Higgs mechanism - Wikipedia

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href="#Consequences_for_fermions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Consequences for fermions</span> </div> </a> <ul id="toc-Consequences_for_fermions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History_of_research" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History_of_research"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>History of research</span> </div> </a> <button aria-controls="toc-History_of_research-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History of research subsection</span> </button> <ul id="toc-History_of_research-sublist" class="vector-toc-list"> <li id="toc-Background" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Background"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Background</span> </div> </a> <ul id="toc-Background-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discovery" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discovery"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Discovery</span> </div> </a> <ul id="toc-Discovery-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_explanation_of_the_theory,_from_its_origins_in_superconductivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_explanation_of_the_theory,_from_its_origins_in_superconductivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Simple explanation of the theory, from its origins in superconductivity</span> </div> </a> <ul id="toc-Simple_explanation_of_the_theory,_from_its_origins_in_superconductivity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Landau_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Landau_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Landau model</span> </div> </a> <ul id="toc-Landau_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abelian_Higgs_mechanism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abelian_Higgs_mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Abelian Higgs mechanism</span> </div> </a> <ul id="toc-Abelian_Higgs_mechanism-sublist" class="vector-toc-list"> <li id="toc-Mathematical_details_of_the_abelian_Higgs_mechanism" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mathematical_details_of_the_abelian_Higgs_mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Mathematical details of the abelian Higgs mechanism</span> </div> </a> <ul id="toc-Mathematical_details_of_the_abelian_Higgs_mechanism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-Abelian_Higgs_mechanism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-Abelian_Higgs_mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Non-Abelian Higgs mechanism</span> </div> </a> <ul id="toc-Non-Abelian_Higgs_mechanism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_Higgs_mechanism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_Higgs_mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Affine Higgs mechanism</span> </div> </a> <ul id="toc-Affine_Higgs_mechanism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span 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Available in 29 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A2%D9%84%D9%8A%D8%A9_%D9%87%D9%8A%D8%BA%D8%B2" title="آلية هيغز – Arabic" lang="ar" hreflang="ar" data-title="آلية هيغز" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A6%BF%E0%A6%97%E0%A6%B8_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="হিগস ক্ষেত্র – Bangla" lang="bn" hreflang="bn" data-title="হিগস ক্ষেত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%B7%D0%BC_%D0%A5%D1%96%D0%B3%D1%81%D0%B0" title="Механізм Хігса – Belarusian" lang="be" hreflang="be" data-title="Механізм Хігса" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D1%8A%D0%BC_%D0%BD%D0%B0_%D0%A5%D0%B8%D0%B3%D1%81" title="Механизъм на Хигс – Bulgarian" lang="bg" hreflang="bg" data-title="Механизъм на Хигс" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mecanisme_de_Brout-Englert-Higgs" title="Mecanisme de Brout-Englert-Higgs – Catalan" lang="ca" hreflang="ca" data-title="Mecanisme de Brout-Englert-Higgs" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Higgs-Mechanismus" title="Higgs-Mechanismus – German" lang="de" hreflang="de" data-title="Higgs-Mechanismus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CE%A7%CE%B9%CE%B3%CE%BA%CF%82" title="Μηχανισμός Χιγκς – Greek" lang="el" hreflang="el" data-title="Μηχανισμός Χιγκς" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mecanismo_de_Higgs" title="Mecanismo de Higgs – Spanish" lang="es" hreflang="es" data-title="Mecanismo de Higgs" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D8%A7%D8%B2%D9%88%DA%A9%D8%A7%D8%B1_%D9%87%DB%8C%DA%AF%D8%B2" title="سازوکار هیگز – Persian" lang="fa" hreflang="fa" data-title="سازوکار هیگز" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9canisme_de_Brout-Englert-Higgs-Hagen-Guralnik-Kibble" title="Mécanisme de Brout-Englert-Higgs-Hagen-Guralnik-Kibble – French" lang="fr" hreflang="fr" data-title="Mécanisme de Brout-Englert-Higgs-Hagen-Guralnik-Kibble" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9E%89%EC%8A%A4_%EB%A9%94%EC%BB%A4%EB%8B%88%EC%A6%98" title="힉스 메커니즘 – Korean" lang="ko" hreflang="ko" data-title="힉스 메커니즘" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanisme_Brout-Englert-Higgs" title="Mekanisme Brout-Englert-Higgs – Indonesian" lang="id" hreflang="id" data-title="Mekanisme Brout-Englert-Higgs" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanismo_di_Brout-Englert-Higgs" title="Meccanismo di Brout-Englert-Higgs – Italian" lang="it" hreflang="it" data-title="Meccanismo di Brout-Englert-Higgs" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A0%D7%92%D7%A0%D7%95%D7%9F_%D7%94%D7%99%D7%92%D7%A1" title="מנגנון היגס – Hebrew" lang="he" hreflang="he" data-title="מנגנון היגס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Brout-Englert-Higgsveld" title="Brout-Englert-Higgsveld – Dutch" lang="nl" hreflang="nl" data-title="Brout-Englert-Higgsveld" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%92%E3%83%83%E3%82%B0%E3%82%B9%E6%A9%9F%E6%A7%8B" title="ヒッグス機構 – Japanese" lang="ja" hreflang="ja" data-title="ヒッグス機構" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B9%E0%A8%BF%E0%A8%97%E0%A8%9C%E0%A8%BC_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%9C%E0%A8%BC%E0%A8%AE" title="ਹਿਗਜ਼ ਮਕੈਨਿਜ਼ਮ – Punjabi" lang="pa" hreflang="pa" data-title="ਹਿਗਜ਼ ਮਕੈਨਿਜ਼ਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pole_Higgsa" title="Pole Higgsa – Polish" lang="pl" hreflang="pl" data-title="Pole Higgsa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mecanismo_de_Higgs" title="Mecanismo de Higgs – Portuguese" lang="pt" hreflang="pt" data-title="Mecanismo de Higgs" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D0%BC_%D0%A5%D0%B8%D0%B3%D0%B3%D1%81%D0%B0" title="Механизм Хиггса – Russian" lang="ru" hreflang="ru" data-title="Механизм Хиггса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Higgs_field" title="Higgs field – Simple English" lang="en-simple" hreflang="en-simple" data-title="Higgs field" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Broutov-Englertov-Higgsov_mechanizmus" title="Broutov-Englertov-Higgsov mechanizmus – Slovak" lang="sk" hreflang="sk" data-title="Broutov-Englertov-Higgsov mechanizmus" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D0%B3%D1%81%D0%BE%D0%B2_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D0%B0%D0%BC" title="Хигсов механизам – Serbian" lang="sr" hreflang="sr" data-title="Хигсов механизам" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Higgsmekanismen" title="Higgsmekanismen – Swedish" lang="sv" hreflang="sv" data-title="Higgsmekanismen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Higgs_mekanizmas%C4%B1" title="Higgs mekanizması – Turkish" lang="tr" hreflang="tr" data-title="Higgs mekanizması" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%B7%D0%BC_%D0%A5%D1%96%D0%B3%D0%B3%D1%81%D0%B0" title="Механізм Хіггса – Ukrainian" lang="uk" hreflang="uk" data-title="Механізм Хіггса" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_ch%E1%BA%BF_Higgs" title="Cơ chế Higgs – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ chế Higgs" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Mekanismo_Higgs" title="Mekanismo Higgs – Waray" lang="war" hreflang="war" data-title="Mekanismo Higgs" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B8%8C%E6%A0%BC%E6%96%AF%E6%9C%BA%E5%88%B6" title="希格斯机制 – Chinese" lang="zh" hreflang="zh" data-title="希格斯机制" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q747702#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> 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a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a></th></tr><tr><td class="sidebar-image"><figure class="skin-invert-image noresize mw-ext-imagemap-desc-bottom-right" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/240px-Standard_Model_of_Elementary_Particles.svg.png" decoding="async" width="240" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/360px-Standard_Model_of_Elementary_Particles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/480px-Standard_Model_of_Elementary_Particles.svg.png 2x" data-file-width="1390" data-file-height="1330" usemap="#ImageMap_af1b7d82cba73703" resource="/wiki/File:Standard_Model_of_Elementary_Particles.svg" /></span><map name="ImageMap_af1b7d82cba73703"><area href="/wiki/Up_quark" shape="rect" coords="17,48,59,89" alt="Up quark" title="Up quark" /><area href="/wiki/Charm_quark" shape="rect" coords="60,48,102,89" alt="Charm quark" title="Charm quark" /><area href="/wiki/Top_quark" shape="rect" coords="104,48,145,89" alt="Top quark" title="Top quark" /><area href="/wiki/Gluon" shape="rect" coords="149,48,190,89" alt="Gluon" title="Gluon" /><area href="/wiki/Higgs_boson" shape="rect" coords="194,48,235,89" alt="Higgs boson" title="Higgs boson" /><area href="/wiki/Down_quark" shape="rect" coords="17,91,59,132" alt="Down quark" title="Down quark" /><area href="/wiki/Strange_quark" shape="rect" coords="60,91,102,132" alt="Strange quark" title="Strange quark" /><area href="/wiki/Bottom_quark" shape="rect" coords="104,91,145,132" alt="Bottom quark" title="Bottom quark" /><area href="/wiki/Photon" shape="rect" coords="149,91,190,132" alt="Photon" title="Photon" /><area href="/wiki/Electron" shape="rect" coords="17,137,59,178" alt="Electron" title="Electron" /><area href="/wiki/Muon" shape="rect" coords="60,137,102,178" alt="Muon" title="Muon" /><area href="/wiki/Tau_(particle)" shape="rect" coords="104,137,145,178" alt="Tau (particle)" title="Tau (particle)" /><area href="/wiki/W_and_Z_bosons#Z_bosons}Z_boson" shape="rect" coords="149,137,190,178" alt="W and Z bosons#Z bosons}Z boson" title="W and Z bosons#Z bosons}Z boson" /><area href="/wiki/Electron_neutrino" shape="rect" coords="17,180,59,221" alt="Electron neutrino" title="Electron neutrino" /><area href="/wiki/Muon_neutrino" shape="rect" coords="60,180,102,221" alt="Muon neutrino" title="Muon neutrino" /><area href="/wiki/Tau_neutrino" shape="rect" coords="104,180,145,221" alt="Tau neutrino" title="Tau neutrino" /><area href="/wiki/W_and_Z_bosons" shape="rect" coords="149,180,190,221" alt="W and Z bosons" title="W and Z bosons" /><area href="/wiki/Standard_Model" shape="rect" coords="16,8,224,19" alt="Standard Model" title="Standard Model" /><area href="/wiki/Fermion" shape="rect" coords="17,24,145,38" alt="Fermion" title="Fermion" /><area href="/wiki/Boson" shape="rect" coords="149,24,235,38" alt="Boson" title="Boson" /><area href="/wiki/Quark" shape="rect" coords="6,93,14,133" alt="Quark" title="Quark" /><area href="/wiki/Lepton" shape="rect" coords="6,176,12,219" alt="Lepton" title="Lepton" /><area href="/wiki/Scalar_boson" shape="rect" coords="227,92,233,172" alt="Scalar boson" title="Scalar boson" /><area href="/wiki/Gauge_boson" shape="rect" coords="196,143,202,219" alt="Gauge boson" title="Gauge boson" /><area href="/wiki/Vector_boson" shape="rect" coords="205,165,209,219" alt="Vector boson" title="Vector boson" /></map><figcaption></figcaption></figure><div class="sidebar-caption"><a href="/wiki/Elementary_particle" title="Elementary particle">Elementary particles</a> of the Standard Model</div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a><br /><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a><br /><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <br /> <a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a> <br /> <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a><br /> <a class="mw-selflink selflink">Higgs mechanism</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Constituents</div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a><br /> <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a><br /> <a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">CKM matrix</a><br /><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">Standard Model mathematics</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Limitations</div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Strong_CP_problem" title="Strong CP problem">Strong CP problem</a><br /><a href="/wiki/Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a><br /><a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillations</a><br /><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Physics beyond the Standard Model</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Rutherford</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/James_Chadwick" title="James Chadwick">Chadwick</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/E._C._George_Sudarshan" title="E. C. George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Raymond_Davis_Jr." title="Raymond Davis Jr.">Davis Jr</a></li> <li><a href="/wiki/Carl_David_Anderson" title="Carl David Anderson">Anderson</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Carlo_Rubbia" title="Carlo Rubbia">Rubbia</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Richard_E._Taylor" title="Richard E. Taylor">Taylor</a></li> <li><a href="/wiki/Jerome_Isaac_Friedman" title="Jerome Isaac Friedman">Friedman</a></li> <li><a href="/wiki/C._F._Powell" title="C. F. Powell">Powell</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Leon_M._Lederman" title="Leon M. Lederman">Lederman</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Simon_van_der_Meer" title="Simon van der Meer">Meer</a></li> <li><a href="/wiki/Clyde_Cowan" title="Clyde Cowan">Cowan</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Owen_Chamberlain" title="Owen Chamberlain">Chamberlain</a></li> <li><a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Cabibbo</a></li> <li><a href="/wiki/Melvin_Schwartz" title="Melvin Schwartz">Schwartz</a></li> <li><a href="/wiki/Martin_Lewis_Perl" title="Martin Lewis Perl">Perl</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Makoto_Kobayashi_(physicist)" class="mw-redirect" title="Makoto Kobayashi (physicist)">Kobayashi</a></li> <li><a href="/wiki/Toshihide_Maskawa" title="Toshihide Maskawa">Maskawa</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/Martinus_J._G._Veltman" title="Martinus J. G. Veltman">Veltman</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Abraham_Pais" title="Abraham Pais">Pais</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Hugh_David_Politzer" title="Hugh David Politzer">Politzer</a></li> <li><a href="/wiki/Frederick_Reines" title="Frederick Reines">Reines</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/James_Cronin" title="James Cronin">Cronin</a></li> <li><a href="/wiki/Val_Logsdon_Fitch" title="Val Logsdon Fitch">Fitch</a></li> <li><a href="/wiki/John_Hasbrouck_Van_Vleck" title="John Hasbrouck Van Vleck">Vleck</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Tom_Kibble" title="Tom Kibble">Kibble</a></li> <li><a href="/wiki/Santiago_Ant%C3%BAnez_de_Mayolo" title="Santiago Antúnez de Mayolo">de Mayolo</a></li> <li><a href="/wiki/C%C3%A9sar_Lattes" title="César Lattes">Lattes</a></li> <li><a href="/wiki/George_Zweig" title="George Zweig">Zweig</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output 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class="nv-talk"><a href="/wiki/Template_talk:Standard_model_of_particle_physics" title="Template talk:Standard model of particle physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Standard_model_of_particle_physics" title="Special:EditPage/Template:Standard model of particle physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/Feynman_diagram" title="Feynman diagram"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/211px-Feynmann_Diagram_Gluon_Radiation.svg.png" decoding="async" width="211" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/317px-Feynmann_Diagram_Gluon_Radiation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/422px-Feynmann_Diagram_Gluon_Radiation.svg.png 