Cabibbo–Kobayashi–Maskawa matrix

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class="vector-toc-list"> <li id="toc-Predecessor_–_the_Cabibbo_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Predecessor_–_the_Cabibbo_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Predecessor – the Cabibbo matrix</span> </div> </a> <ul id="toc-Predecessor_–_the_Cabibbo_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CKM_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CKM_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>CKM matrix</span> </div> </a> <ul id="toc-CKM_matrix-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_case_construction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#General_case_construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>General case construction</span> </div> </a> <button aria-controls="toc-General_case_construction-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 General case construction subsection</span> </button> <ul id="toc-General_case_construction-sublist" class="vector-toc-list"> <li id="toc-N_=_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#N_=_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span><span><span><span>N</span></span></span> = 2</span> </div> </a> <ul id="toc-N_=_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-N_=_3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#N_=_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span><span><span><span>N</span></span></span> = 3</span> </div> </a> <ul id="toc-N_=_3-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Observations_and_predictions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Observations_and_predictions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Observations and predictions</span> </div> </a> <ul id="toc-Observations_and_predictions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_universality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Weak_universality"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Weak universality</span> </div> </a> <ul id="toc-Weak_universality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_unitarity_triangles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_unitarity_triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>The unitarity triangles</span> </div> </a> <ul id="toc-The_unitarity_triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parameterizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parameterizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Parameterizations</span> </div> </a> <button aria-controls="toc-Parameterizations-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 Parameterizations subsection</span> </button> <ul id="toc-Parameterizations-sublist" class="vector-toc-list"> <li id="toc-KM_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#KM_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>KM parameters</span> </div> </a> <ul id="toc-KM_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-"Standard"_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#"Standard"_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>"Standard" parameters</span> </div> </a> <ul id="toc-"Standard"_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wolfenstein_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wolfenstein_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Wolfenstein parameters</span> </div> </a> <ul id="toc-Wolfenstein_parameters-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nobel_Prize" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nobel_Prize"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Nobel 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class="vector-toc-numb">10</span> <span>Further reading and external links</span> </div> </a> <ul id="toc-Further_reading_and_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" 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Available in 16 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-16" 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">16 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/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D9%83%D8%A7%D8%A8%D9%8A%D8%A8%D9%88-%D9%83%D9%88%D8%A8%D8%A7%D9%8A%D8%A7%D8%B4%D9%8A-%D9%85%D8%A7%D8%B3%D9%83%D8%A7%D9%88%D8%A7" title="مصفوفة كابيبو-كوباياشي-ماسكاوا – Arabic" lang="ar" hreflang="ar" data-title="مصفوفة كابيبو-كوباياشي-ماسكاوا" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/CKM-%D0%BC%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="CKM-матрыца – Belarusian" lang="be" hreflang="be" data-title="CKM-матрыца" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/CKM-Matrix" title="CKM-Matrix – German" lang="de" hreflang="de" data-title="CKM-Matrix" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_de_Cabibbo-Kobayashi-Maskawa" title="Matriz de Cabibbo-Kobayashi-Maskawa – Spanish" lang="es" hreflang="es" data-title="Matriz de Cabibbo-Kobayashi-Maskawa" 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/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%D8%B3%DB%8C%E2%80%8C%DA%A9%DB%8C%E2%80%8C%D8%A7%D9%85" title="ماتریس سی‌کی‌ام – Persian" lang="fa" hreflang="fa" data-title="ماتریس سی‌کی‌ام" 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/Matrice_CKM" title="Matrice CKM – French" lang="fr" hreflang="fr" data-title="Matrice CKM" 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/%EC%BF%BC%ED%81%AC_%EC%84%9E%EC%9E%84" title="쿼크 섞임 – Korean" lang="ko" hreflang="ko" data-title="쿼크 섞임" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice_CKM" title="Matrice CKM – Italian" lang="it" hreflang="it" data-title="Matrice CKM" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Cabibbo%E2%80%93Kobajasi%E2%80%93Maszkava-m%C3%A1trix" title="Cabibbo–Kobajasi–Maszkava-mátrix – Hungarian" lang="hu" hreflang="hu" data-title="Cabibbo–Kobajasi–Maszkava-mátrix" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%83%93%E3%83%9C%E3%83%BB%E5%B0%8F%E6%9E%97%E3%83%BB%E7%9B%8A%E5%B7%9D%E8%A1%8C%E5%88%97" title="カビボ・小林・益川行列 – Japanese" lang="ja" hreflang="ja" data-title="カビボ・小林・益川行列" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz_Cabibbo-Kobayashiego-Maskawy" title="Macierz Cabibbo-Kobayashiego-Maskawy – Polish" lang="pl" hreflang="pl" data-title="Macierz Cabibbo-Kobayashiego-Maskawy" 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/Matriz_CKM" title="Matriz CKM – Portuguese" lang="pt" hreflang="pt" data-title="Matriz CKM" 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/CKM-%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="CKM-матрица – Russian" lang="ru" hreflang="ru" data-title="CKM-матрица" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/CKM_matrisi" title="CKM matrisi – Turkish" lang="tr" hreflang="tr" data-title="CKM matrisi" 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/CKM-%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F" title="CKM-матриця – Ukrainian" lang="uk" hreflang="uk" data-title="CKM-матриця" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%A1%E6%AF%94%E5%8D%9A-%E5%B0%8F%E6%9E%97-%E7%9B%8A%E5%B7%9D%E7%9F%A9%E9%98%B5" title="卡比博-小林-益川矩阵 – Chinese" lang="zh" hreflang="zh" data-title="卡比博-小林-益川矩阵" 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/Q253728#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div 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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Unitary matrix containing information on the weak interaction</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks" style="width:16em;"><tbody><tr><th class="sidebar-title" style="background:#ddddff;"><a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">Flavour</a> in<br /><a href="/wiki/Particle_physics" title="Particle physics">particle physics</a></th></tr><tr><th class="sidebar-heading" style="background: #efefef;"> Flavour <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a></th></tr><tr><td class="sidebar-content" style="text-align:left;"> <ul><li><a href="/wiki/Isospin" title="Isospin">Isospin</a>: <b>I</b> or <i>I</i><sub>3</sub></li> <li><a href="/wiki/Charm_(quantum_number)" title="Charm (quantum number)">Charm</a>: <i>C</i></li> <li><a href="/wiki/Strangeness" title="Strangeness">Strangeness</a>: <i>S</i></li> <li><a href="/wiki/Topness" title="Topness">Topness</a>: <i>T</i></li> <li><a href="/wiki/Bottomness" title="Bottomness">Bottomness</a>: <i>B</i>′</li></ul></td> </tr><tr><th class="sidebar-heading" style="background: #efefef;"> Related quantum numbers</th></tr><tr><td class="sidebar-content" style="text-align:left;"> <ul><li><a href="/wiki/Baryon_number" title="Baryon number">Baryon number</a>: <i>B</i></li> <li><a href="/wiki/Lepton_number" title="Lepton number">Lepton number</a>: <i>L</i></li> <li><a href="/wiki/Weak_isospin" title="Weak isospin">Weak isospin</a>: <b>T</b> or <i>T</i><sub>3</sub></li> <li><a href="/wiki/Electric_charge" title="Electric charge">Electric charge</a>: <i>Q</i></li> <li><a href="/wiki/X_(charge)" title="X (charge)">X-charge</a>: <i>X</i></li></ul></td> </tr><tr><th class="sidebar-heading" style="background: #efefef;"> Combinations</th></tr><tr><td class="sidebar-content" style="text-align:left;"> <ul><li><a href="/wiki/Hypercharge" title="Hypercharge">Hypercharge</a>: <i>Y</i> <ul><li><i>Y</i> = (<i>B</i> + <i>S</i> + <i>C</i> + <i>B</i>′ + <i>T</i>)</li> <li><i>Y</i> = 2 (<i>Q</i> − <i>I</i><sub>3</sub>)</li></ul></li> <li><a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a>: <i>Y</i><sub>W</sub> <ul><li><i>Y</i><sub>W</sub> = 2 (<i>Q</i> − <i>T</i><sub>3</sub>)</li> <li><i>X</i> + 2<i>Y</i><sub>W</sub> = 5 (<a href="/wiki/B_%E2%88%92_L" title="B − L"><i>B</i> − <i>L</i></a>)</li></ul></li></ul></td> </tr><tr><th class="sidebar-heading" style="background: #efefef;"> Flavour mixing</th></tr><tr><td class="sidebar-content" style="text-align:left;"> <ul><li><a class="mw-selflink selflink">CKM matrix</a></li> <li><a href="/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">PMNS matrix</a></li> <li><a href="/wiki/Quark%E2%80%93lepton_complementarity" title="Quark–lepton complementarity">Flavour complementarity</a></li></ul></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 .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Flavour_quantum_numbers" title="Template:Flavour quantum numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Flavour_quantum_numbers" title="Template talk:Flavour quantum numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Flavour_quantum_numbers" title="Special:EditPage/Template:Flavour quantum numbers"><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>Cabibbo–Kobayashi–Maskawa matrix</b>, <b>CKM matrix</b>, <b>quark mixing matrix</b>, or <b>KM matrix</b> is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> which contains information on the strength of the <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavour</a>-changing <a href="/wiki/Weak_interaction" title="Weak interaction">weak interaction</a>. Technically, it specifies the mismatch of <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a> of <a href="/wiki/Quark" title="Quark">quarks</a> when they propagate freely and when they take part in the <a href="/wiki/Weak_interaction" title="Weak interaction">weak interactions</a>. It is important in the understanding of <a href="/wiki/CP_violation" title="CP violation">CP violation</a>. This matrix was introduced for three generations of quarks by <a href="/wiki/Makoto_Kobayashi_(physicist)" class="mw-redirect" title="Makoto Kobayashi (physicist)">Makoto Kobayashi</a> and <a href="/wiki/Toshihide_Maskawa" title="Toshihide Maskawa">Toshihide Maskawa</a>, adding one <a href="/wiki/Generation_(particle_physics)" title="Generation (particle physics)">generation</a> to the matrix previously introduced by <a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Nicola Cabibbo</a>. This matrix is also an extension of the <a href="/wiki/GIM_mechanism" title="GIM mechanism">GIM mechanism</a>, which only includes two of the three current families of quarks. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_matrix">The matrix</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=1" title="Edit section: The matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Predecessor_–_the_Cabibbo_matrix"><span id="Predecessor_.E2.80.93_the_Cabibbo_matrix"></span>Predecessor – the Cabibbo matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=2" title="Edit section: Predecessor – the Cabibbo matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cabibbo_angle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Cabibbo_angle.svg/270px-Cabibbo_angle.svg.png" decoding="async" width="270" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Cabibbo_angle.svg/405px-Cabibbo_angle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Cabibbo_angle.svg/540px-Cabibbo_angle.svg.png 2x" data-file-width="596" data-file-height="570" /></a><figcaption>The Cabibbo angle represents the rotation of the mass eigenstate vector space formed by the mass eigenstates <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\rangle ,\,|s\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |d\rangle ,\,|s\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46cf22289821c1feea028f12980d964c7a631a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.83ex; height:2.843ex;" alt="{\displaystyle |d\rangle ,\,|s\rangle }"></span> into the weak eigenstate vector space formed by the weak eigenstates <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'\rangle \,,~|\,s'\rangle ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |d'\rangle \,,~|\,s'\rangle ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5457e5f245d9e430a8b2ce7c3f447174d68b4fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.397ex; height:3.009ex;" alt="{\displaystyle |d'\rangle \,,~|\,s'\rangle ~.}"></span> <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">θ</span><sub>c</sub> = 13.02° .