A critical appraisal and evaluation of modern PDFs | The European Physical Journal C

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>A critical appraisal and evaluation of modern PDFs | The European Physical Journal C </title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="A critical appraisal and evaluation of modern PDFs"/> <meta name="twitter:description" content="The European Physical Journal C - We review the present status of the determination of parton distribution functions (PDFs) in the light of the precision requirements for the LHC in Run 2 and other..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig1_HTML.gif"/> <meta name="journal_id" content="10052"/> <meta name="dc.title" content="A critical appraisal and evaluation of modern PDFs"/> <meta name="dc.source" content="The European Physical Journal C 2016 76:8"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2016-08-23"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2016 The Author(s)"/> <meta name="dc.rights" content="2016 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="We review the present status of the determination of parton distribution functions (PDFs) in the light of the precision requirements for the LHC in Run 2 and other future hadron colliders. We provide brief reviews of all currently available PDF sets and use them to compute cross sections for a number of benchmark processes, including Higgs boson production in gluon–gluon fusion at the LHC. We show that the differences in the predictions obtained with the various PDFs are due to particular theory assumptions made in the fits of those PDFs. We discuss PDF uncertainties in the kinematic region covered by the LHC and on averaging procedures for PDFs, such as advocated by the PDF4LHC15 sets, and provide recommendations for the usage of PDF sets for theory predictions at the LHC."/> <meta name="prism.issn" content="1434-6052"/> <meta name="prism.publicationName" content="The European Physical Journal C"/> <meta name="prism.publicationDate" content="2016-08-23"/> <meta name="prism.volume" content="76"/> <meta name="prism.number" content="8"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="46"/> <meta name="prism.copyright" content="2016 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1140/epjc/s10052-016-4285-4"/> <meta name="prism.doi" content="doi:10.1140/epjc/s10052-016-4285-4"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1140/epjc/s10052-016-4285-4.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1140/epjc/s10052-016-4285-4"/> <meta name="citation_journal_title" content="The European Physical Journal C"/> <meta name="citation_journal_abbrev" content="Eur. Phys. J. C"/> <meta name="citation_publisher" content="Springer Berlin Heidelberg"/> <meta name="citation_issn" content="1434-6052"/> <meta name="citation_title" content="A critical appraisal and evaluation of modern PDFs"/> <meta name="citation_volume" content="76"/> <meta name="citation_issue" content="8"/> <meta name="citation_publication_date" content="2016/08"/> <meta name="citation_online_date" content="2016/08/23"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="46"/> <meta name="citation_article_type" content="Regular Article - Experimental Physics"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1140/epjc/s10052-016-4285-4"/> <meta name="DOI" content="10.1140/epjc/s10052-016-4285-4"/> <meta name="size" content="2073460"/> <meta name="citation_doi" content="10.1140/epjc/s10052-016-4285-4"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1140/epjc/s10052-016-4285-4&api_key="/> <meta name="description" content="We review the present status of the determination of parton distribution functions (PDFs) in the light of the precision requirements for the LHC in Run 2 a"/> <meta name="dc.creator" content="Accardi, A."/> <meta name="dc.creator" content="Alekhin, S."/> <meta name="dc.creator" content="Blümlein, J."/> <meta name="dc.creator" content="Garzelli, M. 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We provide brief reviews of all currently available PDF sets and use them to compute cross sections for a number of benchmark processes, including Higgs boson production in gluon–gluon fusion at the LHC. We show that the differences in the predictions obtained with the various PDFs are due to particular theory assumptions made in the fits of those PDFs. 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critical appraisal and evaluation of modern PDFs </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1140/epjc/s10052-016-4285-4.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-A_-Accardi-Aff1-Aff2" data-author-popup="auth-A_-Accardi-Aff1-Aff2" data-author-search="Accardi, A.">A. Accardi</a><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-S_-Alekhin-Aff3-Aff4" data-author-popup="auth-S_-Alekhin-Aff3-Aff4" data-author-search="Alekhin, S.">S. Alekhin</a><sup class="u-js-hide"><a href="#Aff3">3</a>,<a href="#Aff4">4</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-J_-Bl_mlein-Aff5" data-author-popup="auth-J_-Bl_mlein-Aff5" data-author-search="Blümlein, J.">J. Blümlein</a><sup class="u-js-hide"><a href="#Aff5">5</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-M__V_-Garzelli-Aff3" data-author-popup="auth-M__V_-Garzelli-Aff3" data-author-search="Garzelli, M. V.">M. V. Garzelli</a><sup class="u-js-hide"><a href="#Aff3">3</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-K_-Lipka-Aff6" data-author-popup="auth-K_-Lipka-Aff6" data-author-search="Lipka, K.">K. Lipka</a><sup class="u-js-hide"><a href="#Aff6">6</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-W_-Melnitchouk-Aff2" data-author-popup="auth-W_-Melnitchouk-Aff2" data-author-search="Melnitchouk, W.">W. Melnitchouk</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-S_-Moch-Aff3" data-author-popup="auth-S_-Moch-Aff3" data-author-search="Moch, S." data-corresp-id="c1">S. Moch<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff3">3</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-J__F_-Owens-Aff7" data-author-popup="auth-J__F_-Owens-Aff7" data-author-search="Owens, J. F.">J. F. Owens</a><sup class="u-js-hide"><a href="#Aff7">7</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-R_-Pla_akyt_-Aff6" data-author-popup="auth-R_-Pla_akyt_-Aff6" data-author-search="Plačakytė, R.">R. Plačakytė</a><sup class="u-js-hide"><a href="#Aff6">6</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-E_-Reya-Aff8" data-author-popup="auth-E_-Reya-Aff8" data-author-search="Reya, E.">E. Reya</a><sup class="u-js-hide"><a href="#Aff8">8</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-N_-Sato-Aff2" data-author-popup="auth-N_-Sato-Aff2" data-author-search="Sato, N.">N. Sato</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-A_-Vogt-Aff9" data-author-popup="auth-A_-Vogt-Aff9" data-author-search="Vogt, A.">A. Vogt</a><sup class="u-js-hide"><a href="#Aff9">9</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 13 authors for this article" title="Show all 13 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-O_-Zenaiev-Aff6" data-author-popup="auth-O_-Zenaiev-Aff6" data-author-search="Zenaiev, O.">O. Zenaiev</a><sup class="u-js-hide"><a href="#Aff6">6</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>5787 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-altmetric-medium"></use> </svg>1 <span class="app-article-metrics-bar__label">Altmetric</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1140/epjc/s10052-016-4285-4/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> <div class="u-mb-8 c-status-message c-status-message--boxed c-status-message--info"><span class="c-status-message__icon"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-info-filled-medium"></use></svg></span><p class="u-mt-0">A <a href="https://arxiv.org/abs/1603.08906" class="relation-link" data-track="click" data-track-label="link" data-track-action="preprint">preprint version</a> of the article is available at arXiv.</p></div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>We review the present status of the determination of parton distribution functions (PDFs) in the light of the precision requirements for the LHC in Run 2 and other future hadron colliders. We provide brief reviews of all currently available PDF sets and use them to compute cross sections for a number of benchmark processes, including Higgs boson production in gluon–gluon fusion at the LHC. We show that the differences in the predictions obtained with the various PDFs are due to particular theory assumptions made in the fits of those PDFs. We discuss PDF uncertainties in the kinematic region covered by the LHC and on averaging procedures for PDFs, such as advocated by the PDF4LHC15 sets, and provide recommendations for the usage of PDF sets for theory predictions at the LHC.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-021-09602-x/MediaObjects/10052_2021_9602_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1140/epjc/s10052-021-09602-x?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1140/epjc/s10052-021-09602-x">Correlation of theoretical uncertainties in PDF fits and theoretical uncertainties in predictions </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">18 September 2021</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1140%2Fepjp%2Fs13360-024-05865-x/MediaObjects/13360_2024_5865_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1140/epjp/s13360-024-05865-x?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1140/epjp/s13360-024-05865-x">New results in the CTEQ-TEA global analysis of parton distributions in the nucleon </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">04 December 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2FJHEP04%282015%29040/MediaObjects/13130_2015_1078_Figa_HTML.gif" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/JHEP04(2015)040?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/JHEP04(2015)040">Parton distributions for the LHC run II </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">08 April 2015</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1740221327, embedded_user: 'null' } }); </script> <section aria-labelledby="content-related-subjects" data-test="subject-content"> <h3 id="content-related-subjects" class="c-article__sub-heading">Explore related subjects</h3> <span class="u-sans-serif u-text-s u-display-block u-mb-24">Discover the latest articles, news and stories from top researchers in related subjects.</span> <ul class="c-article-subject-list" role="list"> <li class="c-article-subject-list__subject"> <a href="/subject/experimental-particle-physics" data-track="select_related_subject_1" data-track-context="related subjects from content page" data-track-label="Experimental Particle Physics">Experimental Particle Physics</a> </li> </ul> </section> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=10052" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>In Run 2 of the Large Hadron Collider (LHC), the very details of the Standard Model (SM), including cross sections of different processes and Higgs bosons properties, are being measured with very high precision. At the same time, the new data at the highest center-of-mass collision energies ever achieved (<span class="mathjax-tex">$\sqrt{s}=13$</span> TeV) are used to search for physics phenomena beyond the SM (BSM). The experimental data used to perform those measurements are generally expected to have percent-level accuracy, depending on details such as the final states and the acceptance and efficiency of the detectors in particular kinematics ranges.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Table 1 Summary of major hard processes used in the various PDF analyses and the confidence level criteria employed. Detailed references to the different specific data sets used by the various groups are given in Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e701">1</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e704">7</a>] and also the specific statistical analysis applied is described in these papers. Note that different analyses use partly different data sets for some processes</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/1" aria-label="Full size table 1"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>To further test the SM and to identify signals for new physics, measurements need to be compared to precise theoretical predictions, which need to incorporate higher order radiative corrections in quantum chromodynamics (QCD) and, possibly, the electroweak sector of the SM. In order to reach the benchmark precision set by the accuracy of the experimental data, next-to-next-to-leading order (NNLO) corrections in QCD are often required. At next-to-leading order (NLO) in QCD, the residual theoretical uncertainty from truncating the perturbative expansion commonly estimated by variations of the renormalization and factorization scales <span class="mathjax-tex">$\mu _r$</span> and <span class="mathjax-tex">$\mu _f$</span> are often too large compared to the experimental accuracy. Nonetheless, for observables with complex final states, and indeed for many BSM signals, one must still contend with NLO calculations, which will continue to require corresponding NLO fits.</p><p>Parton distribution functions (PDFs) in the proton serve as an essential input for any cross section prediction at hadron colliders and have been measured with increasing precision over the last three decades. Likewise, the strong coupling constant <span class="mathjax-tex">$\alpha _s(M_Z)$</span> at the <i>Z</i> boson mass scale <span class="mathjax-tex">$M_Z$</span> and the masses <span class="mathjax-tex">$m_h$</span> of the heavy quarks <span class="mathjax-tex">$h=c, b, t$</span> are well constrained by existing data and their determination is accurate at least to NNLO. However, despite steady improvements in the accuracy of PDF determinations over the years, the uncertainties associated with PDFs, the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span>, and quark masses still dominate many calculations of cross sections for SM processes at the LHC. A particularly prominent example is the cross section for the production of a SM Higgs boson in the gluon–gluon fusion channel.</p><p>The currently available PDF sets are CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e1816">1</a>], accurate to NLO in QCD, as well as ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e1819">2</a>], CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e1822">3</a>], HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e1825">4</a>], JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e1828">5</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e1832">6</a>], and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e1835">7</a>] to NNLO in QCD. These provide a detailed description of the parton content of the proton, which depends on the chosen sets of experimental data as well as on the theory assumptions and the underlying physics models used in the analyses. Both theoretical and experimental inputs have direct impact on the obtained nonperturbative parameters, namely, the fitted PDFs, the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the quark masses. Moreover, they can lead to large systematic shifts compared to the uncertainties of the experimental data used in the fit. For precision predictions in Run 2 of the LHC it is therefore very important to quantify those effects in detailed validations of the individual PDF sets in order to reduce the uncertainties in those nonperturbative input parameters. Moreover, this will allow one to pinpoint problems with the determination of certain PDFs. Any approach to determine the parton luminosities at the LHC which implies mixing or averaging of various PDFs or of their respective uncertainties, such as that advocated in the recent PDF4LHC recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e1882">8</a>], is therefore potentially dangerous in the context of precision measurements, in particular, or when studying processes at kinematic edges such as at large values of Bjorken <i>x</i> or small scales <span class="mathjax-tex">$Q^2$</span>. The precision measurements of the LHC experiments themselves help to constrain the different sets of PDFs and may even indicate deviations from SM, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss, T.A. Morgan, The NNLO QCD corrections to Z boson production at large transverse momentum. 
 arXiv:1605.04295
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR9" id="ref-link-section-d90629110e1913">9</a>] for an example. It is thus of central importance that comparisons for all available PDF sets are performed in a quantitative manner and with the best available accuracy.</p><p>In this paper we briefly discuss the available world data used to constrain PDFs in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec2">2</a> and stress the need to include only compatible data sets in any analysis. The data analysis relies on comparison with precise theoretical predictions, with many of these implemented in software tools. In this respect, we underline in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a> the importance of open-source code to provide benchmarks and to facilitate theory improvements through indication and reduction of possible errors. In addition, Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a> is devoted to a discussion of a number of crucial theory aspects in PDF fits. These include the treatment of heavy quarks and their masses, QCD corrections for <span class="mathjax-tex">$W^\pm $</span>- and <i>Z</i>-boson production applied in the fit of light-flavor PDFs, and the importance of nuclear corrections in scattering data off nuclei. The strong coupling constant is correlated with the PDFs and is therefore an important parameter to be determined simultaneously with the PDFs. The state of the art is reviewed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec19">4</a>. The need to address PDF uncertainties for cross section predictions is illustrated in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec20">5</a>, with the Higgs boson cross section in the gluon–gluon fusion channel being the most prominent case. Other examples include the production of heavy quarks at the LHC in different kinematic regimes. Our observations illustrate important shortcomings of the recent PDF4LHC recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e1963">8</a>] which are addressed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec28">6</a>, where alternative recommendations for the usage of sets of PDFs for theory predictions at the LHC are provided. Finally, we conclude in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec31">7</a>.</p></div></div></section><section data-title="Data sets and results for PDF fits"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Data sets and results for PDF fits</h2><div class="c-article-section__content" id="Sec2-content"><p>We begin with an overview of the currently available data which can be used to determine PDFs and present the fit results of the various groups.</p><h3 class="c-article__sub-heading" id="Sec3"><span class="c-article-section__title-number">2.1 </span>Data sets used in PDF fits</h3><p>The data used in the various PDF fits overlap to a large extent, as indicated in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab1">1</a>. However, there are also substantial differences which are related to the accuracy required in the analysis, the feasibility of efficiently implementing the corresponding theoretical computations, or the subjective evaluation of the data quality, to name a few.</p><p>The core of all PDF fits comprises the deep-inelastic scattering (DIS) data obtained at the HERA electron–proton (<i>ep</i>) collider and in fixed-target experiments. While the former has used only a proton target, the latter have collected large amounts of data for the deuteron and heavier targets as well. The analysis of nuclear-target data requires an accurate account of nuclear effects. This is challenging already in the case of the loosely-bound deuteron (cf. Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>), and even more so for heavier targets. Therefore, in general, data sets for DIS on targets heavier than deuteron are not used. Nonetheless, different combinations of data sets for the neutrino-induced DIS off iron and lead targets obtained by the CCFR/NuTeV, CDHSW, and CHORUS experiments are included in the CT14, MMHT14, and NNPDF3.0 analyses, but are not used by other groups to avoid any influence of nuclear correction uncertainties. One can also point out the abnormal dependence of the DIS structure functions on the beam energy in the NuTeV experiment [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="H. Paukkunen, C.A. Salgado, Agreement of neutrino deep-inelastic scattering data with global fits of parton distributions. Phys. Rev. Lett. 110, 212301 (2013). 
 arXiv:1302.2001
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR11" id="ref-link-section-d90629110e1999">11</a>] and the poor agreement of the CDHSW data with the QCD predictions on the <span class="mathjax-tex">$Q^2$</span> slope of structure functions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="G. Altarelli, in QCD and experiment: status of 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 . eds. by: P.M. Zerwas, H.A. Kastrup, Workshop on QCD: 20 Years Later Aachen, Germany, June 9–13, pp. 172–204 (1992)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR12" id="ref-link-section-d90629110e2026">12</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="J.P. Berge et al., A measurement of differential cross sections and nucleon structure functions in charged current neutrino interactions on iron. Z. Phys. C 49, 187 (1991)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR14" id="ref-link-section-d90629110e2030">14</a>] as an additional motivation to exclude these data sets.</p><p>The kinematic cuts applied to the commonly used DIS data also differ in various analyses in order to minimize the influence of higher twist contributions. Another important feature of the DIS data analyses in PDF fits concerns the use of data for the DIS structure function <span class="mathjax-tex">$F_2$</span> instead of the data for the measured cross sections. These aspects will be discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>.</p><p>The inclusive DIS data are often supplemented by the semi-inclusive data on the neutral-current and charged-current DIS charm-quark production. The neutral-current sample collected by the HERA experiments provides a valuable tool to study the heavy-quark production mechanism. This is vital for pinning down PDFs, in particular the gluon PDF at small <i>x</i>, relevant for important phenomenological applications at the LHC (cf. Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec20">5</a>). The charged-current charm production data help to constrain the strange sea PDF, which is strongly mixed with contributions from non-strange PDFs in other observables (cf. Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>).</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-2"><figure><figcaption class="c-article-table__figcaption"><b id="Tab2" data-test="table-caption">Table 2 Compilation of precise data on <i>W</i>- and <i>Z</i>-boson production in <i>pp</i> and <span class="mathjax-tex">$p\bar{p}$</span> collisions and the <span class="mathjax-tex">$\chi ^2$</span> values per number of data points obtained for these data sets in different PDF analyses using their individual definitions of <span class="mathjax-tex">$\chi ^2$</span>. The NNLO fit results are quoted as a default, while the NLO values are given for the CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e2181">1</a>] and <span class="u-monospace">HERAFitter</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="HERAFitter developers’ Team Collaboration, S. Camarda et al., QCD analysis of 
 
 
 
 $$W$$
 
 
 W
 
 
 - and 
 
 
 
 $$Z$$
 
 
 Z
 
 
 -boson production at the Tevatron. 
 arXiv:1503.05221
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR31" id="ref-link-section-d90629110e2187">31</a>] PDFs. Missing table entries indicate that the respective data sets have not been used in the analysis. Other data sets of lower accuracy, which have become obsolete and data sets superseded are listed in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab3">3</a> </b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/2" aria-label="Full size table 2"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The Drell–Yan (DY) data are also a necessary ingredient of any PDF analysis since DIS data alone do not allow for a comprehensive disentangling of the quark and anti-quark distributions. Historically, for a long time only fixed-target DY data were available for PDF fits. In particular, this did not allow for a model-independent separation of the valence and sea quarks at small-<i>x</i>. The high precision DY data obtained in proton-proton (<i>pp</i>) and proton–anti-proton (<span class="mathjax-tex">$p\bar{p}$</span>) collisions from the LHC and the Tevatron open new possibilities to study the PDFs at small and large <i>x</i>. The LHC experiments are quickly accumulating statistics and are currently providing data samples at <span class="mathjax-tex">$\sqrt{s}=$</span> 7 and 8 TeV for <i>W</i>- and <i>Z</i>-boson production with typical luminosities of over 20 <span class="mathjax-tex">$\text {fb}^{-1}$</span> per experiment. The rapid progress in experimental measurements causes a greatly non-uniform coverage of the recent DY data in various PDF fits (cf. Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab2">2</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab3">3</a>) and leads to corresponding differences in the accuracy of the extracted PDFs. Another issue here is the theoretical accuracy achieved for the description of the DY data. This varies substantially and will be discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-3"><figure><figcaption class="c-article-table__figcaption"><b id="Tab3" data-test="table-caption">Table 3 Same as Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab2">2</a> for the Tevatron and LHC data sets of lower accuracy and those, which have become superseded but are still used in various PDF analyses</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/3" aria-label="Full size table 3"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>Often, jet production in <i>pp</i> and <span class="mathjax-tex">$p\bar{p}$</span> collisions is used as an additional process to constrain the large-<i>x</i> gluon PDF. Here, the QCD corrections are known to NLO and the calculation of the NNLO ones is in progress [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, J. Pires, Second order QCD corrections to jet production at hadron colliders: the all-gluon contribution. Phys. Rev. Lett. 110, 162003 (2013). 
 arXiv:1301.7310
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR15" id="ref-link-section-d90629110e7876">15</a>]. The incomplete knowledge of the latter is problematic in view of a consistent PDF analysis at NNLO when including those jet data. This will be discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec19">4</a> in connection with the determination of the value of the strong coupling constant <span class="mathjax-tex">$\alpha _s$</span>.</p><p>In addition to these major categories of data commonly used to constrain PDFs, some complementary processes are also employed in some cases, as indicated in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab1">1</a>. These comprise the hadro-production of top-quark pairs from <i>pp</i> and <span class="mathjax-tex">$p\bar{p}$</span> collisions and the associated production of <i>W</i> bosons with charm quarks in <i>pp</i> collisions. Sometimes, also jet production in <i>ep</i> collisions and prompt photon (<span class="mathjax-tex">$\gamma $</span>+jet) production from <i>pp</i> and <span class="mathjax-tex">$p\bar{p}$</span> collisions is considered. Except for <span class="mathjax-tex">$t\bar{t}$</span> production, the necessary QCD corrections are known to NLO only, so that the same arguments as in the case of jet hadro-production data apply, if those data are included in a fit at NNLO accuracy. For <span class="mathjax-tex">$t\bar{t}$</span> production, only the inclusive cross section is considered at the moment in the available PDFs and there is a significant correlation with PDFs, especially of the gluon PDF with the top-quark mass.</p><p>Taken together, the set of these data has a number of data points (NDP) of the order of few thousand, and provides sufficient information to describe the PDFs with an ansatz of about <span class="mathjax-tex">$\mathcal{O}(30)$</span> free parameters. The parameters can include the strong coupling constant <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the heavy-quark masses <span class="mathjax-tex">$m_c$</span>, <span class="mathjax-tex">$m_b$</span> and <span class="mathjax-tex">$m_t$</span>, which are correlated with the PDFs, as will be discussed in Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec19">4</a>. This provides sufficient flexibility for all PDF groups and it is routinely checked that no additional terms are required to improve the quality of the fit. The exception is the NNPDF group, which typically uses <span class="mathjax-tex">$\mathcal{O}(250)$</span> free parameters in the neural network.</p><p>Apart from those considerations there is the general problem of the quality of the experimental data, that is to say whether or not the PDFs are extracted from a consistent data set. The various groups have different approaches, which roughly fall into two classes according to the different confidence level (c.l.) criteria for the value of <span class="mathjax-tex">$\chi ^2$</span> in the goodness-of-fit test. One approach is to fit to a very wide (or even the widest possible) set of data, while the other one rejects inconsistent data sets. In the former case, a tolerance criterion for <span class="mathjax-tex">$\Delta \chi ^2$</span> is introduced (e.g. <span class="mathjax-tex">$\Delta \chi ^2=100$</span>), while the latter approach maintains that <span class="mathjax-tex">$\Delta \chi ^2=1$</span>. For the various PDF groups this information is listed in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab1">1</a>.</p><p>For further reference, we quote here the definition of <span class="mathjax-tex">$\chi ^2$</span> used in data comparisons (Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab10">10</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab11">11</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab12">12</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab14">14</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab15">15</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab16">16</a>). It follows the definition described in Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="H1 Collaboration, F.D. Aaron et al., Inclusive deep-inelastic scattering at high 
 
 
 
 $$Q^2$$
 
 
 
 Q
 2
 
 
 
 with longitudinally polarised lepton beams at HERA. JHEP 09, 061 (2012). 
 arXiv:1206.7007
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR16" id="ref-link-section-d90629110e8464">16</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="ZEUS Collaboration, S. Chekanov et al., A ZEUS next-to-leading-order QCD analysis of data on deep inelastic scattering, Phys. Rev. D 67, 012007 (2003). 
 arXiv:hep-ex/0208023
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR18" id="ref-link-section-d90629110e8467">18</a>] and is expressed as follows:</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \chi ^2= & {} \sum _i \frac{\left[ {\mu _i} - m_i \left( 1 - \sum _j \gamma ^i_j b_j \right) \right] ^2}{ \textstyle \delta ^2_{i,\mathrm{unc}}m_i^2 + \delta ^2_{i,\mathrm{stat}}\, {\mu _i} m_i } + \sum _j b^2_j\nonumber \\&+ \sum _i \ln \frac{ \textstyle \delta ^2_{i,\mathrm{unc}}m_i^2 + \delta ^2_{i,\mathrm{stat}}\, {\mu _i} m_i }{ \textstyle \delta ^2_{i,\mathrm{unc}}\mu _i^2 + \delta ^2_{i,\mathrm{stat}}\mu _i^2 }, \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>where <span class="mathjax-tex">$\mu _i$</span> represents the measurement at the point <i>i</i>, <span class="mathjax-tex">$m_i$</span> is the corresponding theoretical prediction and <span class="mathjax-tex">$\delta _{i,\mathrm{stat}}$</span>, <span class="mathjax-tex">$\delta _{i,\mathrm{unc}}$</span> are the relative statistical and uncorrelated systematic uncertainties, respectively. <span class="mathjax-tex">$\gamma ^i_j$</span> denotes the sensitivity of the measurement to the correlated systematic source <i>j</i> and <span class="mathjax-tex">$b_j$</span> their shifts, with a penalty term <span class="mathjax-tex">$\sum _j b_j^2$</span> added. In addition, a logarithmic term is introduced arising from the likelihood transition to <span class="mathjax-tex">$\chi ^2$</span> when scaling of the errors is applied [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="H1 Collaboration, F.D. Aaron et al., Inclusive deep-inelastic scattering at high 
 
 
 
 $$Q^2$$
 
 
 