2x" data-file-width="400" data-file-height="250" /></a></span><div class="sidebar-caption"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above"> <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Field_(physics)" title="Field (physics)">Field theory</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li> <li><a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li> <li><a href="/wiki/Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li> <li><a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li> <li><a href="/wiki/Topological_charge" class="mw-redirect" title="Topological charge">Topological charge</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Tools</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li> <li><a href="/wiki/Background_field_method" title="Background field method">Background field method</a></li> <li><a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a></li> <li><a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li> <li><a href="/wiki/Crossing_(physics)" title="Crossing (physics)">Crossing</a></li> <li><a href="/wiki/Effective_action" title="Effective action">Effective action</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective field theory</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li> <li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li> <li><a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li> <li><a href="/wiki/Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li> <li><a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li> <li><a href="/wiki/Quantization_(physics)" title="Quantization (physics)">Quantization</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li> <li><a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a></li> <li><a href="/w/index.php?title=Wightman_Axioms&action=edit&redlink=1" class="new" title="Wightman Axioms (page does not exist)">Wightman Axioms</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca equations</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li> <li><a href="/wiki/Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a class="mw-selflink selflink">Higgs mechanism</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Incomplete theories</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li> <li><a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li> <li><a href="/wiki/Carlo_Becchi" title="Carlo Becchi">Becchi</a></li> <li><a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Belavin</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/James_Bjorken" title="James Bjorken">Bjorken</a></li> <li><a href="/wiki/Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li> <li><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li> <li><a href="/wiki/Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li> <li><a href="/wiki/Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li> <li><a href="/wiki/Curtis_Callan" title="Curtis Callan">Callan</a></li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a></li> <li><a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a></li> <li><a href="/wiki/Roger_Dashen" title="Roger Dashen">Dashen</a></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li> <li><a href="/wiki/Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Markus_Fierz" title="Markus Fierz">Fierz</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton</a></li> <li><a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li> <li><a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li> <li><a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li> <li><a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/James_Glimm" title="James Glimm">Glimm</a></li> <li><a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li> <li><a href="/wiki/Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Klaus_Hepp" title="Klaus Hepp">Hepp</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li> <li><a href="/wiki/Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li> <li><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li> <li><a href="/wiki/Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Res_Jost" title="Res Jost">Jost</a></li> <li><a href="/wiki/Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li> <li><a href="/wiki/Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li> <li><a href="/wiki/Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li> <li><a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li> <li><a href="/wiki/Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li> <li><a href="/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li> <li><a href="/wiki/Francis_E._Low" title="Francis E. Low">Low</a></li> <li><a href="/wiki/Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Alexander_Arkadyevich_Migdal" title="Alexander Arkadyevich Migdal">Migdal</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li> <li><a href="/wiki/Mark_Naimark" title="Mark Naimark">Naimark</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li> <li><a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li> <li><a href="/wiki/Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin</a></li> <li><a href="/wiki/Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li> <li><a href="/wiki/Victor_Popov" title="Victor Popov">Popov</a></li> <li><a href="/wiki/Alexandru_Proca" title="Alexandru Proca">Proca</a></li> <li><a href="/wiki/Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">Ruelle</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Robert_Schrader" title="Robert Schrader">Schrader</a></li> <li><a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Irving_Segal" title="Irving Segal">Segal</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li> <li><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li> <li><a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li> <li><a href="/wiki/Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li> <li><a href="/wiki/Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li> <li><a href="/wiki/Raymond_Stora" title="Raymond Stora">Stora</a></li> <li><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li> <li><a href="/wiki/George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li> <li><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring</a></li> <li><a href="/wiki/Shin%27ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li> <li><a href="/wiki/Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li> <li><a href="/wiki/Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li> <li><a href="/wiki/Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Gian_Carlo_Wick" title="Gian Carlo Wick">Wick</a></li> <li><a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee</a></li> <li><a href="/wiki/Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li> <li><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li> <li><a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> <p><br /> </p> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_field_theory" title="Template:Quantum field theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_field_theory" title="Template talk:Quantum field theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theory" title="Special:EditPage/Template:Quantum field theory"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, the <b>Higgs mechanism</b> is essential to explain the <a href="/wiki/Mass_generation" title="Mass generation">generation mechanism</a> of the property "<a href="/wiki/Mass" title="Mass">mass</a>" for <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a>. Without the Higgs mechanism, all <a href="/wiki/Boson" title="Boson">bosons</a> (one of the two classes of particles, the other being <a href="/wiki/Fermion" title="Fermion">fermions</a>) would be considered <a href="/wiki/Massless_particle" title="Massless particle">massless</a>, but measurements show that the <a href="/wiki/W_boson" class="mw-redirect" title="W boson">W<sup>+</sup>, W<sup>−</sup></a>, and <a href="/wiki/Z_boson" class="mw-redirect" title="Z boson">Z<sup>0</sup> bosons</a> actually have relatively large masses of around 80 GeV/<i>c</i><sup>2</sup>. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a <a href="/wiki/Quantum_field" class="mw-redirect" title="Quantum field">quantum field</a> (the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs field</a>) which permeates all of space to the Standard Model. Below some extremely high temperature, the field causes <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a> during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons with which it interacts to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W<sup>±</sup>, and Z</a> <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak</a> gauge bosons through <a href="/wiki/Electroweak_interaction" title="Electroweak interaction">electroweak</a> symmetry breaking.<sup id="cite_ref-PDG_1-0" class="reference"><a href="#cite_note-PDG-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">Large Hadron Collider</a> at <a href="/wiki/CERN" title="CERN">CERN</a> announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature. </p><p>The view of the Higgs mechanism as involving spontaneous symmetry breaking of a gauge symmetry is technically incorrect since by <a href="/wiki/Elitzur%27s_theorem" title="Elitzur's theorem">Elitzur's theorem</a> gauge symmetries can never be spontaneously broken. Rather, the <a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a>–Morchio–Strocchi mechanism reformulates the Higgs mechanism in an entirely gauge invariant way, generally leading to the same results.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The mechanism was proposed in 1962 by <a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Philip Warren Anderson</a>,<sup id="cite_ref-Anderson_3-0" class="reference"><a href="#cite_note-Anderson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> following work in the late 1950s on symmetry breaking in <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a> and a 1960 paper by <a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Yoichiro Nambu</a> that discussed its application within <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>. </p><p>A theory able to finally explain <a href="/wiki/Mass_generation" title="Mass generation">mass generation</a> without "breaking" gauge theory <a href="/wiki/1964_PRL_symmetry_breaking_papers" title="1964 PRL symmetry breaking papers">was published almost simultaneously</a> by three independent groups in 1964: by <a href="/wiki/Robert_Brout" title="Robert Brout">Robert Brout</a> and <a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">François Englert</a>;<sup id="cite_ref-Englert_4-0" class="reference"><a href="#cite_note-Englert-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> by <a href="/wiki/Peter_Higgs" title="Peter Higgs">Peter Higgs</a>;<sup id="cite_ref-Higgs_5-0" class="reference"><a href="#cite_note-Higgs-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and by <a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Gerald Guralnik</a>, <a href="/wiki/C._R._Hagen" title="C. R. Hagen">C. R. Hagen</a>, and <a href="/wiki/Tom_Kibble" title="Tom Kibble">Tom Kibble</a>.<sup id="cite_ref-GHK_6-0" class="reference"><a href="#cite_note-GHK-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Guralnik_7-0" class="reference"><a href="#cite_note-Guralnik-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Scholarpedia_history_8-0" class="reference"><a href="#cite_note-Scholarpedia_history-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The Higgs mechanism is therefore also called the <b>Brout–Englert–Higgs mechanism</b>, or <b>Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism</b>,<sup id="cite_ref-Scholarpedia_9-0" class="reference"><a href="#cite_note-Scholarpedia-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <b>Anderson–Higgs mechanism</b>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <b>Anderson–Higgs–Kibble mechanism</b>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <b>Higgs–Kibble mechanism</b> by <a href="/wiki/Abdus_Salam" title="Abdus Salam">Abdus Salam</a><sup id="cite_ref-frank_close_infinity_puzzle_12-0" class="reference"><a href="#cite_note-frank_close_infinity_puzzle-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and <b>ABEGHHK'tH mechanism</b> (for Anderson, Brout, Englert, Guralnik, Hagen, Higgs, Kibble, and <a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a>) by Peter Higgs.<sup id="cite_ref-frank_close_infinity_puzzle_12-1" class="reference"><a href="#cite_note-frank_close_infinity_puzzle-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The Higgs mechanism in electrodynamics was also discovered independently by <a href="/wiki/Joseph_H._Eberly" title="Joseph H. Eberly">Eberly</a> and Reiss in reverse as the "gauge" Dirac field mass gain due to the artificially displaced electromagnetic field as a Higgs field.<sup id="cite_ref-Eberly_13-0" class="reference"><a href="#cite_note-Eberly-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>On 8 October 2013, following the discovery at CERN's Large Hadron Collider of a new particle that appeared to be the long-sought <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a> predicted by the theory, it was announced that Peter Higgs and François Englert had been awarded the 2013 <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Standard_Model">Standard Model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=1" title="Edit section: Standard Model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Higgs mechanism was incorporated into modern particle physics by <a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> and <a href="/wiki/Abdus_Salam" title="Abdus Salam">Abdus Salam</a>, and is an essential part of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>. </p><p>In the Standard Model, at temperatures high enough that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature, the Higgs field develops a <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a>; some theories suggest the symmetry is spontaneously broken by <a href="/wiki/Tachyon_condensation" title="Tachyon condensation">tachyon condensation</a>, and the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a> acquire masses (also called "electroweak symmetry breaking", or <i>EWSB</i>). In the history of the universe, this is believed to have happened about a <a href="/wiki/Picosecond" title="Picosecond">picosecond</a> <span class="nowrap">(10<sup>−12</sup> s)</span> after the hot big bang, when the universe was at a temperature 159.5 ± 1.5 <a href="/wiki/GeV" class="mw-redirect" title="GeV">GeV</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Fermions, such as the <a href="/wiki/Lepton" title="Lepton">leptons</a> and <a href="/wiki/Quark" title="Quark">quarks</a> in the Standard Model, can also acquire mass as a result of their interaction with the Higgs field, but not in the same way as the gauge bosons. </p> <div class="mw-heading mw-heading3"><h3 id="Structure_of_the_Higgs_field">Structure of the Higgs field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=2" title="Edit section: Structure of the Higgs field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the standard model, the Higgs field is an <a href="/wiki/Special_unitary_group" title="Special unitary group"><b>SU</b>(2)</a> <a href="/wiki/Doublet_state" title="Doublet state">doublet</a> (i.e. the standard representation with two complex components called isospin), which is a <a href="/wiki/Scalar_field_theory" title="Scalar field theory">scalar</a> under Lorentz transformations. Its electric charge is zero; its <a href="/wiki/Weak_isospin" title="Weak isospin">weak isospin</a> is <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> and the third component of weak isospin is −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>; and its <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharge</a> (the charge for the <a href="/wiki/Unitary_group" title="Unitary group"><b>U</b>(1)</a> gauge group defined up to an arbitrary multiplicative constant) is 1. Under <b>U</b>(1) rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other, combining to the standard two-component complex representation of the group <b>U</b>(2). </p><p>The Higgs field, through the interactions specified (summarized, represented, or even simulated) by its potential, induces spontaneous breaking of three out of the four generators ("directions") of the gauge group <b>U</b>(2). This is often written as <b>SU</b>(2)<sub>L</sub> × <b>U</b>(1)<sub>Y</sub>, (which is strictly speaking only the same on the level of infinitesimal symmetries) because the diagonal phase factor also acts on other fields – <a href="/wiki/Quark" title="Quark">quarks</a> in particular. Three out of its four components would ordinarily resolve as <a href="/wiki/Goldstone_boson" title="Goldstone boson">Goldstone bosons</a>, if they were not coupled to gauge fields. </p><p>However, after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three <a href="/wiki/W_and_Z_boson" class="mw-redirect" title="W and Z boson">W and Z bosons</a> (<span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>W<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span>, <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>W<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> and <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>Z<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">0</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span>), and are only observable as components of these <a href="/wiki/Weak_boson" class="mw-redirect" title="Weak boson">weak bosons</a>, which are made massive by their inclusion; only the single remaining degree of freedom becomes a new scalar particle: the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>. The components that do not mix with Goldstone bosons form a massless photon. </p> <div class="mw-heading mw-heading3"><h3 id="The_photon_as_the_part_that_remains_massless">The photon as the part that remains massless</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=3" title="Edit section: The photon as the part that remains massless"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> of the electroweak part of the standard model is <b>SU</b>(2)<sub>L</sub> × <b>U</b>(1)<sub>Y</sub>. The group <b>SU</b>(2) is the group of all 2-by-2 unitary matrices with unit determinant; all the orthonormal changes of coordinates in a complex two dimensional vector space. </p><p>Rotating the coordinates so that the second basis vector points in the direction of the Higgs boson makes the <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a> of <i>H</i> the spinor<span class="nowrap"> <span class="texhtml">( 0, <i>v</i> )</span> .