adamson,+P.&author=Andreopoulos,+C.&author=Arms,+K.E.&author=Armstrong,+R.&author=Auty,+D.J.&author=Ayres,+D.S.&author=Baller,+B.&author=Barr,+G.&author=Barrett,+W.L.&author=Becker,+B.R.&publication_year=2008&journal=Phys.+Rev.+Lett.&volume=101&pages=151601&doi=10.1103/PhysRevLett.101.151601" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.101.151601" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B50-universe-10-00357' class='html-xx' data-content='50.'>Adamson, P.; Andreopoulos, C.; Arms, K.E.; Armstrong, R.; Auty, D.J.; Ayres, D.S.; Baller, B.; Barr, G.; Barrett, W.L.; Becker, B.R.; et al. A Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2010</b>, <span class='html-italic'>105</span>, 151601. [<a href="https://scholar.google.com/scholar_lookup?title=A+Search+for+Lorentz+Invariance+and+CPT+Violation+with+the+MINOS+Far+Detector&author=Adamson,+P.&author=Andreopoulos,+C.&author=Arms,+K.E.&author=Armstrong,+R.&author=Auty,+D.J.&author=Ayres,+D.S.&author=Baller,+B.&author=Barr,+G.&author=Barrett,+W.L.&author=Becker,+B.R.&publication_year=2010&journal=Phys.+Rev.+Lett.&volume=105&pages=151601&doi=10.1103/PhysRevLett.105.151601&pmid=21230890" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.105.151601" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/21230890" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B51-universe-10-00357' class='html-xx' data-content='51.'>Adamson, P.; Andreopoulos, C.; Arms, K.E.; Armstrong, R.; Auty, D.J.; Ayres, D.S.; Baller, B.; Barr, G.; Barrett, W.L.; Becker, B.R.; et al. Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector. <span class='html-italic'>Phys. Rev. D</span> <b>2012</b>, <span class='html-italic'>85</span>, 031101. [<a href="https://scholar.google.com/scholar_lookup?title=Search+for+Lorentz+invariance+and+CPT+violation+with+muon+antineutrinos+in+the+MINOS+Near+Detector&author=Adamson,+P.&author=Andreopoulos,+C.&author=Arms,+K.E.&author=Armstrong,+R.&author=Auty,+D.J.&author=Ayres,+D.S.&author=Baller,+B.&author=Barr,+G.&author=Barrett,+W.L.&author=Becker,+B.R.&publication_year=2012&journal=Phys.+Rev.+D&volume=85&pages=031101&doi=10.1103/PhysRevD.85.031101" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.85.031101" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B52-universe-10-00357' class='html-xx' data-content='52.'>Dighe, A.; Ray, S. CPT violation in long baseline neutrino experiments: A Three flavor analysis. <span class='html-italic'>Phys. Rev. D</span> <b>2008</b>, <span class='html-italic'>78</span>, 036002. [<a href="https://scholar.google.com/scholar_lookup?title=CPT+violation+in+long+baseline+neutrino+experiments:+A+Three+flavor+analysis&author=Dighe,+A.&author=Ray,+S.&publication_year=2008&journal=Phys.+Rev.+D&volume=78&pages=036002&doi=10.1103/PhysRevD.78.036002" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.78.036002" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B53-universe-10-00357' class='html-xx' data-content='53.'>Barenboim, G.; Lykken, J.D. MINOS and CPT-violating neutrinos. <span class='html-italic'>Phys. Rev. D</span> <b>2009</b>, <span class='html-italic'>80</span>, 113008. [<a href="https://scholar.google.com/scholar_lookup?title=MINOS+and+CPT-violating+neutrinos&author=Barenboim,+G.&author=Lykken,+J.D.&publication_year=2009&journal=Phys.+Rev.+D&volume=80&pages=113008&doi=10.1103/PhysRevD.80.113008" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.80.113008" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B54-universe-10-00357' class='html-xx' data-content='54.'>Rebel, B.; Mufson, S. The Search for Neutrino-Antineutrino Mixing Resulting from Lorentz Invariance Violation using Neutrino Interactions in MINOS. <span class='html-italic'>Astropart. Phys.</span> <b>2013</b>, <span class='html-italic'>48</span>, 78–81. [<a href="https://scholar.google.com/scholar_lookup?title=The+Search+for+Neutrino-Antineutrino+Mixing+Resulting+from+Lorentz+Invariance+Violation+using+Neutrino+Interactions+in+MINOS&author=Rebel,+B.&author=Mufson,+S.&publication_year=2013&journal=Astropart.+Phys.&volume=48&pages=78%E2%80%9381&doi=10.1016/j.astropartphys.2013.07.006" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.astropartphys.2013.07.006" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B55-universe-10-00357' class='html-xx' data-content='55.'>de Gouvêa, A.; Kelly, K.J. Neutrino vs. Antineutrino Oscillation Parameters at DUNE and Hyper-Kamiokande. <span class='html-italic'>Phys. Rev. D</span> <b>2017</b>, <span class='html-italic'>96</span>, 095018. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+vs.+Antineutrino+Oscillation+Parameters+at+DUNE+and+Hyper-Kamiokande&author=de+Gouv%C3%AAa,+A.&author=Kelly,+K.J.&publication_year=2017&journal=Phys.+Rev.+D&volume=96&pages=095018&doi=10.1103/PhysRevD.96.095018" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.96.095018" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B56-universe-10-00357' class='html-xx' data-content='56.'>Barenboim, G.; Ternes, C.A.; Tórtola, M. Neutrinos, DUNE and the world best bound on CPT invariance. <span class='html-italic'>Phys. Lett. B</span> <b>2018</b>, <span class='html-italic'>780</span>, 631–637. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrinos,+DUNE+and+the+world+best+bound+on+CPT+invariance&author=Barenboim,+G.&author=Ternes,+C.A.&author=T%C3%B3rtola,+M.&publication_year=2018&journal=Phys.+Lett.+B&volume=780&pages=631%E2%80%93637&doi=10.1016/j.physletb.2018.03.060" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2018.03.060" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B57-universe-10-00357' class='html-xx' data-content='57.'>Majhi, R.; Chembra, S.; Mohanta, R. Exploring the effect of Lorentz invariance violation with the currently running long-baseline experiments. <span class='html-italic'>Eur. Phys. J. C</span> <b>2020</b>, <span class='html-italic'>80</span>, 364. [<a href="https://scholar.google.com/scholar_lookup?title=Exploring+the+effect+of+Lorentz+invariance+violation+with+the+currently+running+long-baseline+experiments&author=Majhi,+R.&author=Chembra,+S.&author=Mohanta,+R.&publication_year=2020&journal=Eur.+Phys.+J.+C&volume=80&pages=364&doi=10.1140/epjc/s10052-020-7963-1" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-020-7963-1" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B58-universe-10-00357' class='html-xx' data-content='58.'>Majhi, R.; Singha, D.K.; Ghosh, M.; Mohanta, R. Distinguishing nonstandard interaction and Lorentz invariance violation at the Protvino to super-ORCA experiment. <span class='html-italic'>Phys. Rev. D</span> <b>2023</b>, <span class='html-italic'>107</span>, 075036. [<a href="https://scholar.google.com/scholar_lookup?title=Distinguishing+nonstandard+interaction+and+Lorentz+invariance+violation+at+the+Protvino+to+super-ORCA+experiment&author=Majhi,+R.&author=Singha,+D.K.&author=Ghosh,+M.&author=Mohanta,+R.&publication_year=2023&journal=Phys.+Rev.+D&volume=107&pages=075036&doi=10.1103/PhysRevD.107.075036" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.107.075036" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B59-universe-10-00357' class='html-xx' data-content='59.'>Abe, K.; Amey, J.; Andreopoulos, C.; Antonova, M.; Aoki, S.; Ariga, A.; Assylbekov, S.; Autiero, D.; Ban, S.; Barbato, F.C.T.; et al. Search for Lorentz and CPT violation using sidereal time dependence of neutrino flavor transitions over a short baseline. <span class='html-italic'>Phys. Rev. D</span> <b>2017</b>, <span class='html-italic'>95</span>, 111101. [<a href="https://scholar.google.com/scholar_lookup?title=Search+for+Lorentz+and+CPT+violation+using+sidereal+time+dependence+of+neutrino+flavor+transitions+over+a+short+baseline&author=Abe,+K.&author=Amey,+J.&author=Andreopoulos,+C.&author=Antonova,+M.&author=Aoki,+S.&author=Ariga,+A.&author=Assylbekov,+S.&author=Autiero,+D.&author=Ban,+S.&author=Barbato,+F.C.T.&publication_year=2017&journal=Phys.+Rev.+D&volume=95&pages=111101&doi=10.1103/PhysRevD.95.111101" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.95.111101" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B60-universe-10-00357' class='html-xx' data-content='60.'>Aguilar-Arevalo, A.A.; Anderson, C.E.; Bazarko, A.O.; Brice, S.J.; Brown, B.C.; Bugel, L.; Cao, J.; Coney, L.; Conrad, J.M.; Cox, D.C.; et al. Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses. <span class='html-italic'>Phys. Lett. B</span> <b>2013</b>, <span class='html-italic'>718</span>, 1303–1308. [<a href="https://scholar.google.com/scholar_lookup?title=Test+of+Lorentz+and+CPT+violation+with+Short+Baseline+Neutrino+Oscillation+Excesses&author=Aguilar-Arevalo,+A.A.&author=Anderson,+C.E.&author=Bazarko,+A.O.&author=Brice,+S.J.&author=Brown,+B.C.&author=Bugel,+L.&author=Cao,+J.&author=Coney,+L.&author=Conrad,+J.M.&author=Cox,+D.C.&publication_year=2013&journal=Phys.+Lett.+B&volume=718&pages=1303%E2%80%931308&doi=10.1016/j.physletb.2012.12.020" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2012.12.020" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B61-universe-10-00357' class='html-xx' data-content='61.'>Giunti, C.; Laveder, M. Hint of CPT Violation in Short-Baseline Electron Neutrino Disappearance. <span class='html-italic'>Phys. Rev. D</span> <b>2010</b>, <span class='html-italic'>82</span>, 113009. [<a href="https://scholar.google.com/scholar_lookup?title=Hint+of+CPT+Violation+in+Short-Baseline+Electron+Neutrino+Disappearance&author=Giunti,+C.&author=Laveder,+M.&publication_year=2010&journal=Phys.+Rev.+D&volume=82&pages=113009&doi=10.1103/PhysRevD.82.113009" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.82.113009" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B62-universe-10-00357' class='html-xx' data-content='62.'>Abe, Y.; Aberle, C.; dos Anjos, J.C.; Bergevin, M.; Bernstein, A.; Bezerra, T.J.C.; Bezrukhov, L.; Blucher, E.; Bowden, N.S.; Buck, C.; et al. First Test of Lorentz Violation with a Reactor-based Antineutrino Experiment. <span class='html-italic'>Phys. Rev. D</span> <b>2012</b>, <span class='html-italic'>86</span>, 112009. [<a href="https://scholar.google.com/scholar_lookup?title=First+Test+of+Lorentz+Violation+with+a+Reactor-based+Antineutrino+Experiment&author=Abe,+Y.&author=Aberle,+C.&author=dos+Anjos,+J.C.&author=Bergevin,+M.&author=Bernstein,+A.&author=Bezerra,+T.J.C.&author=Bezrukhov,+L.&author=Blucher,+E.&author=Bowden,+N.S.&author=Buck,+C.&publication_year=2012&journal=Phys.+Rev.+D&volume=86&pages=112009&doi=10.1103/PhysRevD.86.112009" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.86.112009" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B63-universe-10-00357' class='html-xx' data-content='63.'>Diaz, J.S.; Schwetz, T. Limits on CPT violation from solar neutrinos. <span class='html-italic'>Phys. Rev. D</span> <b>2016</b>, <span class='html-italic'>93</span>, 093004. [<a href="https://scholar.google.com/scholar_lookup?title=Limits+on+CPT+violation+from+solar+neutrinos&author=Diaz,+J.S.&author=Schwetz,+T.&publication_year=2016&journal=Phys.+Rev.+D&volume=93&pages=093004&doi=10.1103/PhysRevD.93.093004" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.93.093004" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B64-universe-10-00357' class='html-xx' data-content='64.'>Hooper, D.; Morgan, D.; Winstanley, E. Lorentz and CPT invariance violation in high-energy neutrinos. <span class='html-italic'>Phys. Rev. D</span> <b>2005</b>, <span class='html-italic'>72</span>, 065009. [<a href="https://scholar.google.com/scholar_lookup?title=Lorentz+and+CPT+invariance+violation+in+high-energy+neutrinos&author=Hooper,+D.&author=Morgan,+D.&author=Winstanley,+E.&publication_year=2005&journal=Phys.+Rev.+D&volume=72&pages=065009&doi=10.1103/PhysRevD.72.065009" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.72.065009" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B65-universe-10-00357' class='html-xx' data-content='65.'>Tomar, G.; Mohanty, S.; Pakvasa, S. Lorentz Invariance Violation and IceCube Neutrino Events. <span class='html-italic'>J. High Energy Phys.</span> <b>2015</b>, <span class='html-italic'>11</span>, 22. [<a href="https://scholar.google.com/scholar_lookup?title=Lorentz+Invariance+Violation+and+IceCube+Neutrino+Events&author=Tomar,+G.&author=Mohanty,+S.&author=Pakvasa,+S.&publication_year=2015&journal=J.+High+Energy+Phys.&volume=11&pages=22&doi=10.1007/JHEP11(2015)022" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP11(2015)022" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B66-universe-10-00357' class='html-xx' data-content='66.'>Liao, J.; Marfatia, D. IceCube’s astrophysical neutrino energy spectrum from CPT violation. <span class='html-italic'>Phys. Rev. D</span> <b>2018</b>, <span class='html-italic'>97</span>, 041302. [<a href="https://scholar.google.com/scholar_lookup?title=IceCube%E2%80%99s+astrophysical+neutrino+energy+spectrum+from+CPT+violation&author=Liao,+J.&author=Marfatia,+D.&publication_year=2018&journal=Phys.+Rev.+D&volume=97&pages=041302&doi=10.1103/PhysRevD.97.041302" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.97.041302" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B67-universe-10-00357' class='html-xx' data-content='67.'>Abbasi, R.; Abdou, Y.; Abu-Zayyad, T.; Adams, J.; Aguilar, J.A.; Ahlers, M.; Andeen, K.; Auffenberg, J.; Bai, X.; Baker, M.; et al. Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube. <span class='html-italic'>Phys. Rev. D</span> <b>2010</b>, <span class='html-italic'>82</span>, 112003. [<a href="https://scholar.google.com/scholar_lookup?title=Search+for+a+Lorentz-violating+sidereal+signal+with+atmospheric+neutrinos+in+IceCube&author=Abbasi,+R.&author=Abdou,+Y.&author=Abu-Zayyad,+T.&author=Adams,+J.&author=Aguilar,+J.A.&author=Ahlers,+M.&author=Andeen,+K.&author=Auffenberg,+J.&author=Bai,+X.&author=Baker,+M.&publication_year=2010&journal=Phys.+Rev.+D&volume=82&pages=112003&doi=10.1103/PhysRevD.82.112003" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.82.112003" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B68-universe-10-00357' class='html-xx' data-content='68.'>Chatterjee, A.; Gandhi, R.; Singh, J. Probing Lorentz and CPT Violation in a Magnetized Iron Detector using Atmospheric Neutrinos. <span class='html-italic'>J. High Energy Phys.</span> <b>2014</b>, <span class='html-italic'>6</span>, 45. [<a href="https://scholar.google.com/scholar_lookup?title=Probing+Lorentz+and+CPT+Violation+in+a+Magnetized+Iron+Detector+using+Atmospheric+Neutrinos&author=Chatterjee,+A.&author=Gandhi,+R.&author=Singh,+J.&publication_year=2014&journal=J.+High+Energy+Phys.&volume=6&pages=45&doi=10.1007/JHEP06(2014)045" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP06(2014)045" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B69-universe-10-00357' class='html-xx' data-content='69.'>Sahoo, S.; Kumar, A.; Agarwalla, S.K. Probing Lorentz Invariance Violation with atmospheric neutrinos at INO-ICAL. <span class='html-italic'>J. High Energy Phys.</span> <b>2022</b>, <span class='html-italic'>3</span>, 50. [<a href="https://scholar.google.com/scholar_lookup?title=Probing+Lorentz+Invariance+Violation+with+atmospheric+neutrinos+at+INO-ICAL&author=Sahoo,+S.&author=Kumar,+A.&author=Agarwalla,+S.K.&publication_year=2022&journal=J.+High+Energy+Phys.&volume=3&pages=50&doi=10.1007/JHEP03(2022)050" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP03(2022)050" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B70-universe-10-00357' class='html-xx' data-content='70.'>Datta, A.; Gandhi, R.; Mehta, P.; Sankar, S.U. Atmospheric neutrinos as a probe of CPT and Lorentz violation. <span class='html-italic'>Phys. Lett. B</span> <b>2004</b>, <span class='html-italic'>597</span>, 356–361. [<a href="https://scholar.google.com/scholar_lookup?title=Atmospheric+neutrinos+as+a+probe+of+CPT+and+Lorentz+violation&author=Datta,+A.&author=Gandhi,+R.&author=Mehta,+P.&author=Sankar,+S.U.&publication_year=2004&journal=Phys.+Lett.+B&volume=597&pages=356%E2%80%93361&doi=10.1016/j.physletb.2004.07.035" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2004.07.035" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B71-universe-10-00357' class='html-xx' data-content='71.'>Abe, K.; Haga, Y.; Hayato, Y.; Ikeda, M.; Iyogi, K.; Kameda, J.; Kishimoto, Y.; Miura, M.; Moriyama, S.; Nakahata, M.; et al. Test of Lorentz invariance with atmospheric neutrinos. <span class='html-italic'>Phys. Rev. D</span> <b>2015</b>, <span class='html-italic'>91</span>, 052003. [<a href="https://scholar.google.com/scholar_lookup?title=Test+of+Lorentz+invariance+with+atmospheric+neutrinos&author=Abe,+K.&author=Haga,+Y.&author=Hayato,+Y.&author=Ikeda,+M.&author=Iyogi,+K.&author=Kameda,+J.&author=Kishimoto,+Y.&author=Miura,+M.&author=Moriyama,+S.&author=Nakahata,+M.&publication_year=2015&journal=Phys.+Rev.+D&volume=91&pages=052003&doi=10.1103/PhysRevD.91.052003" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.91.052003" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B72-universe-10-00357' class='html-xx' data-content='72.'>Argüelles, C.A.; Aurisano, A.J.; Batell, B.; Berger, J.; Bishai, M.; Boschi, T.; Byrnes, N.; Chatterjee, A.; Chodos, A.; Coan, T.; et al. New opportunities at the next-generation neutrino experiments I: BSM neutrino physics and dark matter. <span class='html-italic'>Rept. Prog. Phys.</span> <b>2020</b>, <span class='html-italic'>83</span>, 124201. [<a href="https://scholar.google.com/scholar_lookup?title=New+opportunities+at+the+next-generation+neutrino+experiments+I:+BSM+neutrino+physics+and+dark+matter&author=Arg%C3%BCelles,+C.A.&author=Aurisano,+A.J.&author=Batell,+B.&author=Berger,+J.&author=Bishai,+M.&author=Boschi,+T.&author=Byrnes,+N.&author=Chatterjee,+A.&author=Chodos,+A.&author=Coan,+T.&publication_year=2020&journal=Rept.+Prog.+Phys.&volume=83&pages=124201&doi=10.1088/1361-6633/ab9d12" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1361-6633/ab9d12" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B73-universe-10-00357' class='html-xx' data-content='73.'>Argüelles, C.A.; Barenboim, G.; Bustamante, M.; Coloma, P.; Denton, P.B.; Esteban, I.; Farzan, Y.; Martínez, E.F.; Forero, D.V.; Gago, A.M.; et al. Snowmass white paper: Beyond the standard model effects on neutrino flavor: Submitted to the proceedings of the US community study on the future of particle physics (Snowmass 2021). <span class='html-italic'>Eur. Phys. J. C</span> <b>2023</b>, <span class='html-italic'>83</span>, 15. [<a href="https://scholar.google.com/scholar_lookup?title=Snowmass+white+paper:+Beyond+the+standard+model+effects+on+neutrino+flavor:+Submitted+to+the+proceedings+of+the+US+community+study+on+the+future+of+particle+physics+(Snowmass+2021)&author=Arg%C3%BCelles,+C.A.&author=Barenboim,+G.&author=Bustamante,+M.&author=Coloma,+P.&author=Denton,+P.B.&author=Esteban,+I.&author=Farzan,+Y.&author=Mart%C3%ADnez,+E.F.&author=Forero,+D.V.&author=Gago,+A.M.&publication_year=2023&journal=Eur.+Phys.+J.+C&volume=83&pages=15&doi=10.1140/epjc/s10052-022-11049-7" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-022-11049-7" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B74-universe-10-00357' class='html-xx' data-content='74.'>NuFit 5.2. 2022. Available online: <a href="http://www.nu-fit.org" target="_blank" rel="noopener noreferrer">www.nu-fit.org</a> (accessed on 15 March 2024).</li><li id='B75-universe-10-00357' class='html-xx' data-content='75.'>Huber, P.; Lindner, M.; Winter, W. Simulation of long-baseline neutrino oscillation experiments with GLoBES (General Long Baseline Experiment Simulator). <span class='html-italic'>Comput. Phys. Commun.</span> <b>2005</b>, <span class='html-italic'>167</span>, 195. [<a href="https://scholar.google.com/scholar_lookup?title=Simulation+of+long-baseline+neutrino+oscillation+experiments+with+GLoBES+(General+Long+Baseline+Experiment+Simulator)&author=Huber,+P.&author=Lindner,+M.&author=Winter,+W.&publication_year=2005&journal=Comput.+Phys.+Commun.&volume=167&pages=195&doi=10.1016/j.cpc.2005.01.003" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.cpc.2005.01.003" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B76-universe-10-00357' class='html-xx' data-content='76.'>Huber, P.; Kopp, J.; Lindner, M.; Rolinec, M.; Winter, W. New features in the simulation of neutrino oscillation experiments with GLoBES 3.0: General Long Baseline Experiment Simulator. <span class='html-italic'>Comput. Phys. Commun.</span> <b>2007</b>, <span class='html-italic'>177</span>, 432–438. [<a href="https://scholar.google.com/scholar_lookup?title=New+features+in+the+simulation+of+neutrino+oscillation+experiments+with+GLoBES+3.0:+General+Long+Baseline+Experiment+Simulator&author=Huber,+P.&author=Kopp,+J.&author=Lindner,+M.&author=Rolinec,+M.&author=Winter,+W.&publication_year=2007&journal=Comput.+Phys.+Commun.&volume=177&pages=432%E2%80%93438&doi=10.1016/j.cpc.2007.05.004" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.cpc.2007.05.004" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B77-universe-10-00357' class='html-xx' data-content='77.'>Kopp, J.; Lindner, M.; Ota, T.; Sato, J. Non-standard neutrino interactions in reactor and superbeam experiments. <span class='html-italic'>Phys. Rev. D</span> <b>2008</b>, <span class='html-italic'>77</span>, 013007. [<a href="https://scholar.google.com/scholar_lookup?title=Non-standard+neutrino+interactions+in+reactor+and+superbeam+experiments&author=Kopp,+J.&author=Lindner,+M.&author=Ota,+T.&author=Sato,+J.&publication_year=2008&journal=Phys.+Rev.+D&volume=77&pages=013007&doi=10.1103/PhysRevD.77.013007" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.77.013007" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B78-universe-10-00357' class='html-xx' data-content='78.'>Huber, P.; Lindner, M.; Winter, W. Superbeams versus neutrino factories. <span class='html-italic'>Nucl. Phys. B</span> <b>2002</b>, <span class='html-italic'>645</span>, 3–48. [<a href="https://scholar.google.com/scholar_lookup?title=Superbeams+versus+neutrino+factories&author=Huber,+P.&author=Lindner,+M.&author=Winter,+W.&publication_year=2002&journal=Nucl.+Phys.+B&volume=645&pages=3%E2%80%9348&doi=10.1016/S0550-3213(02)00825-8" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0550-3213(02)00825-8" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B79-universe-10-00357' class='html-xx' data-content='79.'>Fogli, G.L.; Lisi, E.; Marrone, A.; Montanino, D.; Palazzo, A. Getting the most from the statistical analysis of solar neutrino oscillations. <span class='html-italic'>Phys. Rev. D</span> <b>2002</b>, <span class='html-italic'>66</span>, 053010. [<a href="https://scholar.google.com/scholar_lookup?title=Getting+the+most+from+the+statistical+analysis+of+solar+neutrino+oscillations&author=Fogli,+G.L.&author=Lisi,+E.&author=Marrone,+A.&author=Montanino,+D.&author=Palazzo,+A.&publication_year=2002&journal=Phys.+Rev.