 Q
 2
 
 
 
 with longitudinally polarised lepton beams at HERA. JHEP 09, 061 (2012). 
 arXiv:1206.7007
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR16" id="ref-link-section-d90629110e9078">16</a>].</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig1_HTML.gif?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig1_HTML.gif" alt="figure 1" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>The <i>u</i>-valence, <i>d</i>-valence, gluon and sea quark (<span class="mathjax-tex">$x\Sigma = 2x(\bar{u}+\bar{c}+\bar{d}+\bar{s})$</span>) PDFs with their 1 <span class="mathjax-tex">$\sigma $</span> uncertainty bands of ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e9218">2</a>], HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e9222">4</a>] and JR14 (set <span class="u-monospace">JR14NNLO08VF</span>) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e9228">5</a>] at NNLO at the scale <span class="mathjax-tex">$Q^2=4~\,\mathrm {GeV}^2$</span>; absolute results (<i>left</i>) and ratio with respect to ABM12 (<i>right</i>)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig2_HTML.gif?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig2_HTML.gif" alt="figure 2" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig1">1</a> for the CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e9307">3</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e9310">6</a>] and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e9313">7</a>] PDF sets with their 1 <span class="mathjax-tex">$\sigma $</span> uncertainty bands at NNLO; absolute results (<i>left</i>) and ratio with respect to CT14 (<i>right</i>)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig3_HTML.gif?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig3_HTML.gif" alt="figure 3" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig1">1</a> at the scale <span class="mathjax-tex">$Q^2=100~\,\mathrm {GeV}^2$</span> with the sea <span class="mathjax-tex">$ x\Sigma = 2x(\bar{u}+\bar{c}+\bar{d}+\bar{s}+\bar{b})$</span> </p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Fig. 4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig4_HTML.gif?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig4_HTML.gif" alt="figure 4" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a> at the scale <span class="mathjax-tex">$Q^2=100~\,\mathrm {GeV}^2$</span> with the sea <span class="mathjax-tex">$ x\Sigma = 2x(\bar{u}+\bar{c}+\bar{d}+\bar{s}+\bar{b})$</span> </p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="Fig. 5"><figure><figcaption><b id="Fig5" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 5</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/5" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig5_HTML.gif?as=webp"><img aria-describedby="Fig5" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig5_HTML.gif" alt="figure 5" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-5-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig3">3</a> at the scale <span class="mathjax-tex">$Q^2=M_Z^2$</span> </p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/5" data-track-dest="link:Figure5 Full size image" aria-label="Full size image figure 5" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-6" data-title="Fig. 6"><figure><figcaption><b id="Fig6" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 6</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/6" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig6_HTML.gif?as=webp"><img aria-describedby="Fig6" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig6_HTML.gif" alt="figure 6" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-6-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig4">4</a> at the scale <span class="mathjax-tex">$Q^2=M_Z^2$</span> </p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/6" data-track-dest="link:Figure6 Full size image" aria-label="Full size image figure 6" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>It is important to note that the <span class="mathjax-tex">$\chi ^2$</span> values obtained with Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ1">1</a>) will not necessarily correspond to numbers quoted by PDF groups due to different <span class="mathjax-tex">$\chi ^2$</span> definitions, data treatment and other parameters, see also Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab1">1</a>.</p><h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">2.2 </span>Results for PDFs</h3><p>Before we start a detailed discussion of the theoretical aspects of the PDF determinations we would like to illustrate the present status of PDF sets at NNLO in QCD and discuss briefly some differences, which are clearly visible. The currently available sets at NNLO in QCD are shown in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig1">1</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig3">3</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig4">4</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig5">5</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig6">6</a>. The light-quark (<i>u</i>, <i>d</i>) valence PDFs together with the gluon and the quark sea distributions (<span class="mathjax-tex">$x\Sigma = 2x(\bar{u}+\bar{c}+\bar{d}+\bar{s})$</span> for four active flavors) with the respective uncertainty bands are displayed in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig1">1</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig3">3</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig5">5</a> at the scales <span class="mathjax-tex">$Q^2=4~\,\mathrm {GeV}^2$</span>, <span class="mathjax-tex">$100~\,\mathrm {GeV}^2$</span> and <span class="mathjax-tex">$M_Z^2$</span> in the range <span class="mathjax-tex">$10^{-4} \le x \le 1$</span> for the sets ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e10198">2</a>], HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e10202">4</a>] and JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e10205">5</a>]. Likewise, Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig4">4</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig6">6</a> show the sets CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e10217">3</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e10221">6</a>] and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e10224">7</a>].</p><p>The main features of the present NNLO PDFs in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig1">1</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig3">3</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig4">4</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig5">5</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig6">6</a> in the main kinematic region of <i>x</i> and <span class="mathjax-tex">$Q^2$</span> relevant for hard scattering events at Tevatron and the LHC can be characterized as follows. The agreement in the distributions <span class="mathjax-tex">$xu_v$</span>, and to a slightly lesser extent <span class="mathjax-tex">$\Sigma $</span>, is very good for ABM12, JR14 and HERAPDF2.0, as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig1">1</a>. For the valence PDF <span class="mathjax-tex">$xd_v$</span> there is also an overall reasonable agreement, but the distribution deviates by more than <span class="mathjax-tex">$1 \sigma $</span> at <span class="mathjax-tex">$x \gtrsim 0.1$</span> in the case of HERAPDF2.0. One should note that <span class="mathjax-tex">$xd_v$</span> is more difficult to measure in <span class="mathjax-tex">$e^{\pm }p$</span> DIS at HERA than <span class="mathjax-tex">$xu_v$</span> and additional constraints from deuteron data are important to fix the details of this PDF, as discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a> below.</p><p>The results on the gluon momentum distribution <i>xg</i> are clearly different at low values of <i>x</i>. Here, JR14 obtains the largest values, followed by ABM12 and HERAPDF2.0, with the latter displaying a valence-like shape below <span class="mathjax-tex">$x = 10^{-3}$</span>. For CT14, MMHT14 and NNPDF3.0 there is very good agreement for <span class="mathjax-tex">$xu_v$</span>, cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a>. Some differences are visible in case of <span class="mathjax-tex">$xd_v$</span>, where CT14 reports larger values than NNPDF3.0 at <span class="mathjax-tex">$x \gtrsim 5 \cdot 10^{-3}$</span> and vice versa for smaller <i>x</i>. The spread in <span class="mathjax-tex">$\Sigma $</span> for the sets in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a> is much greater than those by ABM12, JR14 and HERAPDF2.0. This is true as well for the gluon PDF <i>xg</i> with the CT14 uncertainty band for the gluon PDF also covering the predictions for the distributions by ABM12, and HERAPDF2.0. Note that the error bands for CT14 in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig2">2</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig4">4</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig6">6</a> correspond to the c.l. of 68 %.</p><p>The disagreement in <span class="mathjax-tex">$xd_v$</span> between HERAPDF2.0 and ABM12 or JR14 persists through the evolution from <span class="mathjax-tex">$Q^2 = 4~\,\mathrm {GeV}^2$</span> to <span class="mathjax-tex">$Q^2 = M_Z^2$</span>, cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig3">3</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig5">5</a>. Likewise, the spread in <span class="mathjax-tex">$xd_v$</span> between CT14, MMHT14 and NNPDF3.0 becomes more pronounced, as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig4">4</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig6">6</a>. On the other hand, differences in the singlet PDFs <span class="mathjax-tex">$\Sigma $</span> and <i>xg</i>, while still somewhat visible at <span class="mathjax-tex">$Q^2 = 100~\,\mathrm {GeV}^2$</span>, largely wash out at scales <span class="mathjax-tex">$Q^2 = M_Z^2$</span> which govern the physics of central rapidity events at the LHC. Those remaining differences persist at large scales (as in the case of the gluon PDFs at large <span class="mathjax-tex">$x > 0.1$</span>) and will have a significant impact. The crucial test for all PDF sets comes through a detailed comparison of cross section predictions to data. This will be discussed in the remainder of the paper, in particular in Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec20">5</a>.</p></div></div></section><section data-title="Theory for PDF fits"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5"><span class="c-article-section__title-number">3 </span>Theory for PDF fits</h2><div class="c-article-section__content" id="Sec5-content"><p>In the following we describe the basic theoretical issues for a consistent determination of the twist-two PDFs from DIS and other hard scattering data, on the basis of perturbative QCD at NNLO using the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span> scheme for renormalization and factorization.</p><h3 class="c-article__sub-heading" id="Sec6"><span class="c-article-section__title-number">3.1 </span>Theory for analyses of DIS data</h3><p>The world DIS data are provided in terms of reduced cross sections by the different experiments. QED and electroweak radiative corrections [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="A. Kwiatkowski, H. Spiesberger, H.J. Möhring, Heracles: an event generator for 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR46" id="ref-link-section-d90629110e11035">46</a>] are applied, which requires careful study of different kinematic variables [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="A. Arbuzov, D.Yu. Bardin, J. Blümlein, L. Kalinovskaya, T. Riemann, Hector 1.00: a program for the calculation of QED, QCD and electroweak corrections to 
 
 
 
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 scattering. Phys. Lett. B 553, 242 (2003). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR49" id="ref-link-section-d90629110e11041">49</a>]. In this way also the contributions from the exchange of more than one gauge boson to the partonic twist-2 terms are taken care of. In part, also the very small QED corrections to the hadronic tensor are already accounted for. These have a flat kinematic behavior and amount to <span class="mathjax-tex">$\mathcal{O}(1\,\%)$</span> or less [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="J. Kripfganz, H. Perlt, Electroweak radiative corrections and quark mass singularities. Z. Phys. C 41, 319 (1988)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR50" id="ref-link-section-d90629110e11081">50</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 53" title="M. Roth, S. Weinzierl, QED corrections to the evolution of parton distributions. Phys. Lett. B 590, 190 (2004). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR53" id="ref-link-section-d90629110e11084">53</a>].</p><p>The reduced cross sections are differential in either two of the kinematic variables in the set <span class="mathjax-tex">$\{x,y,Q^2\}$</span>. The virtuality <span class="mathjax-tex">$Q^2 = -q^2$</span> of the process is given by the 4-momentum transfer <i>q</i> to the hadronic system. The Bjorken variable is defined as <span class="mathjax-tex">$x = Q^2/(s y)$</span>, with <span class="mathjax-tex">$y = 2 p \cdot q/s$</span>, and <span class="mathjax-tex">$s = (p+l)^2$</span> the squared center-of-mass energy, where <i>p</i> and <i>l</i> denote the 4-momenta of the nucleon and the lepton. At energies much greater than the nucleon mass <i>M</i>, in the nucleon rest frame <i>y</i> is the fractional energy of the lepton transferred to the nucleon. The double differential cross sections used in the QCD analyses are given by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="A. Arbuzov, D.Yu. Bardin, J. Blümlein, L. Kalinovskaya, T. Riemann, Hector 1.00: a program for the calculation of QED, QCD and electroweak corrections to 
 
 
 
 $$ep$$
 
 
 
 e
 p
 
 
 
 and 
 
 
 
 $$l^\pm N$$
 
 
 
 
 l
 ±
 
 N
 
 
 
 deep-inelastic neutral and charged current scattering. Comput. Phys. Commun. 94, 128 (1996). 
 arXiv:hep-ph/9511434
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR46" id="ref-link-section-d90629110e11315">46</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="J. Blümlein, The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 28 (2013). 
 arXiv:1208.6087
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR54" id="ref-link-section-d90629110e11319">54</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e11322">55</a>]</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{d^2 \sigma _\mathrm{NC}^{l^\pm N}}{dx dy}= & {} \frac{2 \pi \alpha ^2 s}{Q^4} \Biggl \{ \left[ 2(1-y) - 2xy \frac{M^2}{s} \right] \nonumber \\&\times F^{NC}_2(x,Q^2) + Y_- xF^{NC}_3(x,Q^2) \nonumber \\&+\, y^2\left( 1 - \frac{2m_l^2}{Q^2}\right) 2xF^{NC}_1(x,Q^2) \Biggr \}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2) </div></div> <div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{d^2 \sigma _\mathrm{NC}^{\nu (\bar{\nu })N}}{dx dy}= & {} \frac{G_F^2 s}{16 \pi } \left[ \frac{M_Z^2}{Q^2 +M_Z^2}\right] ^2 \left\{ Y_+ W_2^\mathrm{NC}(x,Q^2) \right. \nonumber \\&\left. \pm Y_- xW_3^\mathrm{NC}(x,Q^2) - y^2 W_L^\mathrm{NC}(x,Q^2)\right\} \, , \end{aligned}$$</span></div><div class="c-article-equation__number"> (3) </div></div> <div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{d^2 \sigma _\mathrm{CC}}{dx dy}= & {} \frac{G_F^2 s}{4 \pi } \left[ \frac{M_W^2}{Q^2 + M_W^2}\right] ^2 \left\{ Y_+ W_2^\mathrm{CC}(x,Q^2) \right. \nonumber \\&\left. \pm Y_- xW_3^\mathrm{CC}(x,Q^2) - y^2 W_L^\mathrm{CC}(x,Q^2)\right\} , \end{aligned}$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>where <span class="mathjax-tex">$\alpha $</span> and <span class="mathjax-tex">$G_F$</span> denote the fine-structure and Fermi constants, <span class="mathjax-tex">$Y_\pm = 1 \pm (1-y)^2$</span> and we keep the dependence on the masses of the nucleon (<i>M</i>), the <i>W</i> and <i>Z</i> boson (<span class="mathjax-tex">$M_{W}$</span>, <span class="mathjax-tex">$M_{Z}$</span>) and the lepton (<span class="mathjax-tex">$m_l$</span>).</p><p>The structure functions <span class="mathjax-tex">$F^{NC}_i$</span> and <span class="mathjax-tex">$W_i$</span> are nonperturbative quantities defining the hadronic tensor. They can be measured by varying <i>y</i> at fixed <span class="mathjax-tex">$Q^2$</span> and <i>x</i> and form the input to the subsequent analysis. Note that in some previous experiments, assumptions were made about the longitudinal structure functions <span class="mathjax-tex">$F^{NC}_L$</span> and <span class="mathjax-tex">$W_L$</span>, where (in the massless limit)</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F^{NC}_L(x,Q^2) = F^{NC}_2(x,Q^2) - 2x F^{NC}_1(x,Q^2)\, , \end{aligned}$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>since at the time of the data analysis the corresponding QCD corrections were still missing. Therefore, it is important to use the differential cross sections in Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ2">2</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ4">4</a>) and to add the correct longitudinal structure functions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="S. Moch, J.A.M. Vermaseren, A. Vogt, The longitudinal structure function at the third order. Phys. Lett. B 606, 123 (2005). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR56" id="ref-link-section-d90629110e12771">56</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="J.A.M. Vermaseren, A. Vogt, S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange. Nucl. Phys. B 724, 3 (2005). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR57" id="ref-link-section-d90629110e12775">57</a>], cf. also [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 58" title="J. Blümlein, H. Böttcher, A. Guffanti, Non-singlet QCD analysis of deep-inelastic world data at 
 
 
 
 $$(\alpha _s^3)$$
 
 
 
 (
 
 α
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 3
 
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 . Nucl. Phys. B 774, 182 (2007). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR58" id="ref-link-section-d90629110e12778">58</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 59" title="S. Alekhin, J. Blümlein, S. Moch, Higher order constraints on the Higgs production rate from fixed-target DIS data. Eur. Phys. J. C 71, 1723 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR59" id="ref-link-section-d90629110e12781">59</a>]. The structure functions are measured for DIS off massive proton and deuteron targets and are, therefore, subject to target mass corrections, which play an important role in the region of lower values of <span class="mathjax-tex">$Q^2$</span> and larger values of <i>x</i>. They are available in Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="J. Blümlein, The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 28 (2013). 
 arXiv:1208.6087
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR54" id="ref-link-section-d90629110e12811">54</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 60" title="H. Georgi, H.D. Politzer, Freedom at moderate energies: masses in color dynamics. Phys. Rev. D 14, 1829 (1976)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR60" id="ref-link-section-d90629110e12815">60</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 61" title="F.M. Steffens, M.D. Brown, W. Melnitchouk, S. Sanches, Parton distributions in the presence of target mass corrections. Phys. Rev. C 86, 065208 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR61" id="ref-link-section-d90629110e12818">61</a>].</p><p>The neutral- and charged-current structure functions <span class="mathjax-tex">$F^{NC}_i$</span>, <span class="mathjax-tex">$W^{NC}_i$</span> and <span class="mathjax-tex">$W^{CC}_i$</span> consist of a sum of several terms, each weighted by powers of the QED and electroweak couplings, and <span class="mathjax-tex">$F^{NC}_i$</span> also include the <span class="mathjax-tex">$\gamma -Z$</span> mixing, which has to be accounted for, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="A. Arbuzov, D.Yu. Bardin, J. Blümlein, L. Kalinovskaya, T. Riemann, Hector 1.00: a program for the calculation of QED, QCD and electroweak corrections to 
 
 
 
 $$ep$$
 
 
 
 e
 p
 
 
 
 and 
 
 
 
 $$l^\pm N$$
 
 
 
 
 l
 ±
 
 N
 
 
 
 deep-inelastic neutral and charged current scattering. Comput. Phys. Commun. 94, 128 (1996). 
 arXiv:hep-ph/9511434
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR46" id="ref-link-section-d90629110e12976">46</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="J. Blümlein, The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 28 (2013). 
 arXiv:1208.6087
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR54" id="ref-link-section-d90629110e12979">54</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e12982">55</a>]. Then, considering one specific gauge boson exchange, one arrives at a representation for the individual structure functions <span class="mathjax-tex">$F_i$</span>, which are only subject to QCD corrections. For example, for pure photon exchange, they are given by</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_i(x,Q^2) = F_i^{\tau =2}(x,Q^2) + \sum _{k=2}^\infty \frac{C^{\tau =2k}_i(x,Q^2)}{Q^{2(k-1)}}~, \end{aligned}$$</span></div><div class="c-article-equation__number"> (6) </div></div><p>where <span class="mathjax-tex">$F_i^{\tau =2}$</span> denotes the leading-twist term and the coefficients <span class="mathjax-tex">$C^\tau _i$</span> parametrize the higher twist contributions. The latter terms are of relevance for many DIS data sets, see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec2">2</a>.</p><p>Present day QCD analyses are aimed at determining the leading-twist contributions to the structure functions. There are two ways to account for the higher twist terms:</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(i)</span> <p>One is fitting the higher twist terms in <span class="mathjax-tex">$F_i$</span>. A rigorous approach requires the knowledge of their scaling violations (term by term) and of the various Wilson coefficients to higher orders in <span class="mathjax-tex">$\alpha _s$</span>, see e.g. Sect. 16 in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="J. Blümlein, The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 28 (2013). 
 arXiv:1208.6087
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR54" id="ref-link-section-d90629110e13328">54</a>]. Since at present this is practically out of reach, such fits remain rather phenomenological. Moreover, the size of the (non-singlet) higher twist contributions to the structure function <span class="mathjax-tex">$F_2$</span> vary strongly with the correction applied to the leading-twist term up to next-to-next-to-next-to-leading order (N<span class="mathjax-tex">$^3$</span>LO), as shown in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 58" title="J. Blümlein, H. Böttcher, A. Guffanti, Non-singlet QCD analysis of deep-inelastic world data at 
 
 
 
 $$(\alpha _s^3)$$
 
 
 
 (
 
 α
 s
 3
 
 )
 
 
 
 . Nucl. Phys. B 774, 182 (2007). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR58" id="ref-link-section-d90629110e13380">58</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 62" title="J. Blümlein, H. Böttcher, Higher twist contributions to the structure functions 
 
 
 
 $$F_2^p(x, Q^2)$$
 
 
 
 
 F
 2
 p
 
 
 (
 x
 ,
 
 Q
 2
 
 )
 
 
 
 
 and 
 
 
 
 $$F_2^d(x, Q^2)$$
 
 
 
 
 F
 2
 d
 
 
 (
 x
 ,
 
 Q
 2
 
 )
 
 
 
 
 at large 
 
 
 
 $$x$$
 
 
 x
 
 
 and higher orders. Phys. Lett. B 662, 336 (2008). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR62" id="ref-link-section-d90629110e13383">62</a>]. Also, the non-singlet and singlet higher twist contributions are different [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 63" title="S.I. Alekhin, Global fit to the charged leptons DIS data: 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR63" id="ref-link-section-d90629110e13386">63</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e13389">64</a>].</p> </li> <li> <span class="u-custom-list-number">(ii)</span> <p>One has to find appropriate cuts to sufficiently reduce the higher twist terms. For instance, in the flavor non-singlet analysis of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 58" title="J. Blümlein, H. Böttcher, A. Guffanti, Non-singlet QCD analysis of deep-inelastic world data at 
 
 
 
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 α
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR58" id="ref-link-section-d90629110e13403">58</a>] the cuts are taken to be <span class="mathjax-tex">$Q^2 \ge 4~\,\mathrm {GeV}^2$</span>, <span class="mathjax-tex">$W^2 = M^2 + Q^2(1-x)/x \ge 12.5~\,\mathrm {GeV}^2$</span>. In the combined singlet and non-singlet analysis of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e13539">64</a>], <span class="mathjax-tex">$Q^2 \ge 10~\,\mathrm {GeV}^2$</span>, <span class="mathjax-tex">$W^2 \ge 12.5~\,\mathrm {GeV}^2$</span> have been used. These bounds are found empirically by cutting on <span class="mathjax-tex">$W^2$</span> and/or <span class="mathjax-tex">$Q^2$</span> starting from larger values. Applying these cuts severely limits the amount of large-<i>x</i> DIS data to be fitted, and usually leads to an increase of the errors of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and other fitted fundamental parameters and distributions.</p> </li> </ol><p>Both methods (i) and (ii) allow to access the leading-twist contributions to the DIS structure functions, with some qualifications, however. The cuts suggested in (ii) remove the large-<i>x</i> region potentially sensitive to the higher twist terms. However, they do not affect the data at <span class="mathjax-tex">$x \lesssim 0.1$</span>, where higher twist terms still play an important role [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e13765">64</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 65" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, The impact of the final HERA combined data on PDFs obtained from a global fit. 
 arXiv:1601.03413
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR65" id="ref-link-section-d90629110e13768">65</a>]. To some extent, the influence of higher twist can be dampened by using the DIS data for the structure function <span class="mathjax-tex">$F_2$</span> instead of the cross section, since in this case the contribution to the structure function <span class="mathjax-tex">$F_L$</span> need not be considered. It should be kept in mind, though, that the experimental separation of the structure functions <span class="mathjax-tex">$F_{2}$</span> and <span class="mathjax-tex">$F_{L}$</span> in the full phase space of common DIS experiments is very difficult without dedicated longitudinal–transverse cross section separations. Therefore, the data on <span class="mathjax-tex">$F_{2}$</span> and <span class="mathjax-tex">$F_{3}$</span> are typically extracted from the cross section once a certain model for the structure function <span class="mathjax-tex">$F_L$</span> is taken. This approach is justified only at large <i>x</i>, however, where the contribution of <span class="mathjax-tex">$F_L$</span> is small and even large uncertainties in the modeling of <span class="mathjax-tex">$F_L$</span> cannot affect the extracted values of <span class="mathjax-tex">$F_{2}$</span> and <span class="mathjax-tex">$F_{3}$</span>. The procedure is not applicable for HERA kinematics, on the other hand, and introduces a bias into the analysis of the data taken by the New Muon Collaboration (NMC), in particular, a shift in the value of <span class="mathjax-tex">$\alpha _s$</span> preferred by the fit [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 59" title="S. Alekhin, J. Blümlein, S. Moch, Higher order constraints on the Higgs production rate from fixed-target DIS data. Eur. Phys. J. C 71, 1723 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR59" id="ref-link-section-d90629110e14066">59</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 66" title="J. Gao, M. Guzzi, J. Huston, H.-L. Lai, Z. Li, P. Nadolsky, J. Pumplin, D. Stump, C.P. Yuan, CT10 next-to-next-to-leading order global analysis of QCD. Phys. Rev. D 89, 033009 (2014). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR66" id="ref-link-section-d90629110e14069">66</a>], cf. Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec19">4</a>. Nonetheless, the MMHT14 analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e14075">6</a>] is still based on the DIS structure function data, as are the CJ15 and CT14 analyses [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e14078">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e14082">3</a>]. The latter two use cross section data for HERA, and for HERA and NMC, respectively, and structure function data elsewhere. While CT14 performed this important change for the HERA and NMC data, the authors of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 67" title="R.S. Thorne, G. Watt, PDF dependence of Higgs cross sections at the tevatron and LHC: response to recent criticism. JHEP 08, 100 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR67" id="ref-link-section-d90629110e14085">67</a>] report that the change has little impact. Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 59" title="S. Alekhin, J. Blümlein, S. Moch, Higher order constraints on the Higgs production rate from fixed-target DIS data. Eur. Phys. J. C 71, 1723 (2011). 
 arXiv:1101.5261
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR59" id="ref-link-section-d90629110e14088">59</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e14091">64</a>], on the other hand, disagree with this claim.</p><p>The deep-inelastic structure functions are inclusive quantities and contain massless parton and heavy-quark contributions,</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_i^{\tau = 2}(x,Q^2) = F_i^\mathrm{massless}(x,Q^2) + F_i^\mathrm{massive}(x,Q^2)\, . \end{aligned}$$</span></div><div class="c-article-equation__number"> (7) </div></div><p>Here the massless terms are given by</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F_i^\mathrm{massless}(x,Q^2) = \sum _{j}\, C_{i,j}\left( x,\frac{Q^2}{\mu ^2}\right) \otimes f_j(x,\mu ^2)\, , \end{aligned}$$</span></div><div class="c-article-equation__number"> (8) </div></div><p>where <span class="mathjax-tex">$C_{i,j}$</span> denote the massless Wilson coefficients, <span class="mathjax-tex">$f_j$</span> the massless PDFs and <span class="mathjax-tex">$\mu ^2$</span> is the factorization scale. The Mellin convolution is abbreviated by <span class="mathjax-tex">$\otimes $</span> and the sum over <i>j</i> is over all contributing partons. The renormalization group equation for <span class="mathjax-tex">$F_i^\mathrm{massless}$</span> allows one to eliminate the dependence on <span class="mathjax-tex">$\mu ^2$</span> order-by-order in perturbation theory. This also applies to <span class="mathjax-tex">$F_i^{\tau = 2}$</span>. Through the massive contributions <span class="mathjax-tex">$F_i^\mathrm{massive}$</span> there is a dependence on the heavy-quark masses <span class="mathjax-tex">$m_c$</span> and <span class="mathjax-tex">$m_b$</span> in the present world DIS data. Note that <span class="mathjax-tex">$F_i^\mathrm{massive}$</span> is <i>not</i> the structure function of a tagged heavy-flavor sample, which would be infrared sensitive [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 68" title="A. Chuvakin, J. Smith, W.L. van Neerven, Comparison between variable flavor number schemes for charm quark electroproduction. Phys. Rev. D 61, 096004 (2000). 
 arXiv:hep-ph/9910250
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR68" id="ref-link-section-d90629110e14682">68</a>]. Rather, <span class="mathjax-tex">$F_i^\mathrm{massive}$</span> is just given as the difference of the complete structure function <span class="mathjax-tex">$F_i^{\tau = 2}$</span> and the massless one in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ8">8</a>).</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec7"><span class="c-article-section__title-number">3.1.1 </span>Massless PDFs</h4><p>For all QCD calculations we use perturbation theory. The factorized representation in terms of Wilson coefficients and PDFs is obtained using the light-cone expansion [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 69" title="K.G. Wilson, Nonlagrangian models of current algebra. Phys. Rev. 179, 1499 (1969)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR69" id="ref-link-section-d90629110e14757">69</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 72" title="Y. Frishman, Operator products at almost light like distances. Ann. Phys. 66, 373 (1971)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR72" id="ref-link-section-d90629110e14760">72</a>]. For a proper definition of the Wilson coefficients and the PDFs one has to use the LSZ formalism and refer to asymptotic states at large times <span class="mathjax-tex">$t \rightarrow \pm \infty $</span>, given by massless partons. We first describe the massless contributions in Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ7">7</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ8">8</a>), and then discuss the contribution of heavy quarks. The Wilson coefficients in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ8">8</a>) have a perturbative expansion in the strong coupling constant. At one- [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 73" title="W. Furmanski, R. Petronzio, Lepton–hadron processes beyond leading order in quantum chromodynamics. Z. Phys. C 11, 293 (1982)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR73" id="ref-link-section-d90629110e14802">73</a>], two- [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 74" title="E.B. Zijlstra, W.L. van Neerven, Contribution of the second order gluonic Wilson coefficient to the deep-inelastic structure function. Phys. Lett. B 273, 476 (1991)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR74" id="ref-link-section-d90629110e14805">74</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 81" title="S. Moch, J.A.M. Vermaseren, Deep-inelastic structure functions at two loops. Nucl. Phys. B 573, 853 (2000). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR81" id="ref-link-section-d90629110e14808">81</a>], and three-loop order [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="S. Moch, J.A.M. Vermaseren, A. Vogt, The longitudinal structure function at the third order. Phys. Lett. B 606, 123 (2005). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR56" id="ref-link-section-d90629110e14811">56</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="J.A.M. Vermaseren, A. Vogt, S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange. Nucl. Phys. B 724, 3 (2005). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR57" id="ref-link-section-d90629110e14814">57</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 82" title="S. Moch, M. Rogal, Charged current deep-inelastic scattering at three loops. Nucl. Phys. B 782, 51 (2007). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR82" id="ref-link-section-d90629110e14818">82</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 84" title="S. Moch, J.A.M. Vermaseren, A. Vogt, Third-order QCD corrections to the charged-current structure function 
 
 
 
 $$F_3$$
 
 
 
 F
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 . Nucl. Phys. B 813, 220 (2009). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR84" id="ref-link-section-d90629110e14821">84</a>] they have been calculated for the neutral-current structure functions <span class="mathjax-tex">$F_{i}$</span>, with <span class="mathjax-tex">$i=1,2,3$</span>, except for the <span class="mathjax-tex">$\gamma -Z$</span> mixing contribution at three loops.</p><p>The structure functions in general depend on the following three non-singlet and singlet combinations of parton densities:</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} q_{jk}^{\pm }= & {} f_j \pm \bar{f}_j - (f_k \pm \bar{f}_k), \quad q^{v} = \sum _{l=1}^{n_f} (f_l - \bar{f}_l), \nonumber \\ q^{s}= & {} \sum _{l=1}^{n_f} (f_l + \bar{f}_l), \end{aligned}$$</span></div><div class="c-article-equation__number"> (9) </div></div><p>with the light-quark distributions <span class="mathjax-tex">$f_i$</span> of flavor <i>i</i> and <span class="mathjax-tex">$n_f$</span> the number of massless flavors. These combinations evolve in <span class="mathjax-tex">$\mu ^2$</span> from an initial scale <span class="mathjax-tex">$\mu _0^2$</span> by the QCD evolution equations, where the singlet distribution <span class="mathjax-tex">$q^{s}(x,\mu ^2)$</span> mixes with the gluon distribution <span class="mathjax-tex">$g(x,\mu ^2)$</span>,</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\frac{d}{d \ln (\mu ^2)} q^{i}(x,\mu ^2) = P^{i}(x) \otimes q^{i}(x,\mu ^2), \quad \nonumber \\&\quad i = \pm , v\, , \end{aligned}$$</span></div><div class="c-article-equation__number"> (10) </div></div> <div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\frac{d}{d \ln (\mu ^2)} \left( \begin{array}{c} q^{s}(x,\mu ^2)\\ g(x,\mu ^2) \end{array} \right) = \left( \begin{array}{cc} P_{qq}(x) &{} P_{qg}(x) \\ P_{gq}(x) &{} P_{gg}(x) \end{array} \right) \nonumber \\&\quad \otimes \left( \begin{array}{c} q^{s}(x,\mu ^2)\\ g(x,\mu ^2) \end{array} \right) . \end{aligned}$$</span></div><div class="c-article-equation__number"> (11) </div></div><p>The non-singlet splitting functions are given by</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P^{\pm } (x)= & {} P_{qq}(x) \pm P_{q\bar{q}}(x)\, ,\end{aligned}$$</span></div><div class="c-article-equation__number"> (12) </div></div> <div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} P^{v} (x)= & {} P_{qq}(x) - P_{q\bar{q}}(x) + {n_f}\left( P_{qq}^s(x) - P_{q\bar{q}}^s(x)\right) ,\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>while the anomalous dimensions <span class="mathjax-tex">$\gamma _{ij}$</span> corresponding to the splitting functions <span class="mathjax-tex">$P_{ij}$</span> are obtained by a Mellin transform,</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \gamma _{ij}(N) = - \int _0^1 dx\, x^N\, P_{ij}(x), \end{aligned}$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>where we suppress for brevity the dependence of <span class="mathjax-tex">$P_{ij}$</span> and <span class="mathjax-tex">$\gamma _{ij}$</span> on the strong coupling <span class="mathjax-tex">$a_s(\mu ^2) = \alpha _s(\mu ^2)/(4\pi )$</span>. The <span class="mathjax-tex">$P_{ij}$</span> are known as well at one- [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 85" title="D.J. Gross, F. Wilczek, Asymptotically free gauge theories. 1. Phys. Rev. D 8, 3633 (1973)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR85" id="ref-link-section-d90629110e16634">85</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 90" title="G. Altarelli, G. Parisi, Asymptotic freedom in parton language. Nucl. Phys. B 126, 298 (1977)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR90" id="ref-link-section-d90629110e16637">90</a>], two- [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 81" title="S. Moch, J.A.M. Vermaseren, Deep-inelastic structure functions at two loops. Nucl. Phys. B 573, 853 (2000). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR120" id="ref-link-section-d90629110e16973">120</a>] by arranging the solution systematically in powers of the coupling constants <span class="mathjax-tex">$a_s(\mu ^2)$</span> and <span class="mathjax-tex">$a_s(\mu _0^2)$</span>, and even forming factorization-scheme invariant expressions. In case of the <i>x</i>-space solutions this is usually not done due to the necessary iterative solution. In the small-<i>x</i> region the iterative solution usually leads to a pile-up of a few per cent [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 121" title="J. Blümlein, S. Riemersma, W.L. van Neerven, A. Vogt, Theoretical uncertainties in the QCD evolution of structure functions and their impact on 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR126" id="ref-link-section-d90629110e17102">126</a>], the heavy-flavor corrections to the DIS functions are described by Wilson coefficients. The leading order results are of <span class="mathjax-tex">$\mathcal{O}(a_s)$</span>. Higher order corrections in the perturbative expansions are, therefore, of <span class="mathjax-tex">$\mathcal{O}(a_s^2)$</span> at NLO, and of <span class="mathjax-tex">$\mathcal{O}(a_s^3)$</span> at NNLO, similar to the case of the longitudinal structure function [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="S. Moch, J.A.M. Vermaseren, A. Vogt, The longitudinal structure function at the third order. Phys. Lett. B 606, 123 (2005). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR57" id="ref-link-section-d90629110e17221">57</a>]. The corrections in the neutral- and charged-current cases are available in one- [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 127" title="E. Witten, Heavy quark contributions to deep-inelastic scattering. Nucl. Phys. B 104, 445 (1976)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR127" id="ref-link-section-d90629110e17224">127</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 134" title="J. Blümlein, A. Hasselhuhn, P. Kovacikova, S. Moch, 
 