</span> The generators for rotations about the x, y, and z axes are by half the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> <span class="texhtml mvar" style="font-style:italic;">σ</span><sub>x</sub>, <span class="texhtml mvar" style="font-style:italic;">σ</span><sub>y</sub>, and <span class="texhtml mvar" style="font-style:italic;">σ</span><sub>z</sub>, so that a rotation of angle <span class="texhtml mvar" style="font-style:italic;">θ</span> about the z-axis takes the vacuum to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\Bigl (}\ 0\ ,\ v\ e^{-{\tfrac {1}{2}}\ i\ \theta }\ {\Bigr )}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mtext> </mtext> <mn>0</mn> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>v</mi> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mi>i</mi> <mtext> </mtext> <mi>θ</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\Bigl (}\ 0\ ,\ v\ e^{-{\tfrac {1}{2}}\ i\ \theta }\ {\Bigr )}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10e035e82c7327844f30eacb5cb918678956f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.563ex; height:5.509ex;" alt="{\displaystyle \ {\Bigl (}\ 0\ ,\ v\ e^{-{\tfrac {1}{2}}\ i\ \theta }\ {\Bigr )}~.}"></span></dd></dl> <p>While the <span class="texhtml mvar" style="font-style:italic;">T</span><sub>x</sub> and <span class="texhtml mvar" style="font-style:italic;">T</span><sub>y</sub> generators mix up the top and bottom components of the <a href="/wiki/Spinor" title="Spinor">spinor</a>, the <span class="texhtml mvar" style="font-style:italic;">T</span><sub>z</sub> rotations only multiply each by opposite phases. This phase can be undone by a <b>U</b>(1) rotation of angle <span class="nowrap"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span> <span class="texhtml mvar" style="font-style:italic;">θ</span> .</span> Consequently, under both an <b>SU</b>(2) <span class="texhtml mvar" style="font-style:italic;">T</span><sub>z</sub>-rotation and a <b>U</b>(1) rotation by an amount <span class="nowrap"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span> <span class="texhtml mvar" style="font-style:italic;">θ</span> ,</span> the vacuum is invariant. </p><p>This combination of generators </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ Q=T_{3}+{\tfrac {\ 1\ }{2}}\ Y_{\mathsf {W}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Q</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Q=T_{3}+{\tfrac {\ 1\ }{2}}\ Y_{\mathsf {W}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b7b92aaa52d8720dea7f7308d27ec65660853a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.884ex; height:3.509ex;" alt="{\displaystyle \ Q=T_{3}+{\tfrac {\ 1\ }{2}}\ Y_{\mathsf {W}}\ }"></span></dd></dl> <p>defines the unbroken part of the gauge group, where <span class="texhtml mvar" style="font-style:italic;">Q</span> is the electric charge, <span class="texhtml mvar" style="font-style:italic;">T</span><sub>3</sub> is the generator of rotations around the 3-axis in the <b>SU</b>(2) and <span class="texhtml mvar" style="font-style:italic;">Y</span><sub>W</sub> is the <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharge</a> generator of the <b>U</b>(1). This combination of generators (a <i>3</i> rotation in the <b>SU</b>(2) and a simultaneous <b>U</b>(1) rotation by half the angle) preserves the vacuum, and defines the unbroken gauge group in the standard model, namely the electric charge group. The part of the gauge field in this direction stays massless, and amounts to the physical photon. By contrast, the broken trace-orthogonal charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ T_{3}-{\tfrac {\ 1\ }{2}}\ Y_{\mathsf {W}}=2\ T_{3}-Q\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">W</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mtext> </mtext> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>Q</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ T_{3}-{\tfrac {\ 1\ }{2}}\ Y_{\mathsf {W}}=2\ T_{3}-Q\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a09581c1c731ca062cbdd30ce59532c8129f9677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.88ex; height:3.509ex;" alt="{\displaystyle \ T_{3}-{\tfrac {\ 1\ }{2}}\ Y_{\mathsf {W}}=2\ T_{3}-Q\ }"></span> couples to the massive <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>Z<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">0</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> boson. </p> <div class="mw-heading mw-heading3"><h3 id="Consequences_for_fermions">Consequences for fermions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=4" title="Edit section: Consequences for fermions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In spite of the introduction of spontaneous symmetry breaking, the mass terms preclude chiral gauge invariance. For these fields, the mass terms should always be replaced by a gauge-invariant "Higgs" mechanism. One possibility is some kind of <a href="/wiki/Yukawa_coupling" class="mw-redirect" title="Yukawa coupling">Yukawa coupling</a> (see below) between the fermion field <span class="texhtml mvar" style="font-style:italic;">ψ</span> and the Higgs field <span class="texhtml">φ</span>, with unknown couplings <span class="texhtml mvar" style="font-style:italic;">G<sub>ψ</sub></span>, which after symmetry breaking (more precisely: after expansion of the Lagrange density around a suitable ground state) again results in the original mass terms, which are now, however (i.e., by introduction of the Higgs field) written in a gauge-invariant way. The Lagrange density for the Yukawa interaction of a fermion field <span class="texhtml mvar" style="font-style:italic;">ψ</span> and the Higgs field <span class="texhtml">φ</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {L}}_{\mathrm {Fermion} }(\phi ,A,\psi )~=~{\overline {\psi }}\ \gamma ^{\mu }\ D_{\mu }\ \psi ~+~G_{\psi }\ {\overline {\psi }}\ \phi \ \psi \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>A</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mtext> </mtext> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> <mi>ϕ</mi> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}_{\mathrm {Fermion} }(\phi ,A,\psi )~=~{\overline {\psi }}\ \gamma ^{\mu }\ D_{\mu }\ \psi ~+~G_{\psi }\ {\overline {\psi }}\ \phi \ \psi \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d1e9a1f002539227feb92ad4c6c62da00632e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.366ex; height:3.676ex;" alt="{\displaystyle \ {\mathcal {L}}_{\mathrm {Fermion} }(\phi ,A,\psi )~=~{\overline {\psi }}\ \gamma ^{\mu }\ D_{\mu }\ \psi ~+~G_{\psi }\ {\overline {\psi }}\ \phi \ \psi \ ,}"></span></dd></dl> <p>where again the gauge field <span class="texhtml mvar" style="font-style:italic;">A</span> only enters via the gauge covariant derivative operator <span class="texhtml mvar" style="font-style:italic;">D<sub>μ</sub></span> (i.e., it is only indirectly visible). The quantities <span class="texhtml mvar" style="font-style:italic;">γ<sub>μ</sub></span> are the <a href="/wiki/Dirac_matrices" class="mw-redirect" title="Dirac matrices">Dirac matrices</a>, and <span class="texhtml mvar" style="font-style:italic;">G<sub>ψ</sub></span> is the already-mentioned Yukawa coupling parameter for <span class="texhtml mvar" style="font-style:italic;">ψ</span>. Now the mass-generation follows the same principle as above, namely from the existence of a finite expectation value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |\langle \phi \rangle |~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |\langle \phi \rangle |~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe31fe132d7bed870444ebefff2c84b413d8a404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.297ex; height:2.843ex;" alt="{\displaystyle \ |\langle \phi \rangle |~.}"></span> Again, this is crucial for the existence of the property <i>mass</i>. </p> <div class="mw-heading mw-heading2"><h2 id="History_of_research">History of research</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=5" title="Edit section: History of research"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Background">Background</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=6" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a> offered a framework to introduce bosons into relativistic quantum field theories. However, according to <a href="/wiki/Goldstone%27s_theorem" class="mw-redirect" title="Goldstone's theorem">Goldstone's theorem</a>, these bosons should be massless.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The only observed particles which could be approximately interpreted as Goldstone bosons were the <a href="/wiki/Pion" title="Pion">pions</a>, which <a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Yoichiro Nambu</a> related to <a href="/wiki/Chiral_symmetry" class="mw-redirect" title="Chiral symmetry">chiral symmetry</a> breaking. </p><p>A similar problem arises with <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a> (also known as <a href="/wiki/Non-abelian_gauge_theory" class="mw-redirect" title="Non-abelian gauge theory">non-abelian gauge theory</a>), which predicts massless <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>-1 <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a>. Massless weakly-interacting gauge bosons lead to long-range forces, which are only observed for electromagnetism and the corresponding massless <a href="/wiki/Photon" title="Photon">photon</a>. Gauge theories of the <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak force</a> needed a way to describe massive gauge bosons in order to be consistent. </p> <div class="mw-heading mw-heading3"><h3 id="Discovery">Discovery</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=7" title="Edit section: Discovery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Andersonphoto.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Andersonphoto.jpg/220px-Andersonphoto.jpg" decoding="async" width="220" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Andersonphoto.jpg/330px-Andersonphoto.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Andersonphoto.jpg/440px-Andersonphoto.jpg 2x" data-file-width="586" data-file-height="701" /></a><figcaption>Philip W. Anderson, the first to implement the mechanism in 1962.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:AIP-Sakurai-best.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/AIP-Sakurai-best.JPG/220px-AIP-Sakurai-best.JPG" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/AIP-Sakurai-best.JPG/330px-AIP-Sakurai-best.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/AIP-Sakurai-best.JPG/440px-AIP-Sakurai-best.JPG 2x" data-file-width="4256" data-file-height="2832" /></a><figcaption>Five of the six 2010 APS <a href="/wiki/Sakurai_Prize" title="Sakurai Prize">Sakurai Prize</a> Winners – (L to R) Tom Kibble, Gerald Guralnik, Carl Richard Hagen, François Englert, and Robert Brout</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Higgs,_Peter_(1929).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Higgs%2C_Peter_%281929%29.jpg/220px-Higgs%2C_Peter_%281929%29.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Higgs%2C_Peter_%281929%29.jpg/330px-Higgs%2C_Peter_%281929%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0d/Higgs%2C_Peter_%281929%29.jpg 2x" data-file-width="400" data-file-height="300" /></a><figcaption>Peter Higgs in 2009</figcaption></figure> <p>That breaking gauge symmetries did not lead to massless particles was observed in 1961 by <a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Julian Schwinger</a>,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> but he did not demonstrate massive particles would eventuate. This was done in <a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Philip Warren Anderson</a>'s 1962 paper<sup id="cite_ref-Anderson_3-1" class="reference"><a href="#cite_note-Anderson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but only in non-relativistic field theory; it also discussed consequences for particle physics but did not work out an explicit relativistic model. The relativistic model was developed in 1964 by three independent groups: </p> <ul><li><a href="/wiki/Robert_Brout" title="Robert Brout">Robert Brout</a> and <a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">François Englert</a><sup id="cite_ref-Englert_4-1" class="reference"><a href="#cite_note-Englert-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Peter Higgs</a><sup id="cite_ref-Higgs_5-1" class="reference"><a href="#cite_note-Higgs-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Gerald Guralnik</a>, <a href="/wiki/C._R._Hagen" title="C. R. Hagen">Carl Richard Hagen</a>, and <a href="/wiki/Tom_Kibble" title="Tom Kibble">Tom Kibble</a>.<sup id="cite_ref-GHK_6-1" class="reference"><a href="#cite_note-GHK-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Guralnik_7-1" class="reference"><a href="#cite_note-Guralnik-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Scholarpedia_history_8-1" class="reference"><a href="#cite_note-Scholarpedia_history-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <p>Slightly later, in 1965, but independently from the other publications<sup id="cite_ref-Polyakov_19-0" class="reference"><a href="#cite_note-Polyakov-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CernCourier_23-0" class="reference"><a href="#cite_note-CernCourier-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> the mechanism was also proposed by <a href="/wiki/Alexander_Migdal" class="mw-redirect" title="Alexander Migdal">Alexander Migdal</a> and <a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Alexander Polyakov</a>,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> at that time Soviet undergraduate students. However, their paper was delayed by the editorial office of <a href="/wiki/Journal_of_Experimental_and_Theoretical_Physics" title="Journal of Experimental and Theoretical Physics">JETP</a>, and was published late, in 1966. </p><p>The mechanism is closely analogous to phenomena previously discovered by <a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Yoichiro Nambu</a> involving the "vacuum structure" of quantum fields in <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> A similar but distinct effect (involving an affine realization of what is now recognized as the Higgs field), known as the <a href="/wiki/Stueckelberg_action" title="Stueckelberg action">Stueckelberg mechanism</a>, had previously been studied by <a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Ernst Stueckelberg</a>. </p><p>These physicists discovered that when a gauge theory is combined with an additional field that spontaneously breaks the symmetry group, the gauge bosons can consistently acquire a nonzero mass. In spite of the large values involved (see below) this permits a gauge theory description of the weak force, which was independently developed by <a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> and <a href="/wiki/Abdus_Salam" title="Abdus Salam">Abdus Salam</a> in 1967. Higgs's original article presenting the model was rejected by <i><a href="/wiki/Physics_Letters" title="Physics Letters">Physics Letters</a></i>. When revising the article before resubmitting it to <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>, he added a sentence at the end,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete <a href="/wiki/Group_representation" title="Group representation">representations</a> of the symmetry group; these are the Higgs bosons. </p><p>The three papers by Brout and Englert; Higgs; and Guralnik, Hagen, and Kibble were each recognized as "milestone letters" by <i>Physical Review Letters</i> in 2008.