+D&volume=66&pages=053010&doi=10.1103/PhysRevD.66.053010" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.66.053010" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B80-universe-10-00357' class='html-xx' data-content='80.'>Abi, B.; Acciarri, R.; Acero, M.A.; Adamov, G.; Adams, D.; Adinolfi, M.; Ahmad, Z.; Ahmed, J.; Alion, T.; Monsalve, S.A.; et al. Prospects for beyond the Standard Model physics searches at the Deep Underground Neutrino Experiment. <span class='html-italic'>Eur. Phys. J. C</span> <b>2021</b>, <span class='html-italic'>81</span>, 322. [<a href="https://scholar.google.com/scholar_lookup?title=Prospects+for+beyond+the+Standard+Model+physics+searches+at+the+Deep+Underground+Neutrino+Experiment&author=Abi,+B.&author=Acciarri,+R.&author=Acero,+M.A.&author=Adamov,+G.&author=Adams,+D.&author=Adinolfi,+M.&author=Ahmad,+Z.&author=Ahmed,+J.&author=Alion,+T.&author=Monsalve,+S.A.&publication_year=2021&journal=Eur.+Phys.+J.+C&volume=81&pages=322&doi=10.1140/epjc/s10052-021-09007-w" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-021-09007-w" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B81-universe-10-00357' class='html-xx' data-content='81.'>Singh, M.; Bustamante, M.; Agarwalla, S.K. Flavor-dependent long-range neutrino interactions in DUNE & T2HK: Alone they constrain, together they discover. <span class='html-italic'>J. High Energy Phys.</span> <b>2023</b>, <span class='html-italic'>8</span>, 101. [<a href="https://scholar.google.com/scholar_lookup?title=Flavor-dependent+long-range+neutrino+interactions+in+DUNE+%2526+T2HK:+Alone+they+constrain,+together+they+discover&author=Singh,+M.&author=Bustamante,+M.&author=Agarwalla,+S.K.&publication_year=2023&journal=J.+High+Energy+Phys.&volume=8&pages=101&doi=10.1007/JHEP08(2023)101" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP08(2023)101" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B82-universe-10-00357' class='html-xx' data-content='82.'>Berryman, J.M.; de Gouvêa, A.; Kelly, K.J.; Peres, O.L.G.; Tabrizi, Z. Large, Extra Dimensions at the Deep Underground Neutrino Experiment. <span class='html-italic'>Phys. Rev. D</span> <b>2016</b>, <span class='html-italic'>94</span>, 033006. [<a href="https://scholar.google.com/scholar_lookup?title=Large,+Extra+Dimensions+at+the+Deep+Underground+Neutrino+Experiment&author=Berryman,+J.M.&author=de+Gouv%C3%AAa,+A.&author=Kelly,+K.J.&author=Peres,+O.L.G.&author=Tabrizi,+Z.&publication_year=2016&journal=Phys.+Rev.+D&volume=94&pages=033006&doi=10.1103/PhysRevD.94.033006" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.94.033006" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B83-universe-10-00357' class='html-xx' data-content='83.'>Agarwalla, S.K.; Das, S.; Giarnetti, A.; Meloni, D.; Singh, M. Enhancing sensitivity to leptonic CP violation using complementarity among DUNE, T2HK, and T2HKK. <span class='html-italic'>Eur. Phys. J. C</span> <b>2023</b>, <span class='html-italic'>83</span>, 694. [<a href="https://scholar.google.com/scholar_lookup?title=Enhancing+sensitivity+to+leptonic+CP+violation+using+complementarity+among+DUNE,+T2HK,+and+T2HKK&author=Agarwalla,+S.K.&author=Das,+S.&author=Giarnetti,+A.&author=Meloni,+D.&author=Singh,+M.&publication_year=2023&journal=Eur.+Phys.+J.+C&volume=83&pages=694&doi=10.1140/epjc/s10052-023-11863-7" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-023-11863-7" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B84-universe-10-00357' class='html-xx' data-content='84.'>Denton, P.B.; Gehrlein, J.; Pestes, R. <span class='html-italic'>CP</span> -Violating Neutrino Nonstandard Interactions in Long-Baseline-Accelerator Data. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2021</b>, <span class='html-italic'>126</span>, 051801. [<a href="https://scholar.google.com/scholar_lookup?title=CP+-Violating+Neutrino+Nonstandard+Interactions+in+Long-Baseline-Accelerator+Data&author=Denton,+P.B.&author=Gehrlein,+J.&author=Pestes,+R.&publication_year=2021&journal=Phys.+Rev.+Lett.&volume=126&pages=051801&doi=10.1103/PhysRevLett.126.051801" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.126.051801" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B85-universe-10-00357' class='html-xx' data-content='85.'>de Gouvêa, A.; Kelly, K.J. Non-standard Neutrino Interactions at DUNE. <span class='html-italic'>Nucl. Phys. B</span> <b>2016</b>, <span class='html-italic'>908</span>, 318–335. [<a href="https://scholar.google.com/scholar_lookup?title=Non-standard+Neutrino+Interactions+at+DUNE&author=de+Gouv%C3%AAa,+A.&author=Kelly,+K.J.&publication_year=2016&journal=Nucl.+Phys.+B&volume=908&pages=318%E2%80%93335&doi=10.1016/j.nuclphysb.2016.03.013" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.nuclphysb.2016.03.013" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B86-universe-10-00357' class='html-xx' data-content='86.'>Neutrino Non-Standard Interactions: A Status Report. <span class='html-italic'>SciPost Phys. Proc.</span> <b>2019</b>, <span class='html-italic'>2</span>, 1. [<a href="https://doi.org/10.21468/SciPostPhysProc.2.001" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B87-universe-10-00357' class='html-xx' data-content='87.'>Bakhti, P.; Rajaee, M. Sensitivities of future reactor and long-baseline neutrino experiments to NSI. <span class='html-italic'>Phys. Rev. D</span> <b>2021</b>, <span class='html-italic'>103</span>, 075003. [<a href="https://scholar.google.com/scholar_lookup?title=Sensitivities+of+future+reactor+and+long-baseline+neutrino+experiments+to+NSI&author=Bakhti,+P.&author=Rajaee,+M.&publication_year=2021&journal=Phys.+Rev.+D&volume=103&pages=075003&doi=10.1103/PhysRevD.103.075003" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.103.075003" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B88-universe-10-00357' class='html-xx' data-content='88.'>Ge, S.F.; Parke, S.J. Scalar Nonstandard Interactions in Neutrino Oscillation. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2019</b>, <span class='html-italic'>122</span>, 211801. [<a href="https://scholar.google.com/scholar_lookup?title=Scalar+Nonstandard+Interactions+in+Neutrino+Oscillation&author=Ge,+S.F.&author=Parke,+S.J.&publication_year=2019&journal=Phys.+Rev.+Lett.&volume=122&pages=211801&doi=10.1103/PhysRevLett.122.211801&pmid=31283351" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.122.211801" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/31283351" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B89-universe-10-00357' class='html-xx' data-content='89.'>Denton, P.B.; Giarnetti, A.; Meloni, D. How to identify different new neutrino oscillation physics scenarios at DUNE. <span class='html-italic'>J. High Energy Phys.</span> <b>2023</b>, <span class='html-italic'>2</span>, 210. [<a href="https://scholar.google.com/scholar_lookup?title=How+to+identify+different+new+neutrino+oscillation+physics+scenarios+at+DUNE&author=Denton,+P.B.&author=Giarnetti,+A.&author=Meloni,+D.&publication_year=2023&journal=J.+High+Energy+Phys.&volume=2&pages=210&doi=10.1007/JHEP02(2023)210" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP02(2023)210" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B90-universe-10-00357' class='html-xx' data-content='90.'>Aguilar, J.; Anastasopoulos, M.; Baussan, E.; Bhattacharyya, A.K.; Bignami, A.; Blennow, M.; Bogomilov, M.; Bolling, B.; Bouquerel, E.; Bramati, F.; et al. Study of non-standard interaction mediated by a scalar field at ESSnuSB experiment. <span class='html-italic'>arXiv</span> <b>2023</b>, arXiv:2310.10749. [<a href="https://scholar.google.com/scholar_lookup?title=Study+of+non-standard+interaction+mediated+by+a+scalar+field+at+ESSnuSB+experiment&author=Aguilar,+J.&author=Anastasopoulos,+M.&author=Baussan,+E.&author=Bhattacharyya,+A.K.&author=Bignami,+A.&author=Blennow,+M.&author=Bogomilov,+M.&author=Bolling,+B.&author=Bouquerel,+E.&author=Bramati,+F.&publication_year=2023&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B91-universe-10-00357' class='html-xx' data-content='91.'>Gupta, A.; Majumdar, D.; Prakash, S. Neutrino oscillation measurements with JUNO in the presence of scalar NSI. <span class='html-italic'>arXiv</span> <b>2023</b>, arXiv:2306.07343. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillation+measurements+with+JUNO+in+the+presence+of+scalar+NSI&author=Gupta,+A.&author=Majumdar,+D.&author=Prakash,+S.&publication_year=2023&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B92-universe-10-00357' class='html-xx' data-content='92.'>Sarker, A.; Medhi, A.; Bezboruah, D.; Devi, M.M.; Dutta, D. Impact of scalar NSI on the neutrino mass hierarchy sensitivity at DUNE, T2HK and T2HKK. <span class='html-italic'>arXiv</span> <b>2023</b>, arXiv:2309.12249. [<a href="https://scholar.google.com/scholar_lookup?title=Impact+of+scalar+NSI+on+the+neutrino+mass+hierarchy+sensitivity+at+DUNE,+T2HK+and+T2HKK&author=Sarker,+A.&author=Medhi,+A.&author=Bezboruah,+D.&author=Devi,+M.M.&author=Dutta,+D.&publication_year=2023&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B93-universe-10-00357' class='html-xx' data-content='93.'>Grifols, J.A.; Masso, E.; Peris, S. Supernova neutrinos as probes of long range nongravitational interactions of dark matter. <span class='html-italic'>Astropart. Phys.</span> <b>1994</b>, <span class='html-italic'>2</span>, 161–165. [<a href="https://scholar.google.com/scholar_lookup?title=Supernova+neutrinos+as+probes+of+long+range+nongravitational+interactions+of+dark+matter&author=Grifols,+J.A.&author=Masso,+E.&author=Peris,+S.&publication_year=1994&journal=Astropart.+Phys.&volume=2&pages=161%E2%80%93165&doi=10.1016/0927-6505(94)90038-8" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/0927-6505(94)90038-8" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B94-universe-10-00357' class='html-xx' data-content='94.'>Grifols, J.A.; Masso, E.; Toldra, R. Majorana neutrinos and long range forces. <span class='html-italic'>Phys. Lett. B</span> <b>1996</b>, <span class='html-italic'>389</span>, 563–565. [<a href="https://scholar.google.com/scholar_lookup?title=Majorana+neutrinos+and+long+range+forces&author=Grifols,+J.A.&author=Masso,+E.&author=Toldra,+R.&publication_year=1996&journal=Phys.+Lett.+B&volume=389&pages=563%E2%80%93565&doi=10.1016/S0370-2693(96)01304-4" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0370-2693(96)01304-4" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B95-universe-10-00357' class='html-xx' data-content='95.'>Grifols, J.A.; Masso, E. Neutrino oscillations in the sun probe long range leptonic forces. <span class='html-italic'>Phys. Lett. B</span> <b>2004</b>, <span class='html-italic'>579</span>, 123–126. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillations+in+the+sun+probe+long+range+leptonic+forces&author=Grifols,+J.A.&author=Masso,+E.&publication_year=2004&journal=Phys.+Lett.+B&volume=579&pages=123%E2%80%93126&doi=10.1016/j.physletb.2003.10.078" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2003.10.078" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B96-universe-10-00357' class='html-xx' data-content='96.'>Mishra, P.; Majhi, R.; Pusty, S.K.; Ghosh, M.; Mohanta, R. Study of Long Range Force in P2SO and T2HKK. <span class='html-italic'>arXiv</span> <b>2024</b>, arXiv:2402.19178. [<a href="https://scholar.google.com/scholar_lookup?title=Study+of+Long+Range+Force+in+P2SO+and+T2HKK&author=Mishra,+P.&author=Majhi,+R.&author=Pusty,+S.K.&author=Ghosh,+M.&author=Mohanta,+R.&publication_year=2024&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B97-universe-10-00357' class='html-xx' data-content='97.'>Agarwalla, S.K.; Bustamante, M.; Singh, M.; Swain, P. A plethora of long-range neutrino interactions probed by DUNE and T2HK. <span class='html-italic'>arXiv</span> <b>2024</b>, arXiv:2404.02775. [<a href="https://scholar.google.com/scholar_lookup?title=A+plethora+of+long-range+neutrino+interactions+probed+by+DUNE+and+T2HK&author=Agarwalla,+S.K.&author=Bustamante,+M.&author=Singh,+M.&author=Swain,+P.&publication_year=2024&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B98-universe-10-00357' class='html-xx' data-content='98.'>Agarwalla, S.K.; Bustamante, M.; Das, S.; Narang, A. Present and future constraints on flavor-dependent long-range interactions of high-energy astrophysical neutrinos. <span class='html-italic'>J. High Energy Phys.</span> <b>2023</b>, <span class='html-italic'>8</span>, 113. [<a href="https://scholar.google.com/scholar_lookup?title=Present+and+future+constraints+on+flavor-dependent+long-range+interactions+of+high-energy+astrophysical+neutrinos&author=Agarwalla,+S.K.&author=Bustamante,+M.&author=Das,+S.&author=Narang,+A.&publication_year=2023&journal=J.+High+Energy+Phys.&volume=8&pages=113&doi=10.1007/JHEP08(2023)113" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP08(2023)113" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B99-universe-10-00357' class='html-xx' data-content='99.'>Chatterjee, S.S.; Dasgupta, A.; Agarwalla, S.K. Exploring Flavor-Dependent Long-Range Forces in Long-Baseline Neutrino Oscillation Experiments. <span class='html-italic'>J. High Energy Phys.</span> <b>2015</b>, <span class='html-italic'>12</span>, 167. [<a href="https://scholar.google.com/scholar_lookup?title=Exploring+Flavor-Dependent+Long-Range+Forces+in+Long-Baseline+Neutrino+Oscillation+Experiments&author=Chatterjee,+S.S.&author=Dasgupta,+A.&author=Agarwalla,+S.K.&publication_year=2015&journal=J.+High+Energy+Phys.&volume=12&pages=167&doi=10.1007/JHEP12(2015)167" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP12(2015)167" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B100-universe-10-00357' class='html-xxx' data-content='100.'>Coloma, P.; Gonzalez-Garcia, M.C.; Maltoni, M. Neutrino oscillation constraints on U(1)’ models: From non-standard interactions to long-range forces. <span class='html-italic'>J. High Energy Phys.</span> <b>2021</b>, <span class='html-italic'>1</span>, 114, Erratum in: <span class='html-italic'>J. High Energy Phys.</span> <b>2022</b>, <span class='html-italic'>11</span>, 115. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillation+constraints+on+U(1)%E2%80%99+models:+From+non-standard+interactions+to+long-range+forces&author=Coloma,+P.&author=Gonzalez-Garcia,+M.C.&author=Maltoni,+M.&publication_year=2021&journal=J.+High+Energy+Phys.&volume=1&pages=114&doi=10.1007/JHEP11(2022)115" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP11(2022)115" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B101-universe-10-00357' class='html-xxx' data-content='101.'>Smirnov, A.Y.; Xu, X.J. Wolfenstein potentials for neutrinos induced by ultra-light mediators. <span class='html-italic'>J. High Energy Phys.</span> <b>2019</b>, <span class='html-italic'>12</span>, 46. [<a href="https://scholar.google.com/scholar_lookup?title=Wolfenstein+potentials+for+neutrinos+induced+by+ultra-light+mediators&author=Smirnov,+A.Y.&author=Xu,+X.J.&publication_year=2019&journal=J.+High+Energy+Phys.&volume=12&pages=46&doi=10.1007/JHEP12(2019)046" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP12(2019)046" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B102-universe-10-00357' class='html-xxx' data-content='102.'>Chauhan, G.; Xu, X.J. Impact of the cosmic neutrino background on long-range force searches. <span class='html-italic'>J. High Energy Phys.</span> <b>2024</b>, <span class='html-italic'>7</span>, 255. [<a href="https://scholar.google.com/scholar_lookup?title=Impact+of+the+cosmic+neutrino+background+on+long-range+force+searches&author=Chauhan,+G.&author=Xu,+X.J.&publication_year=2024&journal=J.+High+Energy+Phys.&volume=7&pages=255&doi=10.1007/JHEP07(2024)255" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP07(2024)255" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B103-universe-10-00357' class='html-xxx' data-content='103.'>He, X.G.; Joshi, G.C.; Lew, H.; Volkas, R.R. New-<span class='html-italic'>Z</span><sup>′</sup> phenomenology. <span class='html-italic'>Phys. Rev. D</span> <b>1991</b>, <span class='html-italic'>43</span>, R22–R24. [<a href="https://scholar.google.com/scholar_lookup?title=New-Z%E2%80%B2+phenomenology&author=He,+X.G.&author=Joshi,+G.C.&author=Lew,+H.&author=Volkas,+R.R.&publication_year=1991&journal=Phys.+Rev.+D&volume=43&pages=R22%E2%80%93R24&doi=10.1103/PhysRevD.43.R22" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.43.R22" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B104-universe-10-00357' class='html-xxx' data-content='104.'>He, X.G.; Joshi, G.C.; Lew, H.; Volkas, R.R. Simplest <span class='html-italic'>Z</span><sup>′</sup> model. <span class='html-italic'>Phys. Rev. D</span> <b>1991</b>, <span class='html-italic'>44</span>, 2118–2132. [<a href="https://scholar.google.com/scholar_lookup?title=Simplest+Z%E2%80%B2+model&author=He,+X.G.&author=Joshi,+G.C.&author=Lew,+H.&author=Volkas,+R.R.&publication_year=1991&journal=Phys.+Rev.+D&volume=44&pages=2118%E2%80%932132&doi=10.1103/PhysRevD.44.2118" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.44.2118" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B105-universe-10-00357' class='html-xxx' data-content='105.'>Foot, R.; He, X.G.; Lew, H.; Volkas, R.R. Model for a light Z-prime boson. <span class='html-italic'>Phys. Rev. D</span> <b>1994</b>, <span class='html-italic'>50</span>, 4571–4580. [<a href="https://scholar.google.com/scholar_lookup?title=Model+for+a+light+Z-prime+boson&author=Foot,+R.&author=He,+X.G.&author=Lew,+H.&author=Volkas,+R.R.&publication_year=1994&journal=Phys.+Rev.+D&volume=50&pages=4571%E2%80%934580&doi=10.1103/PhysRevD.50.4571" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.50.4571" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B106-universe-10-00357' class='html-xxx' data-content='106.'>Asai, K.; Hamaguchi, K.; Nagata, N.; Tseng, S.Y.; Tsumura, K. Minimal Gauged U(1)<sub>Lα-Lβ</sub> Models Driven into a Corner. <span class='html-italic'>Phys. Rev. D</span> <b>2019</b>, <span class='html-italic'>99</span>, 055029. [<a href="https://scholar.google.com/scholar_lookup?title=Minimal+Gauged+U(1)L%CE%B1-L%CE%B2+Models+Driven+into+a+Corner&author=Asai,+K.&author=Hamaguchi,+K.&author=Nagata,+N.&author=Tseng,+S.Y.&author=Tsumura,+K.&publication_year=2019&journal=Phys.+Rev.+D&volume=99&pages=055029&doi=10.1103/PhysRevD.99.055029" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.99.055029" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B107-universe-10-00357' class='html-xxx' data-content='107.'>Asai, K.; Hamaguchi, K.; Nagata, N. Predictions for the neutrino parameters in the minimal gauged U(1)<sub>Lμ-Lτ</sub> model. <span class='html-italic'>Eur. Phys. J. C</span> <b>2017</b>, <span class='html-italic'>77</span>, 763. [<a href="https://scholar.google.com/scholar_lookup?title=Predictions+for+the+neutrino+parameters+in+the+minimal+gauged+U(1)L%CE%BC-L%CF%84+model&author=Asai,+K.&author=Hamaguchi,+K.&author=Nagata,+N.&publication_year=2017&journal=Eur.+Phys.+J.+C&volume=77&pages=763&doi=10.1140/epjc/s10052-017-5348-x" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1140/epjc/s10052-017-5348-x" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B108-universe-10-00357' class='html-xxx' data-content='108.'>Lou, Y.; Nomura, T. Neutrino observables in gauged <span class='html-italic'>U</span>(1)<sub>Lα-Lβ</sub> models with two Higgs doublet and one singlet scalars. <span class='html-italic'>arXiv</span> <b>2024</b>, arXiv:2406.01030. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+observables+in+gauged+U(1)L%CE%B1-L%CE%B2+models+with+two+Higgs+doublet+and+one+singlet+scalars&author=Lou,+Y.&author=Nomura,+T.&publication_year=2024&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B109-universe-10-00357' class='html-xxx' data-content='109.'>Bustamante, M.; Agarwalla, S.K. Universe’s Worth of Electrons to Probe Long-Range Interactions of High-Energy Astrophysical Neutrinos. <span class='html-italic'>Phys. Rev. Lett.</span> <b>2019</b>, <span class='html-italic'>122</span>, 061103. [<a href="https://scholar.google.com/scholar_lookup?title=Universe%E2%80%99s+Worth+of+Electrons+to+Probe+Long-Range+Interactions+of+High-Energy+Astrophysical+Neutrinos&author=Bustamante,+M.&author=Agarwalla,+S.K.&publication_year=2019&journal=Phys.+Rev.+Lett.&volume=122&pages=061103&doi=10.1103/PhysRevLett.122.061103&pmid=30822075" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevLett.122.061103" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>] [<a href="https://www.ncbi.nlm.nih.gov/pubmed/30822075" class='cross-ref pub_med' data-typ='pmid' target='_blank' rel='noopener noreferrer'>PubMed</a>]</li><li id='B110-universe-10-00357' class='html-xxx' data-content='110.'>Wise, M.B.; Zhang, Y. Lepton Flavorful Fifth Force and Depth-dependent Neutrino Matter Interactions. <span class='html-italic'>J. High Energy Phys.</span> <b>2018</b>, <span class='html-italic'>6</span>, 53. [<a href="https://scholar.google.com/scholar_lookup?title=Lepton+Flavorful+Fifth+Force+and+Depth-dependent+Neutrino+Matter+Interactions&author=Wise,+M.B.&author=Zhang,+Y.&publication_year=2018&journal=J.+High+Energy+Phys.&volume=6&pages=53&doi=10.1007/JHEP06(2018)053" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP06(2018)053" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B111-universe-10-00357' class='html-xxx' data-content='111.'>Khatun, A.; Thakore, T.; Kumar Agarwalla, S. Can INO be Sensitive to Flavor-Dependent Long-Range Forces? <span class='html-italic'>J. High Energy Phys.</span> <b>2018</b>, <span class='html-italic'>4</span>, 23. [<a href="https://scholar.google.com/scholar_lookup?title=Can+INO+be+Sensitive+to+Flavor-Dependent+Long-Range+Forces?&author=Khatun,+A.&author=Thakore,+T.&author=Kumar+Agarwalla,+S.&publication_year=2018&journal=J.+High+Energy+Phys.&volume=4&pages=23&doi=10.1007/JHEP04(2018)023" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP04(2018)023" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B112-universe-10-00357' class='html-xxx' data-content='112.'>Agarwalla, S.K.; Das, S.; Masud, M.; Swain, P. Evolution of neutrino mass-mixing parameters in matter with non-standard interactions. <span class='html-italic'>J. High Energy Phys.</span> <b>2021</b>, <span class='html-italic'>11</span>, 94. [<a href="https://scholar.google.com/scholar_lookup?title=Evolution+of+neutrino+mass-mixing+parameters+in+matter+with+non-standard+interactions&author=Agarwalla,+S.K.&author=Das,+S.&author=Masud,+M.&author=Swain,+P.&publication_year=2021&journal=J.+High+Energy+Phys.&volume=11&pages=94&doi=10.1007/JHEP11(2021)094" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP11(2021)094" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B113-universe-10-00357' class='html-xxx' data-content='113.'