 
 
 $${\cal O}(\alpha _s)$$
 
 
 
 O
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 α
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 O
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 O
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR138" id="ref-link-section-d90629110e17233">138</a>], where the latter corrections were given in semi-analytic form.</p><p>For the neutral-current exchange the heavy-flavor contributions to the structure functions <span class="mathjax-tex">$F_{i}$</span> with <span class="mathjax-tex">$i=2,L$</span> are [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 139" title="A. Behring, I. Bierenbaum, J. Blümlein, A. De Freitas, S. Klein, F. Wißbrock, The logarithmic contributions to the 
 
 
 
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 O
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 α
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 O
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 $$Q^2 \gg m^2$$
 
 
 
 
 Q
 2
 
 ≫
 
 m
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 Q
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR150" id="ref-link-section-d90629110e19092">150</a>]. This provides a good approximation of the NNLO corrections.</p><h3 class="c-article__sub-heading" id="Sec9"><span class="c-article-section__title-number">3.2 </span>Heavy-flavor PDFs</h3><p>An important issue in PDF fits concerns the number of active quark flavors and the theoretical description of heavy quarks such as charm and bottom. Due to the large range of hard scales <i>Q</i> for the scattering processes considered, different effective theories may be applied. At low scales, when <span class="mathjax-tex">$Q \simeq \mathcal{O}(\mathrm{few})~\,\mathrm {GeV}$</span>, one typically works with <span class="mathjax-tex">$n_f = 3$</span> massless quark flavors, setting <span class="mathjax-tex">$n_f = 3$</span> in the hard scattering cross section, the evolution kernels and the anomalous dimensions. In this case, only the light-quark PDFs for up, down and strange are taken into account. At higher scales, e.g., for hadro-production of jets at high transverse momentum <span class="mathjax-tex">$p_t$</span> or top quarks, additional dynamical degrees of freedom lead to theories with <span class="mathjax-tex">$n_f \,{>}\, 3$</span>. By means of the renormalization group and matching these are related to the case with <span class="mathjax-tex">$n_f = 3$</span> massless quarks. Technically, one has to apply decoupling relations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 151" title="T. Appelquist, J. Carazzone, Infrared singularities and massive fields. Phys. Rev. D 11, 2856 (1975)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR151" id="ref-link-section-d90629110e19304">151</a>] at some matching scale <span class="mathjax-tex">$\mu $</span>, for instance in the transition of <span class="mathjax-tex">$\alpha _s^{(n_f)} \rightarrow \alpha _s^{(n_f+1)}$</span>. This introduces some logarithmic dependence on the masses of the heavy quarks <span class="mathjax-tex">$m_c$</span>, <span class="mathjax-tex">$m_b$</span> and <span class="mathjax-tex">$m_t$</span> for charm, bottom and top. One should also note that the matching of the effective theories for <span class="mathjax-tex">$n_f \rightarrow n_f+1$</span> does not need to be smooth. In fact, it introduces discontinuities, such as for the running coupling as a solution of the QCD <span class="mathjax-tex">$\beta $</span>-function at higher order in the perturbative expansion, where <span class="mathjax-tex">$\alpha _s^{(n_f)}(\mu ) \ne \alpha _s^{(n_f+1)}(\mu )$</span> in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme at the matching scale <span class="mathjax-tex">$\mu $</span>, see e.g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 152" title="K.G. Chetyrkin, J.H. Kühn, M. Steinhauser, RunDec: a mathematica package for running and decoupling of the strong coupling and quark masses. Comput. Phys. Commun. 133, 43 (2000). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR152" id="ref-link-section-d90629110e19685">152</a>].</p><p>In a similar manner, PDFs in theories with a fixed number <span class="mathjax-tex">$n_f > 3$</span> of quark flavors are related to those for <span class="mathjax-tex">$n_f = 3$</span> with the help of heavy-quark operator matrix elements (OMEs) <span class="mathjax-tex">$A_{ij}$</span> at a chosen matching scale <span class="mathjax-tex">$\mu $</span>. Potential non-universal non-logarithmic heavy-flavor effects are taken care of by the Wilson coefficients. Starting with the PDFs in a so-called fixed-flavor number scheme (FFNS) with <span class="mathjax-tex">$n_f$</span> fixed, one has <span class="mathjax-tex">$f_i^{(n_f)} \rightarrow f_i^{(n_f+1)}$</span> for the light-quark distributions <span class="mathjax-tex">$f_i$</span> and <span class="mathjax-tex">$(q^{s,\, (n_f)}, g^{(n_f)}) \rightarrow (q^{s,\, (n_f+1)}, g^{(n_f+1)})$</span> for the gluon and the singlet quark distributions with operator mixing in the singlet sector. In particular, one has [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 140" title="I. Bierenbaum, J. Blümlein, S. Klein, Mellin moments of the 
 
 
 
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 heavy flavor contributions to unpolarized deep-inelastic scattering at 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR140" id="ref-link-section-d90629110e20075">140</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 153" title="M. Buza, Y. Matiounine, J. Smith, W.L. van Neerven, Charm electroproduction viewed in the variable flavor number scheme versus fixed order perturbation theory. Eur. Phys. J. C 1, 301–320 (1998). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR153" id="ref-link-section-d90629110e20078">153</a>]</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&f_k(n_f+1, \mu ^2) + f_{\bar{k}}(n_f+1, \mu ^2)\nonumber \\&\quad = A_{qq,h}^{ns}\Big (n_f, \frac{\mu ^2}{m^2}\Big ) \otimes \left[ f_k(n_f, \mu ^2) + f_{\bar{k}}(n_f, \mu ^2)\right] \nonumber \\&\qquad + \frac{1}{n_f} A_{qq,h}^{ps}\Big (n_f, \frac{\mu ^2}{m^2}\Big ) \otimes {q^{s}(n_f, \mu ^2)} \nonumber \\&\qquad + \frac{1}{n_f} A_{qg,h}^{s}\Big (n_f, \frac{\mu ^2}{m^2}\Big ) \otimes {g(n_f, \mu ^2)}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (17) </div></div> <div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&g(n_f+1, \mu ^2) = A_{gq,h}^{s}\Bigl (n_f,\frac{\mu ^2}{m^2}\Bigr ) \otimes q^{s}(n_f,\mu ^2)\nonumber \\&\quad + A_{gg,h}^{s}\Bigl (n_f,\frac{\mu ^2}{m^2}\Bigr ) \otimes g(n_f,\mu ^2), \end{aligned}$$</span></div><div class="c-article-equation__number"> (18) </div></div> <div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&q^{s}(n_f+1,\mu ^2) = \left[ A_{qq,h}^{ns}\left( n_f, \frac{\mu ^2}{m^2}\right) + A_{qq,h}^{ps} \left( n_f, \frac{\mu ^2}{m^2}\right) \right. \nonumber \\&\quad \left. +A_{hq}^{ps} \left( n_f, \frac{\mu ^2}{m^2}\right) \right] \otimes q^{s}(n_f,\mu ^2) \nonumber \\&\quad +\left[ A^{s}_{qg,h}\left( n_f, \frac{\mu ^2}{m^2}\right) + A^{s}_{hg}\left( n_f, \frac{\mu ^2}{m^2}\right) \right] \nonumber \\&\quad \otimes g(n_f,\mu ^2). \end{aligned}$$</span></div><div class="c-article-equation__number"> (19) </div></div><p>PDFs for charm and bottom (<span class="mathjax-tex">$h=c,b$</span>) are then constructed as</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} f_{h+\bar{h}}(n_f+1, \mu ^2)= & {} {A_{hq}^{ps}\left( n_f, \frac{\mu ^2}{m^2}\right) } \otimes {q^{s}(n_f, \mu ^2)}\nonumber \\&+ {A_{hg}^{s}\left( n_f, \frac{\mu ^2}{m^2}\right) } \otimes {g(n_f, \mu ^2)} \end{aligned}$$</span></div><div class="c-article-equation__number"> (20) </div></div><p>at the matching scale <span class="mathjax-tex">$\mu $</span> from the quark singlet and gluon PDFs with <span class="mathjax-tex">$h = {\bar{h}}$</span>.</p><p>The matching conditions are typically imposed at the scale <span class="mathjax-tex">$\mu = m_h$</span>, and <span class="mathjax-tex">$f_{h + {\bar{h}}} = 0$</span> is assumed for scales <span class="mathjax-tex">$\mu \le m_h$</span>. The necessary heavy-quark OMEs <span class="mathjax-tex">$A_{ij}$</span> depend logarithmically on the heavy-quark masses as <span class="mathjax-tex">$\alpha _s^l \ln ^k(\mu ^2/m_h^2)$</span> with <span class="mathjax-tex">$0 \le k \le l$</span> in the perturbative expansion. As discussed above, the OMEs are known to NLO analytically [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 149" title="I. Bierenbaum, J. Blümlein, S. Klein, Two-loop massive operator matrix elements and unpolarized heavy flavor production at asymptotic values 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR155" id="ref-link-section-d90629110e21989">155</a>]. Thus, charm and bottom PDFs can be consistently extracted in QCD with a fixed number <span class="mathjax-tex">$n_f =3,4$</span> or 5.</p><p>It should be stressed, however, that the decoupling relations for PDFs in Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ17">17</a> <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ18">18</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ20">20</a>) assume the presence of one heavy quark at a time upon moving from lower scales to higher ones. Beginning at three-loop order, however, there are graphs containing both charm- and bottom-quark lines, and charm quarks cannot be treated as massless at the scale of the bottom-quark due to <span class="mathjax-tex">$(m_c/m_b)^2 \approx 1/10$</span>. Such terms cannot be attributed to either the charm- or bottom-quark PDFs, but rather one has to decouple charm and bottom quarks together at some large scale. The simultaneous decoupling of bottom and charm quarks in the strong coupling constant <span class="mathjax-tex">$\alpha _s$</span> is discussed, for instance, in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 156" title="A.G. Grozin, M. Höschele, J. Hoff, M. Steinhauser, Simultaneous decoupling of bottom and charm quarks. JHEP 09, 066 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR156" id="ref-link-section-d90629110e22125">156</a>].</p><h3 class="c-article__sub-heading" id="Sec10"><span class="c-article-section__title-number">3.3 </span>Heavy-quarks schemes</h3><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec11"><span class="c-article-section__title-number">3.3.1 </span>Variable-flavor number schemes</h4><p>The hard scattering cross sections also depend on the number of flavors <span class="mathjax-tex">$n_f$</span> and additional parton channels may open up, which have to be included as well. In addition, processes involving massive quarks depend logarithmically on the ratio <span class="mathjax-tex">$Q^2/m_h^2$</span>, where <i>Q</i> is some hard scale associated with the scattering. For the heavy-flavor Wilson coefficients in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ16">16</a>) these logarithms are of the type <span class="mathjax-tex">$\alpha _s^l \ln ^k(Q^2/m_h^2)$</span> with <span class="mathjax-tex">$1 \le k \le l$</span> in perturbation theory. These originate from collinear singularities screened by the heavy-quark mass due to the constrained phase space for gluon emission from massive quark lines, and as a prefactor of these logarithms one has the standard splitting functions. In addition to logarithmic terms, there are also power corrections <span class="mathjax-tex">$(m_h^2/Q^2)^l$</span> in the heavy-flavor Wilson coefficients, usually appearing in form of higher transcendental functions. In the asymptotic regime of <span class="mathjax-tex">$Q^2 \gg m_h^2$</span> the logarithms dominate and the kinematic dependence is measured experimentally, for instance in the tagged flavor case for charm-quark pairs in the structure function <span class="mathjax-tex">$F_2^{c\bar{c}}$</span>. Logarithms of a similar kind are also experimentally observed in differential distributions, e.g. due to the QED corrections proportional to <span class="mathjax-tex">$\ln ^k(Q^2/m_l^2)$</span> with <span class="mathjax-tex">$m_l$</span> being the charged lepton mass, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="A. Kwiatkowski, H. Spiesberger, H.J. Möhring, Heracles: an event generator for 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e23039">162</a>] or RT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e23043">163</a>]. These keep <span class="mathjax-tex">$m_h \ne 0$</span> and are to be distinguished from the zero-mass VFNS (ZM-VFNS), which describes essentially the massless case. Note that presently the GM-VFNS are applied only to one single heavy flavor at the time. That is the sequential transition <span class="mathjax-tex">$n_f \rightarrow n_f+1$</span>, so that the charm or bottom quarks are not considered simultaneously and charm-quark mass effects in the bottom-quark structure function <span class="mathjax-tex">$F_i^b$</span> are neglected as discussed above.</p><p>The various GM-VFNS contain a number of additional assumptions, and some come in more than one variety. The GM-VFNS differ, for instance, in the way the low-<span class="mathjax-tex">$Q^2$</span> region is modeled. This modeling is a necessary undertaking to provide a reasonable behavior of the VFNS in the kinematical regime of present DIS data. Additional assumptions in the GM-VFNS are related to the matching scale <span class="mathjax-tex">$\mu $</span> for the transition <span class="mathjax-tex">$n_f \rightarrow n_f+1$</span> as the adopted choice <span class="mathjax-tex">$\mu = m_h$</span> is not unique, see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 164" title="S. Alekhin, J. Blümlein, S. Moch, Heavy-quark production in deep-inelastic scattering, PoS DIS2013 297 (2013). 
 arXiv:1307.7258
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR164" id="ref-link-section-d90629110e23264">164</a>] for an in-depth discussion.</p><p>Briefly, the problem can be illustrated with the heavy-quark velocity, the leading order formula [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 131" title="M. Glück, E. Hoffmann, E. Reya, Scaling violations and the gluon distribution of the nucleon. Z. Phys. C 13, 119 (1982)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR131" id="ref-link-section-d90629110e23271">131</a>] being</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} v= & {} \sqrt{ 1 - \frac{4 m_h^2}{s}} = \sqrt{ 1 - \frac{4 m_h^2 x}{Q^2(1-x)}}, \nonumber \\ x\le & {} \frac{1}{1+ 4 m_h^2/Q^2}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (21) </div></div><p>The transition <span class="mathjax-tex">$n_f \rightarrow n_f+1$</span> when the corresponding flavor is considered as nearly massless requires light-like velocities <span class="mathjax-tex">$v \simeq 1$</span>. That implies the absence of all power corrections <span class="mathjax-tex">$(m_h^2/Q^2)^l$</span> in the heavy-flavor Wilson coefficients at the matching scale <span class="mathjax-tex">$\mu ^2$</span>. In practice, the matching is often applied at the scale <span class="mathjax-tex">$\mu ^2 = m_h^2$</span> and for kinematics <img src="//media.springernature.com/full/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_IEq375_HTML.gif" alt="">, where this condition is not fulfilled, which implies restrictions on the range in <i>x</i> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ21">21</a>).</p><p>Finally, the logarithmic accuracy of the resummation for large scales <span class="mathjax-tex">$Q^2 \gg m_h^2$</span>, or the order of perturbation theory in current implementations of GM-VFNS, is often not consistent. For example, NNLO evolution [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 103" title="S. Moch, J.A.M. Vermaseren, A. Vogt, The three-loop splitting functions in QCD: The nonsinglet case. Nucl. Phys. B 688, 101–134 (2004). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR103" id="ref-link-section-d90629110e23703">103</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 104" title="A. Vogt, S. Moch, J.A.M. Vermaseren, The three-loop splitting functions in QCD: The singlet case. Nucl. Phys. B 691, 129–181 (2004). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR104" id="ref-link-section-d90629110e23706">104</a>] of the massless PDFs is sometimes combined with the heavy-quark OMEs at NLO [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 149" title="I. Bierenbaum, J. Blümlein, S. Klein, Two-loop massive operator matrix elements and unpolarized heavy flavor production at asymptotic values 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR154" id="ref-link-section-d90629110e23712">154</a>], omitting NNLO results [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 105" title="J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, The 3-loop pure singlet heavy flavor contributions to the structure function 
 
 
 
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 and the anomalous dimension. Nucl. Phys. B 890, 48 (2014). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR105" id="ref-link-section-d90629110e23716">105</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 140" title="I. Bierenbaum, J. Blümlein, S. Klein, Mellin moments of the 
 
 
 
 $${\cal O}(\alpha ^3_s)$$
 
 
 
 O
 (
 
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 s
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 $$Q^2 \gg m^2$$
 
 
 
 
 Q
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 ≫
 
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 and anomalous dimensions. Nucl. Phys. B 820, 417 (2009). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR140" id="ref-link-section-d90629110e23719">140</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 144" title="J. Ablinger, J. Blümlein, S. Klein, C. Schneider, F. Wissbrock, The 
 
 
 
 $${\cal {O}}(\alpha _s^3)$$
 
 
 
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 (
 
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 $${\cal {O}}(n_f)$$
 
 
 
 O
 (
 
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 $$F_2(x, Q^2)$$
 
 
 
 
 F
 2
 
 
 (
 x
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 and transversity. Nucl. Phys. B 844, 26 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR144" id="ref-link-section-d90629110e23722">144</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 147" title="H. Kawamura, N.A. Lo, Presti, S. Moch, A. Vogt, On the next-to-next-to-leading order QCD corrections to heavy-quark production in deep-inelastic scattering. Nucl. Phys. B864, 399 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR147" id="ref-link-section-d90629110e23725">147</a>].</p><p>Altogether, these facts introduce a significant model dependence in any GM-VFNS implementation. A sensitive parameter to test this model dependence is the extraction of the charm- or bottom-quark mass used in different versions of GM-VFNS and subsequent comparison with the Particle Data Group (PDG) results [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e23731">55</a>]. In addition, the quality of the various GM-VFNS can be quantified with the goodness-of-fit for the description of HERA data on DIS charm-quark production obtained from the combination of individual H1 and ZEUS results [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e23734">165</a>].</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-4"><figure><figcaption class="c-article-table__figcaption"><b id="Tab4" data-test="table-caption">Table 4 Values of the charm-quark mass and renormalization scheme used in the PDF fits together with a summary of schemes chosen for the description of the charm-quark structure function <span class="mathjax-tex">$F_2^c$</span> and the theoretical accuracy for the massive quark DIS Wilson coefficients. The values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the DIS charm production cross section data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e23798">165</a>] and HERA inclusive cross section data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e23801">4</a>] are given in two columns with the account of PDF uncertainties (with unc., where CT14 PDF errors scaled from 90 % c.l. to 68 % c.l., i.e., reduced by a factor 1.645) and for the central prediction of each PDF set (nominal). In xFitter [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 166" title="S. Alekhin, O. Behnke, P. Belov, S. Borroni, M. Botje, et al., HERAFitter, Open source QCD fit project. 
 arXiv:1410.4412
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR166" id="ref-link-section-d90629110e23804">166</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 167" title="xFitter, An open source QCD fit framework. 
 http://xFitter.org
 
 [xFitter.org]" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR167" id="ref-link-section-d90629110e23808">167</a>], the values of electroweak parameters like the Fermi constant and <i>W</i>-boson mass are taken from Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e23814">55</a>]. The values for CT14 and for PDF4LHC with the SACOT(<span class="mathjax-tex">$\chi $</span>) scheme have been determined with a cut on <span class="mathjax-tex">$Q^2 \ge 5 \,\mathrm {GeV}^2$</span> on the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e23881">165</a>]</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/4" aria-label="Full size table 4"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec12"><span class="c-article-section__title-number">3.3.2 </span>Validation with DIS charm-quark production</h4><p>The H1 and ZEUS combined data for the DIS charm production cross section are unique for tests of GM-VFNS and span the region of <span class="mathjax-tex">$2.5 \le Q^2 \le 2000~\,\mathrm {GeV}^2$</span> and <span class="mathjax-tex">$3\times 10^{-5} \le x \le 0.05$</span>. Values for the charm-quark mass and <span class="mathjax-tex">$\chi ^2$</span>/NDP for the individual PDF sets ABM12, CJ15, CT14, HERAPDF2.0, JR14, MMHT14, NNPDF3.0 as well as the averaged set PDF4LHC15 are given in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a>, along with the information on the scheme choice for the heavy-quark structure functions and on the theoretical accuracy for the massive quark DIS Wilson coefficients. For reference, Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a> also list the <span class="mathjax-tex">$\chi ^2$</span>/NDP values for the HERA inclusive cross section data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e26868">4</a>]. Comparisons to data for the DIS charm production cross section are shown in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig7">7</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig8">8</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig9">9</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig10">10</a>. Note that Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a> adopt the standard definition of perturbative orders for the heavy-quark structure functions. This is not shared by CT14, MMHT14 and NNPDF3.0 in their GM-VFNS. There the Born contribution to the heavy-quark Wilson coefficients for <span class="mathjax-tex">$ep \rightarrow c{\bar{c}}$</span>, which is proportional to <span class="mathjax-tex">$\mathcal{O}(\alpha _s)$</span>, is referred to as being “NLO”. Analogously, the one-loop corrections of order <span class="mathjax-tex">$\mathcal{O}(\alpha _s^2)$</span> are denoted by “NNLO”.</p><p>Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a> and Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig7">7</a> show that the ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e27015">2</a>] and JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e27018">5</a>] PDFs at NNLO, using charm-quark masses in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme, provide a good description of the data. Both ABM12 and JR14 use the approximate massive three-loop Wilson coefficients as obtained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 147" title="H. Kawamura, N.A. Lo, Presti, S. Moch, A. Vogt, On the next-to-next-to-leading order QCD corrections to heavy-quark production in deep-inelastic scattering. Nucl. Phys. B864, 399 (2012). 
 arXiv:1205.5727
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR147" id="ref-link-section-d90629110e27052">147</a>] by interpolating between existing <span class="mathjax-tex">$\mathcal{O}(\alpha _s^3)$</span> soft-gluon threshold resummation results and the <span class="mathjax-tex">$\mathcal{O}(\alpha _s^3)$</span> asymptotic <span class="mathjax-tex">$(Q^2 \gg m_c^2)$</span> coefficients [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 140" title="I. Bierenbaum, J. Blümlein, S. Klein, Mellin moments of the 
 
 
 
 $${\cal O}(\alpha ^3_s)$$
 
 
 
 O
 (
 
 α
 s
 3
 
 )
 
 
 
 heavy flavor contributions to unpolarized deep-inelastic scattering at 
 
 
 
 $$Q^2 \gg m^2$$
 
 
 
 
 Q
 2
 
 ≫
 
 m
 2
 
 
 
 
 and anomalous dimensions. Nucl. Phys. B 820, 417 (2009). 
 arXiv:0904.3563
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR140" id="ref-link-section-d90629110e27177">140</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 144" title="J. Ablinger, J. Blümlein, S. Klein, C. Schneider, F. Wissbrock, The 
 
 
 
 $${\cal {O}}(\alpha _s^3)$$
 
 
 
 O
 (
 
 α
 s
 3
 
 )
 
 
 
 massive operator matrix elements of 
 
 
 
 $${\cal {O}}(n_f)$$
 
 
 
 O
 (
 
 n
 f
 
 )
 
 
 
 for the structure function 
 
 
 
 $$F_2(x, Q^2)$$
 
 
 
 
 F
 2
 
 
 (
 x
 ,
 
 Q
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 and transversity. Nucl. Phys. B 844, 26 (2011). 
 arXiv:1008.3347
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR144" id="ref-link-section-d90629110e27180">144</a>]. This is referred to as <span class="mathjax-tex">$\mathcal{O}(\alpha _s^3)_\mathrm{approx}$</span> in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a>. The HERAPDF2.0 fit [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e27235">4</a>] also obtains a good description of the data, cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig8">8</a>. This is the only set which has fitted also to the HERA inclusive cross section data of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e27241">4</a>]. On the other hand, the SACOT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 160" title="M. Krämer, F.I. Olness, D.E. Soper, Treatment of heavy quarks in deeply inelastic scattering. Phys. Rev. D 62, 096007 (2000). 
 arXiv:hep-ph/0003035
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR160" id="ref-link-section-d90629110e27244">160</a>] GM-VFNS at NLO used by CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
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 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e27248">1</a>] does not describe the data too well, although we should note that the HERA charm data were not included in the CJ15 fit itself.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-7" data-title="Fig. 7"><figure><figcaption><b id="Fig7" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 7</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/7" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig7_HTML.gif?as=webp"><img aria-describedby="Fig7" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig7_HTML.gif" alt="figure 7" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-7-desc"><p>Comparison of HERA data for the DIS pair-production of charm quarks [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e27261">165</a>] to the QCD predictions at NNLO in the FFNS using ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e27264">2</a>] and JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e27267">5</a>] PDFs with a running charm-quark mass</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/7" data-track-dest="link:Figure7 Full size image" aria-label="Full size image figure 7" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-8" data-title="Fig. 8"><figure><figcaption><b id="Fig8" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 8</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/8" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig8_HTML.gif?as=webp"><img aria-describedby="Fig8" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig8_HTML.gif" alt="figure 8" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-8-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig7">7</a> with QCD predictions at NLO and NNLO in the RT optimal [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
 arXiv:1201.6180
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e27292">163</a>] VFNS using the HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e27295">4</a>] PDF sets at NLO and NNLO</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/8" data-track-dest="link:Figure8 Full size image" aria-label="Full size image figure 8" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-9" data-title="Fig. 9"><figure><figcaption><b id="Fig9" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 9</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/9" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig9_HTML.gif?as=webp"><img aria-describedby="Fig9" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig9_HTML.gif" alt="figure 9" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-9-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig7">7</a> with QCD predictions at NLO and different versions of VFNS using the PDFs CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e27320">3</a>] (SACOT-<span class="mathjax-tex">$\chi $</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 161" title="W.-K. Tung, S. Kretzer, C. Schmidt, Open heavy flavor production in QCD: Conceptual framework and implementation issues. J. Phys. G28, 983 (2002). 
 arXiv:hep-ph/0110247
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR161" id="ref-link-section-d90629110e27343">161</a>]), MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e27346">6</a>] (RT optimal [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
 arXiv:1201.6180
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e27350">163</a>]), and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e27353">7</a>] (FONLL-B [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 162" title="S. Forte, E. Laenen, P. Nason, J. Rojo, Heavy quarks in deep-inelastic scattering. Nucl. Phys. B 834, 116 (2010). 
 arXiv:1001.2312
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e27356">162</a>])</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/9" data-track-dest="link:Figure9 Full size image" aria-label="Full size image figure 9" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-10" data-title="Fig. 10"><figure><figcaption><b id="Fig10" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/10" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig10_HTML.gif?as=webp"><img aria-describedby="Fig10" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig10_HTML.gif" alt="figure 10" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-10-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig7">7</a> with QCD predictions at NLO using the <span class="u-monospace">PDF4LHC_100</span> PDF set and various VFNS: FONLL-B [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 162" title="S. Forte, E. Laenen, P. Nason, J. Rojo, Heavy quarks in deep-inelastic scattering. Nucl. Phys. B 834, 116 (2010). 
 arXiv:1001.2312
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e27384">162</a>], RT optimal [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
 arXiv:1201.6180
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e27387">163</a>] and SACOT-<span class="mathjax-tex">$\chi $</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 161" title="W.-K. Tung, S. Kretzer, C. Schmidt, Open heavy flavor production in QCD: Conceptual framework and implementation issues. J. Phys. G28, 983 (2002). 
 arXiv:hep-ph/0110247
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR161" id="ref-link-section-d90629110e27411">161</a>]</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/10" data-track-dest="link:Figure10 Full size image" aria-label="Full size image figure 10" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The remarkable fact in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a> and Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig9">9</a> is, however, that the GM-VFNS SACOT(<span class="mathjax-tex">$\chi $</span>) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 161" title="W.-K. Tung, S. Kretzer, C. Schmidt, Open heavy flavor production in QCD: Conceptual framework and implementation issues. J. Phys. G28, 983 (2002). 
 arXiv:hep-ph/0110247
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR161" id="ref-link-section-d90629110e27452">161</a>] of CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e27455">3</a>] and RT optimal [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
 arXiv:1201.6180
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e27459">163</a>] of MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e27462">6</a>] have difficulties in describing the DIS charm production data. Note that MMHT14 models the heavy-quark Wilson coefficient functions at <span class="mathjax-tex">$\mathcal{O}(\alpha _s^3)$</span> for low <span class="mathjax-tex">$Q^2$</span> as described in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
 arXiv:1201.6180
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e27528">163</a>] using known leading threshold logarithms [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 168" title="E. Laenen, S. Moch, Soft gluon resummation for heavy quark electroproduction. Phys. Rev. D 59, 034027 (1999). 
 arXiv:hep-ph/9809550
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR168" id="ref-link-section-d90629110e27531">168</a>] and <span class="mathjax-tex">$\ln (1/x)$</span> terms [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 148" title="S. Catani, M. Ciafaloni, F. Hautmann, High-energy factorization and small x heavy flavor production. Nucl. Phys. B 366, 135 (1991)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR148" id="ref-link-section-d90629110e27571">148</a>], which have been shown not to be leading. This is indicated as <span class="mathjax-tex">$\mathcal{O}(\alpha _s^2)$</span> in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a>. Note that CT14 has applied a universal cut of <span class="mathjax-tex">$Q^2 \ge 4~\,\mathrm {GeV}^2$</span> on all DIS data, excluding the bin at <span class="mathjax-tex">$Q^2 = 2.5~\,\mathrm {GeV}^2$</span> in the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e27711">165</a>] from the fit (cf. the upper left plot in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig9">9</a>). We have checked that including the low <span class="mathjax-tex">$Q^2$</span> bin leads to a dramatic deterioration of the fit quality.</p><p>In addition, the schemes SACOT(<span class="mathjax-tex">$\chi $</span>) and RT optimal as well as FONLL-C [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 162" title="S. Forte, E. Laenen, P. Nason, J. Rojo, Heavy quarks in deep-inelastic scattering. Nucl. Phys. B 834, 116 (2010). 
 arXiv:1001.2312
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e27764">162</a>] of NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e27767">7</a>] do not improve the fit quality when comparing NLO and NNLO fits. We note in this context that those fits do not include the exact asymptotic [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 105" title="J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, The 3-loop pure singlet heavy flavor contributions to the structure function 
 