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> While each of these seminal papers took similar approaches, the contributions and differences among the <a href="/wiki/1964_PRL_symmetry_breaking_papers" title="1964 PRL symmetry breaking papers">1964 PRL symmetry breaking papers</a> are noteworthy. All six physicists were jointly awarded the 2010 <a href="/wiki/Sakurai_Prize" title="Sakurai Prize">J. J. Sakurai Prize for Theoretical Particle Physics</a> for this work.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Benjamin_W._Lee" title="Benjamin W. Lee">Benjamin W. Lee</a> is often credited with first naming the "Higgs-like" mechanism, although there is debate around when this first occurred.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> One of the first times the <i>Higgs</i> name appeared in print was in 1972 when <a href="/wiki/Gerardus_%27t_Hooft" class="mw-redirect" title="Gerardus 't Hooft">Gerardus 't Hooft</a> and <a href="/wiki/Martinus_J._G._Veltman" title="Martinus J. G. Veltman">Martinus J. G. Veltman</a> referred to it as the "Higgs–Kibble mechanism" in their Nobel winning paper.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Simple_explanation_of_the_theory,_from_its_origins_in_superconductivity"><span id="Simple_explanation_of_the_theory.2C_from_its_origins_in_superconductivity"></span>Simple explanation of the theory, from its origins in superconductivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=8" title="Edit section: Simple explanation of the theory, from its origins in superconductivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The proposed Higgs mechanism arose as a result of theories proposed to explain observations in <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>. A superconductor does not allow penetration by external magnetic fields (the <a href="/wiki/Meissner_effect" title="Meissner effect">Meissner effect</a>). This strange observation implies that the electromagnetic field somehow becomes short ranged during this phenomenon. Successful theories arose to explain this during the 1950s, first for fermions (<a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a>, 1950), and then for bosons (<a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a>, 1957). </p><p>In these theories, superconductivity is interpreted as arising from a <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">charged condensate</a>. Initially, the condensate value does not have any preferred direction. This implies it is scalar, but its <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> is capable of defining a gauge, in gauge based field theories. To do this, the field must be charged. A charged scalar field must also be complex (or described another way, it contains at least two components, and a symmetry capable of rotating each into the other(s)). In naïve gauge theory, a gauge transformation of a condensate usually rotates the phase. However, in these circumstances, it instead fixes a preferred choice of phase. However it turns out that fixing the choice of gauge so that the condensate has the same phase everywhere, also causes the electromagnetic field to gain an extra term. This extra term causes the electromagnetic field to become short range. </p><p><a href="/wiki/Goldstone%27s_theorem" class="mw-redirect" title="Goldstone's theorem">Goldstone's theorem</a> also plays a role in such theories. The connection is technically, when a condensate breaks a symmetry, then the state reached by acting with a symmetry generator on the condensate has the same energy as before. This means that some kinds of oscillation will not involve change of energy. Oscillations with unchanged energy imply that excitations (particles) associated with the oscillation are massless. </p><p>Once attention was drawn to this theory within particle physics, the parallels were clear. A change of the usually long range electromagnetic field to become short ranged, within a gauge invariant theory, was exactly the needed effect sought for the weak force bosons (because a long range force has massless gauge bosons, and a short ranged force implies massive gauge bosons, suggesting that a result of this interaction is that the field's gauge bosons acquired mass, or a similar and equivalent effect). The features of a field required to do this was also quite well defined – it would have to be a charged scalar field, with at least two components, and complex in order to support a symmetry able to rotate these into each other. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. In the non-relativistic context this is a <a href="/wiki/Superconductor" class="mw-redirect" title="Superconductor">superconductor</a>, more formally known as the <a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Landau model</a> of a charged <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a>. In the relativistic condensate, the condensate is a scalar field that is relativistically invariant. </p> <div class="mw-heading mw-heading3"><h3 id="Landau_model">Landau model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=10" title="Edit section: Landau model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Higgs mechanism is a type of <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a> which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged, or, in field language, when a charged field has a nonzero vacuum expectation value. Interaction with the quantum fluid filling the space prevents certain forces from propagating over long distances (as it does inside a superconductor; e.g., in the <a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a>). </p><p>A superconductor expels all magnetic fields from its interior, a phenomenon known as the <a href="/wiki/Meissner_effect" title="Meissner effect">Meissner effect</a>. This was mysterious for a long time, because it implies that electromagnetic forces somehow become short-range inside the superconductor. Contrast this with the behavior of an ordinary metal. In a metal, the conductivity shields electric fields by rearranging charges on the surface until the total field cancels in the interior. </p><p>But magnetic fields can penetrate to any distance, and if a <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopole</a> (an isolated magnetic pole) is surrounded by a metal the field can escape without collimating into a string. In a superconductor, however, electric charges move with no dissipation, and this allows for permanent surface currents, not just surface charges. When magnetic fields are introduced at the boundary of a superconductor, they produce surface currents which exactly neutralize them. </p><p>The Meissner effect arises due to currents in a thin surface layer, whose <a href="/wiki/London_penetration_depth" title="London penetration depth">thickness can be calculated</a> from the simple model of Ginzburg–Landau theory, which treats superconductivity as a charged Bose–Einstein condensate. </p><p>Suppose that a superconductor contains bosons with charge <span class="texhtml mvar" style="font-style:italic;">q</span> . The wavefunction of the bosons can be described by introducing a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \psi \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4de3cd6f5e2562efed5567d8c4a20ba00b9ab17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.321ex; height:2.509ex;" alt="{\displaystyle \ \psi \ ,}"></span> which obeys the <a href="/wiki/Schr%C3%B6dinger_field" title="Schrödinger field">Schrödinger equation as a field equation</a>. In units where the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, <span class="texhtml mvar" style="font-style:italic;">ħ</span>, is set to 1: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ i\ {\frac {\partial }{\ \partial t\ }}\ \psi ~=~{\frac {\ \left(\nabla -iqA\right)^{2}}{2m}}\ \psi ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>i</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mtext> </mtext> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>−</mo> <mi>i</mi> <mi>q</mi> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ i\ {\frac {\partial }{\ \partial t\ }}\ \psi ~=~{\frac {\ \left(\nabla -iqA\right)^{2}}{2m}}\ \psi ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da08892c68078d7dc2348d43611c64b9a234def1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.465ex; height:6.176ex;" alt="{\displaystyle \ i\ {\frac {\partial }{\ \partial t\ }}\ \psi ~=~{\frac {\ \left(\nabla -iqA\right)^{2}}{2m}}\ \psi ~.}"></span></dd></dl> <p>The operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \psi (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0fd5b1043a5fef4a1f3c26021b56db431b972b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.813ex; height:2.843ex;" alt="{\displaystyle \ \psi (x)\ }"></span> annihilates a boson at the point <span class="texhtml mvar" style="font-style:italic;">x</span>, while its adjoint <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \psi ^{\dagger }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi ^{\dagger }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4d2bb7afb51f488768675711dc615b04e7fce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.637ex; height:3.009ex;" alt="{\displaystyle \ \psi ^{\dagger }\ }"></span> creates a new boson at the same point. The wavefunction of the Bose–Einstein condensate is then the <a href="/wiki/Expectation_value" class="mw-redirect" title="Expectation value">expectation value</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \langle \psi \rangle \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \langle \psi \rangle \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d743e1bb842b03f3a61f6b835395e30b7a0f5c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.484ex; height:2.843ex;" alt="{\displaystyle \ \langle \psi \rangle \ }"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \psi (x)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi (x)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee63d5eac5a034443313e47ddb8e53c95918aaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle \ \psi (x)\ ,}"></span> which is a classical function that obeys the same equation. The interpretation of the expectation value is that it is the phase that one should give to a newly created boson so that it will coherently superpose with all the other bosons already in the condensate. </p><p>When there is a charged condensate, the electromagnetic interactions are screened. To see this, consider the effect of a <a href="/wiki/Gauge_transformation" class="mw-redirect" title="Gauge transformation">gauge transformation</a> on the field. A gauge transformation rotates the phase of the condensate by an amount which changes from point to point, and shifts the vector potential by a gradient: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi &\rightarrow e^{iq\phi (x)}\psi \\\\A&\rightarrow A+\nabla \phi ~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ψ</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>q</mi> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi &\rightarrow e^{iq\phi (x)}\psi \\\\A&\rightarrow A+\nabla \phi ~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edd718b1cd93da344cb9dc973366e505842ee08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:15.241ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\psi &\rightarrow e^{iq\phi (x)}\psi \\\\A&\rightarrow A+\nabla \phi ~.\end{aligned}}}"></span></dd></dl> <p>When there is no condensate, this transformation only changes the definition of the phase of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \psi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29cfb72b17498c521ba3bb385d48f6dbe5c00c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.674ex; height:2.509ex;" alt="{\displaystyle \ \psi \ }"></span> at every point. But when there is a condensate, the phase of the condensate defines a preferred choice of phase. </p><p>The condensate wave function can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\rho (x)\ e^{i\theta (x)}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\rho (x)\ e^{i\theta (x)}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d56d959cd7c6596c4f186e50201f924d6b840ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.773ex; height:3.343ex;" alt="{\displaystyle \psi (x)=\rho (x)\ e^{i\theta (x)}\ ,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">ρ</span> is real amplitude, which determines the local density of the condensate. If the condensate were neutral, the flow would be along the gradients of <span class="texhtml mvar" style="font-style:italic;">θ</span>, the direction in which the phase of the Schrödinger field changes. If the phase <span class="texhtml mvar" style="font-style:italic;">θ</span> changes slowly, the flow is slow and has very little energy. But now <span class="texhtml mvar" style="font-style:italic;">θ</span> can be made equal to zero just by making a gauge transformation to rotate the phase of the field. </p><p>The energy of slow changes of phase can be calculated from the Schrödinger kinetic energy, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ H={\frac {1}{\ 2\ m\ }}\ {\Bigl |}\left(iqA+\nabla \right)\psi {\Bigr |}^{2}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>q</mi> <mi>A</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ H={\frac {1}{\ 2\ m\ }}\ {\Bigl |}\left(iqA+\nabla \right)\psi {\Bigr |}^{2}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296d2d6c759b710cf04d215b8e8bd04f4f0fcbd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.168ex; height:5.176ex;" alt="{\displaystyle \ H={\frac {1}{\ 2\ m\ }}\ {\Bigl |}\left(iqA+\nabla \right)\psi {\Bigr |}^{2}\ ,}"></span></dd></dl> <p>and taking the density of the condensate <span class="texhtml mvar" style="font-style:italic;">ρ</span> to be constant, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\approx {\frac {~\rho ^{2}\ }{2\ m}}\ \left(qA+\nabla \theta \right)^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn>2</mn> <mtext> </mtext> <mi>m</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mi>A</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\approx {\frac {~\rho ^{2}\ }{2\ m}}\ \left(qA+\nabla \theta \right)^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51ddab324755f1607748ed46c7bfe88fc897a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.133ex; height:5.676ex;" alt="{\displaystyle H\approx {\frac {~\rho ^{2}\ }{2\ m}}\ \left(qA+\nabla \theta \right)^{2}~.}"></span></dd></dl> <p>Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\;q^{2}\rho ^{2}\ }{2\ m}}A^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn>2</mn> <mtext> </mtext> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\;q^{2}\rho ^{2}\ }{2\ m}}A^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742043a8012b7ff1cc9bd569c60c2ba8ff908bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.477ex; height:5.676ex;" alt="{\displaystyle {\frac {\;q^{2}\rho ^{2}\ }{2\ m}}A^{2}~.}"></span></dd></dl> <p>When this term is present, electromagnetic interactions become short-ranged. Every field mode, no matter how long the wavelength, oscillates with a nonzero frequency. The lowest frequency can be read off from the energy of a long wavelength <span class="texhtml mvar" style="font-style:italic;">A</span> mode, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\approx {\frac {\;{\dot {A}}^{2}}{2}}+{\frac {\ q^{2}\rho ^{2}\ }{2\ m}}\ A^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn>2</mn> <mtext> </mtext> <mi>m</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\approx {\frac {\;{\dot {A}}^{2}}{2}}+{\frac {\ q^{2}\rho ^{2}\ }{2\ m}}\ A^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f839878d0a26ecbcb5e0f1def1f23c93d59c754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.018ex; height:6.343ex;" alt="{\displaystyle E\approx {\frac {\;{\dot {A}}^{2}}{2}}+{\frac {\ q^{2}\rho ^{2}\ }{2\ m}}\ A^{2}~.