>Joshipura, A.S.; Mohanty, S. Constraints on flavor dependent long range forces from atmospheric neutrino observations at super-Kamiokande. <span class='html-italic'>Phys. Lett. B</span> <b>2004</b>, <span class='html-italic'>584</span>, 103–108. [<a href="https://scholar.google.com/scholar_lookup?title=Constraints+on+flavor+dependent+long+range+forces+from+atmospheric+neutrino+observations+at+super-Kamiokande&author=Joshipura,+A.S.&author=Mohanty,+S.&publication_year=2004&journal=Phys.+Lett.+B&volume=584&pages=103%E2%80%93108&doi=10.1016/j.physletb.2004.01.057" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2004.01.057" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B114-universe-10-00357' class='html-xxx' data-content='114.'>Bandyopadhyay, A.; Dighe, A.; Joshipura, A.S. Constraints on flavor-dependent long range forces from solar neutrinos and KamLAND. <span class='html-italic'>Phys. Rev. D</span> <b>2007</b>, <span class='html-italic'>75</span>, 093005. [<a href="https://scholar.google.com/scholar_lookup?title=Constraints+on+flavor-dependent+long+range+forces+from+solar+neutrinos+and+KamLAND&author=Bandyopadhyay,+A.&author=Dighe,+A.&author=Joshipura,+A.S.&publication_year=2007&journal=Phys.+Rev.+D&volume=75&pages=093005&doi=10.1103/PhysRevD.75.093005" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.75.093005" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B115-universe-10-00357' class='html-xxx' data-content='115.'>Gonzalez-Garcia, M.C.; Maltoni, M. Determination of matter potential from global analysis of neutrino oscillation data. <span class='html-italic'>J. High Energy Phys.</span> <b>2013</b>, <span class='html-italic'>9</span>, 152. [<a href="https://scholar.google.com/scholar_lookup?title=Determination+of+matter+potential+from+global+analysis+of+neutrino+oscillation+data&author=Gonzalez-Garcia,+M.C.&author=Maltoni,+M.&publication_year=2013&journal=J.+High+Energy+Phys.&volume=9&pages=152&doi=10.1007/JHEP09(2013)152" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP09(2013)152" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B116-universe-10-00357' class='html-xxx' data-content='116.'>Honda, M.; Kao, Y.; Okamura, N.; Pronin, A.; Takeuchi, T. Constraints on New Physics from Long Baseline Neutrino Oscillation Experiments. <span class='html-italic'>arXiv</span> <b>2007</b>, arXiv:0707.4545. [<a href="https://scholar.google.com/scholar_lookup?title=Constraints+on+New+Physics+from+Long+Baseline+Neutrino+Oscillation+Experiments&author=Honda,+M.&author=Kao,+Y.&author=Okamura,+N.&author=Pronin,+A.&author=Takeuchi,+T.&publication_year=2007&journal=arXiv" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li><li id='B117-universe-10-00357' class='html-xxx' data-content='117.'>Farzan, Y.; Palomares-Ruiz, S. Flavor of cosmic neutrinos preserved by ultralight dark matter. <span class='html-italic'>Phys. Rev. D</span> <b>2019</b>, <span class='html-italic'>99</span>, 051702. [<a href="https://scholar.google.com/scholar_lookup?title=Flavor+of+cosmic+neutrinos+preserved+by+ultralight+dark+matter&author=Farzan,+Y.&author=Palomares-Ruiz,+S.&publication_year=2019&journal=Phys.+Rev.+D&volume=99&pages=051702&doi=10.1103/PhysRevD.99.051702" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.99.051702" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B118-universe-10-00357' class='html-xxx' data-content='118.'>Aartsen, M.G.; Abbasi, R.; Ackermann, M.; Adams, J.; Aguilar, J.A.; Ahlers, M.; Ahrens, M.; Alispach, C.; Allison, P.; Amin, N.M.; et al. IceCube-Gen2: The window to the extreme Universe. <span class='html-italic'>J. Phys. G</span> <b>2021</b>, <span class='html-italic'>48</span>, 060501. [<a href="https://scholar.google.com/scholar_lookup?title=IceCube-Gen2:+The+window+to+the+extreme+Universe&author=Aartsen,+M.G.&author=Abbasi,+R.&author=Ackermann,+M.&author=Adams,+J.&author=Aguilar,+J.A.&author=Ahlers,+M.&author=Ahrens,+M.&author=Alispach,+C.&author=Allison,+P.&author=Amin,+N.M.&publication_year=2021&journal=J.+Phys.+G&volume=48&pages=060501&doi=10.1088/1361-6471/abbd48" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1361-6471/abbd48" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B119-universe-10-00357' class='html-xxx' data-content='119.'>Heeck, J.; Rodejohann, W. Gauged <span class='html-italic'>L</span><sub>μ</sub>-<span class='html-italic'>L</span><sub>τ</sub> and different Muon Neutrino and Anti-Neutrino Oscillations: MINOS and beyond. <span class='html-italic'>J. Phys. G</span> <b>2011</b>, <span class='html-italic'>38</span>, 085005. [<a href="https://scholar.google.com/scholar_lookup?title=Gauged+L%CE%BC-L%CF%84+and+different+Muon+Neutrino+and+Anti-Neutrino+Oscillations:+MINOS+and+beyond&author=Heeck,+J.&author=Rodejohann,+W.&publication_year=2011&journal=J.+Phys.+G&volume=38&pages=085005&doi=10.1088/0954-3899/38/8/085005" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/0954-3899/38/8/085005" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B120-universe-10-00357' class='html-xxx' data-content='120.'>McMillan, P.J. Mass models of the Milky Way. <span class='html-italic'>Mon. Not. R. Astron. Soc.</span> <b>2011</b>, <span class='html-italic'>414</span>, 2446–2457. [<a href="https://scholar.google.com/scholar_lookup?title=Mass+models+of+the+Milky+Way&author=McMillan,+P.J.&publication_year=2011&journal=Mon.+Not.+R.+Astron.+Soc.&volume=414&pages=2446%E2%80%932457&doi=10.1111/j.1365-2966.2011.18564.x" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1111/j.1365-2966.2011.18564.x" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B121-universe-10-00357' class='html-xxx' data-content='121.'>Miller, M.J.; Bregman, J.N. The Structure of the Milky Way’s Hot Gas Halo. <span class='html-italic'>Astrophys. J.</span> <b>2013</b>, <span class='html-italic'>770</span>, 118. [<a href="https://scholar.google.com/scholar_lookup?title=The+Structure+of+the+Milky+Way%E2%80%99s+Hot+Gas+Halo&author=Miller,+M.J.&author=Bregman,+J.N.&publication_year=2013&journal=Astrophys.+J.&volume=770&pages=118&doi=10.1088/0004-637X/770/2/118" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/0004-637X/770/2/118" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B122-universe-10-00357' class='html-xxx' data-content='122.'>Baryakhtar, M.; Lasenby, R.; Teo, M. Black Hole Superradiance Signatures of Ultralight Vectors. <span class='html-italic'>Phys. Rev. D</span> <b>2017</b>, <span class='html-italic'>96</span>, 035019. [<a href="https://scholar.google.com/scholar_lookup?title=Black+Hole+Superradiance+Signatures+of+Ultralight+Vectors&author=Baryakhtar,+M.&author=Lasenby,+R.&author=Teo,+M.&publication_year=2017&journal=Phys.+Rev.+D&volume=96&pages=035019&doi=10.1103/PhysRevD.96.035019" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.96.035019" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B123-universe-10-00357' class='html-xxx' data-content='123.'>Arkani-Hamed, N.; Motl, L.; Nicolis, A.; Vafa, C. The String landscape, black holes and gravity as the weakest force. <span class='html-italic'>J. High Energy Phys.</span> <b>2007</b>, <span class='html-italic'>6</span>, 60. [<a href="https://scholar.google.com/scholar_lookup?title=The+String+landscape,+black+holes+and+gravity+as+the+weakest+force&author=Arkani-Hamed,+N.&author=Motl,+L.&author=Nicolis,+A.&author=Vafa,+C.&publication_year=2007&journal=J.+High+Energy+Phys.&volume=6&pages=60&doi=10.1088/1126-6708/2007/06/060" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1126-6708/2007/06/060" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B124-universe-10-00357' class='html-xxx' data-content='124.'>de Gouvêa, A. Neutrino Mass Models. <span class='html-italic'>Ann. Rev. Nucl. Part. Sci.</span> <b>2016</b>, <span class='html-italic'>66</span>, 197–217. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+Mass+Models&author=de+Gouv%C3%AAa,+A.&publication_year=2016&journal=Ann.+Rev.+Nucl.+Part.+Sci.&volume=66&pages=197%E2%80%93217&doi=10.1146/annurev-nucl-102115-044600" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1146/annurev-nucl-102115-044600" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B125-universe-10-00357' class='html-xxx' data-content='125.'>King, S.F. Neutrino mass models. <span class='html-italic'>Rept. Prog. Phys.</span> <b>2004</b>, <span class='html-italic'>67</span>, 107–158. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+mass+models&author=King,+S.F.&publication_year=2004&journal=Rept.+Prog.+Phys.&volume=67&pages=107%E2%80%93158&doi=10.1088/0034-4885/67/2/R01" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/0034-4885/67/2/R01" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B126-universe-10-00357' class='html-xxx' data-content='126.'>Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R.; March-Russell, J. Neutrino masses from large extra dimensions. <span class='html-italic'>Phys. Rev. D</span> <b>2001</b>, <span class='html-italic'>65</span>, 024032. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+masses+from+large+extra+dimensions&author=Arkani-Hamed,+N.&author=Dimopoulos,+S.&author=Dvali,+G.R.&author=March-Russell,+J.&publication_year=2001&journal=Phys.+Rev.+D&volume=65&pages=024032&doi=10.1103/PhysRevD.65.024032" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.65.024032" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B127-universe-10-00357' class='html-xxx' data-content='127.'>Dienes, K.R.; Dudas, E.; Gherghetta, T. Neutrino oscillations without neutrino masses or heavy mass scales: A Higher dimensional seesaw mechanism. <span class='html-italic'>Nucl. Phys. B</span> <b>1999</b>, <span class='html-italic'>557</span>, 25. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillations+without+neutrino+masses+or+heavy+mass+scales:+A+Higher+dimensional+seesaw+mechanism&author=Dienes,+K.R.&author=Dudas,+E.&author=Gherghetta,+T.&publication_year=1999&journal=Nucl.+Phys.+B&volume=557&pages=25&doi=10.1016/S0550-3213(99)00377-6" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0550-3213(99)00377-6" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B128-universe-10-00357' class='html-xxx' data-content='128.'>Dvali, G.R.; Smirnov, A.Y. Probing large extra dimensions with neutrinos. <span class='html-italic'>Nucl. Phys. B</span> <b>1999</b>, <span class='html-italic'>563</span>, 63–81. [<a href="https://scholar.google.com/scholar_lookup?title=Probing+large+extra+dimensions+with+neutrinos&author=Dvali,+G.R.&author=Smirnov,+A.Y.&publication_year=1999&journal=Nucl.+Phys.+B&volume=563&pages=63%E2%80%9381&doi=10.1016/S0550-3213(99)00574-X" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0550-3213(99)00574-X" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B129-universe-10-00357' class='html-xxx' data-content='129.'>Mohapatra, R.N.; Perez-Lorenzana, A. Three flavor neutrino oscillations in models with large extra dimensions. <span class='html-italic'>Nucl. Phys. B</span> <b>2001</b>, <span class='html-italic'>593</span>, 451–470. [<a href="https://scholar.google.com/scholar_lookup?title=Three+flavor+neutrino+oscillations+in+models+with+large+extra+dimensions&author=Mohapatra,+R.N.&author=Perez-Lorenzana,+A.&publication_year=2001&journal=Nucl.+Phys.+B&volume=593&pages=451%E2%80%93470&doi=10.1016/S0550-3213(00)00634-9" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0550-3213(00)00634-9" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B130-universe-10-00357' class='html-xxx' data-content='130.'>Barbieri, R.; Creminelli, P.; Strumia, A. Neutrino oscillations from large extra dimensions. <span class='html-italic'>Nucl. Phys. B</span> <b>2000</b>, <span class='html-italic'>585</span>, 28–44. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrino+oscillations+from+large+extra+dimensions&author=Barbieri,+R.&author=Creminelli,+P.&author=Strumia,+A.&publication_year=2000&journal=Nucl.+Phys.+B&volume=585&pages=28%E2%80%9344&doi=10.1016/S0550-3213(00)00348-5" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0550-3213(00)00348-5" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B131-universe-10-00357' class='html-xxx' data-content='131.'>Davoudiasl, H.; Langacker, P.; Perelstein, M. Constraints on large extra dimensions from neutrino oscillation experiments. <span class='html-italic'>Phys. Rev. D</span> <b>2002</b>, <span class='html-italic'>65</span>, 105015. [<a href="https://scholar.google.com/scholar_lookup?title=Constraints+on+large+extra+dimensions+from+neutrino+oscillation+experiments&author=Davoudiasl,+H.&author=Langacker,+P.&author=Perelstein,+M.&publication_year=2002&journal=Phys.+Rev.+D&volume=65&pages=105015&doi=10.1103/PhysRevD.65.105015" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.65.105015" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B132-universe-10-00357' class='html-xxx' data-content='132.'>Antoniadis, I.; Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. New dimensions at a millimeter to a Fermi and superstrings at a TeV. <span class='html-italic'>Phys. Lett. B</span> <b>1998</b>, <span class='html-italic'>436</span>, 257–263. [<a href="https://scholar.google.com/scholar_lookup?title=New+dimensions+at+a+millimeter+to+a+Fermi+and+superstrings+at+a+TeV&author=Antoniadis,+I.&author=Arkani-Hamed,+N.&author=Dimopoulos,+S.&author=Dvali,+G.R.&publication_year=1998&journal=Phys.+Lett.+B&volume=436&pages=257%E2%80%93263&doi=10.1016/S0370-2693(98)00860-0" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0370-2693(98)00860-0" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B133-universe-10-00357' class='html-xxx' data-content='133.'>Antoniadis, I.; Kiritsis, E.; Rizos, J.; Tomaras, T.N. D-branes and the standard model. <span class='html-italic'>Nucl. Phys. B</span> <b>2003</b>, <span class='html-italic'>660</span>, 81–115. [<a href="https://scholar.google.com/scholar_lookup?title=D-branes+and+the+standard+model&author=Antoniadis,+I.&author=Kiritsis,+E.&author=Rizos,+J.&author=Tomaras,+T.N.&publication_year=2003&journal=Nucl.+Phys.+B&volume=660&pages=81%E2%80%93115&doi=10.1016/S0550-3213(03)00256-6" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0550-3213(03)00256-6" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B134-universe-10-00357' class='html-xxx' data-content='134.'>Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. The Hierarchy problem and new dimensions at a millimeter. <span class='html-italic'>Phys. Lett. B</span> <b>1998</b>, <span class='html-italic'>429</span>, 263–272. [<a href="https://scholar.google.com/scholar_lookup?title=The+Hierarchy+problem+and+new+dimensions+at+a+millimeter&author=Arkani-Hamed,+N.&author=Dimopoulos,+S.&author=Dvali,+G.R.&publication_year=1998&journal=Phys.+Lett.+B&volume=429&pages=263%E2%80%93272&doi=10.1016/S0370-2693(98)00466-3" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/S0370-2693(98)00466-3" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B135-universe-10-00357' class='html-xxx' data-content='135.'>Arkani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity. <span class='html-italic'>Phys. Rev. D</span> <b>1999</b>, <span class='html-italic'>59</span>, 086004. [<a href="https://scholar.google.com/scholar_lookup?title=Phenomenology,+astrophysics+and+cosmology+of+theories+with+submillimeter+dimensions+and+TeV+scale+quantum+gravity&author=Arkani-Hamed,+N.&author=Dimopoulos,+S.&author=Dvali,+G.R.&publication_year=1999&journal=Phys.+Rev.+D&volume=59&pages=086004&doi=10.1103/PhysRevD.59.086004" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.59.086004" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B136-universe-10-00357' class='html-xxx' data-content='136.'>Esmaili, A.; Peres, O.L.G.; Tabrizi, Z. Probing Large Extra Dimensions with IceCube. <span class='html-italic'>J. Cosmol. Astropart. Phys.</span> <b>2014</b>, <span class='html-italic'>12</span>, 2. [<a href="https://scholar.google.com/scholar_lookup?title=Probing+Large+Extra+Dimensions+with+IceCube&author=Esmaili,+A.&author=Peres,+O.L.G.&author=Tabrizi,+Z.&publication_year=2014&journal=J.+Cosmol.+Astropart.+Phys.&volume=12&pages=2&doi=10.1088/1475-7516/2014/12/002" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1088/1475-7516/2014/12/002" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B137-universe-10-00357' class='html-xxx' data-content='137.'>Machado, P.A.N.; Nunokawa, H.; Zukanovich Funchal, R. Testing for Large Extra Dimensions with Neutrino Oscillations. <span class='html-italic'>Phys. Rev. D</span> <b>2011</b>, <span class='html-italic'>84</span>, 013003. [<a href="https://scholar.google.com/scholar_lookup?title=Testing+for+Large+Extra+Dimensions+with+Neutrino+Oscillations&author=Machado,+P.A.N.&author=Nunokawa,+H.&author=Zukanovich+Funchal,+R.&publication_year=2011&journal=Phys.+Rev.+D&volume=84&pages=013003&doi=10.1103/PhysRevD.84.013003" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.84.013003" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B138-universe-10-00357' class='html-xxx' data-content='138.'>Machado, P.A.N.; Nunokawa, H.; dos Santos, F.A.P.; Funchal, R.Z. Bulk Neutrinos as an Alternative Cause of the Gallium and Reactor Anti-neutrino Anomalies. <span class='html-italic'>Phys. Rev. D</span> <b>2012</b>, <span class='html-italic'>85</span>, 073012. [<a href="https://scholar.google.com/scholar_lookup?title=Bulk+Neutrinos+as+an+Alternative+Cause+of+the+Gallium+and+Reactor+Anti-neutrino+Anomalies&author=Machado,+P.A.N.&author=Nunokawa,+H.&author=dos+Santos,+F.A.P.&author=Funchal,+R.Z.&publication_year=2012&journal=Phys.+Rev.+D&volume=85&pages=073012&doi=10.1103/PhysRevD.85.073012" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.85.073012" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B139-universe-10-00357' class='html-xxx' data-content='139.'>Basto-Gonzalez, V.S.; Esmaili, A.; Peres, O.L.G. Kinematical Test of Large Extra Dimension in Beta Decay Experiments. <span class='html-italic'>Phys. Lett. B</span> <b>2013</b>, <span class='html-italic'>718</span>, 1020–1023. [<a href="https://scholar.google.com/scholar_lookup?title=Kinematical+Test+of+Large+Extra+Dimension+in+Beta+Decay+Experiments&author=Basto-Gonzalez,+V.S.&author=Esmaili,+A.&author=Peres,+O.L.G.&publication_year=2013&journal=Phys.+Lett.+B&volume=718&pages=1020%E2%80%931023&doi=10.1016/j.physletb.2012.11.048" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2012.11.048" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B140-universe-10-00357' class='html-xxx' data-content='140.'>Girardi, I.; Meloni, D. Constraining new physics scenarios in neutrino oscillations from Daya Bay data. <span class='html-italic'>Phys. Rev. D</span> <b>2014</b>, <span class='html-italic'>90</span>, 073011. [<a href="https://scholar.google.com/scholar_lookup?title=Constraining+new+physics+scenarios+in+neutrino+oscillations+from+Daya+Bay+data&author=Girardi,+I.&author=Meloni,+D.&publication_year=2014&journal=Phys.+Rev.+D&volume=90&pages=073011&doi=10.1103/PhysRevD.90.073011" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.90.073011" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B141-universe-10-00357' class='html-xxx' data-content='141.'>Rodejohann, W.; Zhang, H. Signatures of Extra Dimensional Sterile Neutrinos. <span class='html-italic'>Phys. Lett. B</span> <b>2014</b>, <span class='html-italic'>737</span>, 81–89. [<a href="https://scholar.google.com/scholar_lookup?title=Signatures+of+Extra+Dimensional+Sterile+Neutrinos&author=Rodejohann,+W.&author=Zhang,+H.&publication_year=2014&journal=Phys.+Lett.+B&volume=737&pages=81%E2%80%9389&doi=10.1016/j.physletb.2014.08.035" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.physletb.2014.08.035" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B142-universe-10-00357' class='html-xxx' data-content='142.'>Carena, M.; Li, Y.Y.; Machado, C.S.; Machado, P.A.N.; Wagner, C.E.M. Neutrinos in Large Extra Dimensions and Short-Baseline <span class='html-italic'>ν</span><sub>e</sub> Appearance. <span class='html-italic'>Phys. Rev. D</span> <b>2017</b>, <span class='html-italic'>96</span>, 095014. [<a href="https://scholar.google.com/scholar_lookup?title=Neutrinos+in+Large+Extra+Dimensions+and+Short-Baseline+%CE%BDe+Appearance&author=Carena,+M.&author=Li,+Y.Y.&author=Machado,+C.S.&author=Machado,+P.A.N.&author=Wagner,+C.E.M.&publication_year=2017&journal=Phys.+Rev.+D&volume=96&pages=095014&doi=10.1103/PhysRevD.96.095014" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.96.095014" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B143-universe-10-00357' class='html-xxx' data-content='143.'>Stenico, G.V.; Forero, D.V.; Peres, O.L.G. A Short Travel for Neutrinos in Large Extra Dimensions. <span class='html-italic'>J. High Energy Phys.</span> <b>2018</b>, <span class='html-italic'>11</span>, 155. [<a href="https://scholar.google.com/scholar_lookup?title=A+Short+Travel+for+Neutrinos+in+Large+Extra+Dimensions&author=Stenico,+G.V.&author=Forero,+D.V.&author=Peres,+O.L.G.&publication_year=2018&journal=J.+High+Energy+Phys.&volume=11&pages=155&doi=10.1007/JHEP11(2018)155" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1007/JHEP11(2018)155" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B144-universe-10-00357' class='html-xxx' data-content='144.'>Basto-Gonzalez, V.S.; Forero, D.V.; Giunti, C.; Quiroga, A.A.; Ternes, C.A. Short-baseline oscillation scenarios at JUNO and TAO. <span class='html-italic'>Phys. Rev. D</span> <b>2022</b>, <span class='html-italic'>105</span>, 075023. [<a href="https://scholar.google.com/scholar_lookup?title=Short-baseline+oscillation+scenarios+at+JUNO+and+TAO&author=Basto-Gonzalez,+V.S.&author=Forero,+D.V.&author=Giunti,+C.&author=Quiroga,+A.A.&author=Ternes,+C.A.&publication_year=2022&journal=Phys.+Rev.+D&volume=105&pages=075023&doi=10.1103/PhysRevD.105.075023" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.105.075023" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B145-universe-10-00357' class='html-xxx' data-content='145.'>Forero, D.V.; Giunti, C.; Ternes, C.A.; Tyagi, O. Large extra dimensions and neutrino experiments. <span class='html-italic'>Phys. Rev. D</span> <b>2022</b>, <span class='html-italic'>106</span>, 035027. [<a href="https://scholar.google.com/scholar_lookup?title=Large+extra+dimensions+and+neutrino+experiments&author=Forero,+D.V.&author=Giunti,+C.&author=Ternes,+C.A.&author=Tyagi,+O.&publication_year=2022&journal=Phys.+Rev.+D&volume=106&pages=035027&doi=10.1103/PhysRevD.106.035027" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.106.035027" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li><li id='B146-universe-10-00357' class='html-xxx' data-content='146.'>Roy, S. Capability of the proposed long-baseline experiments to probe large extra dimension. <span class='html-italic'>Phys. Rev. D</span> <b>2023</b>, <span class='html-italic'>108</span>, 055015. [<a href="https://scholar.google.com/scholar_lookup?title=Capability+of+the+proposed+long-baseline+experiments+to+probe+large+extra+dimension&author=Roy,+S.&publication_year=2023&journal=Phys.+Rev.