 
 
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 (
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 $${\cal {O}}(\alpha _s^3)$$
 
 
 
 O
 (
 
 α
 s
 3
 
 )
 
 
 
 massive operator matrix elements of 
 
 
 
 $${\cal {O}}(n_f)$$
 
 
 
 O
 (
 
 n
 f
 
 )
 
 
 
 for the structure function 
 
 
 
 $$F_2(x, Q^2)$$
 
 
 
 
 F
 2
 
 
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 Q
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR144" id="ref-link-section-d90629110e27777">144</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 146" title="J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wißbrock, The 3-loop non-singlet heavy flavor contributions and anomalous dimensions for the structure function 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR146" id="ref-link-section-d90629110e27780">146</a>] and approximate [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 147" title="H. Kawamura, N.A. Lo, Presti, S. Moch, A. Vogt, On the next-to-next-to-leading order QCD corrections to heavy-quark production in deep-inelastic scattering. Nucl. Phys. B864, 399 (2012). 
 arXiv:1205.5727
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR147" id="ref-link-section-d90629110e27783">147</a>] <span class="mathjax-tex">$O(\alpha _s^3)$</span> results for the heavy-quark Wilson coefficients in their theory predictions. The averaged set PDF4LHC15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e27824">8</a>], shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig10">10</a>, mixes PDFs derived with different mass schemes (ACOT, FONNL and RT) and does not describe the data very well for virtualities up to <span class="mathjax-tex">$Q^2 \lesssim 20~\,\mathrm {GeV}^2$</span>.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec13"><span class="c-article-section__title-number">3.3.3 </span>Charm-quark mass</h4><p>Dedicated studies of the charm-quark mass dependence have been performed by several groups. In the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme, the value of <span class="mathjax-tex">$m_c(m_c)=1.24~^{+~0.04}_{-~0.08}~$</span>GeV has been obtained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 169" title="S. Alekhin, J. Blümlein, K. Daum, K. Lipka, S. Moch, Precise charm-quark mass from deep-inelastic scattering. Phys. Lett. B 720, 172 (2013). 
 arXiv:1212.2355
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR169" id="ref-link-section-d90629110e27989">169</a>] together with <span class="mathjax-tex">$\chi ^2/$</span>NDP=61/52 for the description of the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e28023">165</a>] as a variant of the ABM11 fit [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e28027">64</a>]. Other groups, which keep a fixed value of <span class="mathjax-tex">$m_c$</span> in the analyses, cf. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a>, have studied the effects of varying <span class="mathjax-tex">$m_c$</span> in predefined ranges. This has been done, for example, in the older NNPDF2.1 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e28081">170</a>] and MSTW analyses [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 171" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Heavy-quark mass dependence in global PDF analyses and 3- and 4-flavour parton distributions. Eur. Phys. J. C 70, 51 (2010). 
 arXiv:1007.2624
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR171" id="ref-link-section-d90629110e28084">171</a>] as well as for the MMHT PDFs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e28088">172</a>]. The latter yields a pole mass of <span class="mathjax-tex">$m_c^\mathrm{pole}=1.25~\,\mathrm {GeV}$</span> as the best fit with <span class="mathjax-tex">$\chi ^2$</span>/NDP = 75/52, while the nominal fit uses <span class="mathjax-tex">$m_c^\mathrm{pole}=1.4~\,\mathrm {GeV}$</span> at the price of a deterioration in the value of <span class="mathjax-tex">$\chi ^2$</span>/NDP = 82/52. HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
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 e
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 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e28225">4</a>] has performed a scan of the values of <span class="mathjax-tex">$\chi ^2$</span>/NDP leading to <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.43~\,\mathrm {GeV}$</span> at NNLO quoted in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a> as the best fit. NNPDF3.0 computes heavy-quark structure functions with expressions for the pole mass definition, but adopts numerical values for the charm-quark pole mass, <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.275~\,\mathrm {GeV}$</span>, which corresponds to the current PDG value for the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>mass. This value is different from the one used in NNPDF2.3, namely <span class="mathjax-tex">$m_c^\mathrm{pole} = \sqrt{2}~\,\mathrm {GeV}$</span>. Within the framework of the CT10 PDFs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 173" title="H.L. Lai, M. Guzzi, J. Huston, Z. Li, P.M. Nadolsky, J. Pumplin, C.-P. Yuan, New parton distributions for collider physics. Phys. Rev. D 82, 074024 (2010). 
 arXiv:1007.2241
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR173" id="ref-link-section-d90629110e28417">173</a>] the charm-quark mass in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme has been determined in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 174" title="J. Gao, M. Guzzi, P.M. Nadolsky, Charm quark mass dependence in a global QCD analysis. Eur. Phys. J. C 73, 2541 (2013). 
 arXiv:1304.3494
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR174" id="ref-link-section-d90629110e28450">174</a>] using the SACOT(<span class="mathjax-tex">$\chi $</span>) scheme at order <span class="mathjax-tex">$\mathcal{O}(\alpha _s^2)$</span>, although with a significant spread in the central values reported (<span class="mathjax-tex">$m_c(m_c)=1.12 - 1.24~$</span>GeV) depending on assumption in the fit.</p><p>In this context, it is worth to point out that the running mass <span class="mathjax-tex">$m_c(\mu )$</span> in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme is free from renormalon ambiguities and can therefore be determined with high precision. The PDG [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e28638">55</a>] quotes <span class="mathjax-tex">$m_c(m_c) = 1.275 \pm 0.025~\,\mathrm {GeV}$</span> based on the averaging different mass determination in various kinematics. DIS charm-quark production analyzed in the FFNS <span class="mathjax-tex">$(n_f=3)$</span> leads to <span class="mathjax-tex">$m_c(m_c) = 1.24 \pm 0.03~^{+~0.03}_{-~0.03}$</span> GeV at NNLO [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 169" title="S. Alekhin, J. Blümlein, K. Daum, K. Lipka, S. Moch, Precise charm-quark mass from deep-inelastic scattering. Phys. Lett. B 720, 172 (2013). 
 arXiv:1212.2355
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR169" id="ref-link-section-d90629110e28813">169</a>], while measurements of the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>mass in <span class="mathjax-tex">$e^+e^-$</span> annihilation give, for instance, <span class="mathjax-tex">$m_c(m_c) = 1.279 \pm 0.013$</span> GeV [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 175" title="K.G. Chetyrkin, J.H. Kühn, A. Maier, P. Maierhöfer, P. Marquard, M. Steinhauser, C. Sturm, Charm and bottom quark masses: an update. Phys. Rev. D 80, 074010 (2009). 
 arXiv:0907.2110
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR175" id="ref-link-section-d90629110e28932">175</a>] and <span class="mathjax-tex">$m_c(m_c) = 1.288 \pm 0.020$</span> GeV [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 176" title="B. Dehnadi, A.H. Hoang, V. Mateu, Bottom and charm mass determinations with a convergence test. JHEP 08, 155 (2015). 
 arXiv:1504.07638
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR176" id="ref-link-section-d90629110e28988">176</a>]. The determination from quarkonium 1<i>S</i> energy levels yields <span class="mathjax-tex">$m_c(m_c) = 1.246 \pm 0.023$</span> GeV [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 177" title="Y. Kiyo, G. Mishima, Y. Sumino, Determination of 
 
 
 
 $$m_c$$
 
 
 
 m
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 m
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR177" id="ref-link-section-d90629110e29046">177</a>]. All these values are consistent with each other within the uncertainties.</p><p>In contrast, the accuracy of the pole mass <span class="mathjax-tex">$m_c^\mathrm{pole}$</span> is limited to be of the order of the QCD scale <span class="mathjax-tex">$\Lambda _\mathrm{QCD}$</span> and, moreover, the conversion from the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>mass <span class="mathjax-tex">$m_c(m_c)$</span> at low scales to the pole mass <span class="mathjax-tex">$m_c^\mathrm{pole}$</span> does not converge. Using <span class="mathjax-tex">$\alpha _s(M_Z)=0.1184$</span>, for example, the conversion yields for the central value of the PDG <span class="mathjax-tex">$m_c^\mathrm{pole}=1.47~\,\mathrm {GeV}$</span> at one loop, <span class="mathjax-tex">$m_c^\mathrm{pole}=1.67~\,\mathrm {GeV}$</span> at two loops, <span class="mathjax-tex">$m_c^\mathrm{pole}=1.93~\,\mathrm {GeV}$</span> at three loops, and <span class="mathjax-tex">$m_c^\mathrm{pole}=2.39~\,\mathrm {GeV}$</span> at four-loops [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 142" title="P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Quark mass relations to four-loop order in perturbative QCD. Phys. Rev. Lett. 114, 142002 (2015). 
 arXiv:1502.01030
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR142" id="ref-link-section-d90629110e29422">142</a>]. The PDG quotes <span class="mathjax-tex">$m_c^\mathrm{pole}=1.67 \pm 0.07~\,\mathrm {GeV}$</span> for conversion at two loops.</p><p>The low values for the pole mass of the charm quark assumed or obtained in some PDF fits as shown in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a> are thus not compatible with other determinations and with the world average. The rigorous determination of the charm-quark mass discussed, for instance, in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 169" title="S. Alekhin, J. Blümlein, K. Daum, K. Lipka, S. Moch, Precise charm-quark mass from deep-inelastic scattering. Phys. Lett. B 720, 172 (2013). 
 arXiv:1212.2355
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR169" id="ref-link-section-d90629110e29478">169</a>] provides a more controlled way of determining <span class="mathjax-tex">$m_c$</span> from the world DIS data, taking also into account its correlation with <span class="mathjax-tex">$\alpha _s(M_Z)$</span>.</p><h3 class="c-article__sub-heading" id="Sec14"><span class="c-article-section__title-number">3.4 </span>Light-flavor PDFs</h3><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec15"><span class="c-article-section__title-number">3.4.1 </span>Up- and down-quark distributions</h4><p>The total quark contribution to nucleon matrix elements is known fairly well due to constraints from the available DIS data obtained in the fixed-target and collider experiments in the <i>x</i>-range <span class="mathjax-tex">$10^{-4} \lesssim x \lesssim 0.8$</span>. However, a thorough disentangling of the quark flavor structure is still a challenging task in any PDF analysis. At moderate and large <i>x</i> values this has been routinely achieved by using a combination of the DIS data obtained on proton and deuteron targets. However, uncertainties in the modeling of nuclear corrections in the deuteron introduce a controllable source of theoretical uncertainty on the <i>d</i>-quark PDF obtained in this way, especially at large <i>x</i>, as discussed below.</p><p>An alternative way to resolve the <i>u</i>- and <i>d</i>-quark contributions is to use data on <i>W</i>- and <i>Z</i>-boson production obtained in <i>pp</i> and <span class="mathjax-tex">$p\bar{p}$</span> collisions at the LHC and Tevatron, respectively. Those experiments probe the <i>W</i> and <i>Z</i> rapidity distributions up to rapidities of <span class="mathjax-tex">$y=3-4$</span>, depending on details of the experiments, with an integrated luminosity of <span class="mathjax-tex">$O(1)~\mathrm{fb}^{-1}$</span> achieved in each run. Such data samples are quite competitive in accuracy with the ones obtained in fixed-target DIS experiments, and provide simultaneously constraints on the quark and anti-quark PDFs at large and small <i>x</i>. Furthermore, the <i>d</i>-quark PDF extracted from a combination of the existing data on DIS off protons and <i>W</i> / <i>Z</i>-boson production in <span class="mathjax-tex">$pp (p\bar{p})$</span> collisions is not sensitive to nuclear corrections. Moreover, if DIS data with small hadronic invariant masses <span class="mathjax-tex">$W^2$</span> are not used in the analyses in order to reduce the sensitivity to higher twist contributions, the statistical potential of the deuteron data is reduced and they become less competitive as compared to the collider data, cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig11">11</a>.</p><p>As mentioned above, one can further constrain the <i>u</i>- and <i>d</i>-quark flavor separated distributions by utilizing fixed-target deuteron DIS data. However, nuclear effects need to be accounted for in cross sections and structure functions in order to access the underlying PDFs. The theoretical uncertainty inherent in this nuclear correction procedure should be added to the statistical PDF uncertainties. Nonetheless, the reduction of the uncertainties due to the increased number of fitted data points is even greater, leading to an overall smaller <i>d</i>-quark PDF uncertainty than in fits performed without deuterium data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="S. Alekhin, J. Blümlein, S. Moch, R. Placakyte, Iso-spin asymmetry of quark distributions and implications for single top-quark production at the LHC. 
 arXiv:1508.07923
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR30" id="ref-link-section-d90629110e29856">30</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 191" title="J.F. Owens, A. Accardi, W. Melnitchouk, Global parton distributions with nuclear and finite-
 
 
 
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 corrections. Phys. Rev. D 87, 094012 (2013). 
 arXiv:1212.1702
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR191" id="ref-link-section-d90629110e29859">191</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 192" title="A. Accardi, PDFs from nucleons to nuclei. PoS DIS2015, 001 (2015). 
 arXiv:1602.02035
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR192" id="ref-link-section-d90629110e29863">192</a>]. Furthermore, as shown in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e29866">1</a>], and discussed in more detail below, it is possible to significantly reduce the nuclear correction uncertainty by exploiting the interplay of the deuteron DIS data and the recent high-statistics DØ data on the reconstructed <span class="mathjax-tex">$W^\pm $</span> boson charge asymmetry at large rapidity, which is equally sensitive to the <i>d</i> / <i>u</i> ratio but is not affected by nuclear corrections.</p><p>The <i>W</i> / <i>Z</i>-boson collider data also provide a valuable constraint on the small-<i>x</i> quark PDFs. In particular, the charge asymmetry of leptons originating from the <i>W</i> decays is sensitive to the SU(2) flavor asymmetry of the non-strange sea, also referred to as the “isospin” asymmetry <span class="mathjax-tex">$I(x)=[\bar{d}(x)-\bar{u}(x)]$</span> at small <i>x</i>. This asymmetry is constrained by the DY data from fixed-targets with protons and deuterons collected by the Fermilab experiment E866 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 193" title="NuSea Collaboration, R.S. Towell et al., Improved measurement of the 
 
 
 
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 d
 
 
 ¯
 
 
 /
 
 
 u
 
 
 ¯
 
 
 
 
 
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 arXiv:hep-ex/0103030
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR193" id="ref-link-section-d90629110e30004">193</a>]. However, the E866 data are not sensitive to the value of <i>I</i>(<i>x</i>) at small <i>x</i> (<span class="mathjax-tex">$x \lesssim 0.2$</span>). Therefore, <i>I</i>(<i>x</i>) is sometimes parametrized in a Regge-like form as <span class="mathjax-tex">$I(x) \sim x^{0.5}$</span> such that it vanishes at <span class="mathjax-tex">$x=0$</span> (cf. the MMHT results in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig12">12</a>).</p><p>The large-rapidity tail of the <i>W</i> / <i>Z</i>-boson production data allows for a model-independent check of <i>I</i>(<i>x</i>) at small <i>x</i>. The asymmetry preferred by the combination of the currently available LHC and Tevatron data turns out to be negative at <span class="mathjax-tex">$x<0.01$</span>, while the Regge-like limit with a vanishing <i>I</i>(<i>x</i>) can still be recovered at <span class="mathjax-tex">$x \lesssim 10^{-5}$</span>, cf. the ABMP15 results in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig12">12</a>. The CT14 analysis only includes the Tevatron forward DY data, but also confirms the negative trend in <i>I</i>(<i>x</i>) at small <i>x</i>, with errors in <i>I</i>(<i>x</i>) being substantially larger than those from ABMP15.</p><p>Finally, an important issue is the theoretical accuracy which is employed in the description of the DY data. There are significant differences as shown in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab5">5</a> and these cause an additional spread in the fit quality and the results for the PDFs when comparing different NNLO PDF sets.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec16"><span class="c-article-section__title-number">3.4.2 </span>Strange-quark distribution</h4><p>The main information on the strange sea distribution comes from charm-quark production in neutrino-induced charged-current DIS experiments. The publication of data from CHORUS and NOMAD has recently enlarged the statistics available for those experiments. As a net result, the uncertainty in the strange PDF is now reduced down to a few percent at <span class="mathjax-tex">$x\sim 0.1$</span> (cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig13">13</a>). However, at small <i>x</i> the strange sea distribution is still poorly known in view of the restricted kinematics of the production of charm quarks from fixed targets. Furthermore, since neutrino DIS experiments usually involve nuclear targets, care needs to be taken when extracting free-nucleon PDFs from the nuclear cross sections. Nuclear effects in neutrino DIS and possible differences between those in charged-lepton DIS have been discussed recently in Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 192" title="A. Accardi, PDFs from nucleons to nuclei. PoS DIS2015, 001 (2015). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR194" id="ref-link-section-d90629110e30279">194</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 195" title="K. Kovarik, I. Schienbein, F.I. Olness, J.Y. Yu, C. Keppel, J.G. Morfin, J.F. Owens, T. Stavreva, Nuclear corrections in neutrino-nucleus DIS and their compatibility with global NPDF analyses. Phys. Rev. Lett. 106, 122301 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR195" id="ref-link-section-d90629110e30283">195</a>], for instance. Supplementary information on the strange sea at small <i>x</i>, independent of nuclear effects, can be obtained from the associated production of charm quarks and <i>W</i> bosons in the <i>pp</i> collisions at the LHC. A constraint from collider data on <span class="mathjax-tex">$W+c$</span> is potentially less sensitive to the <i>c</i>-quark fragmentation model compared to the one from semi-leptonic decays of charm, which plays major role in the existing fixed-target DIS experiments. The <span class="mathjax-tex">$W+c$</span> data collected by ATLAS and CMS prefer a somewhat enhanced strange sea as compared to the fixed-target determination, cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig13">13</a>. However, the NNLO QCD corrections to this process are still unknown. They are not taken into account in the analysis of <span class="mathjax-tex">$W+c$</span> data so far and may have a substantial influence on the fit. The strange sea extracted by ATLAS from an analysis of the combined inclusive data on the <i>W</i>- and <i>Z</i>-boson production is even further enhanced, which suggests a restoration of SU(3) flavor symmetry in the sea distributions. However, the accuracy of this determination is poor due to a limited potential of the inclusive data in disentangling the quark flavors. Therefore, the ATLAS result is in fact comparable with other determinations within uncertainties.</p><p>In general, the existing experimental constraints on the strange PDF are relatively poor. Therefore, the results of various determinations demonstrate a significant spread, which is mainly driven by the data selection. An additional spread between results of earlier PDF analyses appears due to implementation issues. In particular, the strong strange-sea suppression observed in the NNPDF2.1 analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e30389">170</a>] was related to an error in the DIS charm-quark production cross section being off by a factor of two for low scales due to an additional factor of <span class="mathjax-tex">$(1+m_c^2/Q^2)$</span> in Eq. (34) of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e30442">170</a>]. This is now correct in NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e30445">7</a>]. The CT10 analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 66" title="J. Gao, M. Guzzi, J. Huston, H.-L. Lai, Z. Li, P. Nadolsky, J. Pumplin, D. Stump, C.P. Yuan, CT10 next-to-next-to-leading order global analysis of QCD. Phys. Rev. D 89, 033009 (2014). 
 arXiv:1302.6246
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR66" id="ref-link-section-d90629110e30448">66</a>], which reported an enhanced strange sea, may be flawed due to a wrong sign of the photon-<i>Z</i> interference for massive quarks in the structure function <span class="mathjax-tex">$xF_3$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e30484">3</a>]. This error also concerns the earlier results on the strange–anti-strange asymmetry [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 196" title="H.L. Lai, P.M. Nadolsky, J. Pumplin, D. Stump, W.K. Tung, C.P. Yuan, The strange parton distribution of the nucleon: global analysis and applications. JHEP 04, 089 (2007). 
 arXiv:hep-ph/0702268
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR196" id="ref-link-section-d90629110e30487">196</a>] and has now been corrected in CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e30490">3</a>]. Finally, the MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 197" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Parton distributions for the LHC. Eur. Phys. J. C 63, 189 (2009). 
 arXiv:0901.0002
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR197" id="ref-link-section-d90629110e30493">197</a>] analysis suffered from an error in the NLO QCD correction for the charged-current DIS charm-quark production as it had omitted a part of the gluon Wilson coefficient at NLO, which was corrected in MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e30497">6</a>].</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-11" data-title="Fig. 11"><figure><figcaption><b id="Fig11" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 11</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/11" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig11_HTML.gif?as=webp"><img aria-describedby="Fig11" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig11_HTML.gif" alt="figure 11" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-11-desc"><p>The <span class="mathjax-tex">$1\sigma $</span> band for the <i>d</i> / <i>u</i> ratio for the 4-flavor scheme and at the factorization scale <span class="mathjax-tex">$\mu =2~\,\mathrm {GeV}$</span> obtained in the PDF analyses including forward <span class="mathjax-tex">$W^{\pm }$</span> data (CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e30604">3</a>]: <i>red right-tilted hatch</i>, ABMP15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="S. Alekhin, J. Blümlein, S. Moch, R. Placakyte, Iso-spin asymmetry of quark distributions and implications for single top-quark production at the LHC. 
 arXiv:1508.07923
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR30" id="ref-link-section-d90629110e30610">30</a>]: <i>gray shaded area</i>, CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e30616">1</a>]: <i>black dotted lines</i>) in comparison to those including the central <i>W</i>, <i>Z</i> data only and a cut of <span class="mathjax-tex">$W^2 \gtrsim 13~\,\mathrm {GeV}^2$</span> imposed on the deuteron DIS data (MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e30676">6</a>]: <i>blue dashed lines</i>, NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e30682">7</a>]: <i>green left-tilted hatch</i>)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/11" data-track-dest="link:Figure11 Full size image" aria-label="Full size image figure 11" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-12" data-title="Fig. 12"><figure><figcaption><b id="Fig12" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 12</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/12" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig12_HTML.gif?as=webp"><img aria-describedby="Fig12" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig12_HTML.gif" alt="figure 12" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-12-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig11">11</a> for the SU(2) flavor asymmetry of the light-quark sea, or the “isospin” asymmetry, <span class="mathjax-tex">$I(x)=[\bar{d}(x)-\bar{u}(x)]$</span> </p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/12" data-track-dest="link:Figure12 Full size image" aria-label="Full size image figure 12" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec17"><span class="c-article-section__title-number">3.5 </span>Nuclear corrections</h3><p>Many global PDF analyses make use of data with deuterium targets, such as lepton-deuteron DIS and proton-deuteron DY, as a way of obtaining stronger constraints on the flavor dependence of PDFs that are not possible with proton data alone. The use of deuterium data requires that one takes into account differences between PDFs in the deuteron and those in the free nucleon, which arise from effects such as nuclear Fermi motion and binding of the nucleons in the nucleus, as well as nucleon off-shell corrections and nuclear shadowing. While some analyses assume that nuclear corrections in the deuteron are negligible, a number of recent global PDF studies have incorporated nuclear effects into their analyses [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e30815">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e30818">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e30821">64</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 191" title="J.F. Owens, A. Accardi, W. Melnitchouk, Global parton distributions with nuclear and finite-
 
 
 
 $$Q^2$$
 
 
 
 Q
 2
 
 
 
 corrections. Phys. Rev. D 87, 094012 (2013). 
 arXiv:1212.1702
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR191" id="ref-link-section-d90629110e30824">191</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 198" title="S. Alekhin, J. Blümlein, S. Klein, S. Moch, The 3, 4, and 5-flavor NNLO parton from deep-inelastic scattering data and at hadron colliders. Phys. Rev. D 81, 014032 (2010). 
 arXiv:0908.2766
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR198" id="ref-link-section-d90629110e30827">198</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 200" title="A. Accardi, W. Melnitchouk, J.F. Owens, M.E. Christy, C.E. Keppel, L. Zhu, J.G. Morfin, Uncertainties in determining parton distributions at large 
 
 
 
 $$x$$
 
 
 x
 
 
 . Phys. Rev. D 84, 014008 (2011). 
 arXiv:1102.3686
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR200" id="ref-link-section-d90629110e30831">200</a>].</p><p>Generally, the nuclear effects become increasingly important at large values of <i>x</i> (<span class="mathjax-tex">$x \gtrsim 0.4$</span>), as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig14">14</a> illustrates for the ratio of the deuteron to isoscalar nucleon structure functions. In this region the nuclear PDFs can be computed through convolutions of the bound nucleon PDFs and nuclear smearing functions describing the momentum distributions of nucleons in the deuteron. The latter can be expressed in terms of deuteron wave functions, calculated from modern potentials based on high-precision fits to nucleon–nucleon scattering data. These potentials differ primarily in their treatment of the short range <i>NN</i> interaction, and the different strengths of the high-momentum tails of the wave functions translate directly to the magnitude of the nuclear corrections at large <i>x</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 199" title="A. Accardi, M.E. Christy, C.E. Keppel, P. Monaghan, W. Melnitchouk, J.G. Morfin, J.F. Owens, New parton distributions from large-
 
 
 
 $$x$$
 
 
 x
 
 
 and low-
 
 
 
 $$Q^2$$
 
 
 
 Q
 2
 
 
 
 data. Phys. Rev. D 81, 034016 (2010). 
 arXiv:0911.2254
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR199" id="ref-link-section-d90629110e30876">199</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 201" title="J. Arrington, J.G. Rubin, W. Melnitchouk, How well do we know the neutron structure function? Phys. Rev. Lett. 108, 252001 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR201" id="ref-link-section-d90629110e30879">201</a>].</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-5"><figure><figcaption class="c-article-table__figcaption"><b id="Tab5" data-test="table-caption">Table 5 Description of the ATLAS data at <span class="mathjax-tex">$\sqrt{s} = 7$</span> TeV for <span class="mathjax-tex">$W^\pm \rightarrow l^\pm \nu $</span>, <span class="mathjax-tex">$Z\rightarrow l^+l^-$</span> (Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="ATLAS Collaboration, G. Aad et al., Measurement of the inclusive 
 
 
 
 $$W^\pm $$
 
 
 
 W
 ±
 
 
 
 and Z/
 
 
 
 $$\gamma $$
 
 
 γ
 
 
 cross sections in the electron and muon decay channels in 
 
 
 
 $$pp$$
 
 
 
 p
 p
 
 
 
 collisions at 
 
 
 
 $$\sqrt{s}=7$$
 
 
 
 
 s
 
 =
 7
 
 
 
 TeV with the ATLAS detector. Phys. Rev. D 85, 072004 (2012). 
 arXiv:1109.5141
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR19" id="ref-link-section-d90629110e31000">19</a>]) used in the PDF fits. The columns indicate the QCD accuracy of the theoretical predictions along with the tools used to obtain them</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/5" aria-label="Full size table 5"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-13" data-title="Fig. 13"><figure><figcaption><b id="Fig13" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 13</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/13" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig13_HTML.gif?as=webp"><img aria-describedby="Fig13" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig13_HTML.gif" alt="figure 13" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-13-desc"><p>The <span class="mathjax-tex">$1\sigma $</span> band for the strange sea suppression factor <span class="mathjax-tex">$r_s=(s+\bar{s})/(2\bar{d})$</span> as a function of Bjorken <i>x</i> obtained in the variants of the ABM analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 184" title="S. Alekhin, J. Blümlein, L. Caminadac, K. Lipka, K. Lohwasser, S. Moch, R. Petti, R. Placakyte, Determination of strange sea quark distributions from fixed-target and collider data. Phys. Rev. D 91, 094002 (2015). 
 arXiv:1404.6469
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR184" id="ref-link-section-d90629110e31404">184</a>] based on the combination of the data by NuTeV/CCFR [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 185" title="NuTeV Collaboration, M. Goncharov et al., Precise measurement of dimuon production cross sections in muon neutrino Fe and muon anti-neutrino Fe deep-inelastic scattering at the Tevatron. Phys. Rev. D64, 112006 (2001). 
 arXiv:hep-ex/0102049
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR185" id="ref-link-section-d90629110e31407">185</a>], CHORUS [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 186" title="A. Kayis-Topaksu et al., Measurement of charm production in neutrino charged-current interactions. New J. Phys. 13, 093002 (2011). 
 arXiv:1107.0613
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR186" id="ref-link-section-d90629110e31411">186</a>] and NOMAD [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 187" title="NOMAD Collaboration, O. Samoylov et al., A precision measurement of charm dimuon production in neutrino interactions from the NOMAD experiment, Nucl. Phys. B876, 339 (2013). 
 arXiv:1308.4750
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR187" id="ref-link-section-d90629110e31414">187</a>] (<i>shaded area</i>), and CHORUS [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 186" title="A. Kayis-Topaksu et al., Measurement of charm production in neutrino charged-current interactions. New J. Phys. 13, 093002 (2011). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR186" id="ref-link-section-d90629110e31420">186</a>], CMS [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 188" title="CMS Collaboration, S. Chatrchyan et al., Measurement of associated 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR188" id="ref-link-section-d90629110e31423">188</a>] and ATLAS [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 189" title="ATLAS Collaboration, G. Aad et al., Measurement of the production of a 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR189" id="ref-link-section-d90629110e31426">189</a>] (<i>dashed lines</i>), compared with the results obtained by the CMS analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="CMS Collaboration, S. Chatrchyan et al., Measurement of the muon charge asymmetry in inclusive 
 
 
 
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 7 TeV and an improved determination of light parton distribution functions. Phys. Rev. D 90(3), 032004 (2014). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR21" id="ref-link-section-d90629110e31433">21</a>] (<i>hatched area</i>) and by the ATLAS <i>epWZ</i>-fit [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 189" title="ATLAS Collaboration, G. Aad et al., Measurement of the production of a 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR189" id="ref-link-section-d90629110e31442">189</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 190" title="ATLAS Collaboration, G. Aad et al., Determination of the strange quark density of the proton from ATLAS measurements of the 
 
 
 
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 and 
 
 
 