}"></span></dd></dl> <p>This is a harmonic oscillator with frequency </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {{\frac {1}{\ m\ }}\ q^{2}\ \rho ^{2}\ }}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>m</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {{\frac {1}{\ m\ }}\ q^{2}\ \rho ^{2}\ }}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f5cdc3468d966eae3664e89c81ed7ceb6ad916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.721ex; height:6.176ex;" alt="{\displaystyle {\sqrt {{\frac {1}{\ m\ }}\ q^{2}\ \rho ^{2}\ }}~.}"></span></dd></dl> <p>The quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left|\psi (x)\right|^{2}=\rho ^{2}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left|\psi (x)\right|^{2}=\rho ^{2}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a20b5ae593d4464b4326a1092830602e0e69e51f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.516ex; height:3.343ex;" alt="{\displaystyle \ \left|\psi (x)\right|^{2}=\rho ^{2}\ }"></span> is the density of the condensate of superconducting particles. </p><p>In an actual superconductor, the charged particles are electrons, which are fermions not bosons. So in order to have superconductivity, the electrons need to somehow bind into <a href="/wiki/Cooper_pair" title="Cooper pair">Cooper pairs</a>. The charge of the condensate <span class="texhtml mvar" style="font-style:italic;">q</span> is therefore twice the electron charge <span class="texhtml mvar" style="font-style:italic;">−e</span>. The pairing in a normal superconductor is due to lattice vibrations, and is in fact very weak; this means that the pairs are very loosely bound. The description of a Bose–Einstein condensate of loosely bound pairs is actually more difficult than the description of a condensate of elementary particles, and was only worked out in 1957 by <a href="/wiki/John_Bardeen" title="John Bardeen">John Bardeen</a>, <a href="/wiki/Leon_Cooper" title="Leon Cooper">Leon Cooper</a>, and <a href="/wiki/John_Robert_Schrieffer" title="John Robert Schrieffer">John Robert Schrieffer</a> in the famous <a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Abelian_Higgs_mechanism">Abelian Higgs mechanism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=11" title="Edit section: Abelian Higgs mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gauge invariance means that certain transformations of the gauge field do not change the energy at all. If an arbitrary gradient is added to <span class="texhtml mvar" style="font-style:italic;">A</span>, the energy of the field is exactly the same. This makes it difficult to add a mass term, because a mass term tends to push the field toward the value zero. But the zero value of the vector potential is not a gauge invariant idea. What is zero in one gauge is nonzero in another. </p><p>So in order to give mass to a gauge theory, the gauge invariance must be broken by a condensate. The condensate will then define a preferred phase, and the phase of the condensate will define the zero value of the field in a gauge-invariant way. The gauge-invariant definition is that a gauge field is zero when the phase change along any path from parallel transport is equal to the phase difference in the condensate wavefunction. </p><p>The condensate value is described by a quantum field with an expectation value, just as in the <a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau model</a>. </p><p>In order for the phase of the vacuum to define a gauge, the field must have a phase (also referred to as 'to be charged'). In order for a scalar field <span class="texhtml">Φ</span> to have a phase, it must be complex, or (equivalently) it should contain two fields with a symmetry which rotates them into each other. The vector potential changes the phase of the quanta produced by the field when they move from point to point. In terms of fields, it defines how much to rotate the real and imaginary parts of the fields into each other when comparing field values at nearby points. </p><p>The only <a href="/wiki/Renormalization" title="Renormalization">renormalizable</a> model where a complex scalar field <span class="texhtml">Φ</span> acquires a nonzero value is the 'Mexican-hat' model, where the field energy has a minimum away from zero. The action for this model is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S(\phi )=\int d^{4}x\left[{\tfrac {1}{2}}\left|\partial \phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right]\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>λ</mi> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mi>ϕ</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S(\phi )=\int d^{4}x\left[{\tfrac {1}{2}}\left|\partial \phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right]\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb1f05c7dda6c344f045fa84d6fe7ed6b2b8414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.053ex; height:6.176ex;" alt="{\displaystyle \ S(\phi )=\int d^{4}x\left[{\tfrac {1}{2}}\left|\partial \phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right]\ ,}"></span></dd></dl> <p>which results in the Hamiltonian </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ H(\phi )={\tfrac {1}{2}}\left(\left|{\dot {\phi }}\right|^{2}+\left|\nabla \phi \right|^{2}\right)+V\left(\left|\phi \right|\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>H</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mo>|</mo> <mi>ϕ</mi> <mo>|</mo> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ H(\phi )={\tfrac {1}{2}}\left(\left|{\dot {\phi }}\right|^{2}+\left|\nabla \phi \right|^{2}\right)+V\left(\left|\phi \right|\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605ec295d464951a02ccca53215019a6319095ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.201ex; height:4.843ex;" alt="{\displaystyle \ H(\phi )={\tfrac {1}{2}}\left(\left|{\dot {\phi }}\right|^{2}+\left|\nabla \phi \right|^{2}\right)+V\left(\left|\phi \right|\right)~.}"></span></dd></dl> <p>The first term is the kinetic energy of the field. The second term is the extra potential energy when the field varies from point to point. The third term is the potential energy when the field has any given magnitude. </p><p>This potential energy, the <b>Higgs potential</b>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~V\left(z,\Phi \right)=\lambda \left(\left|z\right|^{2}-\Phi ^{2}\right)^{2}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>,</mo> <mi mathvariant="normal">Φ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~V\left(z,\Phi \right)=\lambda \left(\left|z\right|^{2}-\Phi ^{2}\right)^{2}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed53edecc6642694f5fdb132ebe1e120e34a814b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.884ex; height:5.176ex;" alt="{\displaystyle ~V\left(z,\Phi \right)=\lambda \left(\left|z\right|^{2}-\Phi ^{2}\right)^{2}\ ,}"></span><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Mexican_hat_potential" class="mw-redirect" title="Mexican hat potential">has a graph</a> which looks like a <a href="/wiki/Sombrero" title="Sombrero">Mexican hat</a>, which gives the model its name. In particular, the minimum energy value is not at <span class="nowrap"><span class="texhtml"> <i>z</i> = 0</span> ,</span> but on the circle of points where the magnitude of <span class="texhtml mvar" style="font-style:italic;">z</span> <span class="nowrap">is <span class="texhtml">Φ</span> .</span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mecanismo_de_Higgs_PH.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Mecanismo_de_Higgs_PH.png/220px-Mecanismo_de_Higgs_PH.png" decoding="async" width="220" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Mecanismo_de_Higgs_PH.png/330px-Mecanismo_de_Higgs_PH.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Mecanismo_de_Higgs_PH.png/440px-Mecanismo_de_Higgs_PH.png 2x" data-file-width="464" data-file-height="455" /></a><figcaption>Higgs potential <span class="texhtml mvar" style="font-style:italic;">V</span>. For a fixed value of <span class="texhtml mvar" style="font-style:italic;">λ</span>, the potential is presented upwards against the real and imaginary parts <span class="nowrap">of <span class="texhtml">Φ</span> .</span> The <i><a href="/wiki/Sombrero" title="Sombrero">Mexican-hat</a></i> or <i>champagne-bottle profile</i> at the ground should be noted.</figcaption></figure> <p>When the field <span class="texhtml"> Φ(<i>x</i>) </span> is not coupled to electromagnetism, the Mexican-hat potential has flat directions. Starting in any one of the circle of vacua and changing the phase of the field from point to point costs very little energy. Mathematically, if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi (x)=\Phi e^{i\theta (x)}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \phi (x)=\Phi e^{i\theta (x)}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa8a20f8a9e611986abe25e2c046bb18db2dfe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.336ex; height:3.343ex;" alt="{\displaystyle \ \phi (x)=\Phi e^{i\theta (x)}\ }"></span></dd></dl> <p>with a constant prefactor, then the action for the field <span class="nowrap"> <span class="texhtml"><i>θ</i>(<i>x</i>)</span> ,</span> i.e., the "phase" of the Higgs field <span class="nowrap"> <span class="texhtml"> Φ(<i>x</i>)</span> ,</span> has only derivative terms. This is not a surprise: Adding a constant to <span class="texhtml"><i>θ</i>(<i>x</i>)</span> is a symmetry of the original theory, so different values of <span class="texhtml"><i>θ</i>(<i>x</i>)</span> cannot have different energies. This is an example of configuring the model to conform to <a href="/wiki/Goldstone%27s_theorem" class="mw-redirect" title="Goldstone's theorem">Goldstone's theorem</a>: Spontaneously broken continuous symmetries (normally) produce massless excitations. </p><p>The Abelian Higgs model is the Mexican-hat model coupled to <a href="/wiki/Maxwell%27s_theory" class="mw-redirect" title="Maxwell's theory">electromagnetism</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S(\phi ,A)=\int d^{4}x\left[-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+\left|\left(\partial -iqA\right)\phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right]~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∂</mi> <mo>−</mo> <mi>i</mi> <mi>q</mi> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>λ</mi> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mi>ϕ</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S(\phi ,A)=\int d^{4}x\left[-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+\left|\left(\partial -iqA\right)\phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right]~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a78b06ff1a3bd4c032bdf6e0efd03bade51bf83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.63ex; height:6.176ex;" alt="{\displaystyle \ S(\phi ,A)=\int d^{4}x\left[-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+\left|\left(\partial -iqA\right)\phi \right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}\right]~.}"></span></dd></dl> <div style="clear:both;" class=""></div> <table class="wikitable collapsible collapsed"> <tbody><tr> <th>Alternate form of the Abelian Higgs model action </th></tr> <tr> <td> <p>The Abelian Higgs model action can also be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S[\phi ,A]=\int d^{4}x\left[-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+D_{\mu }\phi (D^{\mu }\phi )^{*}-V(\phi )\right]\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ϕ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S[\phi ,A]=\int d^{4}x\left[-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+D_{\mu }\phi (D^{\mu }\phi )^{*}-V(\phi )\right]\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8545950e50c80775baa0d7eb3bead1cee480e318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.571ex; height:5.676ex;" alt="{\displaystyle \ S[\phi ,A]=\int d^{4}x\left[-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+D_{\mu }\phi (D^{\mu }\phi )^{*}-V(\phi )\right]\ ,}"></span></dd></dl> <p>where the potential is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ V(\phi )=\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mi>ϕ</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ V(\phi )=\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73cb2d096d974ed95d3db3e9c4d86f484b3ed62a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.38ex; height:5.176ex;" alt="{\displaystyle \ V(\phi )=\lambda \left(\left|\phi \right|^{2}-\Phi ^{2}\right)^{2}~.}"></span></dd></dl> <p>and the covariant derivative <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267983ed233945e70f42935df28e623cfce15c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.148ex; height:2.843ex;" alt="{\displaystyle D_{\mu }}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ D_{\mu }=\partial _{\mu }-iqA_{\mu }~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi>q</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ D_{\mu }=\partial _{\mu }-iqA_{\mu }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8aa5f11120de1e8f535ae4088dfc689871f331d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.191ex; height:2.843ex;" alt="{\displaystyle \ D_{\mu }=\partial _{\mu }-iqA_{\mu }~.}"></span></dd></dl> <p>For completeness, the tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ffd8a045458dffee5609aff076c4f4f0e3e4015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.298ex; height:2.843ex;" alt="{\displaystyle \ F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\ }"></span> is the Maxwell tensor, also known as the electromagnetic field strength, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathrm {U} (1)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathrm {U} (1)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53298e2925490621b97498948b329638f288b09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.876ex; height:2.843ex;" alt="{\displaystyle \ \mathrm {U} (1)\ }"></span> field strength or more geometrically the curvature of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathrm {U} (1)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathrm {U} (1)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53298e2925490621b97498948b329638f288b09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.876ex; height:2.843ex;" alt="{\displaystyle \ \mathrm {U} (1)\ }"></span> connection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ A_{\mu }~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A_{\mu }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e8e0ea7188839714b7adde9a46b72f5faf59b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle \ A_{\mu }~.}"></span> The four-vector gauge field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ A_{\mu }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A_{\mu }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5205ed955b1e62e46146a18106035e66a3e43499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.128ex; height:2.843ex;" alt="{\displaystyle \ A_{\mu }\ }"></span> is also known as the four-potential. </p><p>This makes the gauge-invariance of the action (and therefore Lagrangian and resulting equations of motion) manifest. The potential makes the non-zero vacuum expectation value evident. </p> </td></tr></tbody></table> <p>The classical vacuum is again at the minimum of the potential, where the magnitude of the complex field <span class="texhtml">φ</span> is equal <span class="nowrap">to <span class="texhtml">Φ</span> .</span> But now the phase of the field is arbitrary, because gauge transformations change it. This means that the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \theta (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \theta (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3211b88ed677f2661484548bb9480df4c513a8c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.391ex; height:2.843ex;" alt="{\displaystyle \ \theta (x)\ }"></span> can be set to zero by a gauge transformation, and does not represent any actual degrees of freedom at all. </p><p>Furthermore, choosing a gauge where the phase of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero. So in the Abelian Higgs model, the gauge field acquires a mass. To calculate the magnitude of the mass, consider a constant value of the vector potential <span class="texhtml mvar" style="font-style:italic;">A</span> in the <span class="texhtml mvar" style="font-style:italic;">x</span>-direction in the gauge where the condensate has constant phase. This is the same as a sinusoidally varying condensate in the gauge where the vector potential is zero. In the gauge where A is zero, the potential energy density in the condensate is the scalar gradient energy: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E={\tfrac {1}{2}}\left|\partial \left(\Phi e^{iqAx}\right)\right|^{2}={\tfrac {1}{2}}q^{2}\Phi ^{2}A^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Φ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>q</mi> <mi>A</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E={\tfrac {1}{2}}\left|\partial \left(\Phi e^{iqAx}\right)\right|^{2}={\tfrac {1}{2}}q^{2}\Phi ^{2}A^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f321ff11381b6c0cad0b629413a6b0a59bbb779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.433ex; height:4.009ex;" alt="{\displaystyle \ E={\tfrac {1}{2}}\left|\partial \left(\Phi e^{iqAx}\right)\right|^{2}={\tfrac {1}{2}}q^{2}\Phi ^{2}A^{2}~.