</span> </figcaption></figure> <p>In 1963, <a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Nicola Cabibbo</a> introduced the <b>Cabibbo angle</b> (<span class="texhtml mvar" style="font-style:italic;">θ</span><sub>c</sub>) to preserve the universality of the <a href="/wiki/Weak_interaction" title="Weak interaction">weak interaction</a>.<sup id="cite_ref-Cabibbo_1-0" class="reference"><a href="#cite_note-Cabibbo-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Cabibbo was inspired by previous work by <a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Murray Gell-Mann</a> and Maurice Lévy,<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> on the effectively rotated nonstrange and strange vector and axial weak currents, which he references.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In light of current concepts (quarks had not yet been proposed), the Cabibbo angle is related to the relative probability that <a href="/wiki/Down_quark" title="Down quark">down</a> and <a href="/wiki/Strange_quark" title="Strange quark">strange quarks</a> decay into <a href="/wiki/Up_quark" title="Up quark">up quarks</a> ( |<span class="texhtml mvar" style="font-style:italic;">V</span><sub>ud</sub>|<sup>2</sup>   and   |<span class="texhtml mvar" style="font-style:italic;">V</span><sub>us</sub>|<sup>2</sup> , respectively). In particle physics terminology, the object that couples to the up quark via charged-current weak interaction is a superposition of down-type quarks, here denoted by <span class="texhtml mvar" style="font-style:italic;">d′</span>.<sup id="cite_ref-Hughes_4-0" class="reference"><a href="#cite_note-Hughes-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Mathematically this 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'=V_{\mathrm {ud} }\;d~~+~~V_{\mathrm {us} }\;s~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>s</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'=V_{\mathrm {ud} }\;d~~+~~V_{\mathrm {us} }\;s~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6515e9937f18939e71078659c66b5f0fd2e7a1ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.553ex; height:2.843ex;" alt="{\displaystyle d'=V_{\mathrm {ud} }\;d~~+~~V_{\mathrm {us} }\;s~,}"></span></dd></dl> <p>or using the Cabibbo angle: </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'=\cos \theta _{\mathrm {c} }\;d~~+~~\sin \theta _{\mathrm {c} }\;s~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>s</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'=\cos \theta _{\mathrm {c} }\;d~~+~~\sin \theta _{\mathrm {c} }\;s~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab43d69b8cc685038f5e377f1c4bdcbc71fe563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.222ex; height:2.843ex;" alt="{\displaystyle d'=\cos \theta _{\mathrm {c} }\;d~~+~~\sin \theta _{\mathrm {c} }\;s~.}"></span></dd></dl> <p>Using the currently accepted values for   |<span class="texhtml mvar" style="font-style:italic;">V</span><sub>ud</sub>|   and   |<span class="texhtml mvar" style="font-style:italic;">V</span><sub>us</sub>|   (see below), the Cabibbo angle can be calculated using </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 \tan \theta _{\mathrm {c} }={\frac {\,|V_{\mathrm {us} }|\,}{|V_{\mathrm {ud} }|}}={\frac {0.22534}{0.97427}}\quad \Rightarrow \quad \theta _{\mathrm {c} }=~13.02^{\circ }~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.22534</mn> <mn>0.97427</mn> </mfrac> </mrow> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mtext> </mtext> <msup> <mn>13.02</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta _{\mathrm {c} }={\frac {\,|V_{\mathrm {us} }|\,}{|V_{\mathrm {ud} }|}}={\frac {0.22534}{0.97427}}\quad \Rightarrow \quad \theta _{\mathrm {c} }=~13.02^{\circ }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d7841e8813d259e9f7906b1109fb5b88ebd612b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.077ex; height:6.509ex;" alt="{\displaystyle \tan \theta _{\mathrm {c} }={\frac {\,|V_{\mathrm {us} }|\,}{|V_{\mathrm {ud} }|}}={\frac {0.22534}{0.97427}}\quad \Rightarrow \quad \theta _{\mathrm {c} }=~13.02^{\circ }~.}"></span></dd></dl> <p>When the <a href="/wiki/Charm_quark" title="Charm quark">charm quark</a> was discovered in 1974, it was noticed that the down and strange quark could transition into either the up or charm quark, leading to two sets of equations: </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'=V_{\mathrm {ud} }\;d~~+~~V_{\mathrm {us} }\;s~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>s</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'=V_{\mathrm {ud} }\;d~~+~~V_{\mathrm {us} }\;s~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6515e9937f18939e71078659c66b5f0fd2e7a1ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.553ex; height:2.843ex;" alt="{\displaystyle d'=V_{\mathrm {ud} }\;d~~+~~V_{\mathrm {us} }\;s~,}"></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 s'=V_{\mathrm {cd} }\;d~~+~~V_{\mathrm {cs} }\;s~;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mi>s</mi> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s'=V_{\mathrm {cd} }\;d~~+~~V_{\mathrm {cs} }\;s~;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d9e05775c86e216b5d5726e61c0bd3b09583dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.058ex; height:2.843ex;" alt="{\displaystyle s'=V_{\mathrm {cd} }\;d~~+~~V_{\mathrm {cs} }\;s~;}"></span></dd></dl> <p>or using the Cabibbo angle: </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'=~~~\cos {\theta _{\mathrm {c} }}\;d~~+~~\sin {\theta _{\mathrm {c} }}\;s~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <mspace width="thickmathspace" /> <mi>s</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'=~~~\cos {\theta _{\mathrm {c} }}\;d~~+~~\sin {\theta _{\mathrm {c} }}\;s~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbed0ee9c8f7d3ba4a14b9c7253325d157f4100c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.351ex; height:2.843ex;" alt="{\displaystyle d'=~~~\cos {\theta _{\mathrm {c} }}\;d~~+~~\sin {\theta _{\mathrm {c} }}\;s~,}"></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 s'=-\sin {\theta _{\mathrm {c} }}\;d~~+~~\cos {\theta _{\mathrm {c} }}\;s~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <mspace width="thickmathspace" /> <mi>s</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s'=-\sin {\theta _{\mathrm {c} }}\;d~~+~~\cos {\theta _{\mathrm {c} }}\;s~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abfde648547aec98ce5b48bb0c1410af9bca06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.289ex; height:2.843ex;" alt="{\displaystyle s'=-\sin {\theta _{\mathrm {c} }}\;d~~+~~\cos {\theta _{\mathrm {c} }}\;s~.}"></span></dd></dl> <p>This can also be written in <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix notation</a> 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 {\begin{bmatrix}d'\\s'\end{bmatrix}}={\begin{bmatrix}V_{\mathrm {ud} }&V_{\mathrm {us} }\\V_{cd}&V_{cs}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>d</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>s</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}d'\\s'\end{bmatrix}}={\begin{bmatrix}V_{\mathrm {ud} }&V_{\mathrm {us} }\\V_{cd}&V_{cs}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa27cc86cfa5846f20ebc1431413bdc2c8601fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.951ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}d'\\s'\end{bmatrix}}={\begin{bmatrix}V_{\mathrm {ud} }&V_{\mathrm {us} }\\V_{cd}&V_{cs}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}}~,}"></span></dd></dl> <p>or using the Cabibbo angle </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{bmatrix}d'\\s'\end{bmatrix}}={\begin{bmatrix}~~\cos {\theta _{\mathrm {c} }}&\sin {\theta _{\mathrm {c} }}\\-\sin {\theta _{\mathrm {c} }}&\cos {\theta _{\mathrm {c} }}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>d</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>s</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mtext> </mtext> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}d'\\s'\end{bmatrix}}={\begin{bmatrix}~~\cos {\theta _{\mathrm {c} }}&\sin {\theta _{\mathrm {c} }}\\-\sin {\theta _{\mathrm {c} }}&\cos {\theta _{\mathrm {c} }}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09455838ade05427fdd640b4b831f2a12f9c7339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.428ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}d'\\s'\end{bmatrix}}={\begin{bmatrix}~~\cos {\theta _{\mathrm {c} }}&\sin {\theta _{\mathrm {c} }}\\-\sin {\theta _{\mathrm {c} }}&\cos {\theta _{\mathrm {c} }}\\\end{bmatrix}}{\begin{bmatrix}d\\s\end{bmatrix}}~,}"></span></dd></dl> <p>where the various |<span class="texhtml mvar" style="font-style:italic;">V<sub>ij</sub></span>|<sup>2</sup> represent the probability that the quark of flavor <span class="texhtml mvar" style="font-style:italic;">j</span> decays into a quark of flavor <span class="texhtml mvar" style="font-style:italic;">i</span>. This 2×2 <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a> is called the "Cabibbo matrix", and was subsequently expanded to the 3×3 CKM matrix. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quark_weak_interactions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Quark_weak_interactions.svg/270px-Quark_weak_interactions.svg.png" decoding="async" width="270" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Quark_weak_interactions.svg/405px-Quark_weak_interactions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Quark_weak_interactions.svg/540px-Quark_weak_interactions.svg.png 2x" data-file-width="271" data-file-height="180" /></a><figcaption>A pictorial representation of the six quarks' decay modes, with mass increasing from left to right.