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 Z
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 cross sections. Phys. Rev. Lett. 109, 012001 (2012). 
 arXiv:1203.4051
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR190" id="ref-link-section-d90629110e31445">190</a>] at different values of <i>x</i> (<i>full circles</i>). All quantities refer to the factorization scale <span class="mathjax-tex">$\mu ^2=1.9~\,\mathrm {GeV}^2$</span> </p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/13" data-track-dest="link:Figure13 Full size image" aria-label="Full size image figure 13" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The nucleon off-shell corrections, on the other hand, are somewhat more model dependent, and several model studies have been performed to estimate their effect on nuclear PDFs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 202" title="W. Melnitchouk, A.W. Schreiber, A.W. Thomas, Deep-inelastic scattering from off-shell nucleons. Phys. Rev. D49, 1183 (1994). 
 arXiv:nucl-th/9311008
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR202" id="ref-link-section-d90629110e31515">202</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 206" title="P.J. Ehlers, A. Accardi, L.T. Brady, W. Melnitchouk, Nuclear effects in the proton-deuteron Drell–Yan process. Phys. Rev. D 90, 014010 (2014). 
 arXiv:1405.2039
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR206" id="ref-link-section-d90629110e31518">206</a>]. Some earlier PDF analyses [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 191" title="J.F. Owens, A. Accardi, W. Melnitchouk, Global parton distributions with nuclear and finite-
 
 
 
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 Q
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 corrections. Phys. Rev. D 87, 094012 (2013). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR191" id="ref-link-section-d90629110e31521">191</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 199" title="A. Accardi, M.E. Christy, C.E. Keppel, P. Monaghan, W. Melnitchouk, J.G. Morfin, J.F. Owens, New parton distributions from large-
 
 
 
 $$x$$
 
 
 x
 
 
 and low-
 
 
 
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 Q
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 data. Phys. Rev. D 81, 034016 (2010). 
 arXiv:0911.2254
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR199" id="ref-link-section-d90629110e31524">199</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 200" title="A. Accardi, W. Melnitchouk, J.F. Owens, M.E. Christy, C.E. Keppel, L. Zhu, J.G. Morfin, Uncertainties in determining parton distributions at large 
 
 
 
 $$x$$
 
 
 x
 
 
 . Phys. Rev. D 84, 014008 (2011). 
 arXiv:1102.3686
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR200" id="ref-link-section-d90629110e31527">200</a>] used specific physics-motivated models for the off-shell corrections, while more recent approaches have fitted the off-shell parameters directly to data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
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 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e31531">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 205" title="S.A. Kulagin, R. Petti, Global study of nuclear structure functions. Nucl. Phys. A 765, 126 (2006). 
 arXiv:hep-ph/0412425
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR205" id="ref-link-section-d90629110e31534">205</a>]. Other analyses [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e31537">6</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 207" title="A.D. Martin, A.J.T.M. Mathijssen, W.J. Stirling, R.S. Thorne, B.J.A. Watt, G. Watt, Extended parameterisations for MSTW PDFs and their effect on lepton charge asymmetry from W decays. Eur. Phys. J. C 73, 2318 (2013). 
 arXiv:1211.1215
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR207" id="ref-link-section-d90629110e31540">207</a>] have attempted to parametrize the entire nuclear correction in terms of a universal, <span class="mathjax-tex">$Q^2$</span>-independent function, without appealing to physical constraints. In this approach, to account for the effects of Fermi smearing a functional form must be used that produces the steep rise in the <span class="mathjax-tex">$F_2^d/F_2^N$</span> ratio at high <i>x</i>, such as with a logarithm raised to a high power [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 207" title="A.D. Martin, A.J.T.M. Mathijssen, W.J. Stirling, R.S. Thorne, B.J.A. Watt, G. Watt, Extended parameterisations for MSTW PDFs and their effect on lepton charge asymmetry from W decays. Eur. Phys. J. C 73, 2318 (2013). 
 arXiv:1211.1215
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR207" id="ref-link-section-d90629110e31612">207</a>].</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-14" data-title="Fig. 14"><figure><figcaption><b id="Fig14" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 14</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/14" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig14_HTML.gif?as=webp"><img aria-describedby="Fig14" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig14_HTML.gif" alt="figure 14" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-14-desc"><p>(<i>Left panel</i>) Ratio of deuteron to isoscalar nucleon structure functions <span class="mathjax-tex">$F_2^d/F_2^N$</span> computed from the CJ15 PDFs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e31673">1</a>] for different values of <span class="mathjax-tex">$Q^2$</span>. The <i>pink envelope</i> represents the fit uncertainties for <span class="mathjax-tex">$Q^2=10$</span> GeV<span class="mathjax-tex">$^2$</span>. The downturn in the ratio at <span class="mathjax-tex">$Q^2=2$</span> GeV<span class="mathjax-tex">$^2$</span> is due to target mass corrections. (<i>Right panel</i>) Impact on the <i>d</i> / <i>u</i> ratio from the CJ15 fit [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e31824">1</a>] (<i>red band</i>) of removing the deuterium nuclear corrections (<i>green band</i>), and omitting all deuterium data (<i>cross-hatched band</i>)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/14" data-track-dest="link:Figure14 Full size image" aria-label="Full size image figure 14" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The effects of the nuclear corrections are most directly visible in the extraction of the <i>d</i>-quark PDF at large <i>x</i>, see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
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 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e31854">1</a>]. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig14">14</a> shows that omitting nuclear smearing effects in the deuteron leads to an overestimated <i>d</i> / <i>u</i> ratio at <span class="mathjax-tex">$x \gtrsim 0.6$</span>. In fact, omitting nuclear corrections induces a strong tension between the SLAC deuteron DIS data (see, e.g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 208" title="S. Dasu, P. de Barbaro, A. Bodek, H. Harada, M. Krasny, et al., Measurement of kinematic and nuclear dependence of 
 
 
 
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 in deep-inelastic electron scattering. Phys. Rev. D 49, 5641 (1994)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR208" id="ref-link-section-d90629110e31893">208</a>]) and the recent high precision <i>W</i>-boson asymmetry data from the DØ collaboration at the Tevatron [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="DØ Collaboration, V.M. Abazov et al., Measurement of the electron charge asymmetry in 
 
 
 
 $${p\bar{p}\rightarrow W+X \rightarrow e\nu +X}$$
 
 
 
 p
 
 
 p
 
 
 ¯
 
 
 →
 W
 +
 X
 →
 e
 ν
 +
 X
 
 
 
 decays in 
 
 
 
 $${p\bar{p}}$$
 
 
 
 p
 
 
 p
 
 
 ¯
 
 
 
 
 
 collisions at 
 
 
 
 $${\sqrt{s}=1.96}$$
 
 
 
 
 s
 
 =
 1.96
 
 
 
 TeV. Phys. Rev. D 91(3), 032007 (2015). 
 arXiv:1412.2862
 
 . [Erratum: Phys. Rev. D 91(7), 079901 (2015)]" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR26" id="ref-link-section-d90629110e31899">26</a>], which are sensitive to the <i>d</i>-quark PDF in a similar large-<i>x</i> range as the SLAC data, but are not affected by nuclear corrections. It also causes an artificial deformation of the <i>d</i>-quark distribution, leading to essentially uncontrolled systematic errors when quark distributions are needed beyond the <i>x</i> range constrained by the data. This illustrates not only the theoretical but also the phenomenological need for such corrections when considering data at large <i>x</i>. Of course, one can choose to avoid nuclear effects altogether by using only proton data; however, doing so increases the uncertainty on the <i>d</i>-quark PDF at both small and large values of <i>x</i>, as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig14">14</a> illustrates. Additional details concerning the role of nuclear corrections when using deuterium target data in global fits can be found in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
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 x
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e31928">1</a>]. The extrapolation of nuclear effects from the deuteron to heavy nuclei is unclear, especially in view of the differences between the off-shell quark deformation fitted using deuteron targets [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
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 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e31931">1</a>] and using the ratio of heavy nuclei to deuteron structure functions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 205" title="S.A. Kulagin, R. Petti, Global study of nuclear structure functions. Nucl. Phys. A 765, 126 (2006). 
 arXiv:hep-ph/0412425
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR205" id="ref-link-section-d90629110e31934">205</a>]. As mentioned in the previous section, in general care should also be exercised when using neutrino-nucleus scattering data to obtain, for example, constraints on strange-quark PDFs, due to the currently poor understanding of the interaction dynamics of the final state heavy quark propagating through the target nucleus [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 192" title="A. Accardi, PDFs from nucleons to nuclei. PoS DIS2015, 001 (2015). 
 arXiv:1602.02035
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR192" id="ref-link-section-d90629110e31937">192</a>].</p><h3 class="c-article__sub-heading" id="Sec18"><span class="c-article-section__title-number">3.6 </span>Software and tools</h3><p>Data used in the PDF fits cover a wide range of kinematics and stem from a large number of different scattering processes. In order to achieve an accurate theoretical description of both the PDF evolution and the hard scattering cross sections, well-tested software and tools are necessary. Benchmark numbers for the PDF evolution have long been established, see e.g., the Les Houches report [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 209" title="W. Giele et al., The QCD / SM Working Group: Summary Report. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR209" id="ref-link-section-d90629110e31948">209</a>], and open-source evolution codes such as <span class="u-monospace">QCDNUM</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 210" title="M. Botje, QCDNUM: fast QCD evolution and convolution. Comput. Phys. Commun. 182, 490 (2011). 
 arXiv:1005.1481
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR210" id="ref-link-section-d90629110e31954">210</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 211" title="M. Botje, Erratum for the time-like evolution in QCDNUM. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR211" id="ref-link-section-d90629110e31957">211</a>] and <span class="u-monospace">Hoppet</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 212" title="G.P. Salam, J. Rojo, A higher order perturbative parton evolution toolkit (HOPPET). Comput. Phys. Commun. 180, 120 (2009). 
 arXiv:0804.3755
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR212" id="ref-link-section-d90629110e31964">212</a>] are available in Bjorken <i>x</i>-space and <span class="u-monospace">QCD-Pegasus</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 123" title="A. Vogt, Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS. Comput. Phys. Commun. 170, 65 (2005). 
 arXiv:hep-ph/0408244
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR123" id="ref-link-section-d90629110e31973">123</a>] in Mellin <i>N</i>-space. This is an important development as it allows to expose the software used in the PDF fits to systematic validation, the need of which can be illustrated with recent theory improvements published by various groups. For example, MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 197" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Parton distributions for the LHC. Eur. Phys. J. C 63, 189 (2009). 
 arXiv:0901.0002
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR197" id="ref-link-section-d90629110e31979">197</a>] has tested its NNLO evolution code against <span class="u-monospace">QCD-Pegasus</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 123" title="A. Vogt, Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS. Comput. Phys. Commun. 170, 65 (2005). 
 arXiv:hep-ph/0408244
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR123" id="ref-link-section-d90629110e31986">123</a>] and corrected the implementation of one of the heavy-quark OMEs.</p><p>For the hard scattering cross sections of the various processes, fast fitting methods like <span class="u-monospace">fastNLO</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 213" title="T. Kluge, K. Rabbertz, M. Wobisch, FastNLO: Fast pQCD calculations for PDF fits. 
 arXiv:hep-ph/0609285
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR213" id="ref-link-section-d90629110e31995">213</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 214" title="D. Britzger, K. Rabbertz, F. Stober, M. Wobisch, New features in version 2 of the fastNLO project. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR214" id="ref-link-section-d90629110e31998">214</a>] and <span class="u-monospace">APPLGrid</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 181" title="T. Carli, D. Clements, A. Cooper-Sarkar, C. Gwenlan, G.P. Salam, F. Siegert, P. Starovoitov, M. Sutton, A posteriori inclusion of parton density functions in NLO QCD final-state calculations at hadron colliders: the APPLGRID Project. Eur. Phys. J. C 66, 503 (2010). 
 arXiv:0911.2985
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR181" id="ref-link-section-d90629110e32004">181</a>] have been developed. In addition, some groups have also published open-source code for the theory predictions of all physical cross sections employed in their analyses. The ABM11 and ABM12 fits [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e32008">2</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e32011">64</a>] use <span class="u-monospace">OPENQCDRAD</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 215" title="OPENQCDRAD. 
 http://www-zeuthen.desy.de/~alekhin/OPENQCDRAD
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR215" id="ref-link-section-d90629110e32017">215</a>] code, which is publicly available. The HERAPDF2.0 fit [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e32020">4</a>] relies on the QCD fit platform <span class="u-monospace">xFitter</span> (formerly known as <span class="u-monospace">HERAFitter</span>) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 166" title="S. Alekhin, O. Behnke, P. Belov, S. Borroni, M. Botje, et al., HERAFitter, Open source QCD fit project. 
 arXiv:1410.4412
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR166" id="ref-link-section-d90629110e32030">166</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 167" title="xFitter, An open source QCD fit framework. 
 http://xFitter.org
 
 [xFitter.org]" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR167" id="ref-link-section-d90629110e32033">167</a>], which is an open-source package that provides a framework for the determination of PDFs and enables the choice of theoretical options for obtaining PDF-dependent cross section predictions. In particular, <span class="u-monospace">xFitter</span> allows for a choice of different available schemes for treatment of heavy quarks in DIS. In Mellin <i>N</i>-space, an efficient method exists [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 216" title="D.A. Kosower, Extracting parton densities from collider data. Nucl. Phys. B 520, 263 (1998). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR216" id="ref-link-section-d90629110e32042">216</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 217" title="M. Stratmann, W. Vogelsang, Towards a global analysis of polarized parton distributions. Phys. Rev. D 64, 114007 (2001). 
 arXiv:hep-ph/0107064
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR217" id="ref-link-section-d90629110e32046">217</a>] which improves on that by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 218" title="D. Graudenz, M. Hampel, A. Vogt, C. Berger, The Mellin transform technique for the extraction of the gluon density. Z. Phys. C 70, 77 (1996). 
 arXiv:hep-ph/9506333
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR218" id="ref-link-section-d90629110e32049">218</a>] and which has been widely used in analyses, e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 217" title="M. Stratmann, W. Vogelsang, Towards a global analysis of polarized parton distributions. Phys. Rev. D 64, 114007 (2001). 
 arXiv:hep-ph/0107064
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR217" id="ref-link-section-d90629110e32052">217</a>]. However, no code has been made publicly available so far.</p><p>Given the increasing precision of PDF analyses, which is driven by the accuracy of the experimental data, there is ongoing demand to provide theoretical predictions that are as precise as possible. This has stimulated recent checks of the analysis software used by various groups and has resulted in a number of documented improvements. The list includes, for example, the corrections to the different parts of the DIS cross section calculations in the NNPDF2.1, MSTW and CT10 PDF analyses as mentioned in the discussion of the PDFs for strange sea above.</p><p>This illustrates that there is a continued need for benchmarking the hard scattering cross sections of relevance for PDF determinations in order to consolidate the accuracy of theory predictions for those observables. In this respect, open-source software may facilitate future theory improvements and may help to establish standards for precision theory predictions.</p></div></div></section><section data-title="Strong coupling constant"><div class="c-article-section" id="Sec19-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec19"><span class="c-article-section__title-number">4 </span>Strong coupling constant</h2><div class="c-article-section__content" id="Sec19-content"><p>The value of the strong coupling constant <span class="mathjax-tex">$\alpha _s(M_Z)$</span> has a direct impact on the size of a number of cross sections at the LHC, such as Higgs boson production, see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec20">5</a>, and is therefore an important parameter. Due to QCD factorization, <span class="mathjax-tex">$\alpha _s$</span> exhibits a significant correlation with the gluon PDF and also with the charm-quark mass, as documented in the published correlation matrices, see for instance [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e32142">64</a>]. Therefore, the strong coupling constant has come to require particular attention in the context of global PDF analyses.</p><p>Current precision determinations of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> require NNLO accuracy in QCD because of the small uncertainties in the experimental data analyzed and the significantly reduced dependence from the variation of the renormalization scale indicating the uncertainty due to the truncation of the perturbative series. Extractions of <span class="mathjax-tex">$\alpha _s$</span> at NLO typically yield <span class="mathjax-tex">$\alpha _s(M_Z) \simeq 0.118$</span>, however, the NLO scale uncertainty is large, giving sizable variations <span class="mathjax-tex">$\Delta \alpha _s(M_Z) = 0.005$</span> for <span class="mathjax-tex">$\mu _r \in [Q/2,2Q]$</span> in DIS analyses. Determinations of <span class="mathjax-tex">$\alpha _s$</span> to NNLO accuracy benefit from a significantly reduced renormalization scale dependence, but generally result in smaller central values for <span class="mathjax-tex">$\alpha _s(M_Z)$</span>, with shifts downwards from NLO to NNLO of a few percent in DIS analyses. Beyond NNLO, the perturbative expansion converges, as illustrated in DIS in a valence analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 58" title="J. Blümlein, H. Böttcher, A. Guffanti, Non-singlet QCD analysis of deep-inelastic world data at 
 
 
 
 $$(\alpha _s^3)$$
 
 
 
 (
 
 α
 s
 3
 
 )
 
 
 
 . Nucl. Phys. B 774, 182 (2007). 
 arXiv:hep-ph/0607200
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR58" id="ref-link-section-d90629110e32439">58</a>] at N<span class="mathjax-tex">$^3$</span>LO which yields <span class="mathjax-tex">$\alpha _s(M_Z) = 0.1141~_{-~0.0022}^{+~0.0020}$</span>, in agreement with the NNLO values listed in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a>.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-6"><figure><figcaption class="c-article-table__figcaption"><b id="Tab6" data-test="table-caption">Table 6 Values of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> obtained or used in the nominal PDF sets of the various groups</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/6" aria-label="Full size table 6"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-7"><figure><figcaption class="c-article-table__figcaption"><b id="Tab7" data-test="table-caption">Table 7 Determinations of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> values at NNLO from QCD analyses of the deep-inelastic world data and, partly, including other hard scattering data. For recent compilations, see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 219" title="S. Bethke et al., Workshop On Precision Measurements of 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 . 
 arXiv:1110.0016
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR219" id="ref-link-section-d90629110e33460">219</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 221" title="D. d’Enterria et al., High-precision 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 measurements from LHC to FCC-ee. 
 arXiv:1512.05194
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR221" id="ref-link-section-d90629110e33463">221</a>]</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/7" aria-label="Full size table 7"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>Of course, measurements of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> are not limited to global fits of PDFs, but stem from a large number of different processes and methods at different scales, see, e.g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 219" title="S. Bethke et al., Workshop On Precision Measurements of 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 . 
 arXiv:1110.0016
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR219" id="ref-link-section-d90629110e34977">219</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 221" title="D. d’Enterria et al., High-precision 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 measurements from LHC to FCC-ee. 
 arXiv:1512.05194
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR221" id="ref-link-section-d90629110e34980">221</a>] for discussions and comparisons. Here we restrict ourselves to issues of <span class="mathjax-tex">$\alpha _s$</span> arising in PDF fits. In Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab6">6</a> we give an overview of the <span class="mathjax-tex">$\alpha _s$</span> values currently used in the PDF analyses. There, two aspects are important. Firstly, some PDF analyses leave <span class="mathjax-tex">$\alpha _s$</span> as a free parameter in their fits, which obviously allows one to control its correlation with other PDF parameters and avoids potential biases. Secondly, among the NNLO values of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> used there exists a large spread of <span class="mathjax-tex">$\alpha _s$</span> values, ranging from <span class="mathjax-tex">$\alpha _s(M_Z) = 0.1132$</span> to 0.1183. Some of those fitted values of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> are significantly smaller than, for example, an average provided by the PDG [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e35224">55</a>] in 2014, which gives <span class="mathjax-tex">$\alpha _s(M_Z) = 0.1185 \pm 0.0006$</span> at NNLO, and is often quoted as a motivation for fixing <span class="mathjax-tex">$\alpha _s(M_Z) = 0.118$</span> as in some entries in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab6">6</a>. In the recent 2015 update, the PDG [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 222" title="Particle Data Group Collaboration. 
 http://pdg.lbl.gov/2015/reviews/rpp2015-rev-qcd
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR222" id="ref-link-section-d90629110e35331">222</a>] reports the value <span class="mathjax-tex">$\alpha _s(M_Z) = 0.1181 \pm 0.0013$</span> with the uncertainty increased by a factor of two.</p><p>While the potential agreement or disagreement with the PDG average is beyond the scope of this study, it is instructive to focus on <span class="mathjax-tex">$\alpha _s(M_Z)$</span> measurements from PDF analyses as listed in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a> which have been performed since the NNLO QCD corrections in DIS first became available. This series of measurements has led to <span class="mathjax-tex">$\alpha _s(M_Z)$</span> values which are not only mostly lower than the PDG average, but also exhibit a large spread in the range <span class="mathjax-tex">$\alpha _s(M_Z) = 0.1120 - 0.1175$</span>. This spread is significant given the small size of the experimental uncertainties in the data. As it turns out, the differences in the values of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> can be traced back to different data sets used or to different theory assumptions applied, as indicated in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a>.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-8"><figure><figcaption class="c-article-table__figcaption"><b id="Tab8" data-test="table-caption">Table 8 The jet data sets and the theory approximations used in the NNLO PDF fits. The threshold corrections of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 237" title="N. Kidonakis, J.F. Owens, Effects of higher order threshold corrections in high 
 
 
 
 $$E_T$$
 
 
 
 E
 T
 
 
 
 jet production. Phys. Rev. D 63, 054019 (2001). 
 arXiv:hep-ph/0007268
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR237" id="ref-link-section-d90629110e35593">237</a>] neglect the dependence on the jet radius <i>R</i>. Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 238" title="S. Carrazza, J. Pires, Perturbative QCD description of jet data from LHC Run-I and Tevatron Run-II. JHEP 10, 145 (2014). 
 arXiv:1407.7031
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR238" id="ref-link-section-d90629110e35599">238</a>] has determined the regime of validity (“safety cuts”) of the threshold approximation of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 240" title="D. de Florian, P. Hinderer, A. Mukherjee, F. Ringer, W. Vogelsang, Approximate next-to-next-to-leading order corrections to hadronic jet production. Phys. Rev. Lett. 112, 082001 (2014). 
 arXiv:1310.7192
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR240" id="ref-link-section-d90629110e35602">240</a>] by comparing to the exact NNLO result for the <i>gg</i> channel [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, J. Pires, Second order QCD corrections to jet production at hadron colliders: the all-gluon contribution. Phys. Rev. Lett. 110, 162003 (2013). 
 arXiv:1301.7310
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR15" id="ref-link-section-d90629110e35609">15</a>]</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/8" aria-label="Full size table 8"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>For instance, the inclusion of data for the hadro-production of jets, e.g., from the LHC, does have an impact on the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and can therefore provide valuable constraints. However, it is important to note that the perturbative QCD corrections to the hard scattering cross section are only known completely to NLO, while the exact NNLO result for the <i>gg</i> channel [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, J. Pires, Second order QCD corrections to jet production at hadron colliders: the all-gluon contribution. Phys. Rev. Lett. 110, 162003 (2013). 
 arXiv:1301.7310
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR15" id="ref-link-section-d90629110e35894">15</a>] and approximations based on soft gluon enhancement [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 237" title="N. Kidonakis, J.F. Owens, Effects of higher order threshold corrections in high 
 
 
 
 $$E_T$$
 
 
 
 E
 T
 
 
 
 jet production. Phys. Rev. D 63, 054019 (2001). 
 arXiv:hep-ph/0007268
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR237" id="ref-link-section-d90629110e35897">237</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 240" title="D. de Florian, P. Hinderer, A. Mukherjee, F. Ringer, W. Vogelsang, Approximate next-to-next-to-leading order corrections to hadronic jet production. Phys. Rev. Lett. 112, 082001 (2014). 
 arXiv:1310.7192
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR240" id="ref-link-section-d90629110e35900">240</a>] indicate corrections as large as 15–20 %. Those corrections and their magnitude depend, of course, on the details of the kinematics, the choice of the scale and on the jet parameters (e.g., jet radius <i>R</i>). For high <span class="mathjax-tex">$p_T$</span> they are dominated by threshold logarithms <span class="mathjax-tex">$\ln (p_T)$</span> accompanied by logarithms <span class="mathjax-tex">$\ln (R)$</span> for small jet radii [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 240" title="D. de Florian, P. Hinderer, A. Mukherjee, F. Ringer, W. Vogelsang, Approximate next-to-next-to-leading order corrections to hadronic jet production. Phys. Rev. Lett. 112, 082001 (2014). 
 arXiv:1310.7192
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR240" id="ref-link-section-d90629110e35996">240</a>].</p><p>The <span class="mathjax-tex">$\alpha _s(M_Z)$</span> values in PDF analyses currently determined with the help of jet data (cf. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a>) are, strictly speaking, valid to NLO accuracy only and therefore subject to significantly larger theory uncertainties due to the variation of the renormalization scale. The various groups employ different approaches in their NNLO analyses to cope with this inconsistency, such as using dynamical scales or applying some variant of threshold corrections, as detailed in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab8">8</a>. As a result of these efforts, the gluon PDF and <span class="mathjax-tex">$\alpha _s$</span> obtained, for example, in the MMHT14 and NNPDF3.0 analyses are in a good agreement.</p><p>Different modeling of important theory aspects, such as whether or not to include target mass corrections, higher twist contributions and nuclear corrections in the description of DIS data, or whether or not to use a VFNS in the description of DIS heavy-quark data, can account for the range of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> values in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a>. With largely similar model assumptions, NNPDF2.1 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 234" title="S. Lionetti, R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Precision determination of 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR234" id="ref-link-section-d90629110e36127">234</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 235" title="R.D. Ball, V. Bertone, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, S. Lionetti, J. Rojo, M. Ubiali, Precision NNLO determination of 
 
 
 
 $$\alpha _s(M_Z)$$
 
 
 
 
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 M
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 using an unbiased global parton set. Phys. Lett. B 707, 66 (2012). 
 arXiv:1110.2483
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR235" id="ref-link-section-d90629110e36130">235</a>], MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 231" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Uncertainties on 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 in global PDF analyses and implications for predicted hadronic cross sections. Eur. Phys. J. C 64, 653 (2009). 
 arXiv:0905.3531
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR231" id="ref-link-section-d90629110e36133">231</a>] and MMHT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 236" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Uncertainties on 
 
 
 
 $$\alpha _S$$
 
 
 
 α
 S
 
 
 
 in the MMHT2014 global PDF analysis and implications for SM predictions. Eur. Phys. J. C 75, 435 (2015). 
 arXiv:1506.05682
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR236" id="ref-link-section-d90629110e36137">236</a>] obtained the range <span class="mathjax-tex">$\alpha _s(M_Z) = 0.1171 - 0.1174$</span>. All these choices can lead to systematic shifts of the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span>. Let us briefly mention some of the issues in detail.</p><p>Higher twist contributions do have a big impact, because these terms are fitted within a combined analysis. Alternatively, the part of the DIS data significantly affected by these terms has to be removed by suitable kinematical cuts on the scale <span class="mathjax-tex">$Q^2$</span> and center-of-mass energies <span class="mathjax-tex">$W^2$</span>. In a variant of the ABM11 analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e36288">64</a>], higher twist terms have been omitted and the cuts <span class="mathjax-tex">$W^2 > 12.5~\,\mathrm {GeV}^2$</span> and <span class="mathjax-tex">$Q^2 > 2.5~\,\mathrm {GeV}^2$</span> as used by MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 231" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Uncertainties on 
 
 
 
 $$\alpha _s$$
 
 
 
 α
 s
 
 
 
 in global PDF analyses and implications for predicted hadronic cross sections. Eur. Phys. J. C 64, 653 (2009). 
 arXiv:0905.3531
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR231" id="ref-link-section-d90629110e36386">231</a>] have been applied. This resulted in a sizable shift upwards to <span class="mathjax-tex">$\alpha _s(M_Z^2) = 0.1191 \pm 0.0016$</span> in line with earlier studies in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 241" title="A. Vogt, Structure function evolution at next-to-leading order and beyond. Nucl. Phys. Proc. Suppl. 79, 102 (1999). 
 arXiv:hep-ph/9906337
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR241" id="ref-link-section-d90629110e36444">241</a>]. Yet more conservative cuts of <span class="mathjax-tex">$W^2 > 12.5~\,\mathrm {GeV}^2$</span> and <span class="mathjax-tex">$Q^2 > 10~\,\mathrm {GeV}^2$</span> in the ABM11 variant with higher twist terms set to zero led to <span class="mathjax-tex">$\alpha _s(M_Z^2) = 0.1134 \pm 0.0008$</span>, well in agreement with the nominal value in the ABM11 analysis, cf. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a>. Thus, in PDF analyses without account of higher twist contributions to DIS data such tight cuts are essential. In this regard we disagree with Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 67" title="R.S. Thorne, G. Watt, PDF dependence of Higgs cross sections at the tevatron and LHC: response to recent criticism. JHEP 08, 100 (2011). 
 arXiv:1106.5789
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR67" id="ref-link-section-d90629110e36600">67</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 232" title="R.S. Thorne, The effect on PDFs and 
 
 
 
 $$\alpha _S(M_Z^2)$$
 
 
 
 
 α
 S
 
 
 (
 
 M
 Z
 2
 
 )
 