}"></span></dd></dl> <p>This energy is the same as a mass term <span class="texhtml"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>m</i><sup>2</sup><i>A</i><sup>2</sup> </span> where <span class="nowrap"><span class="texhtml"><i>m</i> = <i>q</i> Φ</span> .</span> </p> <div class="mw-heading mw-heading4"><h4 id="Mathematical_details_of_the_abelian_Higgs_mechanism">Mathematical details of the abelian Higgs mechanism</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=12" title="Edit section: Mathematical details of the abelian Higgs mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable collapsible collapsed"> <tbody><tr> <th>Lagrangian in explicit symmetry broken form </th></tr> <tr> <td> <p>Start from the Lagrangian </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\tfrac {1}{2}}D_{\mu }\phi (D^{\mu }\phi )^{*}-V(\phi )\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>ϕ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\tfrac {1}{2}}D_{\mu }\phi (D^{\mu }\phi )^{*}-V(\phi )\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f833372cfbf19870e6af386233f769034bb4749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.726ex; height:3.509ex;" alt="{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\tfrac {1}{2}}D_{\mu }\phi (D^{\mu }\phi )^{*}-V(\phi )\ ,}"></span></dd></dl> <p>with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ D_{\mu }\phi =\partial _{\mu }\phi +ieA_{\mu }\phi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mo>+</mo> <mi>i</mi> <mi>e</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ϕ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ D_{\mu }\phi =\partial _{\mu }\phi +ieA_{\mu }\phi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d21f0251b7e3a38e41da0dbd06b6ef9a734d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.714ex; height:2.843ex;" alt="{\displaystyle \ D_{\mu }\phi =\partial _{\mu }\phi +ieA_{\mu }\phi \ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ V(\phi )={\tfrac {1}{4}}\lambda (|\phi |^{2}-v^{2})^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ V(\phi )={\tfrac {1}{4}}\lambda (|\phi |^{2}-v^{2})^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a352257d2901001f3b39fa100dd1215ffc1933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.521ex; height:3.676ex;" alt="{\displaystyle \ V(\phi )={\tfrac {1}{4}}\lambda (|\phi |^{2}-v^{2})^{2}~.}"></span></dd></dl> <p>Guided by the minimum of the potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> being at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |\phi |=v\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>v</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |\phi |=v\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/159d4639f76eb534f1d88f69e665a2986b029bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.713ex; height:2.843ex;" alt="{\displaystyle \ |\phi |=v\ ,}"></span> we write the complex scalar field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \phi (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa38808ae6d30714d4d2fccc39f4a9153974c6ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.686ex; height:2.843ex;" alt="{\displaystyle \ \phi (x)\ }"></span> in terms of real scalar fields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \xi (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \xi (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8e329af8c5bcf02169217a7e993f7e39317db5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle \ \xi (x)\ }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \eta (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \eta (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfd75f5e73f470307f36e58b6ad864bf274130e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.47ex; height:2.843ex;" alt="{\displaystyle \ \eta (x)\ }"></span> as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi (x)=e^{i\xi (x)}(\eta (x)+v)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \phi (x)=e^{i\xi (x)}(\eta (x)+v)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c91d450faf92aded95a25290bab56c862f317362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.348ex; height:3.343ex;" alt="{\displaystyle \ \phi (x)=e^{i\xi (x)}(\eta (x)+v)~.}"></span></dd></dl> <p>The field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \xi (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \xi (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8e329af8c5bcf02169217a7e993f7e39317db5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle \ \xi (x)\ }"></span> is known as the Nambu-Goldstone field, and the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \eta (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \eta (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfd75f5e73f470307f36e58b6ad864bf274130e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.47ex; height:2.843ex;" alt="{\displaystyle \ \eta (x)\ }"></span> is known as the Higgs boson. </p><p>Upon rewriting the Lagrangian in terms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \xi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ξ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \xi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df198676d4870bdb51abcee48913a08eca0d2167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.509ex;" alt="{\displaystyle \ \xi \ }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \eta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>η</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \eta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590ec28d1a1c2fd0d5461c1b8f9f3fe2790a1516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.331ex; height:2.176ex;" alt="{\displaystyle \ \eta \ }"></span> one finds </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\tfrac {1}{2}}\left[\partial _{\mu }\eta \partial ^{\mu }\eta +\left(\eta +v\right)^{2}\ \left(\partial _{\mu }\xi +eA_{\mu }\right)^{2}\right]-\left[\lambda v^{2}\eta ^{2}+\lambda v\eta ^{3}+{\tfrac {1}{4}}\lambda \eta ^{4}\right]~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>η</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>η</mi> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>η</mi> <mo>+</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ξ</mi> <mo>+</mo> <mi>e</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow> <mo>[</mo> <mrow> <mi>λ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>λ</mi> <mi>v</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>λ</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\tfrac {1}{2}}\left[\partial _{\mu }\eta \partial ^{\mu }\eta +\left(\eta +v\right)^{2}\ \left(\partial _{\mu }\xi +eA_{\mu }\right)^{2}\right]-\left[\lambda v^{2}\eta ^{2}+\lambda v\eta ^{3}+{\tfrac {1}{4}}\lambda \eta ^{4}\right]~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c67000145f55fc465f6a6e58525209dcee9f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:83.403ex; height:4.843ex;" alt="{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\tfrac {1}{2}}\left[\partial _{\mu }\eta \partial ^{\mu }\eta +\left(\eta +v\right)^{2}\ \left(\partial _{\mu }\xi +eA_{\mu }\right)^{2}\right]-\left[\lambda v^{2}\eta ^{2}+\lambda v\eta ^{3}+{\tfrac {1}{4}}\lambda \eta ^{4}\right]~.}"></span></dd></dl> <p>At this point the only term which contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \xi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ξ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \xi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df198676d4870bdb51abcee48913a08eca0d2167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.509ex;" alt="{\displaystyle \ \xi \ }"></span> is the term containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \partial _{\mu }\xi +eA_{\mu }~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ξ</mi> <mo>+</mo> <mi>e</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \partial _{\mu }\xi +eA_{\mu }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abbeef8274202afc5707647a3b5201fff836bcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.186ex; height:2.843ex;" alt="{\displaystyle \ \partial _{\mu }\xi +eA_{\mu }~.}"></span> But the dependence on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \xi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ξ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \xi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df198676d4870bdb51abcee48913a08eca0d2167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.509ex;" alt="{\displaystyle \ \xi \ }"></span> can be gauged away by the gauge transformation which sends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ A_{\mu }+{\frac {1}{\ e\ }}\partial _{\mu }\xi \mapsto A_{\mu }~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>e</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ξ</mi> <mo stretchy="false">↦</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A_{\mu }+{\frac {1}{\ e\ }}\partial _{\mu }\xi \mapsto A_{\mu }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277327cffc58ba4065a872089fe92f55136ca0ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.764ex; height:5.176ex;" alt="{\displaystyle \ A_{\mu }+{\frac {1}{\ e\ }}\partial _{\mu }\xi \mapsto A_{\mu }~.}"></span> This is known as the unitary or unitarity gauge. In differential-geometric language, as is spelled out in the following box, the condensate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d08be28f538ae3e63aea13e5aca6d0c62772c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \xi (x)}"></span> has defined a canonical trivialization. </p><p>In unitary gauge, the Lagrangian can be organised into parts which depend on the gauge field and Higgs field </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\tfrac {1}{2}}e^{2}\left(\eta +v\right)^{2}A_{\mu }A^{\mu }+{\tfrac {1}{2}}\partial _{\mu }\eta \partial ^{\mu }\eta -\lambda v^{2}\eta ^{2}-\lambda v\eta ^{3}-{\tfrac {1}{4}}\lambda \eta ^{4}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>η</mi> <mo>+</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>η</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>η</mi> <mo>−</mo> <mi>λ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>λ</mi> <mi>v</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>λ</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\tfrac {1}{2}}e^{2}\left(\eta +v\right)^{2}A_{\mu }A^{\mu }+{\tfrac {1}{2}}\partial _{\mu }\eta \partial ^{\mu }\eta -\lambda v^{2}\eta ^{2}-\lambda v\eta ^{3}-{\tfrac {1}{4}}\lambda \eta ^{4}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8e5dad2920b7f957e7e488a96bca223cf03ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:73.499ex; height:3.676ex;" alt="{\displaystyle \ {\mathcal {L}}=-{\tfrac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\tfrac {1}{2}}e^{2}\left(\eta +v\right)^{2}A_{\mu }A^{\mu }+{\tfrac {1}{2}}\partial _{\mu }\eta \partial ^{\mu }\eta -\lambda v^{2}\eta ^{2}-\lambda v\eta ^{3}-{\tfrac {1}{4}}\lambda \eta ^{4}\ }"></span></dd></dl> <p>or into quadratic and interaction pieces </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {L}}=\left[-{\tfrac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\tfrac {1}{2}}e^{2}v^{2}A_{\mu }A^{\mu }+{\tfrac {1}{2}}\partial _{\mu }\eta \partial ^{\mu }\eta -\lambda v^{2}\eta ^{2}\right]+\left[-\lambda v\eta ^{3}-{\tfrac {1}{4}}\lambda \eta ^{4}+{\tfrac {1}{2}}\left(\eta ^{2}+2v\eta \right)A^{\mu }A_{\mu }\right]~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>η</mi> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>η</mi> <mo>−</mo> <mi>λ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>λ</mi> <mi>v</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>λ</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>v</mi> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {L}}=\left[-{\tfrac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\tfrac {1}{2}}e^{2}v^{2}A_{\mu }A^{\mu }+{\tfrac {1}{2}}\partial _{\mu }\eta \partial ^{\mu }\eta -\lambda v^{2}\eta ^{2}\right]+\left[-\lambda v\eta ^{3}-{\tfrac {1}{4}}\lambda \eta ^{4}+{\tfrac {1}{2}}\left(\eta ^{2}+2v\eta \right)A^{\mu }A_{\mu }\right]~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc4535d0f9c4c05c1e53f9e53333a83e2a9d0f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:96.775ex; height:4.843ex;" alt="{\displaystyle \ {\mathcal {L}}=\left[-{\tfrac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\tfrac {1}{2}}e^{2}v^{2}A_{\mu }A^{\mu }+{\tfrac {1}{2}}\partial _{\mu }\eta \partial ^{\mu }\eta -\lambda v^{2}\eta ^{2}\right]+\left[-\lambda v\eta ^{3}-{\tfrac {1}{4}}\lambda \eta ^{4}+{\tfrac {1}{2}}\left(\eta ^{2}+2v\eta \right)A^{\mu }A_{\mu }\right]~.}"></span></dd></dl> <p>By focusing on the quadratic piece, we see that the gauge field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ A_{\mu }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A_{\mu }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5205ed955b1e62e46146a18106035e66a3e43499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.128ex; height:2.843ex;" alt="{\displaystyle \ A_{\mu }\ }"></span> has acquired a <a href="/wiki/Proca_action" title="Proca action">Proca mass</a>, while the Higgs field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \eta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>η</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \eta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590ec28d1a1c2fd0d5461c1b8f9f3fe2790a1516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.331ex; height:2.176ex;" alt="{\displaystyle \ \eta \ }"></span> has a mass of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\sqrt {2\lambda v^{2}~}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>λ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\sqrt {2\lambda v^{2}~}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b28908407f1d1e6c282e34cbb43724305949822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.412ex; height:3.343ex;" alt="{\displaystyle \ {\sqrt {2\lambda v^{2}~}}~.}"></span> </p><p>This method largely carries over to the case where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathrm {U} (1)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathrm {U} (1)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53298e2925490621b97498948b329638f288b09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.876ex; height:2.843ex;" alt="{\displaystyle \ \mathrm {U} (1)\ }"></span> gauge symmetry is promoted to a non-abelian gauge group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ G~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ G~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f17ede2f58d26202111dd8428e90f7bf20f498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.635ex; height:2.176ex;" alt="{\displaystyle \ G~.}"></span> The Nambu-Goldstone field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \xi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ξ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \xi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df198676d4870bdb51abcee48913a08eca0d2167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.509ex;" alt="{\displaystyle \ \xi \ }"></span> is then promoted to a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d26f35ba7dce9ef81ccc5f780e83de6942874fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.752ex; height:2.009ex;" alt="{\displaystyle \ {\mathfrak {g}}}"></span>-valued field, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathfrak {g}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathfrak {g}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cefa9539aac04413f8ea654ccfa7b7af2e798bb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.333ex; height:2.009ex;" alt="{\displaystyle \ {\mathfrak {g}}\ }"></span> is the Lie algebra of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9950f6d5c2ca81131acf6cfe1ba7d26d5da010d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.054ex; height:2.176ex;" alt="{\displaystyle G~.