+D&volume=108&pages=055015&doi=10.1103/PhysRevD.108.055015" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1103/PhysRevD.108.055015" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li></ol></section><section id='FiguresandTables' type='display-objects'><div class="html-fig-wrap" id="universe-10-00357-f001"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f001"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001.png" alt="Universe 10 00357 g001" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance (left panels) and <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance (right panels) probabilities in the presence of off-diagonal CPT violating and conserving LIV parameters. In particular, the top, middle and bottom panels show the effect of <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math>), <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>) and <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>), respectively. Black lines correspond to the standard oscillation case and red (orange) lines to the probabilities obtained for <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> </mrow> </semantics></math> GeV (<math display='inline'><semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> </mrow> </semantics></math>). Solid and dashed curves depict the effects of LIV phases (generically indicated <math display='inline'><semantics> <mo>Φ</mo> </semantics></math>) when <math display='inline'><semantics> <mrow> <mo>Φ</mo> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>Φ</mo> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, respectively. The grey and green shadowed regions illustrate the standard and the high-energy DUNE flux.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f001"> <div class="html-caption"> <b>Figure 1.</b> <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance (left panels) and <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance (right panels) probabilities in the presence of off-diagonal CPT violating and conserving LIV parameters. In particular, the top, middle and bottom panels show the effect of <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics></math>), <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>) and <math display='inline'><semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math> (<math display='inline'><semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics></math>), respectively. Black lines correspond to the standard oscillation case and red (orange) lines to the probabilities obtained for <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>23</mn> </mrow> </msup> </mrow> </semantics></math> GeV (<math display='inline'><semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> </mrow> </semantics></math>). Solid and dashed curves depict the effects of LIV phases (generically indicated <math display='inline'><semantics> <mo>Φ</mo> </semantics></math>) when <math display='inline'><semantics> <mrow> <mo>Φ</mo> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mo>Φ</mo> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, respectively. The grey and green shadowed regions illustrate the standard and the high-energy DUNE flux.</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001.png" alt="Universe 10 00357 g001" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g001.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00357-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f002"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002.png" alt="Universe 10 00357 g002" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> (dotted), <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> (solid) and <math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> (dashed) allowed contours in the <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>−</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> (left panels) and <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>−</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> planes (right panels) for HE-DUNE. The red curves in the bottom panels depict the effect of the addition of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance channel in the analysis.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f002"> <div class="html-caption"> <b>Figure 2.</b> <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> (dotted), <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> (solid) and <math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> (dashed) allowed contours in the <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>−</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> (left panels) and <math display='inline'><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>c</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>−</mo> </mrow> <msub> <mo>Φ</mo> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> planes (right panels) for HE-DUNE. The red curves in the bottom panels depict the effect of the addition of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance channel in the analysis.</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002.png" alt="Universe 10 00357 g002" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g002.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00357-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f003"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003.png" alt="Universe 10 00357 g003" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Same as <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a>, but for the Long-Range Force case. Left (right) plot shows the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance (<math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance) probability. The red, magenta and orange curves refer to the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> cases, respectively. The potentials <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> have been fixed to <math display='inline'><semantics> <mrow> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math> eV.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f003"> <div class="html-caption"> <b>Figure 3.</b> Same as <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a>, but for the Long-Range Force case. Left (right) plot shows the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>e</mi> </msub> </semantics></math> appearance (<math display='inline'><semantics> <msub> <mi>ν</mi> <mi>μ</mi> </msub> </semantics></math> disappearance) probability. The red, magenta and orange curves refer to the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> cases, respectively. The potentials <math display='inline'><semantics> <msub> <mi>V</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </semantics></math> have been fixed to <math display='inline'><semantics> <mrow> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math> eV.</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003.png" alt="Universe 10 00357 g003" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g003.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00357-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f004"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004.png" alt="Universe 10 00357 g004" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> HE-DUNE sensitivity to the LRF potentials. Red, magenta and orange lines correspond to the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> cases, respectively.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f004"> <div class="html-caption"> <b>Figure 4.</b> HE-DUNE sensitivity to the LRF potentials. Red, magenta and orange lines correspond to the <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>μ</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>τ</mi> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mi>L</mi> <mi>e</mi> </msub> <mo>−</mo> <msub> <mi>L</mi> <mi>μ</mi> </msub> </mrow> </semantics></math> cases, respectively.</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004.png" alt="Universe 10 00357 g004" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g004.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00357-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f005"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005.png" alt="Universe 10 00357 g005" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> The 95% CL excluded regions in the <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mo>−</mo> <msub> <mi>G</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> plane, fixing the LRF potentials to the 95% CL HE-DUNE limits showed in <a href="#universe-10-00357-t003" class="html-table">Table 3</a>. See text for details.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f005"> <div class="html-caption"> <b>Figure 5.</b> The 95% CL excluded regions in the <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <msup> <mi>Z</mi> <mo>′</mo> </msup> </msub> <mo>−</mo> <msub> <mi>G</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> </semantics></math> plane, fixing the LRF potentials to the 95% CL HE-DUNE limits showed in <a href="#universe-10-00357-t003" class="html-table">Table 3</a>. See text for details.</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005.png" alt="Universe 10 00357 g005" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g005.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00357-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f006"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006.png" alt="Universe 10 00357 g006" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> Same as <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a> but in the Large Extra Dimension case. Red (orange) curves have been obtained fixing <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> <math display='inline'><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m and <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math> eV (<math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> eV).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f006"> <div class="html-caption"> <b>Figure 6.</b> Same as <a href="#universe-10-00357-f001" class="html-fig">Figure 1</a> but in the Large Extra Dimension case. Red (orange) curves have been obtained fixing <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> <math display='inline'><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m and <math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math> eV (<math display='inline'><semantics> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> eV).</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006.png" alt="Universe 10 00357 g006" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g006.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00357-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f007"> <img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007.png" alt="Universe 10 00357 g007" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#fig_body_display_universe-10-00357-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> (dotted) <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> (solid) and <math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> (dashed) allowed regions in the <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </semantics></math> plane for HE-DUNE.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00357-f007"> <div class="html-caption"> <b>Figure 7.</b> <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> (dotted) <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math> (solid) and <math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> (dashed) allowed regions in the <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </semantics></math> plane for HE-DUNE.</div> <div class="html-img"><img data-large="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007.png" data-original="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007.png" alt="Universe 10 00357 g007" data-lsrc="/universe/universe-10-00357/article_deploy/html/images/universe-10-00357-g007.png" /></div> </div> <div class="html-table-wrap" id="universe-10-00357-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href='#table_body_display_universe-10-00357-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#table_body_display_universe-10-00357-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Best-fit value of the neutrino oscillation parameters in the standard three-flavor scenario. The values of the mixing angles and the mass splittings and their <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> uncertainty intervals are taken from Ref. [<a href="#B10-universe-10-00357" class="html-bibr">10</a>]. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_universe-10-00357-t001"> <div class="html-caption"><b>Table 1.</b> Best-fit value of the neutrino oscillation parameters in the standard three-flavor scenario. The values of the mixing angles and the mass splittings and their <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> uncertainty intervals are taken from Ref. [<a href="#B10-universe-10-00357" class="html-bibr">10</a>].</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Oscillation Parameters (<math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <mn>3</mn> <mi mathvariant="bold-italic">ν</mi> </mrow> </mstyle> </semantics></math>)</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Normal Ordering (NO)</th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msubsup> <mi>θ</mi> <mrow> <mn>12</mn> </mrow> <mo>°</mo> </msubsup> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>33</mn> <mo>.</mo> <msubsup> <mn>41</mn> <mrow> <mo>−</mo> <mn>0.72</mn> </mrow> <mrow> <mo>+</mo> <mn>0.75</mn> </mrow> </msubsup> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msubsup> <mi>θ</mi> <mrow> <mn>23</mn> </mrow> <mo>°</mo> </msubsup> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>42</mn> <mo>.</mo> <msubsup> <mn>2</mn> <mrow> <mo>−</mo> <mn>0.9</mn> </mrow> <mrow> <mo>+</mo> <mn>1.1</mn> </mrow> </msubsup> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msubsup> <mi>θ</mi> <mrow> <mn>13</mn> </mrow> <mo>°</mo> </msubsup> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>8</mn> <mo>.</mo> <msubsup> <mn>58</mn> <mrow> <mo>−</mo> <mn>0.11</mn> </mrow> <mrow> <mo>+</mo> <mn>0.11</mn> </mrow> </msubsup> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msubsup> <mi>δ</mi> <mrow> <mi>C</mi> <mi>P</mi> </mrow> <mo>°</mo> </msubsup> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msubsup> <mn>232</mn> <mrow> <mo>−</mo> <mn>26</mn> </mrow> <mrow> <mo>+</mo> <mn>36</mn> </mrow> </msubsup> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>21</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> (<math display='inline'><semantics> <msup> <mi>eV</mi> <mn>2</mn> </msup> </semantics></math>)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>7</mn> <mo>.</mo> <msubsup> <mn>41</mn> <mrow> <mo>−</mo> <mn>0.20</mn> </mrow> <mrow> <mo>+</mo> <mn>0.21</mn> </mrow> </msubsup> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'><semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>m</mi> <mrow> <mn>31</mn> </mrow> <mn>2</mn> </msubsup> </mrow> </semantics></math> (<math display='inline'><semantics> <msup> <mi>eV</mi> <mn>2</mn> </msup> </semantics></math>)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mo>+</mo> <mn>2</mn> <mo>.</mo> <msubsup> <mn>507</mn> <mrow> <mo>−</mo> <mn>0.027</mn> </mrow> <mrow> <mo>+</mo> <mn>0.026</mn> </mrow> </msubsup> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics> </math></td></tr></tbody> </table> </div> <div class="html-table-wrap" id="universe-10-00357-t002"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href='#table_body_display_universe-10-00357-t002'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#table_body_display_universe-10-00357-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> The 95% bounds on the LIV parameters obtained for HE-DUNE. The upper limits have been obtained marginalizing over the LIV phases. The numbers in brackets refer to the foreseen improvement due to the addition of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance channel in the analysis. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_universe-10-00357-t002"> <div class="html-caption"><b>Table 2.</b> The 95% bounds on the LIV parameters obtained for HE-DUNE. The upper limits have been obtained marginalizing over the LIV phases. The numbers in brackets refer to the foreseen improvement due to the addition of the <math display='inline'><semantics> <msub> <mi>ν</mi> <mi>τ</mi> </msub> </semantics></math> appearance channel in the analysis.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >95% CL limit (<math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <mi mathvariant="bold">GeV</mi> </mrow> </mstyle> </semantics></math>)</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >95% CL limit (<math display='inline'><semantics> <mstyle mathvariant="bold"> <mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>24</mn> </mrow> </msup> </mrow> </mstyle> </semantics></math>)</th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><5.1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>μ</mi> </mrow> </msub> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><0.43</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msub> <mi>a</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><9.3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msub> <mi>c</mi> <mrow> <mi>e</mi> <mi>τ</mi> </mrow> </msub> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><2.23</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msub> <mi>a</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><1.12 (<1.0)</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>τ</mi> </mrow> </msub> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><0.66 (<0.64)</td></tr></tbody> </table> </div> <div class="html-table-wrap" id="universe-10-00357-t003"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href='#table_body_display_universe-10-00357-t003'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/9/357/display" href="#table_body_display_universe-10-00357-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> The 95% CL limits on the Long-Range Force potentials obtained by HE-DUNE. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_universe-10-00357-t003"> <div class="html-caption"><b>Table 3.</b> The 95% CL limits on the Long-Range Force potentials obtained by HE-DUNE.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <msub> <mi mathvariant="bold-italic">V</mi> <mrow> <mi mathvariant="bold-italic">e</mi> <mi mathvariant="bold-italic">μ</mi> </mrow> </msub> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <msub> <mi mathvariant="bold-italic">V</mi> <mrow> <mi mathvariant="bold-italic">e</mi> <mi mathvariant="bold-italic">τ</mi> </mrow> </msub> </mstyle> </semantics> </math></th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mstyle mathvariant="bold"> <msub> <mi mathvariant="bold-italic">V</mi> <mrow> <mi mathvariant="bold-italic">μ</mi> <mi mathvariant="bold-italic">τ</mi> </mrow> </msub> </mstyle> </semantics> </math></th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >95% CL limit</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><<math display='inline'><semantics> <mrow> <mn>2.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> eV</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><<math display='inline'><semantics> <mrow> <mn>1.58</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> eV</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><<math display='inline'><semantics> <mrow> <mn>1.23</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> eV</td></tr></tbody> </table> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.</div></td></tr></table></section> <section id="html-copyright"><br>© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (<a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" rel="noopener noreferrer">https://creativecommons.org/licenses/by/4.0/</a>).</section> </div> </div> <div class="additional-content"> <h2><a name="cite"></a>Share and Cite</h2> <div class="social-media-links" style="text-align: left;"> <a href="/cdn-cgi/l/email-protection#350a135458450e4640575f5056410873475a581007057871657c100674100705100707704d45595a475c5b521007057b5042100705655d4c465c5646100705425c415d10070571505045100705605b51504752475a405b511007057b504041475c5b5a100705704d4550475c58505b411007057d5c525d18705b5047524c1007057359404d100674100705615d50100705765446501007055a53100705795a47505b414f1007057c5b4354475c545b5650100705635c5a5954415c5a5b1007761007057954475250100705704d414754100705715c58505b465c5a5b46100705545b51100705795a5b521867545b5250100705735a475650461344405a410e135458450e575a514c085d414145460f1a1a4242421b5851455c1b565a581a070c0101010205100674100574100574704d45595a475c5b521007057b5042100705655d4c465c5646100705425c415d10070571505045100705605b51504752475a405b511007057b504041475c5b5a100705704d4550475c58505b411007057d5c525d18705b5047524c1007057359404d100674100705615d50100705765446501007055a53100705795a47505b414f1007057c5b4354475c545b5650100705635c5a5954415c5a5b1007761007057954475250100705704d414754100705715c58505b465c5a5b46100705545b51100705795a5b521867545b5250100705735a47565046100574100574745746414754564110067410070571607b701007055c46100705541007055b504d411852505b504754415c5a5b100705595a5b521857544650595c5b501007055b504041475c5b5a1007055a46565c595954415c5a5b100705504d4550475c58505b411b1007057c411007055c46100705504d455056415051100705415a10070558505446404750100776100705425c415d100705405b454750565051505b415051100705454750565c465c5a5b100776100705415d501007055441585a46455d50475c561007055a46565c595954415c5a5b100705455447545850415047461007761007055c5b565940515c5b52100705415d50100705766518435c5a5954415c5b52100705455d544650100705100703515059415410067776651b100705785a47505a43504710077610070546504350475459100705464140515c50461007055d544350100705464052525046415051100705415d54411007055c4146100705405b5c4440501007055350544140475046100705465d5a4059511007055459595a4210070571607b70100705415a10070545475a5750100705465043504754591007055b5042100705455d4c465c56461007054656505b54475c5a461b1007057c5b100705415d5c46100705425a475e1007761007054250100705504d45595a4750100705415d50100705455047535a4758545b5650461007055a53100705415d5010070571607b701007055354471007055150415056415a471007055c5b100705565a5b464147545c5b5c5b521007055b5042100705455d4c465c56461007055c53100705541007055d5c525d18505b5047524c1007055b504041475c5b5a1007055359404d1007055c46100705505845595a4c505110070510070d7d701871607b7010070c1b100705625010070541545e501007055c5b415a1007055456565a405b41100705415d475050100705515c53535047505b411007054656505b54475c5a46100674100705795a47505b414f1007057c5b4354475c545b5650100705635c5a5954415c5a5b10070510070d797c6310070c100776100705795a5b521867545b5250100705735a4756504610070510070d7967734610070c100705545b511007057954475250100705704d414754100705715c58505b465c5a5b4610070510070d7970714610070c1b1007057a404710070547504640594146100705465d5a42100705415d54411007057d701871607b70100705465d5a405951100705575010070554575950100705415a100705465041100705575a405b5146100705565a584550415c415c4350100705415a100705415d5010070556404747505b411007055a5b5046100705545b511007761007055c5b100705455447415c56405954471007761007055c4110070556545b1007055a4041455047535a4758100705415d501007054641545b5154475110070571607b7010070556544554575c595c415c50461007055c5b100705565a5b464147545c5b5c5b52100705766561185043505b100705797c6310070545544754585041504746100705545b51100705415d50100705565a58455456415c535c5654415c5a5b1007054754515c40461007056770711007055a53100705415d50100705797071100705585a5150591b" title="Email"> <i class="fa fa-envelope-square" style="font-size: 30px;"></i> </a> <a href="https://twitter.com/intent/tweet?text=Exploring+New+Physics+with+Deep+Underground+Neutrino+Experiment+High-Energy+Flux%3A+The+Case+of+Lorentz+Invariance+Violation%2C+Large+Extra+Dimensions+and+Long-Range+Forces&hashtags=mdpiuniverse&url=https%3A%2F%2Fwww.mdpi.com%2F2944470&via=Universe_MDPI" onclick="windowOpen(this.href,600,800); return false" target="_blank" rel="noopener noreferrer"> <i class="fa fa-twitter-x-square" style="font-size: 30px;"></i> </a> <a href=" http://www.linkedin.com/shareArticle?mini=true&url=https%3A%2F%2Fwww.mdpi.com%2F2944470&title=Exploring%20New%20Physics%20with%20Deep%20Underground%20Neutrino%20Experiment%20High-Energy%20Flux%3A%20The%20Case%20of%20Lorentz%20Invariance%20Violation%2C%20Large%20Extra%20Dimensions%20and%20Long-Range%20Forces%26source%3Dhttps%3A%2F%2Fwww.mdpi.com%26summary%3DDUNE%20is%20a%20next-generation%20long-baseline%20neutrino%20oscillation%20experiment.