 
 
 
 due to changes in flavour scheme and higher twist contributions. Eur. Phys. J. C 74, 2958 (2014). 
 arXiv:1402.3536
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR232" id="ref-link-section-d90629110e36603">232</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 243" title="NNPDF Collaboration, R.D. Ball et al., Theoretical issues in PDF determination and associated uncertainties. Phys. Lett. B723 (2013) 330. 
 arXiv:1303.1189
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR243" id="ref-link-section-d90629110e36606">243</a>] which claim higher twist effects to be negligible in the framework of MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 197" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Parton distributions for the LHC. Eur. Phys. J. C 63, 189 (2009). 
 arXiv:0901.0002
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR197" id="ref-link-section-d90629110e36609">197</a>] and NNPDF2.3 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 250" title="R.D. Ball et al., Parton distributions with LHC data. Nucl. Phys. B 867, 244 (2013). 
 arXiv:1207.1303
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR250" id="ref-link-section-d90629110e36612">250</a>]. We also note that NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e36616">7</a>] uses a cut of <span class="mathjax-tex">$Q^2 > 3.5~\,\mathrm {GeV}^2$</span> which is too low to remove the higher twist contributions.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-9"><figure><figcaption class="c-article-table__figcaption"><b id="Tab9" data-test="table-caption">Table 9 The Higgs cross section at NNLO in QCD (computed in the effective theory) at <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV for <span class="mathjax-tex">$m_H=125.0$</span> GeV at the nominal scale <span class="mathjax-tex">$\mu _r=\mu _f=m_H$</span> with the PDF (and, if available, also <span class="mathjax-tex">$\alpha _s$</span>) uncertainties. The columns correspond to different choices for the central value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> using the nominal PDF set. The numbers in parenthesis are obtained using the PDF sets <span class="u-monospace">CT14nnlo_as_0115</span>, <span class="u-monospace">HERAPDF20_NNLO_ALPHAS_115</span>, <span class="u-monospace">MMHT2014nnlo_asmzlargerange</span> and <span class="u-monospace">NNPDF30_nnlo_as</span> <span class="u-monospace">_0115</span> </b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/9" aria-label="Full size table 9"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>Higher order constraints from fixed-target DIS data can also lead to shifts in <span class="mathjax-tex">$\alpha _s(M_Z)$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 59" title="S. Alekhin, J. Blümlein, S. Moch, Higher order constraints on the Higgs production rate from fixed-target DIS data. Eur. Phys. J. C 71, 1723 (2011). 
 arXiv:1101.5261
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR59" id="ref-link-section-d90629110e38483">59</a>]. For instance, NMC has measured the DIS differential cross sections and extracted the DIS structure functions <span class="mathjax-tex">$F_2^\mathrm{NMC}$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 242" title="New Muon Collaboration, M. Arneodo et al., Measurement of the proton and deuteron structure functions, 
 
 
 
 $$F_2^p$$
 
 
 
 F
 2
 p
 
 
 
 and 
 
 
 
 $$F_2^d$$
 
 
 
 F
 2
 d
 
 
 
 , and of the ratio 
 
 
 
 $$\sigma _L/\sigma _T$$
 
 
 
 
 σ
 L
 
 /
 
 σ
 T
 
 
 
 
 . Nucl. Phys. B483, 3 (1997). 
 arXiv:hep-ph/9610231
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR242" id="ref-link-section-d90629110e38513">242</a>]. At the time of the NMC analysis, however, the relevant DIS corrections to <span class="mathjax-tex">$\mathcal{O}(\alpha _s^3)$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="J.A.M. Vermaseren, A. Vogt, S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange. Nucl. Phys. B 724, 3 (2005). 
 arXiv:hep-ph/0504242
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR57" id="ref-link-section-d90629110e38556">57</a>] were not available (see discussion after Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Equ5">5</a>) above). This information is, however, important and has to be taken into account now. In case of fitting <span class="mathjax-tex">$F_2^\mathrm{NMC}$</span> and not describing <span class="mathjax-tex">$F_L(x,Q^2)$</span> at NNLO, much larger values of <span class="mathjax-tex">$\alpha _s(M_Z^2)$</span> are obtained [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 67" title="R.S. Thorne, G. Watt, PDF dependence of Higgs cross sections at the tevatron and LHC: response to recent criticism. JHEP 08, 100 (2011). 
 arXiv:1106.5789
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR67" id="ref-link-section-d90629110e38682">67</a>]. It is therefore strongly recommended to fit the published differential scattering cross sections using <span class="mathjax-tex">$F_L(x,Q^2)$</span> at <span class="mathjax-tex">$\mathcal{O}(\alpha _s^3)$</span>. Presently, the MMHT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 236" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Uncertainties on 
 
 
 
 $$\alpha _S$$
 
 
 
 α
 S
 
 
 
 in the MMHT2014 global PDF analysis and implications for SM predictions. Eur. Phys. J. C 75, 435 (2015). 
 arXiv:1506.05682
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR236" id="ref-link-section-d90629110e38772">236</a>] analysis uses <span class="mathjax-tex">$F_L(x,Q^2)$</span> only at NLO. One should note, however, that the values of <span class="mathjax-tex">$F_L(x,Q^2)$</span> at NNLO are significantly different in the small-<i>x</i> region (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 67" title="R.S. Thorne, G. Watt, PDF dependence of Higgs cross sections at the tevatron and LHC: response to recent criticism. JHEP 08, 100 (2011). 
 arXiv:1106.5789
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR67" id="ref-link-section-d90629110e38873">67</a>]).</p><p>Finally, great care needs to be exercised when DIS data off nuclei are included in global fits, see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>. Details of modeling of nuclear corrections can in fact also cause systematic shifts in the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span>. Therefore, Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab7">7</a> indicates if scattering data on heavy nuclei have been included in the determination. For example, MMHT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 236" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Uncertainties on 
 
 
 
 $$\alpha _S$$
 
 
 
 α
 S
 
 
 
 in the MMHT2014 global PDF analysis and implications for SM predictions. Eur. Phys. J. C 75, 435 (2015). 
 arXiv:1506.05682
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR236" id="ref-link-section-d90629110e38930">236</a>] has reported a comparatively high value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> as a consequence of fitting the NuTeV <span class="mathjax-tex">$\nu $</span>Fe DIS data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 185" title="NuTeV Collaboration, M. Goncharov et al., Precise measurement of dimuon production cross sections in muon neutrino Fe and muon anti-neutrino Fe deep-inelastic scattering at the Tevatron. Phys. Rev. D64, 112006 (2001). 
 arXiv:hep-ex/0102049
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR185" id="ref-link-section-d90629110e38998">185</a>]. In general, determinations of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> should be based upon, or at least cross-checked with, fits using proton and deuteron DIS data only.</p></div></div></section><section data-title="Cross section predictions for the LHC"><div class="c-article-section" id="Sec20-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec20"><span class="c-article-section__title-number">5 </span>Cross section predictions for the LHC</h2><div class="c-article-section__content" id="Sec20-content"><h3 class="c-article__sub-heading" id="Sec21"><span class="c-article-section__title-number">5.1 </span>Higgs boson production</h3><p>The dominant production mechanism for the SM Higgs boson at the LHC is the gluon–gluon fusion process. The large size of the QCD radiative corrections to the inclusive cross section at NLO, see, e.g. Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 244" title="M. Spira, A. Djouadi, D. Graudenz, P.M. Zerwas, Higgs boson production at the LHC. Nucl. Phys. B 453, 17 (1995). 
 arXiv:hep-ph/9504378
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR244" id="ref-link-section-d90629110e39057">244</a>], together with the sizable scale uncertainty have motivated systematic theory improvements. In the effective theory based on the limit of a large top-quark mass (<span class="mathjax-tex">$m_t \rightarrow \infty $</span>, integrating out the top-quark loop, but using the full <span class="mathjax-tex">$m_t$</span> dependence in the Born cross section), this has led to the computation of the corresponding corrections at NNLO [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 245" title="R.V. Harlander, W.B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders. Phys. Rev. Lett. 88, 201801 (2002). 
 arXiv:hep-ph/0201206
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR245" id="ref-link-section-d90629110e39116">245</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 247" title="V. Ravindran, J. Smith, W.L. van Neerven, NNLO corrections to the total cross section for Higgs boson production in hadron hadron collisions. Nucl. Phys. B 665, 325 (2003). 
 arXiv:hep-ph/0302135
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR247" id="ref-link-section-d90629110e39119">247</a>] and even to N<span class="mathjax-tex">$^3$</span>LO in QCD [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 106" title="C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos, B. Mistlberger, High precision determination of the gluon fusion Higgs boson cross section at the LHC. 
 arXiv:1602.00695
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR106" id="ref-link-section-d90629110e39147">106</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 248" title="C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, B. Mistlberger, Higgs boson gluon fusion production in QCD at three loops. Phys. Rev. Lett. 114, 212001 (2015). 
 arXiv:1503.06056
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR248" id="ref-link-section-d90629110e39150">248</a>]. This shows an apparent, if slow, convergence of the perturbative expansion, along with greatly reduced sensitivity to the choice for the renormalization and factorization scales <span class="mathjax-tex">$\mu _r$</span> and <span class="mathjax-tex">$\mu _f$</span>. At N<span class="mathjax-tex">$^3$</span>LO the total scale variation amounts to 3 % and estimates of the four-loop corrections support these findings [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 249" title="D. de Florian, J. Mazzitelli, S. Moch, A. Vogt, Approximate N
 
 
 
 $$^{3}$$
 
 
 
 
 3
 
 
 
 LO Higgs-boson production cross section using physical-kernel constraints. JHEP 10, 176 (2014). 
 arXiv:1408.6277
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR249" id="ref-link-section-d90629110e39228">249</a>].</p><p>This leaves, as the largest remaining source of uncertainties in the predictions of the physical cross section, the input for the strong coupling constant <span class="mathjax-tex">$\alpha _s$</span> and the PDFs. Despite the impressive progress in theory and experiment, the situation resembles that after the completion of the NLO QCD corrections, when it was pointed out in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 244" title="M. Spira, A. Djouadi, D. Graudenz, P.M. Zerwas, Higgs boson production at the LHC. Nucl. Phys. B 453, 17 (1995). 
 arXiv:hep-ph/9504378
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR244" id="ref-link-section-d90629110e39259">244</a>] that one of the main residual uncertainties in the predictions was due to the gluon PDF.</p><p>In Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab9">9</a> we summarize the PDF dependence of the inclusive cross section <span class="mathjax-tex">$\sigma (H)^\mathrm{NNLO}$</span> in the effective theory (i.e., in the limit of <span class="mathjax-tex">$m_t \gg m_H$</span>) at <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV for a Higgs boson mass <span class="mathjax-tex">$m_H=125.0$</span> GeV, <span class="mathjax-tex">$\mu _r = \mu _f = m_H$</span>, and <span class="mathjax-tex">$m_t^\mathrm{pole}=172.5$</span> GeV with uncertainties <span class="mathjax-tex">$\sigma (H)^\mathrm{NNLO} + \Delta \sigma (\mathrm {PDF}+\alpha _s)$</span>, and compare the results for various PDF sets. The PDF uncertainties are typically given at the 1<span class="mathjax-tex">$\sigma $</span> c.l. We list the results for <span class="mathjax-tex">$\sigma (H)^\mathrm{NNLO}$</span> using either the values for the strong coupling constant <span class="mathjax-tex">$\alpha _s(M_Z)$</span> at NNLO, corresponding to the respective PDF set, or fixed values of <span class="mathjax-tex">$\alpha _s(M_Z)=0.115$</span> and <span class="mathjax-tex">$\alpha _s(M_Z)=0.118$</span>. This is done to illustrate the fact that in some PDFs the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> is not obtained from a fit to data (including faithful uncertainties) but fixed beforehand, e.g., to the world average [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e39802">55</a>]. Often the same fixed value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> is chosen at NLO and at NNLO independent of the order of perturbation theory, see also Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec19">4</a>. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab9">9</a> shows a large spread for predictions from different PDFs with a range <span class="mathjax-tex">$\sigma (H)^\mathrm{NNLO} = 38.0 - 42.6$</span> pb using the nominal value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span>. Specifically, the PDF and <span class="mathjax-tex">$\alpha _s$</span> differences between different sets are up to <span class="mathjax-tex">$11\,\%$</span> and are significantly larger than the residual scale uncertainty due to N<span class="mathjax-tex">$^3$</span>LO QCD corrections. In addition, the cross sections shift in the range <span class="mathjax-tex">$\sigma (H)^\mathrm{NNLO} = 39.0 - 44.7$</span> pb if a fixed value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> in the range <span class="mathjax-tex">$\alpha _s(M_Z)=0.115 - 0.118$</span> is used. This amounts to a relative difference of more than <span class="mathjax-tex">$13\,\%$</span> and contradicts the most recent estimates of the combined PDF and <span class="mathjax-tex">$\alpha _s$</span> uncertainties in the inclusive cross section [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 106" title="C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos, B. Mistlberger, High precision determination of the gluon fusion Higgs boson cross section at the LHC. 
 arXiv:1602.00695
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR106" id="ref-link-section-d90629110e40222">106</a>], which quote <span class="mathjax-tex">$3.2\,\%$</span>. In general, the findings underpin the importance of controlling the accuracy and the correlation of the strong coupling constant with the PDF parameters in fits.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-10"><figure><figcaption class="c-article-table__figcaption"><b id="Tab10" data-test="table-caption">Table 10 The values of the charm-quark mass (on-shell scheme <span class="mathjax-tex">$m^\mathrm{pole}$</span>) and the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span> in the MSTW analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 171" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Heavy-quark mass dependence in global PDF analyses and 3- and 4-flavour parton distributions. Eur. Phys. J. C 70, 51 (2010). 
 arXiv:1007.2624
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR171" id="ref-link-section-d90629110e40331">171</a>] using the set <span class="u-monospace">MSTW2008nnlo_mcrange</span> together with the value for <span class="mathjax-tex">$\chi ^2$</span>/NDP for the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e40363">165</a>] and the Higgs cross section at NNLO in QCD (computed in the effective theory) at <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV for <span class="mathjax-tex">$m_H=125.0$</span> GeV at the nominal scale <span class="mathjax-tex">$\mu _r=\mu _f=m_H$</span>. The numbers in parentheses are obtained using the PDF set <span class="u-monospace">MSTW2008nnlo_mcrange_fixasmz</span> with the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> fixed to <span class="mathjax-tex">$\alpha _s(M_Z)=0.1171$</span> </b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/10" aria-label="Full size table 10"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-11"><figure><figcaption class="c-article-table__figcaption"><b id="Tab11" data-test="table-caption">Table 11 Same as Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab10">10</a> for the MMHT14 analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e41335">172</a>] using the set <span class="u-monospace">MMHT2014nnlo_mcrange_nf5</span> and setting <span class="mathjax-tex">$\alpha _s(M_Z)$</span> to the best fit value. The numbers of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 182" title="L. Harland-Lang, R. Thorne, Private communication, Apr 12 and June 6 (2016)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR182" id="ref-link-section-d90629110e41385">182</a>] keep full account of the correlation between the PDFs and <span class="mathjax-tex">$\alpha _s$</span>. The values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e41439">165</a>] are those quoted in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e41442">172</a>] for the best fit value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span>. The numbers in parentheses are obtained with the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> fixed to <span class="mathjax-tex">$\alpha _s(M_Z)=0.118$</span> </b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/11" aria-label="Full size table 11"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>Of particular interest is the impact of additional parameters in the PDF fits, such as the charm-quark mass, on the Higgs cross section. The differences in the treatment of heavy quarks and the consequences for the quality of the description of charm-quark DIS data have already been discussed in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>. ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e42252">2</a>] fits the value of <span class="mathjax-tex">$m_c(m_c)$</span> in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme and the uncertainties in the charm-quark mass are included in the uncertainties quoted in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab9">9</a>. Other groups keep a fixed value of the charm-quark mass in the on-shell scheme, cf. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab4">4</a>, and vary the value of <span class="mathjax-tex">$m_c^\mathrm{pole}$</span> within some range. Such studies have been performed in the past by NNPDF2.1 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e42362">170</a>] and MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 171" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Heavy-quark mass dependence in global PDF analyses and 3- and 4-flavour parton distributions. Eur. Phys. J. C 70, 51 (2010). 
 arXiv:1007.2624
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR171" id="ref-link-section-d90629110e42365">171</a>] and more recently by MMHT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e42368">172</a>].</p><p>In Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab10">10</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab11">11</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab12">12</a> we display the results of these fits together with the values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the DIS charm-quark data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e42408">165</a>], mostly computed with <span class="u-monospace">xFitter</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 166" title="S. Alekhin, O. Behnke, P. Belov, S. Borroni, M. Botje, et al., HERAFitter, Open source QCD fit project. 
 arXiv:1410.4412
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR166" id="ref-link-section-d90629110e42415">166</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 167" title="xFitter, An open source QCD fit framework. 
 http://xFitter.org
 
 [xFitter.org]" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR167" id="ref-link-section-d90629110e42418">167</a>], as well as the corresponding cross section for Higgs boson production to NNLO accuracy. The MSTW analysis in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab10">10</a> shows a linear rise of the cross section for increasing values <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.05 - 1.75~\,\mathrm {GeV}$</span> in the range <span class="mathjax-tex">$\sigma (H) = 40.6 - 43.8~\text{ pb }$</span>, which amounts to a variation of more than <span class="mathjax-tex">$7\,\%$</span>. Even if <span class="mathjax-tex">$\alpha _s(M_Z)=0.1171$</span> is kept fixed, the cross section varies in the range <span class="mathjax-tex">$\sigma (H) = 41.6 - 42.6~\text{ pb }$</span>, which is equivalent to <span class="mathjax-tex">$2\,\%$</span>. The best fit in the MSTW analysis with <span class="mathjax-tex">$\chi ^2$</span>/NDP = 63/52 leads to <span class="mathjax-tex">$m_c^\mathrm{pole}=1.3~\,\mathrm {GeV}$</span> and <span class="mathjax-tex">$\alpha _s(M_Z)=0.1166$</span>, both of which are lower than the ones of the nominal fit with <span class="mathjax-tex">$m_c^\mathrm{pole}=1.4~\,\mathrm {GeV}$</span> and <span class="mathjax-tex">$\alpha _s(M_Z)=0.1171$</span>. In Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab11">11</a> the same study is performed for the MMHT PDFs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e42878">172</a>], where the reduced quark mass range <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.15 - 1.55~\,\mathrm {GeV}$</span> still leads to cross section variations <span class="mathjax-tex">$\sigma (H) = 40.5 - 42.1~\text{ pb }$</span> (i.e., <span class="mathjax-tex">$4\,\%$</span>) for the best fit <span class="mathjax-tex">$\alpha _s(M_Z)$</span>, or <span class="mathjax-tex">$\sigma (H) = 42.1 - 42.6~\text{ pb }$</span> (i.e., <span class="mathjax-tex">$1\,\%$</span>) for a fixed <span class="mathjax-tex">$\alpha _s(M_Z)=0.118$</span>. The latter case leads to a best fit of <span class="mathjax-tex">$m_c^\mathrm{pole}=1.2~\,\mathrm {GeV}$</span> with <span class="mathjax-tex">$\chi ^2$</span>/NDP = 70/52, which is significantly smaller than the nominal fit with <span class="mathjax-tex">$m_c^\mathrm{pole}=1.4~\,\mathrm {GeV}$</span> and <span class="mathjax-tex">$\chi ^2$</span>/NDP = 82/52.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-12"><figure><figcaption class="c-article-table__figcaption"><b id="Tab12" data-test="table-caption">Table 12 Same as Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab10">10</a> for various NNPDF analyses. The values of the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span> have been fixed in those fits. The values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the description of the HERA data have been determined with the FONLL-C [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 162" title="S. Forte, E. Laenen, P. Nason, J. Rojo, Heavy quarks in deep-inelastic scattering. Nucl. Phys. B 834, 116 (2010). 
 arXiv:1001.2312
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e43387">162</a>] scheme</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/12" aria-label="Full size table 12"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>NNPDF has performed a study of the <span class="mathjax-tex">$m_c$</span> dependence in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e44036">170</a>], which shows the same trend as for MSTW and MMHT, i.e., the smaller the chosen value of <span class="mathjax-tex">$m_c^\mathrm{pole}$</span>, the better the goodness-of-fit for the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e44066">165</a>]. In addition, Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab12">12</a> displays the changes in the charm-quark mass values from <span class="mathjax-tex">$m_c^\mathrm{pole} =\sqrt{2}~\,\mathrm {GeV}$</span> to <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.275~\,\mathrm {GeV}$</span> in the evolution of the NNPDF fits from v2.1 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e44160">170</a>] and v2.3 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 250" title="R.D. Ball et al., Parton distributions with LHC data. Nucl. Phys. B 867, 244 (2013). 
 arXiv:1207.1303
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR250" id="ref-link-section-d90629110e44163">250</a>] to v3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e44166">7</a>], with the obvious correlation of smaller cross sections for Higgs boson production with smaller chosen values of <span class="mathjax-tex">$m_c^\mathrm{pole}$</span>.</p><p>As pointed out already in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>, on-shell masses <span class="mathjax-tex">$m_c^\mathrm{pole}=1.2 - 1.3~\,\mathrm {GeV}$</span>, as preferred by the goodness-of-fit analyses in Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab10">10</a>, <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab11">11</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab12">12</a> for the charm-quark data from HERA [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e44258">165</a>], are not compatible with the world average of the PDG [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="K.A. Olive, Review of particle physics. Chin. Phys. C 38, 090001 (2014)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR55" id="ref-link-section-d90629110e44261">55</a>]. Thus, in some PDF fits, the numerical value of the charm-quark mass effectively takes over the role of a “tuning” parameter for the Higgs cross section. Note that the three analyses are based on partly different data sets, theory and methodology.</p><h3 class="c-article__sub-heading" id="Sec22"><span class="c-article-section__title-number">5.2 </span>Hadro-production of heavy quarks</h3><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec23"><span class="c-article-section__title-number">5.2.1 </span>Top-quark hadro-production: inclusive cross section</h4> <div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-13"><figure><figcaption class="c-article-table__figcaption"><b id="Tab13" data-test="table-caption">Table 13 The inclusive cross section for top-quark pair production at NNLO in QCD at <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV for a pole mass of <span class="mathjax-tex">$m_t^\mathrm{pole}=172.0$</span> GeV at the nominal scale <span class="mathjax-tex">$\mu _r=\mu _f=m_t^\mathrm{pole}$</span> with the PDF (and, if available, also <span class="mathjax-tex">$\alpha _s$</span>) uncertainties. The columns correspond to different choices for the central value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> using the nominal PDF set. The numbers in parenthesis are obtained using PDF sets <span class="u-monospace">CT14nnlo_as_0115</span>, <span class="u-monospace">HERAPDF20_NNLO_ALPHAS_115</span>, <span class="u-monospace">MMHT2014nnlo_asmzlargerange</span> and <span class="u-monospace">NNPDF30_nnlo_as</span> <span class="u-monospace">_0115</span> </b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/13" aria-label="Full size table 13"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The cross section for the hadro-production of top-quark pairs has been measured with unprecedented accuracy at the LHC in Run 1 with <span class="mathjax-tex">$\sqrt{s}=7~\,\mathrm {TeV}$</span> and <span class="mathjax-tex">$8~\,\mathrm {TeV}$</span>. The inclusive cross section is known to NNLO in QCD [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 251" title="P. Bärnreuther, M. Czakon, A. Mitov, Percent Level Precision Physics at the Tevatron: First Genuine NNLO QCD Corrections to 
 
 
 
 $$q \bar{q} \rightarrow t \bar{t} + X$$
 
 
 
 q
 
 
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 →
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 +
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 . Phys. Rev. Lett. 109, 132001 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR251" id="ref-link-section-d90629110e46245">251</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 254" title="M. Czakon, P. Fiedler, A. Mitov, Total top-quark pair-production cross section at hadron colliders through 
 
 
 
 $$O(\alpha _s^4)$$
 
 
 
 O
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 . Phys. Rev. Lett. 110, 252004 (2013). 
 arXiv:1303.6254
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR254" id="ref-link-section-d90629110e46248">254</a>], featuring good convergence of the perturbation series and reduced sensitivity to the renormalization and factorization scales <span class="mathjax-tex">$\mu _r$</span> and <span class="mathjax-tex">$\mu _f$</span>. These theory predictions adopt the on-shell renormalization scheme for the heavy-quark mass. The conversion to the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme for the heavy-quark mass has been discussed in Refs. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 255" title="U. Langenfeld, S. Moch, P. Uwer, Measuring the running top-quark mass. Phys. Rev. D 80, 054009 (2009). 
 arXiv:0906.5273
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR255" id="ref-link-section-d90629110e46332">255</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 257" title="M. Dowling, S. Moch, Differential distributions for top-quark hadro-production with a running mass. Eur. Phys. J. C 74, 3167 (2014). 
 arXiv:1305.6422
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR257" id="ref-link-section-d90629110e46335">257</a>]. For observables such as the inclusive cross section which are dominated by hard scales <span class="mathjax-tex">$\mu _r \simeq \mu _f \simeq m_t$</span>, the theory predictions in terms of the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>mass for the top quark show an even better scale stability and perturbative convergence.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-14"><figure><figcaption class="c-article-table__figcaption"><b id="Tab14" data-test="table-caption">Table 14 The values of the charm-quark mass (on-shell scheme <span class="mathjax-tex">$m_c^\mathrm{pole}$</span>) and the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span> in the MMHT14 analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e46497">172</a>] together the inclusive cross section for top-quark pair production at NNLO in QCD computed with the set <span class="u-monospace">MMHT2014nnlo_mcrange_nf5</span> at <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV for a pole mass of <span class="mathjax-tex">$m_t^\mathrm{pole}=172.0$</span> GeV at the nominal scale <span class="mathjax-tex">$\mu _r=\mu _f=m_t^\mathrm{pole}$</span> and setting <span class="mathjax-tex">$\alpha _s(M_Z)$</span> to the best fit value. The numbers of Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 182" title="L. Harland-Lang, R. Thorne, Private communication, Apr 12 and June 6 (2016)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR182" id="ref-link-section-d90629110e46661">182</a>] keep full account of the correlation between the PDFs and <span class="mathjax-tex">$\alpha _s$</span>. The values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the HERA data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e46715">165</a>] are those quoted in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e46718">172</a>] for the best fit value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span>. The numbers in parentheses for the cross section and <span class="mathjax-tex">$\chi ^2$</span>/NDP are obtained using the PDF set with the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> fixed to <span class="mathjax-tex">$\alpha _s(M_Z)=0.118$</span> </b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/14" aria-label="Full size table 14"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>In a similar study as for Higgs boson production in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab9">9</a> we illustrate in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab13">13</a> the PDF dependence of the inclusive cross section <span class="mathjax-tex">$\sigma (t{\bar{t}})^\mathrm{NNLO}$</span> for various sets with uncertainties <span class="mathjax-tex">$\Delta \sigma (\mathrm {PDF}+\alpha _s)$</span>. The computation is performed in the theoretical framework as implemented in the <span class="u-monospace">HATHOR</span> code [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 256" title="M. Aliev, H. Lacker, U. Langenfeld, S. Moch, P. Uwer, M. Wiedermann, HATHOR: HAdronic Top and Heavy quarks crOss section calculatoR. Comput. Phys. Commun. 182, 1034 (2011). 
 arXiv:1007.1327
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR256" id="ref-link-section-d90629110e47686">256</a>]. In Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab13">13</a> we choose <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV and fix the pole mass <span class="mathjax-tex">$m_t^\mathrm{pole}=172.0$</span> GeV and the scales at <span class="mathjax-tex">$\mu _r = \mu _f = m_t^\mathrm{pole}$</span>. For this fixed value of <span class="mathjax-tex">$m_t^\mathrm{pole}$</span>, we show the impact of different values for the strong coupling constant at NNLO. We choose <span class="mathjax-tex">$\alpha _s(M_Z)$</span> either corresponding to the respective PDF set or fixed to the values 0.115 and 0.118. The results in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab13">13</a> display a spread in a range <span class="mathjax-tex">$\sigma (t{\bar{t}})^\mathrm{NNLO} = 715 - 834~\text{ pb }$</span> using the nominal value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> for each PDF set, which amounts to a relative range of more than <span class="mathjax-tex">$15\,\%$</span>. This decreases to about <span class="mathjax-tex">$6\,\%$</span>, if the values of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> are fixed to 0.115 or 0.118.</p><p>The theoretical predictions at leading order depend parametrically on the strong coupling constant and the top-quark mass to second power, as well as on the convolution of the gluon PDFs, <span class="mathjax-tex">$\sigma (t{\bar{t}})^\mathrm{LO} \propto (\alpha _s^2/m_t^2) \left( g \otimes g \right) $</span>. Therefore, it is necessary to fully account for the correlations between the top-quark mass, the gluon PDF and the strong coupling when comparing to experimental data. A number of analyses have considered <span class="mathjax-tex">$t{\bar{t}}$</span> hadro-production data. ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e48227">2</a>] has included data for top-quark pair-production in a variant of the fit to determine the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>mass <span class="mathjax-tex">$m_t(m_t)$</span>, keeping the full correlation with <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the gluon PDF. On the other hand, CMS has determined the top-quark pole mass as well as the strong coupling constant in a fit which kept all other parameters mutually fixed [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 258" title="CMS Collaboration, S. Chatrchyan et al., Determination of the top-quark pole mass and strong coupling constant from the 
 
 
 
 $$t \bar{t}$$
 
 
 
 t
 
 
 t
 
 
 ¯
 
 
 
 
 
 production cross section in 
 
 
 
 $$pp$$
 
 
 
 p
 p
 
 
 
 collisions at 
 
 
 
 $$\sqrt{s}$$
 
 
 
 s
 
 
 