}"></span> </p> </td></tr></tbody></table> <table class="wikitable collapsible collapsed"> <tbody><tr> <th>Spontaneous symmetry breaking and trivializations </th></tr> <tr> <td> <p>A more mathematical or specifically differential-geometric viewpoint is that the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \theta (x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \theta (x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3211b88ed677f2661484548bb9480df4c513a8c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.391ex; height:2.843ex;" alt="{\displaystyle \ \theta (x)\ }"></span> picks out a canonical <a href="/wiki/Trivialization_(mathematics)" class="mw-redirect" title="Trivialization (mathematics)">trivialization</a> which breaks the right-invariance of the <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> that the gauge theory lives on. </p><p>This is realized most easily when the theory is based on flat <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathbb {R} ^{1,3}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathbb {R} ^{1,3}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0566b75a55996142c2fb583285dd8685b01e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.82ex; height:3.009ex;" alt="{\displaystyle \ \mathbb {R} ^{1,3}\ ,}"></span> as then the base spacetime is contractible, and hence any <a href="/wiki/Fibre_bundle" class="mw-redirect" title="Fibre bundle">fibre bundle</a> is trivial. In gauge theory one considers <a href="/wiki/Principal_bundles" class="mw-redirect" title="Principal bundles">principal bundles</a> with the spacetime as its base manifold, where the fibre is a torsor of the gauge group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ G~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ G~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f17ede2f58d26202111dd8428e90f7bf20f498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.635ex; height:2.176ex;" alt="{\displaystyle \ G~.}"></span> Crucially, since the principal bundle must be trivial, there exists a global trivialization. In physics, one generally works under an implicit global trivialization and rarely in the more abstract principal bundle. </p><p>However, there are many choices of global trivialization, which differ from one another by a transition function, which can be written as a function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g:\mathbb {R} ^{1,3}\rightarrow G~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>g</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mi>G</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ g:\mathbb {R} ^{1,3}\rightarrow G~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b244a1fadf88235edf014bc3df7e322312ec2b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.314ex; height:3.009ex;" alt="{\displaystyle \ g:\mathbb {R} ^{1,3}\rightarrow G~.}"></span></dd></dl> <p>From the physical viewpoint, this is known as a gauge transformation. There is a corresponding (choice of) transition function or gauge transformation at the algebra level </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \alpha :\mathbb {R} ^{1,3}\rightarrow {\mathfrak {g}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>α</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \alpha :\mathbb {R} ^{1,3}\rightarrow {\mathfrak {g}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fa4e98c2a656dd9a0a45ca5f134e2972d6ebcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.384ex; height:3.009ex;" alt="{\displaystyle \ \alpha :\mathbb {R} ^{1,3}\rightarrow {\mathfrak {g}}\ }"></span></dd></dl> <p>such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \exp(\alpha (x))=g(x)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \exp(\alpha (x))=g(x)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6224e205f68e34c827850f52c2b8a94be2df3591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.537ex; height:2.843ex;" alt="{\displaystyle \ \exp(\alpha (x))=g(x)\ ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\exp ~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\exp ~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/983f45222949f77cd087f3d8020730eb8e00f232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.488ex; height:2.009ex;" alt="{\displaystyle ~\exp ~}"></span> is the exponential map for Lie algebras. Then we can view the phase function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0d0d32f86079d2585f75430af6125a48802d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle \theta (x)}"></span> as a transition function at the algebra level. It picks out a canonical global trivialization which 'differs from' the initial implicit global trivialization by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \theta (x)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \theta (x)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5489337542737a2d0d9c5db90907b916aef85378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.038ex; height:2.843ex;" alt="{\displaystyle \ \theta (x)~.}"></span> </p><p>This breaks the (right-)invariance of the principal bundle under the action of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ G\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>G</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ G\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e97b8a0015758469fb3a014a270b9915af1db360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.635ex; height:2.509ex;" alt="{\displaystyle \ G\ ,}"></span> as this action does not preserve the canonical trivialization. Mathematically, this is the symmetry which is broken during spontaneous symmetry breaking. For the Abelian Higgs mechanism the relevant gauge group is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathrm {U} (1)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathrm {U} (1)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e1514ad3e260af36830f7e2c1919a29e0872dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.523ex; height:2.843ex;" alt="{\displaystyle \ \mathrm {U} (1)~.}"></span> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Non-Abelian_Higgs_mechanism">Non-Abelian Higgs mechanism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=13" title="Edit section: Non-Abelian Higgs mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Non-Abelian Higgs model has the following action </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\phi ,\mathbf {A} )=\int {1 \over 4g^{2}}\mathop {\textrm {tr}} (F^{\mu \nu }F_{\mu \nu })+|D\phi |^{2}+V(|\phi |),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP"> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mi>ϕ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\phi ,\mathbf {A} )=\int {1 \over 4g^{2}}\mathop {\textrm {tr}} (F^{\mu \nu }F_{\mu \nu })+|D\phi |^{2}+V(|\phi |),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2628884203a6b9081aede6cfd070eab847fbe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.37ex; height:5.843ex;" alt="{\displaystyle S(\phi ,\mathbf {A} )=\int {1 \over 4g^{2}}\mathop {\textrm {tr}} (F^{\mu \nu }F_{\mu \nu })+|D\phi |^{2}+V(|\phi |),}"></span></dd></dl> <p>where now the non-Abelian field <b>A</b> is contained in the covariant derivative <i>D</i> and in the tensor components <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270a5daa364616f695db1f470f3d341755df881a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.909ex; height:2.343ex;" alt="{\displaystyle F^{\mu \nu }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f24e4d8457e87f45e16137c04eae9d3343843caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.589ex; height:2.843ex;" alt="{\displaystyle F_{\mu \nu }}"></span> (the relation between <b>A</b> and those components is well-known from the <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>). </p><p>It is exactly analogous to the Abelian Higgs model. Now the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is in a representation of the gauge group, and the gauge covariant derivative is defined by the rate of change of the field minus the rate of change from parallel transport using the gauge field A as a connection. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\phi =\partial \phi -iA^{k}t_{k}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>ϕ</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> <mo>−</mo> <mi>i</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\phi =\partial \phi -iA^{k}t_{k}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff01c4bc4e511f6a4b3b98f2549c99a4a594b91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.9ex; height:3.009ex;" alt="{\displaystyle D\phi =\partial \phi -iA^{k}t_{k}\phi }"></span></dd></dl> <p>Again, the expectation value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> defines a preferred gauge where the vacuum is constant, and fixing this gauge, fluctuations in the gauge field <i>A</i> come with a nonzero energy cost. </p><p>Depending on the representation of the scalar field, not every gauge field acquires a mass. A simple example is in the renormalizable version of an early electroweak model due to <a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Julian Schwinger</a>. In this model, the gauge group is <b>SO</b>(3) (or <b>SU</b>(2) − there are no spinor representations in the model), and the gauge invariance is broken down to <b>U</b>(1) or <b>SO</b>(2) at long distances. To make a consistent renormalizable version using the Higgs mechanism, introduce a scalar field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79840e5f5459eca8a27674790f658e2533ca1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.676ex;" alt="{\displaystyle \phi ^{a}}"></span> which transforms as a vector (a triplet) of <b>SO</b>(3). If this field has a vacuum expectation value, it points in some direction in field space. Without loss of generality, one can choose the <i>z</i>-axis in field space to be the direction that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is pointing, and then the vacuum expectation value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is <span class="nowrap">(0, 0, <span class="texhtml mvar" style="font-style:italic;">Ã</span>)</span>, where <span class="texhtml mvar" style="font-style:italic;">Ã</span> is a constant with dimensions of mass (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=\hbar =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=\hbar =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da53d95302cc3d484fa56968fced5ddbb324c96c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.673ex; height:2.176ex;" alt="{\displaystyle c=\hbar =1}"></span>). </p><p>Rotations around the <i>z</i>-axis form a <b>U</b>(1) subgroup of <b>SO</b>(3) which preserves the vacuum expectation value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, and this is the unbroken gauge group. Rotations around the <i>x</i> and <i>y</i>-axis do not preserve the vacuum, and the components of the <b>SO</b>(3) gauge field which generate these rotations become massive vector mesons. There are two massive W mesons in the Schwinger model, with a mass set by the mass scale <span class="texhtml mvar" style="font-style:italic;">Ã</span>, and one massless <b>U</b>(1) gauge boson, similar to the photon. </p><p>The Schwinger model predicts <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a> at the electroweak unification scale, and does not predict the Z boson. It doesn't break electroweak symmetry properly as in nature. But historically, a model similar to this (but not using the Higgs mechanism) was the first in which the weak force and the electromagnetic force were unified. </p> <div class="mw-heading mw-heading3"><h3 id="Affine_Higgs_mechanism">Affine Higgs mechanism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=14" title="Edit section: Affine Higgs mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Ernst Stueckelberg</a> discovered<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon. Effectively, <a href="/wiki/Stueckelberg_action" title="Stueckelberg action">Stueckelberg's model</a> is a limit of the regular Mexican hat Abelian Higgs model, where the vacuum expectation value <span class="texhtml"><i>H</i></span> goes to infinity and the charge of the Higgs field goes to zero in such a way that their product stays fixed. The mass of the Higgs boson is proportional to <span class="texhtml"><i>H</i></span>, so the Higgs boson becomes infinitely massive and decouples, so is not present in the discussion. The vector meson mass, however, is equal to the product <span class="texhtml"><i>eH</i></span>, and stays finite. </p><p>The interpretation is that when a <b>U</b>(1) gauge field does not require quantized charges, it is possible to keep only the angular part of the Higgs oscillations, and discard the radial part. The angular part of the Higgs field <span class="texhtml"><i>θ</i></span> has the following gauge transformation law: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\theta &\rightarrow \theta +e\alpha \,\\A&\rightarrow A+\partial \alpha ~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>θ</mi> <mo>+</mo> <mi>e</mi> <mi>α</mi> <mspace width="thinmathspace" /> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo>+</mo> <mi mathvariant="normal">∂</mi> <mi>α</mi> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta &\rightarrow \theta +e\alpha \,\\A&\rightarrow A+\partial \alpha ~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ca58d16935f973735f4808d9640c7caa7d09b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.725ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\theta &\rightarrow \theta +e\alpha \,\\A&\rightarrow A+\partial \alpha ~.\end{aligned}}}"></span></dd></dl> <p>The gauge covariant derivative for the angle (which is actually gauge invariant) is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\theta =\partial \theta -eAH~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>θ</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>θ</mi> <mo>−</mo> <mi>e</mi> <mi>A</mi> <mi>H</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\theta =\partial \theta -eAH~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc7ca2cfb3ffc43a179462a87cbe8e4e946c9e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.48ex; height:2.343ex;" alt="{\displaystyle D\theta =\partial \theta -eAH~.}"></span></dd></dl> <p>In order to keep <span class="texhtml"><i>θ</i></span> fluctuations finite and nonzero in this limit, <span class="texhtml"><i>θ</i></span> should be rescaled <span class="nowrap">by <span class="texhtml"><i>H</i></span> ,</span> so that its kinetic term in the action stays normalized. The action for the theta field is read off from the Mexican hat action by substituting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi =He^{i\theta /H}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ϕ</mi> <mo>=</mo> <mi>H</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \phi =He^{i\theta /H}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072aaa21913e36a41a12f935cff5f19a231182ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.291ex; height:3.176ex;" alt="{\displaystyle \ \phi =He^{i\theta /H}~.}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(D\theta )^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -HeA)^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -mA)^{2}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>θ</mi> <mo>−</mo> <mi>H</mi> <mi>e</mi> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>θ</mi> <mo>−</mo> <mi>m</mi> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(D\theta )^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -HeA)^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -mA)^{2}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45973177e0a3b1ee7793ac4fa86782fa926842f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:82.209ex; height:5.676ex;" alt="{\displaystyle S=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(D\theta )^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -HeA)^{2}{\bigr ]}=\int {\bigl [}{\tfrac {1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta -mA)^{2}{\bigr ]}}"></span></dd></dl> <p>since <span class="texhtml"><i>eH</i></span> is the gauge boson mass. By making a gauge transformation to set <span class="nowrap"><span class="texhtml"> <i>θ</i> = 0</span> ,</span> the gauge freedom in the action is eliminated, and the action becomes that of a massive vector field: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\tfrac {1}{2}}\int {\bigl [}{\tfrac {1}{2}}F^{2}+m^{2}A^{2}{\bigr ]}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\tfrac {1}{2}}\int {\bigl [}{\tfrac {1}{2}}F^{2}+m^{2}A^{2}{\bigr ]}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f2fa3dd1ede35f0ea8322d87bd41e57d813383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.456ex; height:5.676ex;" alt="{\displaystyle S={\tfrac {1}{2}}\int {\bigl [}{\tfrac {1}{2}}F^{2}+m^{2}A^{2}{\bigr ]}\,.