%20It%20is%20expected%20to%20measure%2C%20with%20unprecedented%20precision%2C%20the%20atmospheric%20oscillation%20parameters%2C%20including%20the%20CP-violating%20phase%20%CE%B4CP.%20Moreover%2C%20several%20studies%20have%20suggested%20%5B...%5D" onclick="windowOpen(this.href,600,800); return false" title="LinkedIn" target="_blank" rel="noopener noreferrer"> <i class="fa fa-linkedin-square" style="font-size: 30px;"></i> </a> <a href="https://www.facebook.com/sharer.php?u=https://www.mdpi.com/2944470" title="facebook" target="_blank" rel="noopener noreferrer"> <i class="fa fa-facebook-square" style="font-size: 30px;"></i> </a> <a href="javascript:void(0);" title="Wechat" data-reveal-id="weixin-share-modal"> <i class="fa fa-weixin-square" style="font-size: 26px;"></i> </a> <a href="http://www.reddit.com/submit?url=https://www.mdpi.com/2944470" title="Reddit" target="_blank" rel="noopener noreferrer"> <i class="fa fa-reddit-square" style="font-size: 30px;"></i> </a> <a href="http://www.mendeley.com/import/?url=https://www.mdpi.com/2944470" title="Mendeley" target="_blank" rel="noopener noreferrer"> <i class="fa fa-mendeley-square" style="font-size: 30px;"></i> </a> </div> <div class="in-tab" style="padding-top: 0px!important; margin-top: 15px;"> <div><b>MDPI and ACS Style</b></div> <p> Giarnetti, A.; Marciano, S.; Meloni, D. Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces. <em>Universe</em> <b>2024</b>, <em>10</em>, 357. https://doi.org/10.3390/universe10090357 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Giarnetti A, Marciano S, Meloni D. Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces. <em>Universe</em>. 2024; 10(9):357. https://doi.org/10.3390/universe10090357 </p> <b>Chicago/Turabian Style</b><br> <p> Giarnetti, Alessio, Simone Marciano, and Davide Meloni. 2024. "Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces" <em>Universe</em> 10, no. 9: 357. https://doi.org/10.3390/universe10090357 </p> <b>APA Style</b><br> <p> Giarnetti, A., Marciano, S., & Meloni, D. (2024). Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces. <em>Universe</em>, <em>10</em>(9), 357. https://doi.org/10.3390/universe10090357 </p> </div> </div> <div class="info-box no-margin"> Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details <a target="_blank" href="https://www.mdpi.com/about/announcements/784">here</a>. </div> <h2><a name="metrics"></a>Article Metrics</h2> <div class="row"> <div class="small-12 columns"> <div id="loaded_cite_count" style="display:none">No</div> <div id="framed_div_cited_count" class="in-tab" style="display: none; overflow: auto;"></div> <div id="loaded" style="display:none">No</div> <div id="framed_div" class="in-tab" style="display: none; margin-top: 10px;"></div> </div> <div class="small-12 columns"> <div id="article_stats_div" style="display: none; margin-bottom: 1em;"> <h3>Article Access Statistics</h3> <div id="article_stats_swf" ></div> For more information on the journal statistics, click <a href="/journal/universe/stats">here</a>. <div class="info-box"> Multiple requests from the same IP address are counted as one view. </div> </div> </div> </div> </div> </div> </article> </div> </div></div> <div class="webpymol-controls webpymol-controls-template" style="margin-top: 10px; display: none;"> <a class="bzoom">Zoom</a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="borient"> Orient </a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="blines"> As Lines </a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="bsticks"> As Sticks </a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="bcartoon"> As Cartoon </a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="bsurface"> As Surface </a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="bprevscene">Previous Scene</a> <span style="display: inline-block; margin-left: 5px; margin-right: 5px;">|</span> <a class="bnextscene">Next Scene</a> </div> <div id="scifeed-modal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="modalTitle" aria-hidden="true" role="dialog"> </div> <div id="recommended-articles-modal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="modalTitle" aria-hidden="true" role="dialog"> </div> <div id="author-biographies-modal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="modalTitle" aria-hidden="true" role="dialog"> </div> <div id="cite-modal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="Captcha" aria-hidden="true" role="dialog"> <div class="row"> <div class="small-12 columns"> <h2 style="margin: 0;">Cite</h2> </div> <div class="small-12 columns">  <form style="margin:0; padding:0; display:inline;" name="export-bibtex" method="POST" action="/export"> <input type="hidden" name="articles_ids[]" value="1472235"> <input type="hidden" name="export_format_top" value="bibtex"> <input type="hidden" name="export_submit_top" value=""> </form>  <form style="margin:0; padding:0; display:inline;" name="export-endnote" method="POST" action="/export"> <input type="hidden" name="articles_ids[]" value="1472235"> <input type="hidden" name="export_format_top" value="endnote_no_abstract"> <input type="hidden" name="export_submit_top" value=""> </form>  <form style="margin:0; padding:0; display:inline;" name="export-ris" method="POST" action="/export"> <input type="hidden" name="articles_ids[]" value="1472235"> <input type="hidden" name="export_format_top" value="ris"> <input type="hidden" name="export_submit_top" value=""> </form> <div> Export citation file: <a href="javascript:window.document.forms['export-bibtex'].submit()">BibTeX</a> | <a href="javascript:window.document.forms['export-endnote'].submit()">EndNote</a> | <a href="javascript:window.document.forms['export-ris'].submit()">RIS</a> </div> </div> <div class="small-12 columns"> <div class="in-tab"> <div><b>MDPI and ACS Style</b></div> <p> Giarnetti, A.; Marciano, S.; Meloni, D. Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces. <em>Universe</em> <b>2024</b>, <em>10</em>, 357. https://doi.org/10.3390/universe10090357 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Giarnetti A, Marciano S, Meloni D. Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces. <em>Universe</em>. 2024; 10(9):357. https://doi.org/10.3390/universe10090357 </p> <b>Chicago/Turabian Style</b><br> <p> Giarnetti, Alessio, Simone Marciano, and Davide Meloni. 2024. "Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces" <em>Universe</em> 10, no. 9: 357. https://doi.org/10.3390/universe10090357 </p> <b>APA Style</b><br> <p> Giarnetti, A., Marciano, S., & Meloni, D. (2024). Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces. <em>Universe</em>, <em>10</em>(9), 357. https://doi.org/10.3390/universe10090357 </p> </div> </div> <div class="info-box no-margin"> Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details <a target="_blank" href="https://www.mdpi.com/about/announcements/784">here</a>. </div> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> </div> </div> </div> </div> </section> <div id="footer"> <div class="journal-info"> <span> <em><a class="Var_JournalInfo" href="/journal/universe">Universe</a></em>, EISSN 2218-1997, Published by MDPI </span> <div class="large-right"> <span> <a href="/rss/journal/universe" class="rss-link">RSS</a> </span> <span> <a href="/journal/universe/toc-alert">Content Alert</a> </span> </div> </div> <div class="row full-width footer-links" data-equalizer="footer" data-equalizer-mq="small"> <div class="large-2 large-push-4 medium-3 small-6 columns" data-equalizer-watch="footer"> <h3> Further Information </h3> <a href="/apc"> Article Processing Charges </a> <a href="/about/payment"> Pay an Invoice </a> <a href="/openaccess"> Open Access Policy </a> <a href="/about/contact"> Contact MDPI </a> <a href="https://careers.mdpi.com" target="_blank" rel="noopener noreferrer"> Jobs at MDPI </a> </div> <div class="large-2 large-push-4 medium-3 small-6 columns" data-equalizer-watch="footer"> <h3> Guidelines </h3> <a href="/authors"> For Authors </a> <a href="/reviewers"> For Reviewers </a> <a href="/editors"> For Editors </a> <a href="/librarians"> For Librarians </a> <a href="/publishing_services"> For Publishers </a> <a href="/societies"> For Societies </a> <a href="/conference_organizers"> For Conference Organizers </a> </div> <div class="large-2 large-push-4 medium-3 small-6 columns"> <h3> MDPI Initiatives </h3> <a href="https://sciforum.net" target="_blank" rel="noopener noreferrer"> Sciforum </a> <a href="https://www.mdpi.com/books" target="_blank" rel="noopener noreferrer"> MDPI Books </a> <a href="https://www.preprints.org" target="_blank" rel="noopener noreferrer"> Preprints.org </a> <a href="https://www.scilit.net" target="_blank" rel="noopener noreferrer"> Scilit </a> <a href="https://sciprofiles.com?utm_source=mpdi.com&utm_medium=bottom_menu&utm_campaign=initiative" target="_blank" rel="noopener noreferrer"> SciProfiles </a> <a href="https://encyclopedia.pub" target="_blank" rel="noopener noreferrer"> Encyclopedia </a> <a href="https://jams.pub" target="_blank" rel="noopener noreferrer"> JAMS </a> <a href="/about/proceedings"> Proceedings Series </a> </div> <div class="large-2 large-push-4 medium-3 small-6 right-border-large-without columns UA_FooterFollowMDPI"> <h3> Follow MDPI </h3> <a href="https://www.linkedin.com/company/mdpi" target="_blank" rel="noopener noreferrer"> LinkedIn </a> <a href="https://www.facebook.com/MDPIOpenAccessPublishing" target="_blank" rel="noopener noreferrer"> Facebook </a> <a href="https://twitter.com/MDPIOpenAccess" target="_blank" rel="noopener noreferrer"> Twitter </a> </div> <div id="footer-subscribe" class="large-4 large-pull-8 medium-12 small-12 left-border-large columns"> <div class="footer-subscribe__container"> <img class="show-for-large-up" src="https://pub.mdpi-res.com/img/design/mdpi-pub-logo-white-small.png?71d18e5f805839ab?1732286508" alt="MDPI" title="MDPI Open Access Journals" style="height: 50px; margin-bottom: 10px;"> <form id="newsletter" method="POST" action="/subscribe"> <p> Subscribe to receive issue release notifications and newsletters from MDPI journals </p> <select multiple id="newsletter-journal" class="foundation-select" name="journals[]"> <option value="acoustics">Acoustics</option> <option value="amh">Acta Microbiologica Hellenica</option> <option value="actuators">Actuators</option> <option value="admsci">Administrative Sciences</option> <option value="adolescents">Adolescents</option> <option value="arm">Advances in Respiratory Medicine</option> <option value="aerobiology">Aerobiology</option> <option value="aerospace">Aerospace</option> <option value="agriculture">Agriculture</option> <option value="agriengineering">AgriEngineering</option> <option value="agrochemicals">Agrochemicals</option> <option value="agronomy">Agronomy</option> <option value="ai">AI</option> <option value="air">Air</option> <option value="algorithms">Algorithms</option> <option value="allergies">Allergies</option> <option value="alloys">Alloys</option> <option value="analytica">Analytica</option> <option value="analytics">Analytics</option> <option value="anatomia">Anatomia</option> <option value="anesthres">Anesthesia Research</option> <option value="animals">Animals</option> <option value="antibiotics">Antibiotics</option> <option value="antibodies">Antibodies</option> <option value="antioxidants">Antioxidants</option> <option value="applbiosci">Applied Biosciences</option> <option value="applmech">Applied Mechanics</option> <option value="applmicrobiol">Applied Microbiology</option> <option value="applnano">Applied Nano</option> <option value="applsci">Applied Sciences</option> <option value="asi">Applied System Innovation</option> <option value="appliedchem">AppliedChem</option> <option value="appliedmath">AppliedMath</option> <option value="aquacj">Aquaculture Journal</option> <option value="architecture">Architecture</option> <option value="arthropoda">Arthropoda</option> <option value="arts">Arts</option> <option value="astronomy">Astronomy</option> <option value="atmosphere">Atmosphere</option> <option value="atoms">Atoms</option> <option value="audiolres">Audiology Research</option> <option value="automation">Automation</option> <option value="axioms">Axioms</option> <option value="bacteria">Bacteria</option> <option value="batteries">Batteries</option> <option value="behavsci">Behavioral Sciences</option> <option value="beverages">Beverages</option> <option value="BDCC">Big Data and Cognitive Computing</option> <option value="biochem">BioChem</option> <option value="bioengineering">Bioengineering</option> <option value="biologics">Biologics</option> <option value="biology">Biology</option> <option value="blsf">Biology and Life Sciences Forum</option> <option value="biomass">Biomass</option> <option value="biomechanics">Biomechanics</option> <option value="biomed">BioMed</option> <option value="biomedicines">Biomedicines</option> <option value="biomedinformatics">BioMedInformatics</option> <option value="biomimetics">Biomimetics</option> <option value="biomolecules">Biomolecules</option> <option value="biophysica">Biophysica</option> <option value="biosensors">Biosensors</option> <option value="biotech">BioTech</option> <option value="birds">Birds</option> <option value="blockchains">Blockchains</option> <option value="brainsci">Brain Sciences</option> <option value="buildings">Buildings</option> <option value="businesses">Businesses</option> <option value="carbon">C</option> <option value="cancers">Cancers</option> <option value="cardiogenetics">Cardiogenetics</option> <option value="catalysts">Catalysts</option> <option value="cells">Cells</option> <option value="ceramics">Ceramics</option> <option value="challenges">Challenges</option> <option value="ChemEngineering">ChemEngineering</option> <option value="chemistry">Chemistry</option> <option value="chemproc">Chemistry Proceedings</option> <option value="chemosensors">Chemosensors</option> <option value="children">Children</option> <option value="chips">Chips</option> <option value="civileng">CivilEng</option> <option value="cleantechnol">Clean Technologies</option> <option value="climate">Climate</option> <option value="ctn">Clinical and Translational Neuroscience</option> <option value="clinbioenerg">Clinical Bioenergetics</option> <option value="clinpract">Clinics and Practice</option> <option value="clockssleep">Clocks & Sleep</option> <option value="coasts">Coasts</option> <option value="coatings">Coatings</option> <option value="colloids">Colloids and Interfaces</option> <option value="colorants">Colorants</option> <option value="commodities">Commodities</option> <option value="complications">Complications</option> <option value="compounds">Compounds</option> <option value="computation">Computation</option> <option value="csmf">Computer Sciences & Mathematics Forum</option> <option value="computers">Computers</option> <option value="condensedmatter">Condensed Matter</option> <option value="conservation">Conservation</option> <option value="constrmater">Construction Materials</option> <option value="cmd">Corrosion and Materials Degradation</option> <option value="cosmetics">Cosmetics</option> <option value="covid">COVID</option> <option value="crops">Crops</option> <option value="cryo">Cryo</option> <option value="cryptography">Cryptography</option> <option value="crystals">Crystals</option> <option value="cimb">Current Issues in Molecular Biology</option> <option value="curroncol">Current Oncology</option> <option value="dairy">Dairy</option> <option value="data">Data</option> <option value="dentistry">Dentistry Journal</option> <option value="dermato">Dermato</option> <option value="dermatopathology">Dermatopathology</option> <option value="designs">Designs</option> <option value="diabetology">Diabetology</option> <option value="diagnostics">Diagnostics</option> <option value="dietetics">Dietetics</option> <option value="digital">Digital</option> <option value="disabilities">Disabilities</option> <option value="diseases">Diseases</option> <option value="diversity">Diversity</option> <option value="dna">DNA</option> <option value="drones">Drones</option> <option value="ddc">Drugs and Drug Candidates</option> <option value="dynamics">Dynamics</option> <option value="earth">Earth</option> <option value="ecologies">Ecologies</option> <option value="econometrics">Econometrics</option> <option value="economies">Economies</option> <option value="education">Education Sciences</option> <option value="electricity">Electricity</option> <option value="electrochem">Electrochem</option> <option value="electronicmat">Electronic Materials</option> <option value="electronics">Electronics</option> <option value="ecm">Emergency Care and Medicine</option> <option value="encyclopedia">Encyclopedia</option> <option value="endocrines">Endocrines</option> <option value="energies">Energies</option> <option value="esa">Energy Storage and Applications</option> <option value="eng">Eng</option> <option value="engproc">Engineering Proceedings</option> <option value="entropy">Entropy</option> <option value="environsciproc">Environmental Sciences Proceedings</option> <option value="environments">Environments</option> <option value="epidemiologia">Epidemiologia</option> <option value="epigenomes">Epigenomes</option> <option value="ebj">European Burn Journal</option> <option value="ejihpe">European Journal of Investigation in Health, Psychology and Education</option> <option value="fermentation">Fermentation</option> <option value="fibers">Fibers</option> <option value="fintech">FinTech</option> <option value="fire">Fire</option> <option value="fishes">Fishes</option> <option value="fluids">Fluids</option> <option value="foods">Foods</option> <option value="forecasting">Forecasting</option> <option value="forensicsci">Forensic Sciences</option> <option value="forests">Forests</option> <option value="fossstud">Fossil Studies</option> <option value="foundations">Foundations</option> <option value="fractalfract">Fractal and Fractional</option> <option value="fuels">Fuels</option> <option value="future">Future</option> <option value="futureinternet">Future Internet</option> <option value="futurepharmacol">Future Pharmacology</option> <option value="futuretransp">Future Transportation</option> <option value="galaxies">Galaxies</option> <option value="games">Games</option> <option value="gases">Gases</option> <option value="gastroent">Gastroenterology Insights</option> <option value="gastrointestdisord">Gastrointestinal Disorders</option> <option value="gastronomy">Gastronomy</option> <option value="gels">Gels</option> <option value="genealogy">Genealogy</option> <option value="genes">Genes</option> <option value="geographies">Geographies</option> <option value="geohazards">GeoHazards</option> <option value="geomatics">Geomatics</option> <option value="geometry">Geometry</option> <option value="geosciences">Geosciences</option> <option value="geotechnics">Geotechnics</option> <option value="geriatrics">Geriatrics</option> <option value="glacies">Glacies</option> <option value="gucdd">Gout, Urate, and Crystal Deposition Disease</option> <option value="grasses">Grasses</option> <option value="hardware">Hardware</option> <option value="healthcare">Healthcare</option> <option value="hearts">Hearts</option> <option value="hemato">Hemato</option> <option value="hematolrep">Hematology Reports</option> <option value="heritage">Heritage</option> <option value="histories">Histories</option> <option value="horticulturae">Horticulturae</option> <option value="hospitals">Hospitals</option> <option value="humanities">Humanities</option> <option value="humans">Humans</option> <option value="hydrobiology">Hydrobiology</option> <option value="hydrogen">Hydrogen</option> <option value="hydrology">Hydrology</option> <option value="hygiene">Hygiene</option> <option value="immuno">Immuno</option> <option value="idr">Infectious Disease Reports</option> <option value="informatics">Informatics</option> <option value="information">Information</option> <option value="infrastructures">Infrastructures</option> <option value="inorganics">Inorganics</option> <option value="insects">Insects</option> <option value="instruments">Instruments</option> <option value="iic">Intelligent Infrastructure and Construction</option> <option value="ijerph">International Journal of Environmental Research and Public Health</option> <option value="ijfs">International Journal of Financial Studies</option> <option value="ijms">International Journal of Molecular Sciences</option> <option value="IJNS">International Journal of Neonatal Screening</option> <option value="ijpb">International Journal of Plant Biology</option> <option value="ijt">International Journal of Topology</option> <option value="ijtm">International Journal of Translational Medicine</option> <option value="ijtpp">International Journal of Turbomachinery, Propulsion and Power</option> <option value="ime">International Medical Education</option> <option value="inventions">Inventions</option> <option value="IoT">IoT</option> <option value="ijgi">ISPRS International Journal of Geo-Information</option> <option