 = 7 TeV. Phys. Lett. B 728, 496 (2014). 
 arXiv:1307.1907
 
 . [Erratum: Phys. Lett. B728,526(2014)]" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR258" id="ref-link-section-d90629110e48348">258</a>], while Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 259" title="M. Czakon, M.L. Mangano, A. Mitov, J. Rojo, Constraints on the gluon PDF from top quark pair production at hadron colliders. JHEP 07, 167 (2013). 
 arXiv:1303.7215
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR259" id="ref-link-section-d90629110e48351">259</a>] has explored constraints on the gluon PDF from <span class="mathjax-tex">$t{\bar{t}}$</span> hadro-production data using fixed values for <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the pole mass <span class="mathjax-tex">$m_t^\mathrm{pole}$</span>.</p><p>In the global analyses by MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e48465">6</a>] and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e48468">7</a>] those data were also used to fit <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the gluon PDF. These analyses employ a fixed value for the pole mass <span class="mathjax-tex">$m_t^\mathrm{pole}$</span>, which is motivated by precisely measured top-quark masses from kinematic reconstructions, i.e., Monte Carlo masses, but does not account for the above mentioned correlation with <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the gluon PDF. Moreover, the Monte Carlo mass requires additional calibration [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 260" title="J. Kieseler, K. Lipka, S. Moch, Calibration of the top-quark Monte Carlo mass. 
 arXiv:1511.00841
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR260" id="ref-link-section-d90629110e48587">260</a>].</p><p>For the inclusive top-quark cross section we explore in Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab14">14</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab15">15</a> the implicit dependence of the cross section on the charm-quark mass <span class="mathjax-tex">$m_c$</span> used in the GM-VFNS of the PDF fits and list the corresponding values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the DIS charm-quark data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 165" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C73, 2311 (2013). 
 arXiv:1211.1182
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR165" id="ref-link-section-d90629110e48648">165</a>]. This is analogous to the study for the Higgs cross section in Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab11">11</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab12">12</a>. For MMHT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 172" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Charm and beauty quark masses in the MMHT2014 global PDF analysis. 
 arXiv:1510.02332
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR172" id="ref-link-section-d90629110e48658">172</a>] the best fit with <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.25~\,\mathrm {GeV}$</span> and <span class="mathjax-tex">$\alpha _s(M_Z)=0.1167$</span> leads to an inclusive cross section of <span class="mathjax-tex">$\sigma (t{\bar{t}})^\mathrm{NNLO} = 814$</span> pb, which is <span class="mathjax-tex">$2\,\%$</span> lower than the value obtained for the nominal MMHT fit, cf. Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab13">13</a>. Likewise, the changes in the NNPDF fits from v2.1 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 170" title="R.D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J.I. Latorre, J. Rojo, M. Ubiali, Impact of heavy quark masses on parton distributions and LHC phenomenology. Nucl. Phys. B 849, 296 (2011). 
 arXiv:1101.1300
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR170" id="ref-link-section-d90629110e48839">170</a>] and v2.3 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 250" title="R.D. Ball et al., Parton distributions with LHC data. Nucl. Phys. B 867, 244 (2013). 
 arXiv:1207.1303
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR250" id="ref-link-section-d90629110e48842">250</a>] to v3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e48845">7</a>] are documented in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab15">15</a>. The effects amount to almost <span class="mathjax-tex">$2\,\%$</span> when comparing <span class="mathjax-tex">$\sigma (t{\bar{t}})^\mathrm{NNLO}$</span> for the best fit of NNPDF2.1 with <span class="mathjax-tex">$m_c^\mathrm{pole} = \sqrt{2}~\,\mathrm {GeV}$</span> and <span class="mathjax-tex">$\alpha _s(M_Z)=0.119$</span> to the cross section computed with NNPDF3.0 with <span class="mathjax-tex">$m_c^\mathrm{pole} = 1.275~\,\mathrm {GeV}$</span> and <span class="mathjax-tex">$\alpha _s(M_Z)=0.118$</span>. In both Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab14">14</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab15">15</a> there is a correlation showing decreasing cross sections with decreasing values of <span class="mathjax-tex">$m_c^\mathrm{pole}$</span>, although less pronounced than in the case of the Higgs production cross section. The potential bias in the prediction of the inclusive top-quark pair production cross section due to a particular “tuning” of the value of the charm-quark mass for some PDFs is, however, of the same order of magnitude or larger than the quoted PDF uncertainties. Therefore, this needs to be accounted for as an additional modeling uncertainty.</p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-15"><figure><figcaption class="c-article-table__figcaption"><b id="Tab15" data-test="table-caption">Table 15 Same as Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab14">14</a> for various NNPDF analyses. The values of the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span> have always been fixed in those fits. The values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the description of the HERA data have been determined with the FONLL-C [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 162" title="S. Forte, E. Laenen, P. Nason, J. Rojo, Heavy quarks in deep-inelastic scattering. Nucl. Phys. B 834, 116 (2010). 
 arXiv:1001.2312
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e49231">162</a>] scheme</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/15" aria-label="Full size table 15"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-15" data-title="Fig. 15"><figure><figcaption><b id="Fig15" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 15</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/15" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig15_HTML.gif?as=webp"><img aria-describedby="Fig15" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig15_HTML.gif" alt="figure 15" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-15-desc"><p>(<i>Left panel</i>) Predictions for top-quark pair production cross sections at approximate NNLO as a function of the top-quark rapidity using different PDFs at NNLO with the respective PDF uncertainty (depicted by bands of different style). (<i>Right panel</i>) The acceptance and extrapolation estimators with the respective PDF uncertainties, obtained by using different PDF sets</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/15" data-track-dest="link:Figure15 Full size image" aria-label="Full size image figure 15" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec24"><span class="c-article-section__title-number">5.2.2 </span>Top-quark hadro-production: differential distributions</h4><p>The differential cross section of the top-quark pair production is also known to NNLO in QCD [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 261" title="M. Czakon, D. Heymes, A. Mitov, High-precision differential predictions for top-quark pairs at the LHC. 
 arXiv:1511.00549
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR261" id="ref-link-section-d90629110e49903">261</a>]. Publicly available codes such as <span class="u-monospace">Difftop</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 262" title="M. Guzzi, K. Lipka, S. Moch, Top-quark pair production at hadron colliders: differential cross section and phenomenological applications with DiffTop. JHEP 01, 082 (2015). 
 arXiv:1406.0386
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR262" id="ref-link-section-d90629110e49909">262</a>] provide differential distributions to approximate NNLO accuracy based on soft-gluon threshold resummation results. We use <span class="u-monospace">Difftop</span> to calculate the distribution in the top-quark rapidity <span class="mathjax-tex">$y_t$</span> for proton-proton collisions at <span class="mathjax-tex">$\sqrt{s}=13$</span> TeV at NNLO<span class="mathjax-tex">$_\mathrm{approx}$</span> accuracy using the ABM12, CT14, MMHT14, NNPDF3.0, and the PDF4LHC15 PDF sets at NNLO with their respective <span class="mathjax-tex">$\alpha _s$</span> values. Here, we take the top-quark pole mass to be <span class="mathjax-tex">$m_t^\mathrm{pole}=172.5$</span> GeV, following the preferences in the LHC analyses. The renormalization and factorization scales are set to <span class="mathjax-tex">$m_t^\mathrm{pole}$</span> and the choice of a dynamical scale does not change the following discussions.</p><p>By using differential cross sections, not only the sensitivity of top-quark pair production to the PDFs can be estimated, but also possible effects on the experimental acceptance by changing the PDF choice. In the experimental analysis, the PDF dependent acceptance corrections arise mostly from the PDF dependent normalization of the production cross section and originate from the phase space regions uncovered by the detector. Usually, the acceptances are determined by using Monte Carlo simulations as a ratio of the number of reconstructed events in the fiducial volume of the detector (visible phase space) to the number of events generated in the full phase space. In the case of top-quark pair production, the visible (full) phase space would correspond to the top-quark rapidity range of <span class="mathjax-tex">$|y_t|<2.5$</span> <span class="mathjax-tex">$(|y_t|<3)$</span>. Here, an acceptance estimator and a related extrapolation factor are calculated by using <span class="u-monospace">Difftop</span> predictions for the respective cross section ratios <span class="mathjax-tex">$\sigma _\mathrm{vis}/\sigma _\mathrm{tot}$</span> and <span class="mathjax-tex">$\sigma _\mathrm{unmeasured}/\sigma _\mathrm{tot}$</span>. Such estimators are not expected to describe the true experimental efficiency, but are helpful for drawing conclusions about PDF related effects.</p><p>The predictions of the top-quark rapidity and the acceptance estimates obtained by using <span class="u-monospace">Difftop</span> with different PDFs are shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig15">15</a>. The largest difference in the global normalization of the predicted cross sections is observed if the ABM12 PDFs are used instead of the CT14, NNPDF3.0 or MMHT14 sets. The origin of this effect is again the smaller nominal value of <span class="mathjax-tex">$\alpha _s$</span> in ABM12 in combination with a smaller gluon PDF in the <i>x</i> range relevant to top-quark pair production at <span class="mathjax-tex">$\sqrt{s}=13~\,\mathrm {TeV}$</span>. The corresponding acceptance estimators and their uncertainties, obtained from the error propagation of the corresponding PDF uncertainties at 68 % c.l., however, demonstrate significant differences also in the expected acceptance corrections, obtained by using ABM12 alternative to other PDFs.</p><p>The recent PDF4LHC recommendation [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e50332">8</a>] for calculation of the acceptance corrections for precision observables, such as the top-quark pair-production cross section in the LHC Run 2 data taking period, is to use the set <span class="u-monospace">PDF4LHC15_100</span>, which is obtained by averaging the CT14, MMHT14 and NNPDF3.0 PDFs. While the central prediction obtained by using PDF4LHC15 is indeed very close to those obtained with the CT14, MMHT14 or NNPDF3.0 PDFs, the error on the corresponding acceptance estimator somewhat underestimates the acceptance spread of the individual PDFs with their uncertainties. Furthermore, it does not cover the difference in the acceptances to the one using the ABM12 PDF. Therefore, for the conservative estimate of the acceptance correction and its uncertainty, as demanded in the measurement of SM precision observables, the use of the <span class="u-monospace">PDF4LHC15_100</span> set would lead to a significant underestimation of the uncertainty on the resulting cross section measurement.</p><p>A further conclusion from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig15">15</a> is that in the case of top-quark pair production, once calculational speed is needed, it seems to be sufficient to consider a reduced choice of PDF sets. For instance, instead of using the averaged set <span class="u-monospace">PDF4LHC15_100</span> one can take just one of the three PDFs, CT14, MMHT14 or NNPDF3.0. Alternative PDF choices can then always be studied to some approximation with a reweighting method. In spite of the valiant effort in Ref. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e50351">8</a>] to provide a uniform solution, the PDF choice for measurements of precision observables must be decided on a case-by-case basis for each particular process.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec25"><span class="c-article-section__title-number">5.2.3 </span>Bottom-quark hadro-production</h4><p>Bottom-quark production in proton-proton collisions at the LHC is also dominated by the gluon–gluon fusion process. Therefore, the LHCb measurements of <i>B</i>-meson production in the forward region [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 263" title="LHCb Collaboration, R. Aaij et al., Measurement of B meson production cross sections in proton–proton collisions at 
 
 
 
 $$\sqrt{s} =$$
 
 
 
 
 s
 
 =
 
 
 
 7 TeV. JHEP 08 (2013) 117, 
 arXiv:1306.3663
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR263" id="ref-link-section-d90629110e50365">263</a>] with rapidities <span class="mathjax-tex">$2.0< y < 4.5$</span> at <span class="mathjax-tex">$\sqrt{s}=7$</span> TeV probe the gluon distributions simultaneously at small <i>x</i> up to <span class="mathjax-tex">$x \sim 2 \times 10^{-5}$</span> and at large <span class="mathjax-tex">$x \simeq 1$</span>. The small-<i>x</i> region is not accessible with HERA DIS data, for example. The potential improvements of PDFs near the edges of the currently covered kinematical region, namely, at small <i>x</i> and low scales, was first illustrated in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 264" title="O. Zenaiev, A. Geiser, K. Lipka, J. Blümlein, A. Cooper-Sarkar, et al., Impact of heavy-flavour production cross sections measured by the LHCb experiment on parton distribution functions at low x. 
 arXiv:1503.04581
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR264" id="ref-link-section-d90629110e50503">264</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 265" title="O. Zenaiev, Charm production and QCD analysis at HERA and LHC. PhD thesis, U. Hamburg, Dept. Phys., DESY-THESIS-2015-012 (2015)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR265" id="ref-link-section-d90629110e50506">265</a>] using differential LHCb data on hadro-production of <span class="mathjax-tex">$c\bar{c}$</span> and <span class="mathjax-tex">$b\bar{b}$</span> pairs.</p><p>In the present comparison in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab16">16</a>, the normalized cross sections, <span class="mathjax-tex">$(\mathrm{d}\sigma /\mathrm{d}y) / (\mathrm{d}\sigma /\mathrm{d}y_0)$</span>, for bottom-quark production are calculated from the absolute measurements published by LHCb, with <span class="mathjax-tex">$\mathrm{d}\sigma /\mathrm{d}y_0$</span> being the cross section in the central bin, <span class="mathjax-tex">$3< y_0 < 3.5$</span>, of the measured rapidity range in each <span class="mathjax-tex">$p_T$</span> bin [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 264" title="O. Zenaiev, A. Geiser, K. Lipka, J. Blümlein, A. Cooper-Sarkar, et al., Impact of heavy-flavour production cross sections measured by the LHCb experiment on parton distribution functions at low x. 
 arXiv:1503.04581
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR264" id="ref-link-section-d90629110e50763">264</a>]. In the absence of NNLO QCD corrections, the theoretical predictions are obtained at NLO in QCD [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 266" title="P. Nason, S. Dawson, R.K. Ellis, The total cross section for the production of heavy quarks in hadronic collisions. Nucl. Phys. B 303, 607 (1988)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR266" id="ref-link-section-d90629110e50766">266</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 268" title="P. Nason, S. Dawson, R.K. Ellis, The one particle inclusive differential cross section for heavy quark production in hadronic collisions. Nucl. Phys. B327, 49 (1989). [Erratum: Nucl. Phys.B 335, 260 (1990)]" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR268" id="ref-link-section-d90629110e50769">268</a>] using a fixed number of flavors, <span class="mathjax-tex">$n_f = 3$</span>, for the hard scattering cross sections. Since data for the hadro-production of heavy quarks other than top have not been considered for publicly available PDF fits thus far, issues of any model dependence such as in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 158" title="R. Harlander, M. Krämer, M. Schumacher, Bottom-quark associated Higgs-boson production: reconciling the four- and five-flavour scheme approach. 
 arXiv:1112.3478
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR158" id="ref-link-section-d90629110e50803">158</a>] due to the use of GM-VFNS cannot be quantified. In the calculation of the normalized cross sections, the theoretical uncertainty is strongly reduced, since variations of the renormalization and factorization scales as well as of the fragmentation parameters do not significantly affect the shape of the <i>y</i> distributions for heavy-flavor production, while this shape remains sensitive to PDFs.</p> <div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-16"><figure><figcaption class="c-article-table__figcaption"><b id="Tab16" data-test="table-caption">Table 16 The values of <span class="mathjax-tex">$\chi ^2$</span>/NDP for the normalised bottom-quark cross sections measured at LHCb [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 263" title="LHCb Collaboration, R. Aaij et al., Measurement of B meson production cross sections in proton–proton collisions at 
 
 
 
 $$\sqrt{s} =$$
 
 
 
 
 s
 
 =
 
 
 
 7 TeV. JHEP 08 (2013) 117, 
 arXiv:1306.3663
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR263" id="ref-link-section-d90629110e50847">263</a>] using the NLO PDFs of the individual groups. The left column accounts for the quoted PDF uncertainties (with the CJ15 and CT14 PDF uncertainties rescaled to 68 % c.l.), while the right column uses the central prediction of each PDF set</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1140/epjc/s10052-016-4285-4/tables/16" aria-label="Full size table 16"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-16" data-title="Fig. 16"><figure><figcaption><b id="Fig16" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 16</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/16" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig16_HTML.gif?as=webp"><img aria-describedby="Fig16" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig16_HTML.gif" alt="figure 16" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-16-desc"><p>Theoretical predictions for the total <span class="mathjax-tex">$pp \rightarrow c\bar{c}$</span> cross section as a function of the center-of-mass energy <span class="mathjax-tex">$\sqrt{s}$</span> at NLO (<i>dashed lines</i>) and NNLO (<i>solid lines</i>) QCD accuracy in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>mass scheme with <span class="mathjax-tex">$m_c(m_c)~=~1.27$</span> GeV and scale choice <span class="mathjax-tex">$\mu _R = \mu _F = 2m_c(m_c)$</span> using the central PDF sets (<i>solid lines</i>) of ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e51414">2</a>], CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e51417">1</a>], CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e51420">3</a>] and JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e51424">5</a>] and the respective PDF uncertainties (<i>dashed lines</i>). The predictions for ABM12 (CJ15) use the NNLO (NLO) PDFs independent of the order of perturbation theory. See text for details and references on the experimental data from fixed target experiments and colliders (STAR, PHENIX, ALICE, ATLAS, LHCb)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/16" data-track-dest="link:Figure16 Full size image" aria-label="Full size image figure 16" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-17" data-title="Fig. 17"><figure><figcaption><b id="Fig17" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 17</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/17" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig17_HTML.gif?as=webp"><img aria-describedby="Fig17" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig17_HTML.gif" alt="figure 17" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-17-desc"><p>Same as Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a> using the central PDF sets of HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e51452">4</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e51455">6</a>], NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e51458">7</a>] and PDF4LHC15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e51461">8</a>] together with the respective PDF uncertainties</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/17" data-track-dest="link:Figure17 Full size image" aria-label="Full size image figure 17" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The values for <span class="mathjax-tex">$\chi ^2$</span>/NDP given in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab16">16</a> are computed with the QCD fit platform <span class="u-monospace">xFitter</span> for the individual PDF sets obtained at NLO, namely, ABM11 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="S. Alekhin, J. Blümlein, S. Moch, Parton distribution functions and benchmark cross sections at NNLO. Phys. Rev. D 86, 054009 (2012). 
 arXiv:1202.2281
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR64" id="ref-link-section-d90629110e51508">64</a>], CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e51511">1</a>], CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e51515">3</a>], HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e51518">4</a>], JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e51521">5</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e51524">6</a>], NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e51527">7</a>], as well as the averaged set PDF4LHC15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e51530">8</a>]. All PDFs provide a good description of the data, despite the fact that none of the groups use any data sensitive to the gluons at very low <i>x</i>, in the region directly probed by the LHCb <i>B</i>-meson measurement. Remarkably, one finds that <span class="mathjax-tex">$\chi ^2/\text{ NDP } < 1$</span> for the vast majority of the groups (left column in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab16">16</a>), suggesting that the derived PDF uncertainties at the edges of the so far measured regions might be inflated.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec26"><span class="c-article-section__title-number">5.2.4 </span>Charm-quark hadro-production</h4><p>Charm-quark hadro-production offers another possibility to illustrate the consistency of the theory predictions for the various PDF sets. The exclusive production of charmed mesons in the forward region at LHCb probes the gluon distribution down to small-<i>x</i> values of <span class="mathjax-tex">$x \sim 5 \times 10^{-6}$</span> at <span class="mathjax-tex">$\sqrt{s}=7$</span> TeV, and data can be confronted with QCD predictions at NLO accuracy, see, e.g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 269" title="M.V. Garzelli, S. Moch, G. Sigl, Lepton fluxes from atmospheric charm revisited. JHEP 10, 115 (2015). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR269" id="ref-link-section-d90629110e51665">269</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 270" title="R. Gauld, J. Rojo, L. Rottoli, J. Talbert, Charm production in the forward region: constraints on the small-x gluon and backgrounds for neutrino astronomy. JHEP 11, 009 (2015). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR270" id="ref-link-section-d90629110e51668">270</a>].</p><p>For the inclusive cross section of the reaction <i>pp</i> <span class="mathjax-tex">$\rightarrow $</span> <span class="mathjax-tex">$c\bar{c}$</span> the QCD predictions are known up to NNLO in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme for the charm-quark mass and display good convergence of the perturbative expansion and stability under variation of the renormalization and factorization scales [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 269" title="M.V. Garzelli, S. Moch, G. Sigl, Lepton fluxes from atmospheric charm revisited. JHEP 10, 115 (2015). 
 arXiv:1507.01570
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR269" id="ref-link-section-d90629110e51763">269</a>]. In Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig17">17</a> we compare the theory predictions at NLO and NNLO with <span class="mathjax-tex">$m_c(m_c) = 1.27~\,\mathrm {GeV}$</span> in the <span class="mathjax-tex">$\overline{\mathrm {MS}}\, $</span>scheme, see Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a>, for the scale choice <span class="mathjax-tex">$\mu _r=\mu _f=2m_c(m_c)$</span> as a function of the center-of-mass energy <span class="mathjax-tex">$\sqrt{s}$</span> to available experimental data. These data span a large range in <span class="mathjax-tex">$\sqrt{s}$</span>, which starts with fixed target experiments at energies up to <span class="mathjax-tex">$\sqrt{s} = 50~\,\mathrm {GeV}$</span> summarized in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 271" title="C. Lourenco, H. Wöhri, Heavy flavour hadro-production from fixed-target to collider energies. Phys. Rept. 433, 127 (2006). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR271" id="ref-link-section-d90629110e52009">271</a>] and HERA-B data [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 272" title="HERA-B Collaboration, I. Abt et al., Measurement of 
 
 
 
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 production in fixed target 920-GeV proton-nucleus collisions. Eur. Phys. J. C52, 531 (2007). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR272" id="ref-link-section-d90629110e52012">272</a>] (purple points in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig17">17</a>). At higher energies RHIC data from PHENIX and STAR [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 273" title="PHENIX Collaboration, A. Adare et al., Measurement of high-
 
 
 
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 GeV. Phys. Rev. D 86, 072013 (2012). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR274" id="ref-link-section-d90629110e52025">274</a>] (black points in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig17">17</a>) are available and the LHC contributes measurements at energies <span class="mathjax-tex">$\sqrt{s}=2.76~\,\mathrm {TeV}$</span> from ALICE [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 275" title="ALICE Collaboration, B. Abelev et al., Measurement of charm production at central rapidity in proton–proton collisions at 
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR275" id="ref-link-section-d90629110e52072">275</a>], at <span class="mathjax-tex">$\sqrt{s}=7~\,\mathrm {TeV}$</span> from ALICE [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 275" title="ALICE Collaboration, B. Abelev et al., Measurement of charm production at central rapidity in proton–proton collisions at 
 
 
 
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 2.76
 
 
 
 TeV. JHEP 1207, 191 (2012). 
 arXiv:1205.4007
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR275" id="ref-link-section-d90629110e52112">275</a>], ATLAS [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 276" title="ATLAS Collaboration, Measurement of 
 
 
 
 $$D^{(*)}$$
 
 
 
 D
 
 (
 
 ∗
 )
 
 
 
 
 meson production cross sections in pp collisions at 
 
 
 
 $$\sqrt{s}=7$$
 
 
 
 
 s
 
 =
 7
 
 
 
 TeV with the ATLAS detector, ATLAS-CONF-2011-017, ATLAS-COM-CONF-2011-030" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR276" id="ref-link-section-d90629110e52115">276</a>] and LHCb [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 277" title="LHCb Collaboration, R. Aaij et al., Prompt charm production in pp collisions at 
 
 
 
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 =
 7
 
 
 
 TeV. Nucl. Phys. B871, 1 (2013). 
 arXiv:1302.2864
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR277" id="ref-link-section-d90629110e52118">277</a>], and at the highest available energy <span class="mathjax-tex">$\sqrt{s}=13~\,\mathrm {TeV}$</span> from LHCb [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 278" title="LHCb Collaboration, R. Aaij et al., Measurements of prompt charm production cross sections in 
 
 
 
 $$pp$$
 
 
 
 p
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 = 13 TeV. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR278" id="ref-link-section-d90629110e52159">278</a>] (blue points in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig17">17</a>). The total cross sections of LHCb have been obtained from charmed hadron production measurements in a limited phase space region [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 277" title="LHCb Collaboration, R. Aaij et al., Prompt charm production in pp collisions at 
 
 
 
 $$\sqrt{s}=7$$
 
 
 
 
 s
 
 =
 7
 
 
 
 TeV. Nucl. Phys. B871, 1 (2013). 
 arXiv:1302.2864
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR277" id="ref-link-section-d90629110e52168">277</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 278" title="LHCb Collaboration, R. Aaij et al., Measurements of prompt charm production cross sections in 
 
 
 
 $$pp$$
 
 
 
 p
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 = 13 TeV. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR278" id="ref-link-section-d90629110e52171">278</a>] using extrapolations based on NLO QCD predictions matched with parton shower Monte Carlo generators.</p><p>The theory predictions for the PDF sets ABM12, CJ15, CT14 and JR14 at NLO and NNLO are shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a>, together with the respective PDF uncertainties. For all these PDF sets the perturbative expansion is stable, the theory computations agree well with the data and predictions, e.g., for a future collider with <span class="mathjax-tex">$\sqrt{s} \simeq 100~\,\mathrm {TeV}$</span>, yield positive cross sections. The PDF uncertainties obtained for CT14, however, do increase significantly above energies of <span class="mathjax-tex">$\sqrt{s} \simeq 1~\,\mathrm {TeV}$</span>.</p><p>The same information for the sets HERAPDF2.0, MMHT14, NNPDF3.0 and PDF4LHC15 is displayed in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig17">17</a>. These predictions all agree with data at low energies but start to behave very differently for HERAPDF2.0, MMHT14 or NNPDF3.0 at energies above <span class="mathjax-tex">$\sqrt{s} \simeq \mathcal{O}(10)~\,\mathrm {TeV}$</span> and for PDF4LHC15 above <span class="mathjax-tex">$\sqrt{s} \simeq \mathcal{O}(100)~\,\mathrm {TeV}$</span>. At the same time, the associated PDF uncertainties in this region of phase space become very large, thereby limiting the predictive power. Typically, the PDF uncertainties of the NNLO sets are even larger than at NLO. In the case of MMHT14 the consistency of the NNLO predictions with LHC data from ALICE [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 275" title="ALICE Collaboration, B. Abelev et al., Measurement of charm production at central rapidity in proton–proton collisions at 
 
 
 
 $$\sqrt{s}=2.76$$
 
 
 
 
 s
 
 =
 2.76
 
 
 
 TeV. JHEP 1207, 191 (2012). 
 arXiv:1205.4007
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR275" id="ref-link-section-d90629110e52358">275</a>], ATLAS [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 276" title="ATLAS Collaboration, Measurement of 
 
 
 
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 D
 
 (
 
 ∗
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 $$\sqrt{s}=7$$
 
 
 
 
 s
 
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 TeV with the ATLAS detector, ATLAS-CONF-2011-017, ATLAS-COM-CONF-2011-030" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR276" id="ref-link-section-d90629110e52361">276</a>] and LHCb [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 277" title="LHCb Collaboration, R. Aaij et al., Prompt charm production in pp collisions at 
 
 
 
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 TeV. Nucl. Phys. B871, 1 (2013). 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR277" id="ref-link-section-d90629110e52365">277</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 278" title="LHCb Collaboration, R. Aaij et al., Measurements of prompt charm production cross sections in 
 
 
 
 $$pp$$
 
 
 
 p
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 = 13 TeV. 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR278" id="ref-link-section-d90629110e52368">278</a>] at energies of <span class="mathjax-tex">$\sqrt{s}=7~\,\mathrm {TeV}$</span> and <span class="mathjax-tex">$13~\,\mathrm {TeV}$</span> deteriorates. For NNPDF3.0 the central prediction at NNLO displays a change in slope for energies above <span class="mathjax-tex">$\sqrt{s} \simeq 3~\,\mathrm {TeV}$</span> leading to a steeply rising cross section. The most striking feature, however, are the negative cross sections for HERAPDF2.0, MMHT14 and PDF4LHC15 at energies above <span class="mathjax-tex">$\sqrt{s} \simeq \mathcal{O}(30 - 100)~\,\mathrm {TeV}$</span>, depending on the chosen set. This is an effect of the negative gluon PDF for those sets at values of <i>x</i> within the kinematic reach of current or future hadron colliders up to <span class="mathjax-tex">$\sqrt{s} \simeq 100~\,\mathrm {TeV}$</span>. This results in an instability of the perturbative expansion of the <span class="mathjax-tex">$\sigma _{pp \rightarrow c{\bar{c}}}$</span> cross section at high energies when the contribution from the quark–gluon channel dominates. The reason for a negative gluon PDF in the NNLO set of PDF4LHC15 (being some average of the CT14, MMHT14 and NNPDF3.0 sets) is unclear. In contrast, other PDFs shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig16">16</a> demonstrate stability of the perturbative expansion through NNLO up to very high energies and good consistency of the predictions with the experimental data.</p><h3 class="c-article__sub-heading" id="Sec27"><span class="c-article-section__title-number">5.3 </span> <span class="mathjax-tex">$W'/Z'$</span> production</h3><p>Cross sections sensitive to large-<i>x</i> parton distributions typically fall rapidly with increasing <i>x</i> values, leading to limitations in the quantity and precision of experimental data and the kinematic range over which they can be obtained. Consequently, the precision to which one can constrain large-<i>x</i> PDFs decreases with <i>x</i>, and systematic uncertainties due to extrapolations into unmeasured regions of <i>x</i> (or those excluded by cuts) increase. Similarly, the theoretical uncertainties due to various approximations in the treatment of nuclear corrections for deuterium data, or target mass and higher twist effects, also become larger.</p><p>To illustrate this, consider the production of a heavy <span class="mathjax-tex">$W'$</span> boson as a function of the <span class="mathjax-tex">$W'$</span> rapidity <span class="mathjax-tex">$y_{W'}$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 279" title="L.T. Brady, A. Accardi, W. Melnitchouk, J.F. Owens, Impact of PDF uncertainties at large 
 