}"></span></dd></dl> <p>To have arbitrarily small charges requires that the <b>U</b>(1) is not the circle of unit complex numbers under multiplication, but the real numbers <b><span class="texhtml">ℝ</span></b> under addition, which is only different in the global topology. Such a <b>U</b>(1) group is non-compact. The field <span class="texhtml"><i>θ</i></span> transforms as an affine representation of the gauge group. Among the allowed gauge groups, only non-compact <b>U</b>(1) admits affine representations, and the <b>U</b>(1) of electromagnetism is experimentally known to be compact, since charge quantization holds to extremely high accuracy. </p><p>The Higgs condensate in this model has infinitesimal charge, so interactions with the Higgs boson do not violate charge conservation. The theory of quantum electrodynamics with a massive photon is still a renormalizable theory, one in which electric charge is still conserved, but <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a> are not allowed. For non-Abelian gauge theory, there is no affine limit, and the Higgs oscillations cannot be too much more massive than the vectors. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Electromagnetic_mass" title="Electromagnetic mass">Electromagnetic mass</a></li> <li><a href="/wiki/Higgs_bundle" title="Higgs bundle">Higgs bundle</a></li> <li><a href="/wiki/Quantum_triviality" title="Quantum triviality">Quantum triviality</a></li> <li><a href="/wiki/Weinberg_angle" title="Weinberg angle">Weinberg angle</a></li> <li><a href="/wiki/Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs equations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Englert's co-author Robert Brout had died in 2011; the Nobel Prize is not usually awarded posthumously.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-PDG-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-PDG_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBernardi,_G.Carena,_M.Junk,_T.2007" class="citation web cs1">Bernardi, G.; 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Acta</i> (in German). <b>11</b>: 225.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Helv.+Phys.+Acta&rft.atitle=Die+Wechselwirkungskr%C3%A4fte+in+der+Elektrodynamik+und+in+der+Feldtheorie+der+Kr%C3%A4fte&rft.volume=11&rft.pages=225&rft.date=1938&rft.aulast=Stueckelberg&rft.aufirst=E.+C.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchumm2004" class="citation book cs1">Schumm, Bruce A. (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/deepdownthingsbr00schu"><i>Deep Down Things</i></a></span>. Baltimore, Maryland: Johns Hopkins University Press. Chapter 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801879715" title="Special:BookSources/9780801879715"><bdi>9780801879715</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deep+Down+Things&rft.place=Baltimore%2C+Maryland&rft.pages=Chapter-9&rft.pub=Johns+Hopkins+University+Press&rft.date=2004&rft.isbn=9780801879715&rft.aulast=Schumm&rft.aufirst=Bruce+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdeepdownthingsbr00schu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKibble2009" class="citation journal cs1">Kibble, Tom W. B. (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.6441">"Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism"</a>. <i>Scholarpedia</i>. <b>4</b> (1): 6441. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SchpJ...4.6441K">2009SchpJ...4.6441K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.6441">10.4249/scholarpedia.6441</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Englert%E2%80%93Brout%E2%80%93Higgs%E2%80%93Guralnik%E2%80%93Hagen%E2%80%93Kibble+mechanism&rft.volume=4&rft.issue=1&rft.pages=6441&rft.date=2009&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.6441&rft_id=info%3Abibcode%2F2009SchpJ...4.6441K&rft.aulast=Kibble&rft.aufirst=Tom+W.+B.&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.6441&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOrgantini2016" class="citation journal cs1 cs1-prop-long-vol">Organtini, Giovanni (2016). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.nuclphysbps.2015.09.463">"The Higgs mechanism for undergraduate students"</a>. <i><a href="/wiki/Nuclear_Physics_(journal)" title="Nuclear Physics (journal)">Nuclear and Particle Physics Proceedings</a></i>. 273–275. <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>: 2572–2574. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016NPPP..273.2572O">2016NPPP..273.2572O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.nuclphysbps.2015.09.463">10.1016/j.nuclphysbps.2015.09.463</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11573%2F1072015">11573/1072015</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+and+Particle+Physics+Proceedings&rft.atitle=The+Higgs+mechanism+for+undergraduate+students&rft.volume=273%E2%80%93275&rft.pages=2572-2574&rft.date=2016&rft_id=info%3Ahdl%2F11573%2F1072015&rft_id=info%3Adoi%2F10.1016%2Fj.nuclphysbps.2015.09.463&rft_id=info%3Abibcode%2F2016NPPP..273.2572O&rft.aulast=Organtini&rft.aufirst=Giovanni&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.nuclphysbps.2015.09.463&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higgs_mechanism&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For a pedagogic introduction to electroweak symmetry breaking with step by step derivations, not found in texts, of many key relations, <i>see</i> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.quantumfieldtheory.info/Electroweak_Sym_breaking.pdf">"Electroweak symmetry breaking"</a> <span class="cs1-format">(PDF)</span>. <i>quantumfieldtheory.info</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=quantumfieldtheory.info&rft.atitle=Electroweak+symmetry+breaking&rft_id=http%3A%2F%2Fwww.quantumfieldtheory.info%2FElectroweak_Sym_breaking.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuralnikHagenKibble1964" class="citation journal cs1">Guralnik, G. S.; Hagen, C. R.; Kibble, T. W. B. (1964). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.585">"Global conservation laws and massless particles"</a>. <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>13</b> (20): 585–587. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhRvL..13..585G">1964PhRvL..13..585G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.585">10.1103/PhysRevLett.13.585</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Global+conservation+laws+and+massless+particles&rft.volume=13&rft.issue=20&rft.pages=585-587&rft.date=1964&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.13.585&rft_id=info%3Abibcode%2F1964PhRvL..13..585G&rft.aulast=Guralnik&rft.aufirst=G.+S.&rft.au=Hagen%2C+C.+R.&rft.au=Kibble%2C+T.+W.+B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.13.585&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_D.1999" class="citation arxiv cs1">Mark D., Roberts (1999). "A generalized Higgs model". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9904080">hep-th/9904080</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+generalized+Higgs+model&rft.date=1999&rft_id=info%3Aarxiv%2Fhep-th%2F9904080&rft.aulast=Mark+D.&rft.aufirst=Roberts&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFermi2010" class="citation audio-visual cs1">Fermi, Fred (2010). <a rel="nofollow" class="external text" href="https://www.youtube.com/view_play_list?p=BDA16F52CA3C9B1D"><i>Sakurai Prize – All Events</i></a> (video) – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sakurai+Prize+%E2%80%93+All+Events&rft.date=2010&rft.aulast=Fermi&rft.aufirst=Fred&rft_id=https%3A%2F%2Fwww.youtube.com%2Fview_play_list%3Fp%3DBDA16F52CA3C9B1D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg_(University_of_Texas_at_Austin)2008" class="citation web cs1">Weinberg (University of Texas at Austin), Steven (21 Jan 2008). <a rel="nofollow" class="external text" href="http://cerncourier.com/cws/article/cern/32522">"From BCS to the LHC"</a>. <i>CERN Courier</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CERN+Courier&rft.atitle=From+BCS+to+the+LHC&rft.date=2008-01-21&rft.aulast=Weinberg+%28University+of+Texas+at+Austin%29&rft.aufirst=Steven&rft_id=http%3A%2F%2Fcerncourier.com%2Fcws%2Farticle%2Fcern%2F32522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg2009" class="citation audio-visual cs1"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (2009-06-11). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Zl4W3DYTIKw"><i>Higgs, dark matter and supersymmetry: What the Large Hadron Collider will tell us</i></a> (video). Zl4W3DYTIKw – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Higgs%2C+dark+matter+and+supersymmetry%3A+What+the+Large+Hadron+Collider+will+tell+us&rft.date=2009-06-11&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DZl4W3DYTIKw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuralnik" class="citation audio-visual cs1">Guralnik, Gerry. <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=WLZ78gwWQI0"><i>Guralnik speaks at Brown University about the 1964 PRL papers</i></a> (video). WLZ78gwWQI0 – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Guralnik+speaks+at+Brown+University+about+the+1964+PRL+papers&rft.aulast=Guralnik&rft.aufirst=Gerry&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DWLZ78gwWQI0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuralnik,_Gerald2013" class="citation journal cs1">Guralnik, Gerald (March 2013). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131015095448/http://www.sps.ch/en/articles/milestones_in_physics/heretical_ideas_that_provided_the_cornerstone_for_the_standard_model_of_particle_physics_1/">"Heretical ideas that provided the cornerstone for the standard model of particle physics"</a>. <i>SPG Mitteilungen</i> (39): 14. Archived from <a rel="nofollow" class="external text" href="http://www.sps.ch/en/articles/milestones_in_physics/heretical_ideas_that_provided_the_cornerstone_for_the_standard_model_of_particle_physics_1/">the original</a> on 2013-10-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-05-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SPG+Mitteilungen&rft.atitle=Heretical+ideas+that+provided+the+cornerstone+for+the+standard+model+of+particle+physics&rft.issue=39&rft.pages=14&rft.date=2013-03&rft.au=Guralnik%2C+Gerald&rft_id=http%3A%2F%2Fwww.sps.ch%2Fen%2Farticles%2Fmilestones_in_physics%2Fheretical_ideas_that_provided_the_cornerstone_for_the_standard_model_of_particle_physics_1%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080416064136/http://www.pas.rochester.edu/urpas/news/Hagen_030708">"Steven Weinberg praises teams for Higgs boson theory"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.pas.rochester.edu/urpas/news/Hagen_030708">the original</a> on 2008-04-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Steven+Weinberg+praises+teams+for+Higgs+boson+theory&rft_id=http%3A%2F%2Fwww.pas.rochester.edu%2Furpas%2Fnews%2FHagen_030708&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="http://prl.aps.org/50years/milestones#1964">"50th anniversary milestone papers"</a>. <i>Physical Review Letters</i>. 2014-02-12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=50th+anniversary+milestone+papers&rft.date=2014-02-12&rft_id=http%3A%2F%2Fprl.aps.org%2F50years%2Fmilestones%231964&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_13-6-2008-12-42-20?newsid=38514">"PRL 50th anniversary milestone papers"</a>. Imperial College London. 13 June 2008.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PRL+50th+anniversary+milestone+papers&rft.pub=Imperial+College+London&rft.date=2008-06-13&rft_id=http%3A%2F%2Fwww3.imperial.ac.uk%2Fnewsandeventspggrp%2Fimperialcollege%2Fnewssummary%2Fnews_13-6-2008-12-42-20%3Fnewsid%3D38514&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKibble2009" class="citation journal cs1">Kibble, Tom (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.6441">"Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism"</a>. <i><a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a></i>. <b>4</b> (1): 6441. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SchpJ...4.6441K">2009SchpJ...4.6441K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.6441">10.4249/scholarpedia.6441</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Englert%E2%80%93Brout%E2%80%93Higgs%E2%80%93Guralnik%E2%80%93Hagen%E2%80%93Kibble+mechanism&rft.volume=4&rft.issue=1&rft.pages=6441&rft.date=2009&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.6441&rft_id=info%3Abibcode%2F2009SchpJ...4.6441K&rft.aulast=Kibble&rft.aufirst=Tom&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.6441&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKibble2009" class="citation journal cs1">Kibble, Tom (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.8741">"Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism (history)"</a>. <i><a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a></i>. <b>4</b> (1): 8741. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SchpJ...4.8741K">2009SchpJ...4.8741K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.8741">10.4249/scholarpedia.8741</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Englert%E2%80%93Brout%E2%80%93Higgs%E2%80%93Guralnik%E2%80%93Hagen%E2%80%93Kibble+mechanism+%28history%29&rft.volume=4&rft.issue=1&rft.pages=8741&rft.date=2009&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.8741&rft_id=info%3Abibcode%2F2009SchpJ...4.8741K&rft.aulast=Kibble&rft.aufirst=Tom&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.8741&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://apps3.aps.org/aps/meetings/april10/roser.pdf">"The hunt for the Higgs at the Tevatron"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+hunt+for+the+Higgs+at+the+Tevatron&rft_id=http%3A%2F%2Fapps3.aps.org%2Faps%2Fmeetings%2Fapril10%2Froser.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriest" class="citation audio-visual cs1">Griest, Kim. <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Y-vKh_jKX7Q"><i>The Mystery of Empty Space – A lecture with UCSD physicist Kim Griest (43 minutes)</i></a> (video). University of California Television. Y-vKh_jKX7Q – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Mystery+of+Empty+Space+%E2%80%93+A+lecture+with+UCSD+physicist+Kim+Griest+%2843+minutes%29&rft.pub=University+of+California+Television&rft.aulast=Griest&rft.aufirst=Kim&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DY-vKh_jKX7Q&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHiggs+mechanism" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Standard_model_of_physics" title="Template:Standard model of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Standard_model_of_physics" title="Template talk:Standard model of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Standard_model_of_physics" title="Special:EditPage/Template:Standard model of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Standard_Model" style="font-size:114%;margin:0 4em"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a> <ul><li><a href="/wiki/Fermion" title="Fermion">Fermions</a></li> <li><a href="/wiki/Gauge_boson" title="Gauge boson">Gauge boson</a></li> <li><a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a></li></ul></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Strong_interaction" title="Strong interaction">Strong interaction</a> <ul><li><a href="/wiki/Color_charge" title="Color charge">Color charge</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a> <ul><li><a href="/wiki/Weak_interaction" title="Weak interaction">Weak interaction</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi's interaction</a></li> <li><a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a></li> <li><a href="/wiki/Weak_isospin" title="Weak isospin">Weak isospin</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/150px-Standard_Model_of_Elementary_Particles.svg.png" decoding="async" width="150" height="144" class="mw-file-element" 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formulation of the Standard Model">Mathematical formulation of the Standard Model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the<br />Standard Model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Evidence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li> <li><a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li> <li><a href="/wiki/Cosmological_constant" title="Cosmological constant">Cosmological constant</a> <ul><li><a href="/wiki/Cosmological_constant_problem" title="Cosmological constant problem">problem</a></li></ul></li> <li><a href="/wiki/CP_violation" title="CP violation">Strong CP problem</a></li> <li><a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Split_supersymmetry" title="Split supersymmetry">Split supersymmetry</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a 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<li><a href="/wiki/India-based_Neutrino_Observatory" title="India-based Neutrino Observatory">INO</a></li> <li><a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">LHC</a></li> <li><a href="/wiki/Sudbury_Neutrino_Observatory" title="Sudbury Neutrino Observatory">SNO</a></li> <li><a href="/wiki/Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li> <li><a href="/wiki/Tevatron" title="Tevatron">Tevatron</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 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Higgs mechanism - Wikipedia