value="J">J</option> <option value="jal">Journal of Ageing and Longevity</option> <option value="jcdd">Journal of Cardiovascular Development and Disease</option> <option value="jcto">Journal of Clinical & Translational Ophthalmology</option> <option value="jcm">Journal of Clinical Medicine</option> <option value="jcs">Journal of Composites Science</option> <option value="jcp">Journal of Cybersecurity and Privacy</option> <option value="jdad">Journal of Dementia and Alzheimer's Disease</option> <option value="jdb">Journal of Developmental Biology</option> <option value="jeta">Journal of Experimental and Theoretical Analyses</option> <option value="jfb">Journal of Functional Biomaterials</option> <option value="jfmk">Journal of Functional Morphology and Kinesiology</option> <option value="jof">Journal of Fungi</option> <option value="jimaging">Journal of Imaging</option> <option value="jintelligence">Journal of Intelligence</option> <option value="jlpea">Journal of Low Power Electronics and Applications</option> <option value="jmmp">Journal of Manufacturing and Materials Processing</option> <option value="jmse">Journal of Marine Science and Engineering</option> <option value="jmahp">Journal of Market Access & Health Policy</option> <option value="jmp">Journal of Molecular Pathology</option> <option value="jnt">Journal of Nanotheranostics</option> <option value="jne">Journal of Nuclear Engineering</option> <option value="ohbm">Journal of Otorhinolaryngology, Hearing and Balance Medicine</option> <option value="jop">Journal of Parks</option> <option value="jpm">Journal of Personalized Medicine</option> <option value="jpbi">Journal of Pharmaceutical and BioTech Industry</option> <option value="jor">Journal of Respiration</option> <option value="jrfm">Journal of Risk and Financial Management</option> <option value="jsan">Journal of Sensor and Actuator Networks</option> <option value="joma">Journal of the Oman Medical Association</option> <option value="jtaer">Journal of Theoretical and Applied Electronic Commerce Research</option> <option value="jvd">Journal of Vascular Diseases</option> <option value="jox">Journal of Xenobiotics</option> <option value="jzbg">Journal of Zoological and Botanical Gardens</option> <option value="journalmedia">Journalism and Media</option> <option value="kidneydial">Kidney and Dialysis</option> <option value="kinasesphosphatases">Kinases and Phosphatases</option> <option value="knowledge">Knowledge</option> <option value="labmed">LabMed</option> <option value="laboratories">Laboratories</option> <option value="land">Land</option> <option value="languages">Languages</option> <option value="laws">Laws</option> <option value="life">Life</option> <option value="limnolrev">Limnological Review</option> <option value="lipidology">Lipidology</option> <option value="liquids">Liquids</option> <option value="literature">Literature</option> <option value="livers">Livers</option> <option value="logics">Logics</option> <option value="logistics">Logistics</option> <option value="lubricants">Lubricants</option> <option value="lymphatics">Lymphatics</option> <option value="make">Machine Learning and Knowledge Extraction</option> <option value="machines">Machines</option> <option value="macromol">Macromol</option> <option value="magnetism">Magnetism</option> <option value="magnetochemistry">Magnetochemistry</option> <option value="marinedrugs">Marine Drugs</option> <option value="materials">Materials</option> <option value="materproc">Materials Proceedings</option> <option value="mca">Mathematical and Computational Applications</option> <option value="mathematics">Mathematics</option> <option value="medsci">Medical Sciences</option> <option value="msf">Medical Sciences Forum</option> <option value="medicina">Medicina</option> <option value="medicines">Medicines</option> <option value="membranes">Membranes</option> <option value="merits">Merits</option> <option value="metabolites">Metabolites</option> <option value="metals">Metals</option> <option value="meteorology">Meteorology</option> <option value="methane">Methane</option> <option value="mps">Methods and Protocols</option> <option value="metrics">Metrics</option> <option value="metrology">Metrology</option> <option value="micro">Micro</option> <option value="microbiolres">Microbiology Research</option> <option value="micromachines">Micromachines</option> <option value="microorganisms">Microorganisms</option> <option value="microplastics">Microplastics</option> <option value="minerals">Minerals</option> <option value="mining">Mining</option> <option value="modelling">Modelling</option> <option value="mmphys">Modern Mathematical Physics</option> <option value="molbank">Molbank</option> <option value="molecules">Molecules</option> <option value="mti">Multimodal Technologies and Interaction</option> <option value="muscles">Muscles</option> <option value="nanoenergyadv">Nanoenergy Advances</option> <option value="nanomanufacturing">Nanomanufacturing</option> <option value="nanomaterials">Nanomaterials</option> <option value="ndt">NDT</option> <option value="network">Network</option> <option value="neuroglia">Neuroglia</option> <option value="neurolint">Neurology International</option> <option value="neurosci">NeuroSci</option> <option value="nitrogen">Nitrogen</option> <option value="ncrna">Non-Coding RNA</option> <option value="nursrep">Nursing Reports</option> <option value="nutraceuticals">Nutraceuticals</option> <option value="nutrients">Nutrients</option> <option value="obesities">Obesities</option> <option value="oceans">Oceans</option> <option value="onco">Onco</option> <option value="optics">Optics</option> <option value="oral">Oral</option> <option value="organics">Organics</option> <option value="organoids">Organoids</option> <option value="osteology">Osteology</option> <option value="oxygen">Oxygen</option> <option value="parasitologia">Parasitologia</option> <option value="particles">Particles</option> <option value="pathogens">Pathogens</option> <option value="pathophysiology">Pathophysiology</option> <option value="pediatrrep">Pediatric Reports</option> <option value="pets">Pets</option> <option value="pharmaceuticals">Pharmaceuticals</option> <option value="pharmaceutics">Pharmaceutics</option> <option value="pharmacoepidemiology">Pharmacoepidemiology</option> <option value="pharmacy">Pharmacy</option> <option value="philosophies">Philosophies</option> <option value="photochem">Photochem</option> <option value="photonics">Photonics</option> <option value="phycology">Phycology</option> <option value="physchem">Physchem</option> <option value="psf">Physical Sciences Forum</option> <option value="physics">Physics</option> <option value="physiologia">Physiologia</option> <option value="plants">Plants</option> <option value="plasma">Plasma</option> <option value="platforms">Platforms</option> <option value="pollutants">Pollutants</option> <option value="polymers">Polymers</option> <option value="polysaccharides">Polysaccharides</option> <option value="populations">Populations</option> <option value="poultry">Poultry</option> <option value="powders">Powders</option> <option value="proceedings">Proceedings</option> <option value="processes">Processes</option> <option value="prosthesis">Prosthesis</option> <option value="proteomes">Proteomes</option> <option value="psychiatryint">Psychiatry International</option> <option value="psychoactives">Psychoactives</option> <option value="psycholint">Psychology International</option> <option value="publications">Publications</option> <option value="qubs">Quantum Beam Science</option> <option value="quantumrep">Quantum Reports</option> <option value="quaternary">Quaternary</option> <option value="radiation">Radiation</option> <option value="reactions">Reactions</option> <option value="realestate">Real Estate</option> <option value="receptors">Receptors</option> <option value="recycling">Recycling</option> <option value="rsee">Regional Science and Environmental Economics</option> <option value="religions">Religions</option> <option value="remotesensing">Remote Sensing</option> <option value="reports">Reports</option> <option value="reprodmed">Reproductive Medicine</option> <option value="resources">Resources</option> <option value="rheumato">Rheumato</option> <option value="risks">Risks</option> <option value="robotics">Robotics</option> <option value="ruminants">Ruminants</option> <option value="safety">Safety</option> <option value="sci">Sci</option> <option value="scipharm">Scientia Pharmaceutica</option> <option value="sclerosis">Sclerosis</option> <option value="seeds">Seeds</option> <option value="sensors">Sensors</option> <option value="separations">Separations</option> <option value="sexes">Sexes</option> <option value="signals">Signals</option> <option value="sinusitis">Sinusitis</option> <option value="smartcities">Smart Cities</option> <option value="socsci">Social Sciences</option> <option value="siuj">Société Internationale d’Urologie Journal</option> <option value="societies">Societies</option> <option value="software">Software</option> <option value="soilsystems">Soil Systems</option> <option value="solar">Solar</option> <option value="solids">Solids</option> <option value="spectroscj">Spectroscopy Journal</option> <option value="sports">Sports</option> <option value="standards">Standards</option> <option value="stats">Stats</option> <option value="stresses">Stresses</option> <option value="surfaces">Surfaces</option> <option value="surgeries">Surgeries</option> <option value="std">Surgical Techniques Development</option> <option value="sustainability">Sustainability</option> <option value="suschem">Sustainable Chemistry</option> <option value="symmetry">Symmetry</option> <option value="synbio">SynBio</option> <option value="systems">Systems</option> <option value="targets">Targets</option> <option value="taxonomy">Taxonomy</option> <option value="technologies">Technologies</option> <option value="telecom">Telecom</option> <option value="textiles">Textiles</option> <option value="thalassrep">Thalassemia Reports</option> <option value="therapeutics">Therapeutics</option> <option value="thermo">Thermo</option> <option value="timespace">Time and Space</option> <option value="tomography">Tomography</option> <option value="tourismhosp">Tourism and Hospitality</option> <option value="toxics">Toxics</option> <option value="toxins">Toxins</option> <option value="transplantology">Transplantology</option> <option value="traumacare">Trauma Care</option> <option value="higheredu">Trends in Higher Education</option> <option value="tropicalmed">Tropical Medicine and Infectious Disease</option> <option value="universe">Universe</option> <option value="urbansci">Urban Science</option> <option value="uro">Uro</option> <option value="vaccines">Vaccines</option> <option value="vehicles">Vehicles</option> <option value="venereology">Venereology</option> <option value="vetsci">Veterinary Sciences</option> <option value="vibration">Vibration</option> <option value="virtualworlds">Virtual Worlds</option> <option value="viruses">Viruses</option> <option value="vision">Vision</option> <option value="waste">Waste</option> <option value="water">Water</option> <option value="wild">Wild</option> <option value="wind">Wind</option> <option value="women">Women</option> <option value="world">World</option> <option value="wevj">World Electric Vehicle Journal</option> <option value="youth">Youth</option> <option value="zoonoticdis">Zoonotic Diseases</option> </select> <input name="email" type="email" placeholder="Enter your email address..." required="required" /> <button class="genericCaptcha button button--dark UA_FooterNewsletterSubscribeButton" type="submit">Subscribe</button> </form> </div> </div> </div> <div id="footer-copyright"> <div class="row"> <div class="columns large-6 medium-6 small-12 text-left"> © 1996-2024 MDPI (Basel, Switzerland) unless otherwise stated </div> <div class="columns large-6 medium-6 small-12 small-text-left medium-text-right large-text-right"> <a data-dropdown="drop-view-disclaimer" aria-controls="drop-view-disclaimer" aria-expanded="false" data-options="align:top; is_hover:true; hover_timeout:2000;"> Disclaimer </a> <div id="drop-view-disclaimer" class="f-dropdown label__btn__dropdown label__btn__dropdown--wide text-left" data-dropdown-content aria-hidden="true" tabindex="-1"> Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. </div> <a href="/about/terms-and-conditions"> Terms and Conditions </a> <a href="/about/privacy"> Privacy Policy </a> </div> </div> </div> </div> <div id="cookie-notification" class="js-allow-cookies" style="display: none;"> <div class="columns large-10 medium-10 small-12"> We use cookies on our website to ensure you get the best experience.<br class="show-for-medium-up"/> Read more about our cookies <a href="/about/privacy">here</a>. </div> <div class="columns large-2 medium-2 small-12 small-only-text-left text-right"> <a class="button button--default" href="/accept_cookies">Accept</a> </div> </div> </div> <div id="main-share-modal" class="reveal-modal reveal-modal-new reveal-modal-new--small" data-reveal aria-labelledby="modalTitle" aria-hidden="true" role="dialog"> <div class="row"> <div class="small-12 columns"> <h2 style="margin: 0;">Share Link</h2> </div> <div class="small-12 columns"> <div class="social-media-links UA_ShareModalLinks" style="text-align: left;"> <a href="/cdn-cgi/l/email-protection#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" title="Email"> <i class="fa fa-envelope-square" style="font-size: 30px;"></i> </a> <a href="https://twitter.com/intent/tweet?text=Exploring+New+Physics+with+Deep+Underground+Neutrino+Experiment+High-Energy+Flux%3A+The+Case+of+Lorentz+Invariance+Violation%2C+Large+Extra+Dimensions+and+Long-Range+Forces&hashtags=mdpiuniverse&url=https%3A%2F%2Fwww.mdpi.com%2F2944470&via=Universe_MDPI" onclick="windowOpen(this.href,600,800); return false" title="Twitter" target="_blank" rel="noopener noreferrer"> <i class="fa fa-twitter-x-square" style="font-size: 30px;"></i> </a> <a href=" http://www.linkedin.com/shareArticle?mini=true&url=https%3A%2F%2Fwww.mdpi.com%2F2944470&title=Exploring%20New%20Physics%20with%20Deep%20Underground%20Neutrino%20Experiment%20High-Energy%20Flux%3A%20The%20Case%20of%20Lorentz%20Invariance%20Violation%2C%20Large%20Extra%20Dimensions%20and%20Long-Range%20Forces%26source%3Dhttps%3A%2F%2Fwww.mdpi.com%26summary%3DDUNE%20is%20a%20next-generation%20long-baseline%20neutrino%20oscillation%20experiment.%20It%20is%20expected%20to%20measure%2C%20with%20unprecedented%20precision%2C%20the%20atmospheric%20oscillation%20parameters%2C%20including%20the%20CP-violating%20phase%20%CE%B4CP.%20Moreover%2C%20several%20studies%20have%20suggested%20%5B...%5D" onclick="windowOpen(this.href,600,800); return false" title="LinkedIn" target="_blank" rel="noopener noreferrer"> <i class="fa fa-linkedin-square" style="font-size: 30px;"></i> </a> <a href="https://www.facebook.com/sharer.php?u=https://www.mdpi.com/2944470" title="facebook" target="_blank" rel="noopener noreferrer"> <i class="fa fa-facebook-square" style="font-size: 30px;"></i> </a> <a href="javascript:void(0);" title="Wechat" data-reveal-id="weixin-share-modal"> <i class="fa fa-weixin-square" style="font-size: 26px;"></i> </a> <a href="http://www.reddit.com/submit?url=https://www.mdpi.com/2944470" title="Reddit" target="_blank" rel="noopener noreferrer"> <i class="fa fa-reddit-square" style="font-size: 30px;"></i> </a> <a href="http://www.mendeley.com/import/?url=https://www.mdpi.com/2944470" title="Mendeley" target="_blank" rel="noopener noreferrer"> <i class="fa fa-mendeley-square" style="font-size: 30px;"></i> </a> <a href="http://www.citeulike.org/posturl?url=https://www.mdpi.com/2944470" title="CiteULike" target="_blank" rel="noopener noreferrer"> <i class="fa fa-citeulike-square" style="font-size: 30px;"></i> </a> </div> </div> <div class="small-9 columns"> <input id="js-clipboard-text" type="text" readonly value="https://www.mdpi.com/2944470" /> </div> <div class="small-3 columns text-left"> <a class="button button--color js-clipboard-copy" data-clipboard-target="#js-clipboard-text">Copy</a> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> <div id="weixin-share-modal" class="reveal-modal reveal-modal-new" data-reveal aria-labelledby="weixin-share-modal-title" aria-hidden="true" role="dialog"> <div class="row"> <div class="small-12 columns"> <h2 id="weixin-share-modal-title" style="margin: 0;">Share</h2> </div> <div class="small-12 columns"> <div class="weixin-qr-code-section"> <?xml version="1.0" standalone="no"?> <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"> <svg width="300" height="300" version="1.1" xmlns="http://www.w3.org/2000/svg"> <desc>https://www.mdpi.com/2944470</desc> <g id="elements" fill="black" stroke="none"> <rect x="0" y="0" width="12" height="12" /> <rect x="12" y="0" width="12" height="12" /> <rect x="24" y="0" width="12" height="12" /> <rect x="36" y="0" width="12" height="12" /> <rect x="48" y="0" width="12" height="12" /> <rect x="60" y="0" width="12" height="12" /> <rect x="72" y="0" width="12" height="12" /> <rect x="96" y="0" width="12" height="12" /> <rect x="108" y="0" width="12" height="12" /> <rect x="120" y="0" width="12" height="12" /> <rect x="132" y="0" width="12" height="12" /> <rect x="156" y="0" width="12" height="12" /> <rect x="168" y="0" width="12" height="12" /> <rect x="216" y="0" width="12" height="12" /> <rect x="228" y="0" width="12" height="12" /> <rect x="240" y="0" width="12" height="12" /> <rect x="252" y="0" width="12" height="12" /> <rect x="264" y="0" width="12" height="12" /> <rect x="276" y="0" width="12" height="12" /> <rect x="288" y="0" width="12" height="12" /> <rect x="0" y="12" width="12" height="12" /> <rect x="72" y="12" width="12" height="12" /> <rect x="132" y="12" width="12" height="12" /> <rect x="144" y="12" width="12" height="12" /> <rect x="168" y="12" width="12" height="12" /> <rect x="216" y="12" width="12" height="12" /> <rect x="288" y="12" width="12" height="12" /> <rect x="0" y="24" width="12" height="12" /> <rect x="24" y="24" width="12" height="12" /> <rect x="36" y="24" width="12" height="12" /> <rect x="48" y="24" width="12" height="12" /> <rect x="72" y="24" width="12" height="12" /> <rect x="108" y="24" width="12" height="12" /> <rect x="132" y="24" width="12" height="12" /> <rect x="156" y="24" width="12" height="12" /> <rect x="168" y="24" width="12" height="12" /> <rect x="216" y="24" width="12" height="12" /> <rect x="240" y="24" width="12" height="12" /> <rect x="252" y="24" width="12" height="12" /> <rect x="264" y="24" width="12" height="12" /> <rect x="288" y="24" width="12" height="12" /> <rect x="0" y="36" width="12" height="12" /> <rect x="24" y="36" width="12" height="12" /> <rect x="36" y="36" width="12" height="12" /> <rect x="48" y="36" width="12" height="12" /> <rect x="72" y="36" width="12" height="12" /> <rect x="120" y="36" width="12" height="12" /> <rect x="132" y="36" width="12" height="12" /> <rect x="144" y="36" width="12" height="12" /> <rect x="156" y="36" width="12" height="12" /> <rect x="180" y="36" width="12" height="12" /> <rect x="192" y="36" width="12" height="12" /> <rect x="216" y="36" width="12" height="12" /> <rect x="240" y="36" width="12" height="12" /> <rect x="252" y="36" width="12" height="12" /> <rect x="264" y="36" width="12" height="12" /> <rect x="288" y="36" width="12" height="12" /> <rect x="0" y="48" width="12" height="12" /> <rect x="24" y="48" width="12" height="12" /> <rect x="36" y="48" width="12" height="12" /> <rect x="48" y="48" width="12" height="12" /> <rect x="72" y="48" width="12" height="12" /> <rect x="120" y="48" width="12" height="12" /> <rect x="132" y="48" width="12" height="12" /> <rect x="144" y="48" width="12" height="12" /> <rect x="156" y="48" width="12" height="12" /> <rect x="168" y="48" width="12" height="12" /> <rect x="216" y="48" width="12" height="12" /> <rect x="240" y="48" width="12" height="12" /> <rect x="252" y="48" width="12" height="12" /> <rect x="264" y="48" width="12" height="12" /> <rect x="288" y="48" width="12" height="12" /> <rect x="0" y="60" width="12" height="12" /> <rect x="72" y="60" width="12" height="12" /> <rect x="108" y="60" width="12" height="12" /> <rect x="132" y="60" width="12" height="12" /> <rect x="144" y="60" width="12" height="12" /> <rect x="156" y="60" width="12" height="12" /> <rect x="168" y="60" width="12" height="12" /> <rect x="216" y="60" width="12" height="12" /> <rect x="288" y="60" width="12" height="12" /> <rect x="0" y="72" width="12" height="12" /> <rect x="12" y="72" width="12" height="12" /> <rect x="24" y="72" width="12" height="12" /> <rect x="36" y="72" width="12" height="12" /> <rect x="48" y="72" width="12" height="12" /> <rect x="60" y="72" width="12" height="12" /> <rect x="72" y="72" width="12" height="12" /> <rect x="96" y="72" width="12" height="12" /> <rect x="120" y="72" width="12" height="12" /> <rect x="144" y="72" width="12" height="12" /> <rect x="168" y="72" width="12" height="12" /> <rect x="192" y="72" width="12" height="12" /> <rect x="216" y="72" width="12" height="12" /> <rect x="228" y="72" width="12" height="12" /> <rect x="240" y="72" width="12" height="12" /> <rect x="252" y="72" width="12" height="12" /> <rect x="264" y="72" width="12" height="12" /> <rect x="276" y="72" width="12" height="12" /> <rect x="288" y="72" width="12" height="12" /> <rect x="96" y="84" width="12" height="12" /> <rect x="108" y="84" width="12" height="12" /> <rect x="144" y="84" width="12" height="12" /> <rect x="156" y="84" width="12" height="12" /> <rect x="0" y="96" width="12" height="12" /> <rect x="12" y="96" width="12" height="12" /> <rect x="36" y="96" width="12" height="12" /> <rect x="48" y="96" width="12" height="12" /> <rect x="72" y="96" width="12" height="12" /> <rect x="108" y="96" width="12" height="12" /> <rect x="144" y="96" width="12" height="12" /> <rect x="168" y="96" width="12" height="12" /> <rect x="192" y="96" width="12" height="12" /> <rect x="216" y="96" width="12" height="12" /> <rect x="288" y="96" width="12" height="12" /> <rect x="24" y="108" width="12" height="12" /> <rect x="84" y="108" width="12" height="12" /> <rect x="120" y="108" width="12" height="12" /> <rect x="144" y="108" width="12" height="12" /> <rect x="156" y="108" width="12" height="12" /> <rect x="168" y="108" width="12" height="12" /> <rect x="180" y="108" width="12" height="12" /> <rect x="204" y="108" width="12" height="12" /> <rect x="228" y="108" width="12" height="12" /> <rect x="240" y="108" width="12" height="12" /> <rect x="252" y="108" width="12" height="12" /> <rect x="264" y="108" width="12" height="12" /> <rect x="276" y="108" width="12" height="12" /> <rect x="0" y="120" width="12" height="12" /> <rect