 
 
 $$x$$
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR279" id="ref-link-section-d90629110e52762">279</a>]. Assuming Standard Model couplings, the parton luminosity for a produced negatively charged <span class="mathjax-tex">$W'^-$</span> boson is given by</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&{{ \mathcal{L}_{W'^-} \, = \, }} \frac{2 \pi G_F}{3\sqrt{2}} x_1 x_2 \bigg [ \cos ^2\theta _C \big ( \bar{u}(x_2)d(x_1) + \bar{c}(x_2)s(x_1) \big )\nonumber \\&\quad + \sin ^2\theta _C \big ( \bar{u}(x_2)s(x_1) + \bar{c}(x_2)d(x_1) \big ) \bigg ] + (x_1 \leftrightarrow x_2) \, , \end{aligned}$$</span></div><div class="c-article-equation__number"> (22) </div></div><p>where <span class="mathjax-tex">$G_F$</span> is the Fermi constant and <span class="mathjax-tex">$\theta _C$</span> the Cabibbo angle. The uncertainty <span class="mathjax-tex">$\delta \mathcal{L}_{W'^-}$</span> in the luminosity is shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig18">18</a> for various PDF sets as a function of <span class="mathjax-tex">$y_{W'}$</span>, for several fixed values of the boson mass from the SM <i>W</i> up to <span class="mathjax-tex">$M_{W'}=7$</span> TeV.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-18" data-title="Fig. 18"><figure><figcaption><b id="Fig18" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 18</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/18" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig18_HTML.gif?as=webp"><img aria-describedby="Fig18" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig18_HTML.gif" alt="figure 18" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-18-desc"><p>Relative uncertainty <span class="mathjax-tex">$\delta \mathcal{L}_{W'^-} / \mathcal{L}_{W'^-}$</span> in the <span class="mathjax-tex">$W'^-$</span> luminosity as a function of rapidity <span class="mathjax-tex">$y_{W'}$</span> for the combined PDF4LHC15 set (<i>dotted</i>), the CJ15 (<i>solid</i>), MMHT14 (<i>dot-dashed</i>), and CT14 (<i>dashed</i>) PDFs for various <span class="mathjax-tex">$W'$</span> masses from 80 GeV (SM) to 7.0 TeV. All PDF uncertainties have been scaled to a common 68 % c.l. as provided by the various groups</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/18" data-track-dest="link:Figure18 Full size image" aria-label="Full size image figure 18" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>Note that as the rapidity or mass of the produced boson increases, so does the momentum fraction <span class="mathjax-tex">$x_{1,2} = (M_{W'}/\sqrt{s})\, e^{\pm y_{W'}}$</span> of one or both partons, in which case the luminosity behaves as <span class="mathjax-tex">$\mathcal{L}^- \sim \bar{u}(x_2) d(x_1)$</span>. Except for the highest <span class="mathjax-tex">$M_{W'}$</span> values, the PDF uncertainty typically remains small up to large values of <span class="mathjax-tex">$y_{W'}$</span>, corresponding to <span class="mathjax-tex">$x_1 \approx 0.65$</span>, beyond which it rises dramatically for all <span class="mathjax-tex">$M_{W'}$</span>. This is precisely the region where data constraining the <i>d</i>-quark PDF are scarce, and theoretical assumptions play an important role [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
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 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e53852">1</a>]. This is particularly pronounced for fits that exclude DIS data at low invariant masses, such as the three fits included in the PDF4LHC combination [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e53855">8</a>]. For large <span class="mathjax-tex">$W'$</span> masses, the <span class="mathjax-tex">$\bar{u}$</span> PDF is evaluated at <span class="mathjax-tex">$x_2 \sim 0.2 - 0.5$</span>, where data are either nonexistent or have large errors, giving rise to the increased uncertainties in some of the PDF sets at <span class="mathjax-tex">$y_{W'} \sim 0$</span>.</p><p>The relative uncertainties in the luminosities in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig18">18</a> have been scaled to a common 68 % c.l., as in the tables in the previous sections. One observes a very large range of uncertainties for the various PDF sets, which stems from different tolerance criteria used and different methodologies employed for the treatment of data at high values of <i>x</i>. The smallest uncertainty is obtained for the CJ15 PDF set, which makes use of low invariant mass data to constrain the high-<i>x</i> region, and does not employ additional tolerance factors inflating the uncertainties. The MMHT and CT14 PDF sets have larger errors, due to stronger cuts on low-mass DIS data and larger tolerances, and consequently the averaged PDF4LHC15 set gives similarly large uncertainties.</p><p>This example illustrates the problematic nature of statistically combining PDF sets that have been determined using very different theoretical treatments of the high-<i>x</i> region, leading to an overestimate of the uncertainties at these kinematics. Using the PDF4LHC15 set as the sole basis for background estimates, for example, one could potentially miss signals of new physics in regions such as at high rapidity <span class="mathjax-tex">$y_{W'}$</span>. A more meaningful PDF uncertainty would be obtained when combining PDF sets obtained under similar conditions and inputs; if large differences are found, these should be investigated further rather than simply averaged over.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-19" data-title="Fig. 19"><figure><figcaption><b id="Fig19" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 19</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/19" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig19_HTML.gif?as=webp"><img aria-describedby="Fig19" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-016-4285-4/MediaObjects/10052_2016_4285_Fig19_HTML.gif" alt="figure 19" loading="lazy"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-19-desc"><p>Ratio of central values of the <span class="mathjax-tex">$W'^-$</span> luminosity <span class="mathjax-tex">$\mathcal{L}_{W'^-}$</span> to the PDF4LHC value (dotted, 68 % c.l. shaded band) as a function of rapidity <span class="mathjax-tex">$y_{W'}$</span>. The PDF sets CJ15 (<i>red solid curve</i>), MMHT14 (<i>blue dot-dashed curve</i>), CT14 (<i>blue dashed curve</i>), and NNPDF3.0 (<i>green dashed curve</i>) are compared for a <span class="mathjax-tex">$W'$</span> mass <span class="mathjax-tex">$M_{W'}=3.5$</span> TeV</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1140/epjc/s10052-016-4285-4/figures/19" data-track-dest="link:Figure19 Full size image" aria-label="Full size image figure 19" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>This is also illustrated in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig19">19</a>, where the central values for the <span class="mathjax-tex">$W'^-$</span> luminosity for several PDF sets are compared relative to the luminosity computed from the central PDF4LHC15 distributions. The different theoretical assumptions utilized in the fits produce systematic differences in the large-<i>x</i> PDFs, which give rise to ratios of central values that are of the same order as the overall PDF4LHC15 68 % c.l. uncertainty, and in the case of the NNPDF3.0 set are about twice as large.</p><p>The fact that the uncertainty bands of the individual sets overlap with that of the PDF4LHC15 set is not, however, an indication that the latter is a good estimate of the PDF uncertainties in this extrapolation region. Rather, the PDF4LHC15 band effectively represents a statistical envelope of the systematic theoretical differences between the sets included in the combination. A comparison with the luminosity computed using the CJ15 PDF set, which is not included in the PDF4LHC15 combination, is instructive in this respect. The two main theoretical assumptions affecting the <span class="mathjax-tex">$W'^-$</span> luminosity are the nuclear corrections in deuterium (applied or fitted in the CJ15 and MMHT14 analyses, as well as in JR14 and ABM12), and the parametrization of the <i>d</i>-quark PDF.</p><p>For the latter, the traditional choice has been to assume a behavior <span class="mathjax-tex">$\propto (1-x)^\beta $</span> as <span class="mathjax-tex">$x \rightarrow 1$</span> for both the <i>d</i>- and <i>u</i>-quark PDFs (as, <i>e.g.</i>, in the MMHT14 and NNPDF3.0 analyses), in which case the <i>d</i> / <i>u</i> ratio either vanishes or becomes infinite in the <span class="mathjax-tex">$x \rightarrow 1$</span> limit depending on whether the exponent <span class="mathjax-tex">$\beta $</span> is larger for <i>d</i> or <i>u</i>. Alternatively, including an additive term in the <i>d</i>-quark PDF proportional to <i>u</i>(<i>x</i>) (as in CJ15) or constraining <span class="mathjax-tex">$\beta _u=\beta _d$</span> (as in CT14) allows the <i>d</i> / <i>u</i> ratio to reach a finite, nonzero limiting value at <span class="mathjax-tex">$x \rightarrow 1$</span>. Furthermore, the CJ15 distributions were also fitted to low invariant mass (3.5 GeV<span class="mathjax-tex">$^2< W^2 < 12.5$</span> GeV<span class="mathjax-tex">$^2$</span>) DIS data, which were excluded by kinematic cuts in the MMHT14, CT14 and NNPDF3.0 analyses. Consequently, the following features can be observed in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig19">19</a>:</p><ul class="u-list-style-bullet"> <li> <p>The MMHT14 curve follows CJ15 closely until <span class="mathjax-tex">$y_{W'} \approx 1 (x \approx 0.65)$</span>, after which the <i>d</i>-quark PDF turns upwards relative to CJ15, in the region not constrained by the large-<i>x</i> and low-<span class="mathjax-tex">$W^2$</span> SLAC data utilized in CJ15.</p> </li> <li> <p>The CT14 curve is lower than CJ15 at <span class="mathjax-tex">$y_{W'} \lesssim 0.6$</span> (<span class="mathjax-tex">$x \lesssim 0.45$</span>), and higher at larger <span class="mathjax-tex">$y_{W'}$</span>, because of the neglect of nuclear corrections. At <span class="mathjax-tex">$y_{W'} > 1$</span> the <i>d</i>-quark PDF is essentially unconstrained since neither the low-<span class="mathjax-tex">$W^2$</span> SLAC data nor the reconstructed Tevatron <i>W</i>-boson production data are included in the fit.</p> </li> <li> <p>The NNPDF3.0 fit, which excludes low-<span class="mathjax-tex">$W^2$</span> DIS data and does not utilize nuclear or hadronic corrections, consistently deviates from all others. It is, however, compatible with those within its own uncertainties, which at large <i>x</i> are about four times larger than that of the other fits.</p> </li> </ul><p>In summary, in extreme kinematic regions, such as at large rapidity or for large-mass observables, caution must be exercised when utilizing PDF error bands and nominal confidence levels provided by the various PDF groups for precision calculations and statistically meaningful comparisons to data. Utilizing the PDF4LHC15 band at face value likely overestimates the current uncertainty on large-<i>x</i> PDFs, and could lead to signals of new physics being missed. Calculations performed with the combination set should always be cross-checked with as many individual PDF sets as possible, taking into account the amount and kind of data included in each fit, as well as the different theoretical inputs. The latter explore different physics issues and can vary considerably from one PDF set to another. When differences arise, further scrutiny of the PDF fit results themselves may be needed before drawing any definitive conclusions.</p></div></div></section><section data-title="Recommendations for PDF usage"><div class="c-article-section" id="Sec28-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec28"><span class="c-article-section__title-number">6 </span>Recommendations for PDF usage</h2><div class="c-article-section__content" id="Sec28-content"><p>Recommendations for the usage of PDFs generally aim in providing guidance for estimates of the magnitude and the uncertainties of cross sections in a reliable but also efficient way. First recommendations have been provided by the <i>PDF4LHC Working Group</i> in the <i>Interim Recommendations</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 280" title="M. Botje et al., The PDF4LHC working group interim recommendations. 
 arXiv:1101.0538
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR280" id="ref-link-section-d90629110e54888">280</a>]. There, the MSTW [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 197" title="A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Parton distributions for the LHC. Eur. Phys. J. C 63, 189 (2009). 
 arXiv:0901.0002
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR197" id="ref-link-section-d90629110e54891">197</a>] PDF was used as a central set for predictions at NNLO in QCD and the procedure for calculation of the PDF uncertainties, based on an envelope of several PDF sets, was proposed. This approach has been criticized for being impractical. The 2015 PDF4LHC recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e54894">8</a>] have evolved from related discussions and aim in improving the efficiency of cross section computations by averaging several PDFs along with their respective uncertainties. Here, we briefly recall these suggestions and put them into context of the findings of the previous sections. We comment on several shortcomings of the recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e54898">8</a>] and propose an alternative for the PDF usage at the LHC.</p><h3 class="c-article__sub-heading" id="Sec29"><span class="c-article-section__title-number">6.1 </span>The 2015 PDF4LHC recommendations: A critical appraisal</h3><p>The 2015 PDF4LHC recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e54908">8</a>] distinguish four cases: (i) Comparisons between data and theory for Standard Model measurements, (ii) Searches for Beyond the Standard Model phenomena, (iii) Calculation of PDF uncertainties in situations when computational speed is needed, or a more limited number of error PDFs may be desirable and (iv) Calculation of PDF uncertainties in precision observables.</p><p>For the case (i), the recommendation is to use the individual PDF sets ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e54914">2</a>], CJ12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 191" title="J.F. Owens, A. Accardi, W. Melnitchouk, Global parton distributions with nuclear and finite-
 
 
 
 $$Q^2$$
 
 
 
 Q
 2
 
 
 
 corrections. Phys. Rev. D 87, 094012 (2013). 
 arXiv:1212.1702
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR191" id="ref-link-section-d90629110e54917">191</a>], CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e54920">3</a>], JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e54923">5</a>], HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e54926">4</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e54930">6</a>], and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e54933">7</a>]. It is not clear, why the full account of the PDF dependence should be limited to SM processes only. Deviations observed in the theory predictions obtained with the various PDFs can often be traced back to the differences in the underlying theoretical assumptions and models in the PDF fits. With more LHC data available, tests of the compatibility of those data sets in the individual PDF fits will become more stringent. Studies to quantify the constraining power of processes like hadro-production of <span class="mathjax-tex">$t {\bar{t}}$</span> pairs, jets or <span class="mathjax-tex">$W^\pm $</span> and <i>Z</i> bosons become possible at high precision.</p><p>For the case (ii), it is recommended to employ the PDF4LHC15 sets [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e55002">8</a>], which represent the combination of the CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e55005">3</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e55008">6</a>], and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e55011">7</a>]. The combination is performed using the Monte Carlo approach at different levels of precision, leading to the recommended sets <span class="u-monospace">PDF4LHC15_30</span> and <span class="u-monospace">PDF4LHC15_100</span>. The restriction to CT14, MMHT14 and NNPDF3.0 implies a bias both for the central value and for the PDF uncertainties of BSM cross section predictions. For example, a bias is introduced by fixing the central value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> to an agreed common value, currently chosen to be <span class="mathjax-tex">$\alpha _s(M_Z) = 0.118$</span> at both NLO and NNLO. This choice is in contradiction with the precision determinations of <span class="mathjax-tex">$\alpha _s(M_Z)$</span> at different orders in perturbation theory, as summarized in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec19">4</a>. Further, for searches at the highest energies, the PDFs are probed close to the hadronic threshold near <span class="mathjax-tex">$x \simeq 1$</span>, where nuclear corrections and other hadronic effects, considered for instance in the CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e55187">1</a>] and JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e55190">5</a>] analyses, are important.</p><p>For the case (iii), the <span class="u-monospace">PDF4LHC15_30</span> sets are recommended to use. We would like to note, that here the balance between the computational speed and the precision of the result (in e.g. MC simulation) has to be determined by the analysers. The problem rises from the large deviations between data and theory predictions at low scales and also at the edges of the kinematical ranges of data currently used in PDF fits as illustrated in Sects. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec5">3</a> and <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Sec20">5</a>. The average of various GM-VFNS for heavy quark production, such as ACOT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 159" title="M.A.G. Aivazis, J.C. Collins, F.I. Olness, W.-K. Tung, Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies. Phys. Rev. D 50, 3102 (1994). 
 arXiv:hep-ph/9312319
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR159" id="ref-link-section-d90629110e55205">159</a>], FONLL [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 162" title="S. Forte, E. Laenen, P. Nason, J. Rojo, Heavy quarks in deep-inelastic scattering. Nucl. Phys. B 834, 116 (2010). 
 arXiv:1001.2312
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR162" id="ref-link-section-d90629110e55208">162</a>] and RT [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 163" title="R.S. Thorne, Effect of changes of variable flavor number scheme on parton distribution functions and predicted cross sections. Phys. Rev. D 86, 074017 (2012). 
 arXiv:1201.6180
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR163" id="ref-link-section-d90629110e55212">163</a>], leaves a large degree of arbitrariness in the theory predictions, cf. Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig10">10</a>. Note that the <span class="u-monospace">PDF4LHC15_30</span> sets were updated in December 2015 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 281" title="A. Buckley, J. Rojo, Private communication" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR281" id="ref-link-section-d90629110e55221">281</a>] to account for an extension of their validity range below the original <span class="mathjax-tex">$Q > 8~\,\mathrm {GeV}$</span> as only discussed in the later publication [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 282" title="J. R. Andersen et al. Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report. 
 arXiv:1605.04692
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR282" id="ref-link-section-d90629110e55258">282</a>].</p><p>For the case (iv), the set <span class="u-monospace">PDF4LHC15_100</span> is recommended. Recalling that this case concerns measurements of the precision observables, it is unclear why PDFs should be treated differently than in the case (i). The differences between individual PDF sets propagate the cross section measurements directly through the acceptance corrections or extrapolation factors, as illustrated in Figs. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig15">15</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig17">17</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Fig19">19</a>. Use of the <span class="u-monospace">PDF4LHC15_100</span> is worrysome, since these differences are smeared out in the combination, which, in addition, is limited to only three PDF sets. The SM parameters, determined using the precision observables obtained in this way, may be biased.</p><p>In summary, the recent PDF4LHC recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e55283">8</a>] cannot be viewed as definitive in the case of precision theory predictions, as the advocated averaging procedure introduces bias, artificially inflates the uncertainties, and makes it difficult to quantify potential discrepancies between the individual PDF sets.</p><h3 class="c-article__sub-heading" id="Sec30"><span class="c-article-section__title-number">6.2 </span>New recommendations for the PDF usage at the LHC</h3><p>Based on the considerations above, we propose modifications to the recommendations for PDF usage at the LHC in order to retain the predictive capability of the individual PDF sets. Two cases can be distinguished:</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p> <b>Precise theory predictions</b>, addressing a class of predictions, within or beyond the SM, which encompasses any type of cross section prediction including radiative corrections of any kind, whether at fixed-order or via resummation to some logarithmic accuracy. This class also includes the MC simulations used for the calculation of the acceptance corrections for precision observables, e.g. cross sections which might be used further for determination of SM parameters.</p><ul class="u-list-style-bullet"> <li> <p> <i>Recommendation:</i> Use the individual recent PDF sets, currently ABM12 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="S. Alekhin, J. Blümlein, S. Moch, The ABM parton distributions tuned to LHC data. Phys. Rev. D 89, 054028 (2014). 
 arXiv:1310.3059
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR2" id="ref-link-section-d90629110e55316">2</a>], CJ15 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Constraints on large-
 
 
 
 $$x$$
 
 
 x
 
 
 parton distributions from new weak boson production and deep-inelastic scattering data. 
 arXiv:1602.03154
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR1" id="ref-link-section-d90629110e55319">1</a>], CT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, The CT14 global analysis of quantum chromodynamics. 
 arXiv:1506.07443
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR3" id="ref-link-section-d90629110e55322">3</a>], JR14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="P. Jimenez-Delgado, E. Reya, Delineating parton distributions and the strong coupling. Phys. Rev. D 89, 074049 (2014). 
 arXiv:1403.1852
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR5" id="ref-link-section-d90629110e55325">5</a>], HERAPDF2.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="ZEUS, H1 Collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep-inelastic 
 
 
 
 $$e^{\pm }p$$
 
 
 
 
 e
 ±
 
 p
 
 
 
 scattering cross sections and QCD analysis of HERA data. 
 arXiv:1506.06042
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR4" id="ref-link-section-d90629110e55329">4</a>], MMHT14 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). 
 arXiv:1412.3989
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR6" id="ref-link-section-d90629110e55332">6</a>], and NNPDF3.0 [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="NNPDF Collaboration, R.D. Ball et al., Parton distributions for the LHC Run II. JHEP 04, 040 (2015). 
 arXiv:1410.8849
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR7" id="ref-link-section-d90629110e55335">7</a>] (or as many as possible), together with the respective uncertainties for the chosen PDF set, the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the heavy quark masses <span class="mathjax-tex">$m_c$</span>, <span class="mathjax-tex">$m_b$</span> and <span class="mathjax-tex">$m_t$</span>. Once a PDF set is updated, the most recent version should be used.</p> </li> <li> <p> <i>Rationale:</i> Precise theory predictions as needed for any comparisons between theory and data for processes in the SM or beyond (such as hadro-production of jets, <span class="mathjax-tex">$W^\pm $</span>- or <i>Z</i>-boson production, either singly or in pairs, heavy-quark hadro-production, or generally the production of new massive particles at the TeV scale) often depend on details of the PDF fits and the underlying theory assumptions and schemes used. Differences in the theory predictions based on the individual sets can give an indication of residual systematic uncertainties or shed light on drawbacks and need for potential improvements in the physics models used in the extraction of those PDFs. This applies in particular to measurements used for the determination of SM parameters such as the strong coupling <span class="mathjax-tex">$\alpha _s(M_Z)$</span>, heavy quark masses <span class="mathjax-tex">$m_c$</span>, <span class="mathjax-tex">$m_b$</span> and <span class="mathjax-tex">$m_t$</span> or the <i>W</i>-boson mass, because these parameters are directly correlated to the PDFs used in their extraction from the experimental observables.</p> </li> </ul> </li> <li> <span class="u-custom-list-number">2.</span> <p> <b>Theory predictions for feasibility studies</b>, the complementary class containing all other cross section predictions where high precision is not required, such as those based on Born approximations and/or order of magnitude estimates, or in cases where precision may be sacrificed in favor of computational speed. Here, also studies of novel accelerators and detectors are addressed.</p><ul class="u-list-style-bullet"> <li> <p> <i>Recommendation</i>: Use any of the recent PDF sets (listed in <span class="u-monospace">LHAPDFv6</span> or later versions).</p> </li> <li> <p> <i>Rationale</i>: Often in phenomenological applications for the modern and future facilities one is interested in a quick order of magnitude estimate for the particular cross sections. These are directly proportional to the parton luminosity and to the value of <span class="mathjax-tex">$\alpha _s(M_Z)$</span>. In these cases, one may be willing to sacrifice precision in favor of computational speed. Here, the usage of the sets <span class="u-monospace">PDF4LHC15_30</span> and <span class="u-monospace">PDF4LHC15_100</span> may provide an efficient estimate of PDF uncertainties, although care must be taken in their interpretation depending on the observable and covered kinematic range. Restricting the recommendation to PDFs listed in the <span class="u-monospace">LHAPDF(v6)</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 283" title="A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, G. Watt, LHAPDF6: parton density access in the LHC precision era. Eur. Phys. J. C 75, 132 (2015). 
 arXiv:1412.7420
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR283" id="ref-link-section-d90629110e55703">283</a>] interface excludes parton luminosities with lesser precision in the interpolation of the underlying grids (e.g., in <span class="u-monospace">LHAPDF(v5)</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 284" title="M. R. Whalley, D. Bourilkov, R. C. Group, The Les Houches accord PDFs (LHAPDF) and LHAGLUE, in HERA and the LHC: A workshop on the implications of HERA for LHC physics. in Proceedings, Part B (2005). 
 arXiv:hep-ph/0508110
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR284" id="ref-link-section-d90629110e55709">284</a>]) or “partonometers” [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 285" title="J.P. Ralston, Pocket partonometer. Phys. Lett. B 172, 430 (1986)" href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR285" id="ref-link-section-d90629110e55712">285</a>] with outdated calibration.</p> </li> </ul> </li> </ol><p>In the Monte Carlo generators, for example, <span class="u-monospace">MadGraph5_aMC@NLO</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 286" title="J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.S. Shao, T. Stelzer, P. Torrielli, M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. JHEP 07, 079 (2014). 
 arXiv:1405.0301
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR286" id="ref-link-section-d90629110e55725">286</a>], <span class="u-monospace">POWHEG-BOX (v2)</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 287" title="POWHEG-BOX (v2). 
 http://powhegbox.mib.infn.it/
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR287" id="ref-link-section-d90629110e55731">287</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 288" title="S. Alioli, P. Nason, C. Oleari, E. Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX. JHEP 06, 043 (2010). 
 arXiv:1002.2581
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR288" id="ref-link-section-d90629110e55735">288</a>] and <span class="u-monospace">SHERPA (v2)</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 289" title="SHERPA (v2). 
 https://sherpa.hepforge.org/doc/SHERPA-MC-2.2.0.html
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR289" id="ref-link-section-d90629110e55741">289</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 290" title="T. Gleisberg, S. Höche, F. Krauss, M. Schönherr, S. Schumann, F. Siegert, J. Winter, Event generation with SHERPA 1.1. JHEP 02 (2009) 007. 
 arXiv:0811.4622
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR290" id="ref-link-section-d90629110e55744">290</a>], or other recently developed generators, like <span class="u-monospace">Geneva</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 291" title="S. Alioli, C.W. Bauer, C. Berggren, F.J. Tackmann, J.R. Walsh, Drell–Yan production at NNLL’+NNLO matched to parton showers. Phys. Rev. D 92, 094020 (2015). 
 arXiv:1508.01475
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR291" id="ref-link-section-d90629110e55750">291</a>], different PDF sets can be efficiently studied with reweighting methods. This allows to generate weighted events for a given setup, and to reweight a-posteriori each event in a fast and efficient way, by generating new weights associated with different choices of renormalization and factorization scales and/or PDFs. Please note, that at present, PDF reweighting is performed by assuming the linear PDF weight dependence, which is not correct, since PDFs are also present in the Sudakov form-factor. Efforts to extend the reweighting to the entire Sudakov form-factor and to the full parton shower are ongoing. The reweighting technique turns out to be particularly useful to compute in a fast (although at the moment approximate) way PDF uncertainties affecting the predictions.</p></div></div></section><section data-title="Conclusion"><div class="c-article-section" id="Sec31-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec31"><span class="c-article-section__title-number">7 </span>Conclusion</h2><div class="c-article-section__content" id="Sec31-content"><p>In this report we have reviewed recent developments in the determination of PDFs in global QCD analyses. Thanks to high precision experimental measurements and continuous theoretical improvements, the parton content of the proton is generally well constrained and PDFs, along with the strong coupling constant <span class="mathjax-tex">$\alpha _s(M_Z)$</span> and the heavy-quark masses <span class="mathjax-tex">$m_c$</span>, <span class="mathjax-tex">$m_b$</span> and <span class="mathjax-tex">$m_t$</span>, have been determined with good accuracy, at least at NNLO in QCD. This forms the foundation for precise cross section predictions at the LHC in Run 2.</p><p>We have briefly discussed the available data used in PDF extractions and the kinematic range covered, and emphasized the importance of selecting mutually consistent sets of data in PDF fits in order to achieve acceptable <span class="mathjax-tex">$\chi ^2$</span> values for the goodness-of-fit estimate. The main thrust of the study has been the computation of benchmark cross sections for a variety of processes at hadron colliders, including Higgs boson production in gluon–gluon fusion. We have illustrated how different choices for the theoretical description of the hard scattering process and choices of parameters have an impact on the predicted cross sections, and lead to systematic shifts that are often significantly larger than the associated PDF and <span class="mathjax-tex">$\alpha _s(M_Z)$</span> uncertainties. A particular example has been the treatment of heavy quarks in DIS, where the quality of the various scheme choices has been quantified in terms of <span class="mathjax-tex">$\chi ^2$</span>/NDP values when comparing predicted cross sections to data. We have also pointed out the inconsistently low values for the pole mass of the charm quark used in some fits, and have stressed the correlation of the strong coupling constant <span class="mathjax-tex">$\alpha _s(M_Z)$</span> with the PDF parameters. Ideally, <span class="mathjax-tex">$\alpha _s(M_Z)$</span> should be determined simultaneously with the PDFs, and we have summarized here the state of the art in the context of PDF analyses.</p><p>Our findings expose a number of shortcomings in the recent PDF4LHC recommendations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="J. Butterworth et al., PDF4LHC recommendations for LHC Run II. J. Phys. G43, 023001 (2016). 
 arXiv:1510.03865
 
 
 " href="/article/10.1140/epjc/s10052-016-4285-4#ref-CR8" id="ref-link-section-d90629110e56066">8</a>]. We have shown that these do not provide sufficient control over some theoretical uncertainties, and may therefore be problematic for precision predictions in Run 2 of the LHC. Instead, we suggest new recommendations for the usage of PDFs based on a theoretically consistent procedure necessary to meet the precision requirements of the LHC era.</p></div></div></section> </div> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. 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Alioli, M. Botje, E.W.N. Glover and K. Rabbertz for discussions, K. Rabbertz also for valuable comments on the manuscript, and L. Harland-Lang and R. Thorne for providing us with the Higgs and <span class="mathjax-tex">$t{\bar{t}}$</span> cross sections in Tables <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab11">11</a> and <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1140/epjc/s10052-016-4285-4#Tab14">14</a>. This work has been supported by Bundesministerium für Bildung und Forschung through contract (05H15GUCC1), by the DOE contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab, and by the European Commission through PITN-GA-2012-316704 (<i>HIGGSTOOLS</i>). The work of A.A. and J.F.O. was supported in part by DOE contracts No. DE-SC0008791 and No. DE-FG02-97ER41922, respectively. Two of the authors (J.B. and S.M.) would like to thank the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Hampton University, Hampton, VA, 23668, USA</p><p class="c-article-author-affiliation__authors-list">A. Accardi</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Jefferson Lab, Newport News, VA, 23606, USA</p><p class="c-article-author-affiliation__authors-list">A. Accardi, W. Melnitchouk & N. Sato</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany</p><p class="c-article-author-affiliation__authors-list">S. Alekhin, M. V. Garzelli & S. Moch</p></li><li id="Aff4"><p class="c-article-author-affiliation__address">Institute for High Energy Physics, 142281, Protvino, Moscow region, Russia</p><p class="c-article-author-affiliation__authors-list">S. Alekhin</p></li><li id="Aff5"><p class="c-article-author-affiliation__address">Deutsches Elektronensynchrotron DESY, Platanenallee 6, 15738, Zeuthen, Germany</p><p class="c-article-author-affiliation__authors-list">J. Blümlein</p></li><li id="Aff6"><p class="c-article-author-affiliation__address">Deutsches Elektronensynchrotron DESY, Notkestraße 85, 22607, Hamburg, Germany</p><p class="c-article-author-affiliation__authors-list">K. Lipka, R. Plačakytė & O. Zenaiev</p></li><li id="Aff7"><p class="c-article-author-affiliation__address">Florida State University, Tallahassee, FL, 32306, USA</p><p class="c-article-author-affiliation__authors-list">J. F. Owens</p></li><li id="Aff8"><p class="c-article-author-affiliation__address">Institut für Physik, Technische Universität Dortmund, 44221, Dortmund, Germany</p><p class="c-article-author-affiliation__authors-list">E. Reya</p></li><li id="Aff9"><p class="c-article-author-affiliation__address">Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, UK</p><p class="c-article-author-affiliation__authors-list">A. Vogt</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-A_-Accardi-Aff1-Aff2"><span class="c-article-authors-search__title u-h3 js-search-name">A. 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Moch</a>.</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p> <b>Open Access</b> This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (<a href="http://creativecommons.org/licenses/by/4.0" rel="license">http://creativecommons.org/licenses/by/4.0</a>/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.</p> <p>Funded by SCOAP<sup>3</sup>.</p> <p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=A%20critical%20appraisal%20and%20evaluation%20of%20modern%20PDFs&author=A.%20Accardi%20et%20al&contentID=10.1140%2Fepjc%2Fs10052-016-4285-4&copyright=The%20Author%28s%29&publication=1434-6044&publicationDate=2016-08-23&publisherName=SpringerNature&orderBeanReset=true&oa=CC%20BY">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1140/epjc/s10052-016-4285-4" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-016-4285-4" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Accardi, A., Alekhin, S., Blümlein, J. <i>et al.</i> A critical appraisal and evaluation of modern PDFs. <i>Eur. 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C</i> <b>76</b>, 471 (2016). https://doi.org/10.1140/epjc/s10052-016-4285-4</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1140/epjc/s10052-016-4285-4?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2016-04-05">05 April 2016</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span 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A critical appraisal and evaluation of modern PDFs | The European Physical Journal C