x="24" y="120" width="12" height="12" /> <rect x="60" y="120" width="12" height="12" /> <rect x="72" y="120" width="12" height="12" /> <rect x="84" y="120" width="12" height="12" /> <rect x="96" y="120" width="12" height="12" /> <rect x="108" y="120" width="12" height="12" /> <rect x="180" y="120" width="12" height="12" /> <rect x="216" y="120" width="12" height="12" /> <rect x="228" y="120" width="12" height="12" /> <rect x="252" y="120" width="12" height="12" /> <rect x="288" y="120" width="12" height="12" /> <rect x="0" y="132" width="12" height="12" /> <rect x="36" y="132" width="12" height="12" /> <rect x="48" y="132" width="12" height="12" /> <rect x="96" y="132" width="12" height="12" /> <rect x="132" y="132" width="12" height="12" /> <rect x="168" y="132" width="12" height="12" /> <rect x="192" y="132" width="12" height="12" /> <rect x="228" y="132" width="12" height="12" /> <rect x="252" y="132" width="12" height="12" /> <rect x="264" y="132" width="12" height="12" /> <rect x="276" y="132" width="12" height="12" /> <rect x="288" y="132" width="12" height="12" /> <rect x="0" y="144" width="12" height="12" /> <rect x="12" y="144" width="12" height="12" /> <rect x="24" y="144" width="12" height="12" /> <rect x="60" y="144" width="12" height="12" /> <rect x="72" y="144" width="12" height="12" /> <rect x="84" y="144" width="12" height="12" /> <rect x="96" y="144" width="12" height="12" /> <rect x="108" y="144" width="12" height="12" /> <rect x="120" y="144" width="12" height="12" /> <rect x="168" y="144" width="12" height="12" /> <rect x="180" y="144" width="12" height="12" /> <rect x="216" y="144" width="12" height="12" /> <rect x="288" y="144" width="12" height="12" /> <rect x="0" y="156" width="12" height="12" /> <rect x="60" y="156" width="12" height="12" /> <rect x="96" y="156" width="12" height="12" /> <rect x="108" y="156" width="12" height="12" /> <rect x="120" y="156" width="12" height="12" /> <rect x="144" y="156" width="12" height="12" /> <rect x="156" y="156" width="12" height="12" /> <rect x="168" y="156" width="12" height="12" /> <rect x="180" y="156" width="12" height="12" /> <rect x="192" y="156" width="12" height="12" /> <rect x="204" y="156" width="12" height="12" /> <rect x="240" y="156" width="12" height="12" /> <rect x="276" y="156" width="12" height="12" /> <rect x="0" y="168" width="12" height="12" /> <rect x="12" y="168" width="12" height="12" /> <rect x="24" y="168" width="12" height="12" /> <rect x="36" y="168" width="12" height="12" /> <rect x="48" y="168" width="12" height="12" /> <rect x="60" y="168" width="12" height="12" /> <rect x="72" y="168" width="12" height="12" /> <rect x="84" y="168" width="12" height="12" /> <rect x="96" y="168" width="12" height="12" /> <rect x="132" y="168" width="12" height="12" /> <rect x="144" y="168" width="12" height="12" /> <rect x="156" y="168" width="12" height="12" /> <rect x="180" y="168" width="12" height="12" /> <rect x="192" y="168" width="12" height="12" /> <rect x="204" y="168" width="12" height="12" /> <rect x="216" y="168" width="12" height="12" /> <rect x="240" y="168" width="12" height="12" /> <rect x="252" y="168" width="12" height="12" /> <rect x="264" y="168" width="12" height="12" /> <rect x="276" y="168" width="12" height="12" /> <rect x="288" y="168" width="12" height="12" /> <rect x="0" y="180" width="12" height="12" /> <rect x="36" y="180" width="12" height="12" /> <rect x="60" y="180" width="12" height="12" /> <rect x="84" y="180" width="12" height="12" /> <rect x="96" y="180" width="12" height="12" /> <rect x="120" y="180" width="12" height="12" /> <rect x="144" y="180" width="12" height="12" /> <rect x="168" y="180" width="12" height="12" /> <rect x="204" y="180" width="12" height="12" /> <rect x="228" y="180" width="12" height="12" /> <rect x="252" y="180" width="12" height="12" /> <rect x="264" y="180" width="12" height="12" /> <rect x="288" y="180" width="12" height="12" /> <rect x="0" y="192" width="12" height="12" /> <rect x="36" y="192" width="12" height="12" /> <rect x="60" y="192" width="12" height="12" /> <rect x="72" y="192" width="12" height="12" /> <rect x="96" y="192" width="12" height="12" /> <rect x="108" y="192" width="12" height="12" /> <rect x="132" y="192" width="12" height="12" /> <rect x="144" y="192" width="12" height="12" /> <rect x="180" y="192" width="12" height="12" /> <rect x="192" y="192" width="12" height="12" /> <rect x="204" y="192" width="12" height="12" /> <rect x="216" y="192" width="12" height="12" /> <rect x="228" y="192" width="12" height="12" /> <rect x="240" y="192" width="12" height="12" /> <rect x="264" y="192" width="12" height="12" /> <rect x="276" y="192" width="12" height="12" /> <rect x="96" y="204" width="12" height="12" /> <rect x="192" y="204" width="12" height="12" /> <rect x="240" y="204" width="12" height="12" /> <rect x="264" y="204" width="12" height="12" /> <rect x="276" y="204" width="12" height="12" /> <rect x="0" y="216" width="12" height="12" /> <rect x="12" y="216" width="12" height="12" /> <rect x="24" y="216" width="12" height="12" /> <rect x="36" y="216" width="12" height="12" /> <rect x="48" y="216" width="12" height="12" /> <rect x="60" y="216" width="12" height="12" /> <rect x="72" y="216" width="12" height="12" /> <rect x="108" y="216" width="12" height="12" /> <rect x="120" y="216" width="12" height="12" /> <rect x="144" y="216" width="12" height="12" /> <rect x="192" y="216" width="12" height="12" /> <rect x="216" y="216" width="12" height="12" /> <rect x="240" y="216" width="12" height="12" /> <rect x="288" y="216" width="12" height="12" /> <rect x="0" y="228" width="12" height="12" /> <rect x="72" y="228" width="12" height="12" /> <rect x="132" y="228" width="12" height="12" /> <rect x="144" y="228" width="12" height="12" /> <rect x="180" y="228" width="12" height="12" /> <rect x="192" y="228" width="12" height="12" /> <rect x="240" y="228" width="12" height="12" /> <rect x="288" y="228" width="12" height="12" /> <rect x="0" y="240" width="12" height="12" /> <rect x="24" y="240" width="12" height="12" /> <rect x="36" y="240" width="12" height="12" /> <rect x="48" y="240" width="12" height="12" /> <rect x="72" y="240" width="12" height="12" /> <rect x="96" y="240" width="12" height="12" /> <rect x="108" y="240" width="12" height="12" /> <rect x="144" y="240" width="12" height="12" /> <rect x="156" y="240" width="12" height="12" /> <rect x="180" y="240" width="12" height="12" /> <rect x="192" y="240" width="12" height="12" /> <rect x="204" y="240" width="12" height="12" /> <rect x="216" y="240" width="12" height="12" /> <rect x="228" y="240" width="12" height="12" /> <rect x="240" y="240" width="12" height="12" /> <rect x="276" y="240" width="12" height="12" /> <rect x="288" y="240" width="12" height="12" /> <rect x="0" y="252" width="12" height="12" /> <rect x="24" y="252" width="12" height="12" /> <rect x="36" y="252" width="12" height="12" /> <rect x="48" y="252" width="12" height="12" /> <rect x="72" y="252" width="12" height="12" /> <rect x="96" y="252" width="12" height="12" /> <rect x="108" y="252" width="12" height="12" /> <rect x="132" y="252" width="12" height="12" /> <rect x="144" y="252" width="12" height="12" /> <rect x="180" y="252" width="12" height="12" /> <rect x="192" y="252" width="12" height="12" /> <rect x="216" y="252" width="12" height="12" /> <rect x="276" y="252" width="12" height="12" /> <rect x="288" y="252" width="12" height="12" /> <rect x="0" y="264" width="12" height="12" /> <rect x="24" y="264" width="12" height="12" /> <rect x="36" y="264" width="12" height="12" /> <rect x="48" y="264" width="12" height="12" /> <rect x="72" y="264" width="12" height="12" /> <rect x="132" y="264" width="12" height="12" /> <rect x="168" y="264" width="12" height="12" /> <rect x="192" y="264" width="12" height="12" /> <rect x="204" y="264" width="12" height="12" /> <rect x="240" y="264" width="12" height="12" /> <rect x="252" y="264" width="12" height="12" /> <rect x="264" y="264" width="12" height="12" /> <rect x="276" y="264" width="12" height="12" /> <rect x="288" y="264" width="12" height="12" /> <rect x="0" y="276" width="12" height="12" /> <rect x="72" y="276" width="12" height="12" /> <rect x="96" y="276" width="12" height="12" /> <rect x="108" y="276" width="12" height="12" /> <rect x="144" y="276" width="12" height="12" /> <rect x="168" y="276" width="12" height="12" /> <rect x="228" y="276" width="12" height="12" /> <rect x="240" y="276" width="12" height="12" /> <rect x="264" y="276" width="12" height="12" /> <rect x="276" y="276" width="12" height="12" /> <rect x="288" y="276" width="12" height="12" /> <rect x="0" y="288" width="12" height="12" /> <rect x="12" y="288" width="12" height="12" /> <rect x="24" y="288" width="12" height="12" /> <rect x="36" y="288" width="12" height="12" /> <rect x="48" y="288" width="12" height="12" /> <rect x="60" y="288" width="12" height="12" /> <rect x="72" y="288" width="12" height="12" /> <rect x="96" y="288" width="12" height="12" /> <rect x="108" y="288" width="12" height="12" /> <rect x="132" y="288" width="12" height="12" /> <rect x="168" y="288" width="12" height="12" /> <rect x="192" y="288" width="12" height="12" /> <rect x="204" y="288" width="12" height="12" /> <rect x="252" y="288" width="12" height="12" /> <rect x="288" y="288" width="12" height="12" /> </g> </svg> </div> </div> </div> <a class="close-reveal-modal" aria-label="Close"> <i class="material-icons">clear</i> </a> </div> <a href="#" class="back-to-top"><span class="show-for-medium-up">Back to Top</span><span class="show-for-small">Top</span></a> <script data-cfasync="false" src="/cdn-cgi/scripts/5c5dd728/cloudflare-static/email-decode.min.js"></script><script src="https://pub.mdpi-res.com/assets/js/modernizr-2.8.3.min.js?5227e0738f7f421d?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/jquery-1.12.4.min.js?4f252523d4af0b47?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/foundation-5.5.3.min.js?6b2ec41c18b29054?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/foundation-5.5.3.equalizer.min.js?0f6c549b75ec554c?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/jquery.multiselect.js?0edd3998731d1091?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/jquery.cycle2.min.js?63413052928f97ee?1732286508"></script> <script> // old browser fix - this way the console log rows won't throw (silent) errors in browsers not supporting console log if (!window.console) window.console = {}; if (!window.console.log) window.console.log = function () { }; var currentJournalNameSystem = "universe"; $(document).ready(function() { $('select.foundation-select').multiselect({ search: true, minHeight: 130, maxHeight: 130, }); $(document).foundation({ orbit: { timer_speed: 4000, }, reveal: { animation: 'fadeAndPop', animation_speed: 100, } }); $(".chosen-select").each(function(element) { var maxSelected = (undefined !== $(this).data('maxselectedoptions') ? $(this).data('maxselectedoptions') : 100); $(this).on('chosen:ready', function(event, data) { var select = $(data.chosen.form_field); if (select.attr('id') === 'journal-browser-volume') { $(data.chosen.dropdown).addClass('UI_JournalBrowser_Volume_Options'); } if (select.attr('id') === 'journal-browser-issue') { $(data.chosen.dropdown).addClass('UI_JournalBrowser_Issue_Options'); } }).chosen({ display_disabled_options: false, disable_search_threshold: 7, max_selected_options: maxSelected, width: "100%" }); }); $(".toEncode").each(function(e) { var oldHref = $(this).attr("href"); var newHref = oldHref.replace('.botdefense.please.enable.javascript.','@'); $(this).attr("href", newHref); if (!$(this).hasClass("emailCaptcha")) { $(this).html(newHref.replace('mailto:', '')); } $(this).removeClass("visibility-hidden"); }); $(document).on('opened.fndtn.reveal', '[data-reveal]', function() { $(document).foundation('equalizer', 'reflow'); }); // fix the images that have tag height / width defined // otherwise the default foundation styles overwrite the tag definitions $("img").each(function() { if ($(this).attr('width') != undefined || $(this).attr('height') != undefined) { $(this).addClass("img-fixed"); } }); $("#basic_search, #advanced_search").submit(function(e) { var searchArguments = false; $(this).find("input,select").not("#search,.search-button").each(function() { if (undefined === $(this).val() || "" === $(this).val()) { $(this).attr('name', null); } else { $(this).attr('name'); searchArguments = true; } }); if (!searchArguments) { window.location = $(this).attr('action'); return false; } }); $(".hide-show-desktop-option").click(function(e) { e.preventDefault(); var parentDiv = $(this).closest("div"); $.ajax({ url: $(this).attr('href'), success: function(msg) { parentDiv.removeClass().hide(); } }); }); $(".generic-toggleable-header").click(function(e) { $(this).toggleClass("active"); $(this).next(".generic-toggleable-content").toggleClass("active"); }); /* * handle whole row as a link if the row contains only one visible link */ $("table.new tr").hover(function() { if ($(this).find("td:visible a").length == 1) { $(this).addClass("single-link"); } }, function() { $(this).removeClass("single-link"); }); $("table.new:not(.table-of-tables)").on("click", "tr.single-link", function(e) { var target = $(e.target); if (!e.ctrlKey && !target.is("a")) { $(this).find("td:visible a")[0].click(); } }); $(document).on("click", ".custom-accordion-for-small-screen-link", function(e) { if ($(this).closest("#basic_search").length > 0) { if ($(".search-container__advanced").first().is(":visible")) { openAdvanced() } } if (Foundation.utils.is_small_only()) { if ($(this).hasClass("active")) { $(this).removeClass("active"); $(this).next(".custom-accordion-for-small-screen-content").addClass("show-for-medium-up"); } else { $(this).addClass("active"); $(this).next(".custom-accordion-for-small-screen-content").removeClass("show-for-medium-up"); $(document).foundation('orbit', 'reflow'); } } if (undefined !== $(this).data("callback")) { var customCallback = $(this).data("callback"); func = window[customCallback]; func(); } }); $(document).on("click", ".js-open-small-search", function(e) { e.preventDefault(); $(this).toggleClass("active").closest(".tab-bar").toggleClass("active"); $(".search-container").toggleClass("hide-for-small-down"); }); $(document).on("click", ".js-open-menu", function(e) { $(".search-container").addClass("hide-for-small-down"); }); $(window).on('resize', function() { recalculate_main_browser_position(); recalculate_responsive_moving_containers(); }); updateSearchLabelVisibilities(); recalculate_main_browser_position(); recalculate_responsive_moving_containers(); if (window.document.documentMode == 11) { $("<link/>", { rel: "stylesheet", type: "text/css", href: "https://fonts.googleapis.com/icon?family=Material+Icons"}).appendTo("head"); } }); function recalculate_main_browser_position() { if (Foundation.utils.is_small_only()) { if ($("#js-main-top-container").parent("#js-large-main-top-container").length > 0) { $("#js-main-top-container").appendTo($("#js-small-main-top-container")); } } else { if ($("#js-main-top-container").parent("#js-small-main-top-container").length > 0) { $("#js-main-top-container").appendTo($("#js-large-main-top-container")); } } } function recalculate_responsive_moving_containers() { $(".responsive-moving-container.large").each(function() { var previousParent = $(".responsive-moving-container.active[data-id='"+$(this).data("id")+"']"); var movingContent = previousParent.html(); if (Foundation.utils.is_small_only()) { var currentParent = $(".responsive-moving-container.small[data-id='"+$(this).data("id")+"']"); } else if (Foundation.utils.is_medium_only()) { var currentParent = $(".responsive-moving-container.medium[data-id='"+$(this).data("id")+"']"); } else { var currentParent = $(".responsive-moving-container.large[data-id='"+$(this).data("id")+"']"); } if (previousParent.attr("class") !== currentParent.attr("class")) { currentParent.html(movingContent); previousParent.html(); currentParent.addClass("active"); previousParent.removeClass("active"); } }); } // cookies allowed is checked from a) local storage and b) from server separately so that the footer bar doesn't // get included in the custom page caches function checkCookiesAllowed() { var cookiesEnabled = localStorage.getItem("mdpi_cookies_enabled"); if (null === cookiesEnabled) { $.ajax({ url: "/ajax_cookie_value/mdpi_cookies_accepted", success: function(data) { if (data.value) { localStorage.setItem("mdpi_cookies_enabled", true); checkDisplaySurvey(); } else { $(".js-allow-cookies").show(); } } }); } else { checkDisplaySurvey(); } } function checkDisplaySurvey() { } window.addEventListener('CookiebotOnAccept', function (e) { var CookieDate = new Date; if (Cookiebot.consent.preferences) { CookieDate.setFullYear(CookieDate.getFullYear() + 1); document.cookie = "mdpi_layout_type_v2=mobile; path=/; expires=" + CookieDate.toUTCString() + ";"; $(".js-toggle-desktop-layout-link").css("display", "inline-block"); } }, false); window.addEventListener('CookiebotOnDecline', function (e) { if (!Cookiebot.consent.preferences) { $(".js-toggle-desktop-layout-link").hide(); if ("" === "desktop") { window.location = "/toggle_desktop_layout_cookie"; } } }, false); var hash = $(location).attr('hash'); if ("#share" === hash) { if (1 === $("#main-share-modal").length) { $('#main-share-modal').foundation('reveal', 'open'); } } </script> <script src="https://pub.mdpi-res.com/assets/js/lib.js?f8d3d71b3a772f9d?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/mdpi.js?c267ce58392b15da?1732286508"></script> <script>var banners_url = 'https://serve.mdpi.com';</script> <script type='text/javascript' src='https://pub.mdpi-res.com/assets/js/ifvisible.min.js?c621d19ecb761212?1732286508'></script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/affix.js?ac4ea55275297c15?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/clipboard.min.js?3f3688138a1b9fc4?1732286508"></script> <script type="text/javascript"> $(document).ready(function() { var helpFunctions = $(".middle-column__help__fixed"); var leftColumnAffix = $(".left-column__fixed"); var middleColumn = $("#middle-column"); var clone = null; helpFunctions.affix({ offset: { top: function() { return middleColumn.offset().top - 8 - (Foundation.utils.is_medium_only() ? 30 : 0); }, bottom: function() { return $("#footer").innerHeight() + 74 + (Foundation.utils.is_medium_only() ? 0 : 0); } } }); if (leftColumnAffix.length > 0) { clone = leftColumnAffix.clone(); clone.addClass("left-column__fixed__affix"); clone.insertBefore(leftColumnAffix); clone.css('width', leftColumnAffix.outerWidth() + 50); clone.affix({ offset: { top: function() { return leftColumnAffix.offset().top - 30 - (Foundation.utils.is_medium_only() ? 50 : 0); }, bottom: function() { return $("#footer").innerHeight() + 92 + (Foundation.utils.is_medium_only() ? 0 : 0); } } }); } $(window).on("resize", function() { if (clone !== null) { clone.css('width', leftColumnAffix.outerWidth() + 50); } }); new ClipboardJS('.js-clipboard-copy'); }); </script> <script src="https://pub.mdpi-res.com/assets/js/jquery-ui-1.13.2.min.js?1e2047978946a1d2?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/slick.min.js?d5a61c749e44e471?1732286508"></script> <script> $(document).ready(function() { $(".link-article-menu").click(function(e) { e.preventDefault(); $(this).find('span').toggle(); $(this).next("div").toggleClass("active"); }); $(".js-similarity-related-articles").click(function(e) { e.preventDefault(); if ('' !== $('#recommended-articles-modal').attr('data-url')) { $('#recommended-articles-modal').foundation('reveal', 'open', $('#recommended-articles-modal').attr('data-url')); } }); $.ajax({ url: "/article/1472235/similarity-related/show-link", success: function(result) { if (result.show) { $('#recommended-articles-modal').attr('data-url', result.link); $('.js-article-similarity-container').show(); } } }); $(document).on('opened.fndtn.reveal', '[data-reveal]', function() { var modal = $(this); if (modal.attr('id') === "author-biographies-modal") { modal.find('.multiple-items').slick({ slidesToShow: 1, nextArrow: '<a class="slick-next" href="#"><i class="material-icons">chevron_right</i></a>', prevArrow: '<a class="slick-prev" href="#"><i class="material-icons">chevron_left</i></a>', slidesToScroll: 1, draggable: false, }); modal.find('.multiple-items').slick('refresh'); } }); }); </script> <script> $(document).ready(function() { $(document).on('keyup', function (e) { if (e.keyCode == 27) { var hElem = $(this).find(".annotator-adder"); if (hElem.length){ hElem.css({'visibility':'hidden'}); } else { document.querySelector("hypothesis-adder").shadowRoot.querySelector(".annotator-adder").style.visibility = "hidden"; } } }); }); </script> <script> window.hypothesisConfig = function () { return { sidebarAppUrl: 'https://commenting.mdpi.com/app.html', showHighlights: 'whenSidebarOpen' , openSidebar: false , assetRoot: 'https://commentingres.mdpi.com/hypothesis', services: [{ apiUrl: 'https://commenting.mdpi.com/api/', authority: 'mdpi', grantToken: '', doi: '10.3390/universe10090357' }], }; }; </script> <script async id="hypothesis_frame"></script> <script type="text/javascript"> if (-1 !== window.location.href.indexOf("?src=")) { window.history.replaceState({}, '', `${location.pathname}`); } $(document).ready(function() { var scifeedCounter = 0; var search = window.location.search; var mathjaxReady = false; // late image file loading $("img[data-lsrc]").each(function() { $(this).attr("src", $(this).data("lsrc")); }); // late mathjax initialization var head = document.getElementsByTagName("head")[0]; var script = document.createElement("script"); script.type = "text/x-mathjax-config"; script[(window.opera ? "innerHTML" : "text")] = "MathJax.Hub.processSectionDelay = 0;\n" + "MathJax.Hub.Config({\n" + " \"menuSettings\": {\n" + " CHTMLpreview: false\n" + " },\n" + " \"CHTML-preview\":{\n" + " disabled: true\n" + " },\n" + " \"HTML-CSS\": {\n" + " scale: 90,\n" + " availableFonts: [],\n" + " preferredFont: null,\n" + " preferredFonts: null,\n" + " webFont:\"Gyre-Pagella\",\n" + " imageFont:'TeX',\n" + " undefinedFamily:\"'Arial Unicode MS',serif\",\n" + " linebreaks: { automatic: false }\n" + " },\n" + " \"TeX\": {\n" + " extensions: ['noErrors.js'],\n" + " noErrors: {\n" + " inlineDelimiters: [\"\",\"\"],\n" + " multiLine: true,\n" + " style: {\n" + " 'font-size': '90%',\n" + " 'text-align': 'left',\n" + " 'color': 'black',\n" + " 'padding': '1px 3px',\n" + " 'border': '1px solid'\n" + " }\n" + " }\n" + " }\n" + "});\n" + "MathJax.Hub.Register.StartupHook('End', function() {\n" + " refreshMathjaxWidths();\n" + " mathjaxReady = true;\n" + "});\n" + "MathJax.Hub.Startup.signal.Interest(function (message) {\n" + " if (message == 'End') {\n" + " var hypoLink = document.getElementById('hypothesis_frame');\n" + " if (null !== hypoLink) {\n" + " hypoLink.setAttribute('src', 'https://commenting.mdpi.com/embed.js');\n" + " }\n" + " }\n" + "});"; head.appendChild(script); script = document.createElement("script"); script.type = "text/javascript"; script.src = "https://pub.mdpi-res.com/bundles/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"; head.appendChild(script); // article version checker if (0 === search.indexOf('?type=check_update&version=')) { $.ajax({ url: "/2218-1997/10/9/357" + "/versioncheck" + search, success: function(result) { $(".js-check-update-container").html(result); } }); } $('#feed_option').click(function() { // tracker if ($('#scifeed_clicked').length<1) { $(this).append('<span style="display:none" id="scifeed_clicked">done</span>'); } $('#feed_data').toggle('slide', { direction: 'up'}, '1000'); // slideToggle(700); OR toggle(700) $("#scifeed_error_msg").html('').hide(); $("#scifeed_notice_msg").html('').hide(); }); $('#feed_option').click(function(event) { setTimeout(function(){ var captchaSection = $("#captchaSection"); captchaSection.removeClass('ui-helper-hidden').find('input').prop('disabled', false); // var img = captchaSection.find('img'); // img.attr('src', img.data('url') + "?" + (new Date()).getTime()); // $(".captcha_reload").trigger("click"); var img = document.getElementById('gregwar_captcha_scifeed'); img.src = '/generate-captcha/gcb_captcha?n=' + (new Date()).getTime(); },800); }); $(document).on('click', '.split_feeds', function() { var name = $( this ).attr('name'); var flag = 1 - ($(this).is(":checked")*1); $('.split_feeds').each(function (index) { if ($( this ).attr('name') !== name) { $(this)[0].checked = flag; } }); }); $(document).on('click', '#scifeed_submit, #scifeed_submit1', function(event) { event.preventDefault(); $(".captcha_reload").trigger("click"); $("#scifeed_error_msg").html(""); $("#scifeed_error_msg").hide(); }); $(document).on('click', '.subscription_toggle', function(event) { if ($(this).val() === 'Create SciFeed' && $('#scifeed_hidden_flag').length>0) { event.preventDefault(); // alert('Here there would be a captcha because user is not logged in'); var captchaSection = $("#captchaSection"); if (captchaSection.hasClass('ui-helper-hidden')) { captchaSection.removeClass('ui-helper-hidden').find('input').prop('disabled', false); var img = captchaSection.find('img'); img.attr('src', img.data('url') + "?" + (new Date()).getTime()); $("#reloadCaptcha").trigger("click"); } } }); $(document).on('click', '.scifeed_msg', function(){ $(this).hide(); }); $(document).on('click', '.article-scilit-search', function(e) { e.preventDefault(); var data = $(".article-scilit-search-data").val(); var dataArray = data.split(';').map(function(keyword) { return "(\"" + keyword.trim() + "\")"; }); var searchQuery = dataArray.join(" OR "); var searchUrl = encodeURI("https://www.scilit.net/articles/search?q="+ searchQuery + "&advanced=1&highlight=1"); var win = window.open(searchUrl, '_blank'); if (win) { win.focus(); } else { window.location(searchUrl); } }); display_stats(); citedCount(); follow_goto(); // Select the node that will be observed for mutations const targetNodes = document.getElementsByClassName('hypothesis-count-container'); // Options for the observer (which mutations to observe) const config = { attributes: false, childList: true, subtree: false }; // Callback function to execute when mutations are observed const callback = function(mutationList, observer) { for(const mutation of mutationList) { if (mutation.type === 'childList') { let node = $(mutation.target); if (parseInt(node.html()) > 0) { node.show(); } } } }; // Create an observer instance linked to the callback function const observer = new MutationObserver(callback); // Start observing the target node for configured mutations for(const targetNode of targetNodes) { observer.observe(targetNode, config); } // Select the node that will be observed for mutations const mathjaxTargetNode = document.getElementById('middle-column'); // Callback function to execute when mutations are observed const mathjaxCallback = function(mutationList, observer) { if (mathjaxReady && typeof(MathJax) !== 'undefined') { refreshMathjaxWidths(); } }; // Create an observer instance linked to the callback function const mathjaxObserver = new ResizeObserver(mathjaxCallback); // Start observing the target node for configured mutations mathjaxObserver.observe(mathjaxTargetNode); }); /* END $(document).ready */ function refreshMathjaxWidths() { let width = ($('.html-body').width()*0.9) + "px"; $('.MathJax_Display').css('max-width', width); $('.MJXc-display').css('max-width', width); } function sendScifeedFrom(form) { if (!$('#scifeed_email').val().trim()) { // empty email alert('Please, provide an email for subscribe to this scifeed'); return false; } else if (!$('#captchaSection').hasClass('ui-helper-hidden') && !$('#captchaSection').find('input').val().trim()) { // empty captcha alert('Please, fill the captcha field.'); return false; } else if( ((($('#scifeed_form').find('input:checkbox:checked').length)-($('#split_feeds:checked').length))<1) || ($('#scifeed_kwd_txt').length < 0 && !$('#scifeed_kwd_txt').val().trim()) || ($('#scifeed_author_txt').length<0 &&!$('#scifeed_author_txt').val().trim()) ) { alert('You did not select anything to subscribe'); return false; } else if(($('#scifeed_form').find('input:checkbox:checked').length)-($('#split_feeds2:checked').length)<1){ alert("You did not select anything to subscribe"); return false; } else { var url = $('#scifeed_subscribe_url').html(); var formData = $(form).serializeArray(); $.post(url, formData).done(function (data) { if (JSON.parse(data)) { $('.scifeed_msg').hide(); var res = JSON.parse(data); var successFeeds = 0; var errorFeeds = 0; if (res) { $('.scifeed_msg').html(''); $.each(res, function (index, val) { if (val) { if (val.error) { errorFeeds++; $("#scifeed_error_msg").append(index+' - '+val.error+'<br>'); } if (val.notice) // for successful feed creation { successFeeds++; // $("#scifeed_notice_msg").append(index+' - '+val.notice+'<br>'); $("#scifeed_notice_msg").append('<li>'+index+'</li>'); } } }); if (successFeeds>0) { text = $('#scifeed_notice_msg').html(); text = 'The following feed'+(successFeeds>1?'s have':' has')+ ' been sucessfully created:<br><ul>'+ text + '</ul>' +($('#scifeed_hidden_flag').length>0 ? 'You are not logged in, so you probably need to validate '+ (successFeeds>1?'them':' it')+'.<br>' :'' ) +'Please check your email'+(successFeeds>1?'s':'')+' for more details.'; //(successFeeds>1?' for each of them':'')+'.<br>'; $("#scifeed_notice_msg").html(text); $("#scifeed_notice_msg").show(); } if (errorFeeds>0) { $("#scifeed_error_msg").show();; } } $("#feed_data").hide(); } }); } } function follow_goto() { var hashStr = location.hash.replace("#",""); if(typeof hashStr !== 'undefined') { if( hashStr == 'supplementary') { document.getElementById('suppl_id').scrollIntoView(); } if( hashStr == 'citedby') { document.getElementById('cited_id').scrollIntoView(); } } } function cited() { $("#framed_div").toggle('fast', function(){ if ($(this).css('display') != 'none') { var loaded = document.getElementById("loaded"); if(loaded.innerHTML == "No") { // Load Xref result var container = document.getElementById("framed_div"); // This replace the content container.innerHTML = "<img src=\"https://pub.mdpi-res.com/img/loading_circle.gif?9a82694213036313?1732286508\" height=\"20\" width=\"20\" alt=\"Processing...\" style=\"vertical-align:middle; margin-right:0.6em;\">"; var url = "/citedby/10.3390%252Funiverse10090357/133"; $.post(url, function(result) { if (result.success) { container.innerHTML = result.view; } loaded.innerHTML = "Yes"; }); } } return true; // for not going at the beginning of the page... }) return true; // for not going at the beginning of the page... } function detect_device() { // Added by Bastien (18/08/2014): based on the http://detectmobilebrowsers.com/ detector var check = false; (function(a){if(/(android|bb\d+|meego).+mobile|avantgo|bada\/|blackberry|blazer|compal|elaine|fennec|hiptop|iemobile|ip(hone|od)|iris|kindle|lge |maemo|midp|mmp|mobile.+firefox|netfront|opera m(ob|in)i|palm( os)?|phone|p(ixi|re)\/|plucker|pocket|psp|series(4|6)0|symbian|treo|up\.(browser|link)|vodafone|wap|windows (ce|phone)|xda|xiino/i.test(a)||/1207|6310|6590|3gso|4thp|50[1-6]i|770s|802s|a wa|abac|ac(er|oo|s\-)|ai(ko|rn)|al(av|ca|co)|amoi|an(ex|ny|yw)|aptu|ar(ch|go)|as(te|us)|attw|au(di|\-m|r |s )|avan|be(ck|ll|nq)|bi(lb|rd)|bl(ac|az)|br(e|v)w|bumb|bw\-(n|u)|c55\/|capi|ccwa|cdm\-|cell|chtm|cldc|cmd\-|co(mp|nd)|craw|da(it|ll|ng)|dbte|dc\-s|devi|dica|dmob|do(c|p)o|ds(12|\-d)|el(49|ai)|em(l2|ul)|er(ic|k0)|esl8|ez([4-7]0|os|wa|ze)|fetc|fly(\-|_)|g1 u|g560|gene|gf\-5|g\-mo|go(\.w|od)|gr(ad|un)|haie|hcit|hd\-(m|p|t)|hei\-|hi(pt|ta)|hp( i|ip)|hs\-c|ht(c(\-| |_|a|g|p|s|t)|tp)|hu(aw|tc)|i\-(20|go|ma)|i230|iac( |\-|\/)|ibro|idea|ig01|ikom|im1k|inno|ipaq|iris|ja(t|v)a|jbro|jemu|jigs|kddi|keji|kgt( |\/)|klon|kpt |kwc\-|kyo(c|k)|le(no|xi)|lg( g|\/(k|l|u)|50|54|\-[a-w])|libw|lynx|m1\-w|m3ga|m50\/|ma(te|ui|xo)|mc(01|21|ca)|m\-cr|me(rc|ri)|mi(o8|oa|ts)|mmef|mo(01|02|bi|de|do|t(\-| |o|v)|zz)|mt(50|p1|v )|mwbp|mywa|n10[0-2]|n20[2-3]|n30(0|2)|n50(0|2|5)|n7(0(0|1)|10)|ne((c|m)\-|on|tf|wf|wg|wt)|nok(6|i)|nzph|o2im|op(ti|wv)|oran|owg1|p800|pan(a|d|t)|pdxg|pg(13|\-([1-8]|c))|phil|pire|pl(ay|uc)|pn\-2|po(ck|rt|se)|prox|psio|pt\-g|qa\-a|qc(07|12|21|32|60|\-[2-7]|i\-)|qtek|r380|r600|raks|rim9|ro(ve|zo)|s55\/|sa(ge|ma|mm|ms|ny|va)|sc(01|h\-|oo|p\-)|sdk\/|se(c(\-|0|1)|47|mc|nd|ri)|sgh\-|shar|sie(\-|m)|sk\-0|sl(45|id)|sm(al|ar|b3|it|t5)|so(ft|ny)|sp(01|h\-|v\-|v )|sy(01|mb)|t2(18|50)|t6(00|10|18)|ta(gt|lk)|tcl\-|tdg\-|tel(i|m)|tim\-|t\-mo|to(pl|sh)|ts(70|m\-|m3|m5)|tx\-9|up(\.b|g1|si)|utst|v400|v750|veri|vi(rg|te)|vk(40|5[0-3]|\-v)|vm40|voda|vulc|vx(52|53|60|61|70|80|81|83|85|98)|w3c(\-| )|webc|whit|wi(g |nc|nw)|wmlb|wonu|x700|yas\-|your|zeto|zte\-/i.test(a.substr(0,4)))check = true})(navigator.userAgent||navigator.vendor||window.opera); return check; } function display_stats(){ $("#article_stats_div").toggle(); return false; } /* * Cited By Scopus */ function citedCount(){ $("#framed_div_cited_count").toggle('fast', function(){ if ($(this).css('display') != 'none') { var loaded = document.getElementById("loaded_cite_count"); // to load only once the result! if(loaded.innerHTML == "No") { // Load Xref result var d = document.getElementById("framed_div_cited_count"); // This replace the content d.innerHTML = "<img src=\"https://pub.mdpi-res.com/img/loading_circle.gif?9a82694213036313?1732286508\" height=\"20\" width=\"20\" alt=\"Processing...\" style=\"vertical-align:middle; margin-right:0.6em;\">"; $.ajax({ method : "POST", url : "/cite-count/10.3390%252Funiverse10090357", success : function(data) { if (data.succ) { d.innerHTML = data.view; loaded.innerHTML = "Yes"; follow_goto(); } } }); } } // end else return true; // for not going at the beginning of the page... }) return true; // for not going at the beginning of the page... } </script><script type="text/javascript" src="https://pub.mdpi-res.com/assets/js/third-party/highcharts/highcharts.js?bdd06f45e34c33df?1732286508"></script><script type="text/javascript" src="https://pub.mdpi-res.com/assets/js/third-party/highcharts/modules/exporting.js?944dc938d06de3a8?1732286508"></script><script type="text/javascript" defer="defer"> var advancedStatsData; var selectedStatsType = "abstract"; $(function(){ var countWrapper = $('#counts-wrapper'); $('#author_stats_id #type_links a').on('click', function(e) { e.preventDefault(); selectedStatsType = $(this).data('type'); $('#article_advanced_stats').vectorMap('set', 'values', advancedStatsData[selectedStatsType]); $('#advanced_stats_max').html(advancedStatsData[selectedStatsType].max); $('#type_links a').removeClass('active'); $(this).addClass('active'); }); $.get('/2218-1997/10/9/357/stats', function (result) { if (!result.success) { return; } // process article metrics part in left column var viewNumber = countWrapper.find(".view-number"); viewNumber.html(result.metrics.views); viewNumber.parent().toggleClass("count-div--grey", result.metrics.views == 0); var downloadNumber = countWrapper.find(".download-number"); downloadNumber.html(result.metrics.downloads); downloadNumber.parent().toggleClass("count-div--grey", result.metrics.downloads == 0); var citationsNumber = countWrapper.find(".citations-number"); citationsNumber.html(result.metrics.citations); citationsNumber.parent().toggleClass("count-div--grey", result.metrics.citations == 0); if (result.metrics.views > 0 || result.metrics.downloads > 0 || result.metrics.citations > 0) { countWrapper.find("#js-counts-wrapper__views, #js-counts-wrapper__downloads").addClass("visible").show(); if (result.metrics.citations > 0) { countWrapper.find('.citations-number').html(result.metrics.citations).show(); countWrapper.find("#js-counts-wrapper__citations").addClass("visible").show(); } else { countWrapper.find("#js-counts-wrapper__citations").remove(); } $("[data-id='article-counters']").removeClass("hidden"); } if (result.metrics.altmetrics_score > 0) { $("#js-altmetrics-donut").show(); } // process view chart in main column var jsondata = result.chart; var series = new Array(); $.each(jsondata.elements, function(i, element) { var dataValues = new Array(); $.each(element.values, function(i, value) { dataValues.push(new Array(value.tip, value.value)); }); series[i] = {name: element.text, data:dataValues}; }); Highcharts.setOptions({ chart: { style: { fontFamily: 'Arial,sans-serif' } } }); $('#article_stats_swf').highcharts({ chart: { type: 'line', width: $("#tabs").width() //* 0.91 }, credits: { enabled: false }, exporting: { enabled: true }, title: { text: jsondata.title.text, x: -20 //center }, xAxis: { categories: jsondata.x_axis.labels.labels, offset: jsondata.x_axis.offset, labels:{ step: jsondata.x_axis.labels.steps, rotation: 30 } }, yAxis: { max: jsondata.y_axis.max, min: jsondata.y_axis.min, offset: jsondata.y_axis.offset, labels: { steps: jsondata.y_axis.steps }, title: { enabled: false } }, tooltip: { formatter: function (){ return this.key.replace("#val#", this.y); } }, legend: { align: 'top', itemDistance: 50 }, series: series }); }); $('#supplement_link').click(function() { document.getElementById('suppl_id').scrollIntoView(); }); $('#stats_link').click(function() { document.getElementById('stats_id').scrollIntoView(); }); // open mol viewer for molbank special supplementary files $('.showJmol').click(function(e) { e.preventDefault(); var jmolModal = $("#jmolModal"); var url = "/article/1472235/jsmol_viewer/__supplementary_id__"; url = url.replace(/__supplementary_id__/g, $(this).data('index')); $('#jsmol-content').attr('src', url); jmolModal.find(".content").html($(this).data('description')); jmolModal.foundation("reveal", "open"); }); }); !function() { "use strict"; function e(e) { try { if ("undefined" == typeof console) return; "error"in console ? console.error(e) : console.log(e) } catch (e) {} } function t(e) { return d.innerHTML = '<a href="' + e.replace(/"/g, """) + '"></a>', d.childNodes[0].getAttribute("href") || "" } function n(n, c) { var o = ""; var k = parseInt(n.substr(c + 4, 2), 16); for (var i = c; i < n.length; i += 2) { if (i != c + 4) { var s = parseInt(n.substr(i, 2), 16) ^ k; o += String.fromCharCode(s); } } try { o = decodeURIComponent(escape(o)); } catch (error) { console.error(error); } return t(o); } function c(t) { for (var r = t.querySelectorAll("a"), c = 0; c < r.length; c++) try { var o = r[c] , a = o.href.indexOf(l); a > -1 && (o.href = "mailto:" + n(o.href, a + l.length)) } catch (i) { e(i) } } function o(t) { for (var r = t.querySelectorAll(u), c = 0; c < r.length; c++) try { var o = r[c] , a = o.parentNode , i = o.getAttribute(f); if (i) { var l = n(i, 0) , d = document.createTextNode(l); a.replaceChild(d, o) } } catch (h) { e(h) } } function a(t) { for (var r = t.querySelectorAll("template"), n = 0; n < r.length; n++) try { i(r[n].content) } catch (c) { e(c) } } function i(t) { try { c(t), o(t), a(t) } catch (r) { e(r) } } var l = "/cnd-cgi/l/email-protection#" , u = ".__cf_email__" , f = "data-cfemail" , d = document.createElement("div"); i(document), function() { var e = document.currentScript || document.scripts[document.scripts.length - 1]; e.parentNode.removeChild(e) }() }(); </script><script type="text/javascript"> function setCookie(cname, cvalue, ctime) { ctime = (typeof ctime === 'undefined') ? 10*365*24*60*60*1000 : ctime; // default => 10 years var d = new Date(); d.setTime(d.getTime() + ctime); // ==> 1 hour = 60*60*1000 var expires = "expires="+d.toUTCString(); document.cookie = cname + "=" + cvalue + "; " + expires +"; path=/"; } function getCookie(cname) { var name = cname + "="; var ca = document.cookie.split(';'); for(var i=0; i<ca.length; i++) { var c = ca[i]; while (c.charAt(0)==' ') c = c.substring(1); if (c.indexOf(name) == 0) return c.substring(name.length, c.length); } return ""; } </script><script type="text/javascript" src="https://d1bxh8uas1mnw7.cloudfront.net/assets/embed.js"></script><script> $(document).ready(function() { if ($("#js-similarity-related-data").length > 0) { $.ajax({ url: '/article/1472235/similarity-related', success: function(response) { $("#js-similarity-related-data").html(response); $("#js-related-articles-menu").show(); $(document).foundation('tab', 'reflow'); MathJax.Hub.Queue(["Typeset", MathJax.Hub]); } }); } }); </script><link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/jquery-ui-1.10.4.custom.min.css?80647d88647bf347?1732286508"><link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/magnific-popup.min.css?04d343e036f8eecd?1732286508"><script type="text/javascript" src="https://pub.mdpi-res.com/assets/js/magnific-popup.min.js?2be3d9e7dc569146?1732286508"></script><script> $(function() { $(".js-show-more-academic-editors").on("click", function(e) { e.preventDefault(); $(this).hide(); $(".academic-editor-container").removeClass("hidden"); }); }); </script> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/vmap/jqvmap.min.css?126a06688aa11c13?1732286508"> <script src="https://pub.mdpi-res.com/assets/js/vmap/jquery.vmap.min.js?935f68d33bdd88a1?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/vmap/jquery.vmap.world.js?16677403c0e1bef1?1732286508"></script> <script> function updateSlick() { $('.multiple-items').slick('setPosition'); } $(document).ready(function() { $('.multiple-items').slick({ slidesToShow: 1, nextArrow: '<a class="slick-next" href="#"><i class="material-icons">chevron_right</i></a>', prevArrow: '<a class="slick-prev" href="#"><i class="material-icons">chevron_left</i></a>', slidesToScroll: 1, responsive: [ { breakpoint: 1024, settings: { slidesToShow: 1, slidesToScroll: 1, } }, { breakpoint: 600, settings: { slidesToShow: 1, slidesToScroll: 1, } }, { breakpoint: 480, settings: { slidesToShow: 1, slidesToScroll: 1, } } ] }); $('.multiple-items').show(); $(document).on('click', '.reviewReportSelector', function(e) { let path = $(this).attr('data-path'); handleReviews(path, $(this)); }); $(document).on('click', '.viewReviewReports', function(e) { let versionOne = $('#versionTab_1'); if (!versionOne.hasClass('activeTab')) { let path = $(this).attr('data-path'); handleReviews(path, versionOne); } location.href = "#reviewReports"; }); $(document).on('click', '.reviewersResponse, .authorResponse', function(e) { let version = $(this).attr('data-version'); let targetVersion = $('#versionTab_' + version); if (!targetVersion.hasClass('activeTab')) { let path = targetVersion.attr('data-path'); handleReviews(path, targetVersion); } location.href = $(this).attr('data-link'); }); $(document).on('click', '.tab', function (e) { e.preventDefault(); $('.tab').removeClass('activeTab'); $(this).addClass('activeTab') $('.tab').each(function() { $(this).closest('.tab-title').removeClass('active'); }); $(this).closest('.tab-title').addClass('active') }); }); function handleReviews(path, target) { $.ajax({ url: path, context: this, success: function (data) { $('.activeTab').removeClass('activeTab'); target.addClass('activeTab'); $('#reviewSection').html(data.view); }, error: function (xhr, ajaxOptions, thrownError) { console.log(xhr.status); console.log(thrownError); } }); } </script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/affix.js?v1?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/storage.js?e9b262d3a3476d25?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/jquery-scrollspy.js?09cbaec0dbb35a67?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/magnific-popup.js?4a09c18460afb26c?1732286508"></script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/underscore.js?f893e294cde60c24?1732286508"></script> <script type="text/javascript"> $('document').ready(function(){ $("#left-column").addClass("show-for-large-up"); $("#middle-column").removeClass("medium-9").removeClass("left-bordered").addClass("medium-12"); $(window).on('resize scroll', function() { /* if ($('.button--drop-down').isInViewport($(".top-bar").outerHeight())) { */ if ($('.button--drop-down').isInViewport()) { $("#js-button-download").hide(); } else { $("#js-button-download").show(); } }); }); $(document).on('DOMNodeInserted', function(e) { var element = $(e.target); if (element.hasClass('menu') && element.hasClass('html-nav') ) { element.addClass("side-menu-ul"); } }); </script> <script src="https://pub.mdpi-res.com/assets/js/xmltohtml/articles.js?5118449d9ad8913a?1732286508"></script> <script> repositionOpenSideBar = function() { $('#left-column').addClass("show-for-large-up show-for-medium-up").show(); $('#middle-column').removeClass('large-12').removeClass('medium-12'); $('#middle-column').addClass('large-9'); } repositionCloseSideBar = function() { $('#left-column').removeClass("show-for-large-up show-for-medium-up").hide(); $('#middle-column').removeClass('large-9'); $('#middle-column').addClass('large-12').addClass('medium-12'); } </script>   <script type="text/plain" data-cookieconsent="marketing"> !function(e,t,n,s,u,a){e.twq||(s=e.twq=function(){s.exe?s.exe.apply(s,arguments):s.queue.push(arguments); },s.version='1.1',s.queue=[],u=t.createElement(n),u.async=!0,u.src='//static.ads-twitter.com/uwt.js', a=t.getElementsByTagName(n)[0],a.parentNode.insertBefore(u,a))}(window,document,'script'); // Insert Twitter Pixel ID and Standard Event data below twq('init','o2pip'); twq('track','PageView'); </script>  <script>(function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'8e71e0e92c18ce63',t:'MTczMjM3MjQ1MS4wMDAwMDA='};var a=document.createElement('script');a.nonce='';a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();</script></body> </html>

CINXE.COM

Exploring New Physics with Deep Underground Neutrino Experiment High-Energy Flux: The Case of Lorentz Invariance Violation, Large Extra Dimensions and Long-Range Forces