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="CKM_matrix">CKM matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=3" title="Edit section: CKM matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Weak_Decay_(flipped).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Weak_Decay_%28flipped%29.svg/270px-Weak_Decay_%28flipped%29.svg.png" decoding="async" width="270" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Weak_Decay_%28flipped%29.svg/405px-Weak_Decay_%28flipped%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Weak_Decay_%28flipped%29.svg/540px-Weak_Decay_%28flipped%29.svg.png 2x" data-file-width="659" data-file-height="692" /></a><figcaption>A diagram depicting the decay routes due to the charged weak interaction and some indication of their likelihood. The intensity of the lines is given by the CKM parameters</figcaption></figure> <p>In 1973, observing that <a href="/wiki/CP-violation" class="mw-redirect" title="CP-violation">CP-violation</a> could not be explained in a four-quark model, Kobayashi and Maskawa generalized the Cabibbo matrix into the Cabibbo–Kobayashi–Maskawa matrix (or CKM matrix) to keep track of the weak decays of three generations of quarks:<sup id="cite_ref-KM_5-0" class="reference"><a href="#cite_note-KM-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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{bmatrix}d'\\s'\\b'\end{bmatrix}}={\begin{bmatrix}V_{\mathrm {ud} }&V_{\mathrm {us} }&V_{\mathrm {ub} }\\V_{\mathrm {cd} }&V_{\mathrm {cs} }&V_{\mathrm {cb} }\\V_{\mathrm {td} }&V_{\mathrm {ts} }&V_{\mathrm {tb} }\end{bmatrix}}{\begin{bmatrix}d\\s\\b\end{bmatrix}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>d</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>s</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mtd> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}d'\\s'\\b'\end{bmatrix}}={\begin{bmatrix}V_{\mathrm {ud} }&V_{\mathrm {us} }&V_{\mathrm {ub} }\\V_{\mathrm {cd} }&V_{\mathrm {cs} }&V_{\mathrm {cb} }\\V_{\mathrm {td} }&V_{\mathrm {ts} }&V_{\mathrm {tb} }\end{bmatrix}}{\begin{bmatrix}d\\s\\b\end{bmatrix}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aea14f3ed3ac02f5fcccaa0b900a2a7c89ab2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:33.626ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}d'\\s'\\b'\end{bmatrix}}={\begin{bmatrix}V_{\mathrm {ud} }&V_{\mathrm {us} }&V_{\mathrm {ub} }\\V_{\mathrm {cd} }&V_{\mathrm {cs} }&V_{\mathrm {cb} }\\V_{\mathrm {td} }&V_{\mathrm {ts} }&V_{\mathrm {tb} }\end{bmatrix}}{\begin{bmatrix}d\\s\\b\end{bmatrix}}~.}"></span></dd></dl> <p>On the left are the <a href="/wiki/Weak_interaction" title="Weak interaction">weak interaction</a> doublet partners of down-type quarks, and on the right is the CKM matrix, along with a vector of mass eigenstates of down-type quarks. The CKM matrix describes the probability of a transition from one flavour <span class="texhtml mvar" style="font-style:italic;">j</span> quark to another flavour <span class="texhtml mvar" style="font-style:italic;">i</span> quark. These transitions are proportional to |<span class="texhtml mvar" style="font-style:italic;">V<sub>ij</sub></span>|<sup>2</sup>. </p><p>As of 2023, the best determination of the individual <a href="/wiki/Absolute_value" title="Absolute value">magnitudes</a> of the CKM matrix elements was:<sup id="cite_ref-PDG2023_6-0" class="reference"><a href="#cite_note-PDG2023-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </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{bmatrix}|V_{ud}|&|V_{us}|&|V_{ub}|\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\|V_{td}|&|V_{ts}|&|V_{tb}|\end{bmatrix}}={\begin{bmatrix}.97435\pm 0.00016&0.22500\pm 0.00067&0.00369\pm 0.00011\\0.22486\pm 0.00067&0.97349\pm 0.00016&0.04182_{-0.00074}^{+0.00085}\\0.00857_{-0.00018}^{+0.00020}&0.04110_{-0.00072}^{+0.00083}&0.999118_{-0.000036}^{+0.000031}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>.97435</mn> <mo>±</mo> <mn>0.00016</mn> </mtd> <mtd> <mn>0.22500</mn> <mo>±</mo> <mn>0.00067</mn> </mtd> <mtd> <mn>0.00369</mn> <mo>±</mo> <mn>0.00011</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.22486</mn> <mo>±</mo> <mn>0.00067</mn> </mtd> <mtd> <mn>0.97349</mn> <mo>±</mo> <mn>0.00016</mn> </mtd> <mtd> <msubsup> <mn>0.04182</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.00074</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>0.00085</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mn>0.00857</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.00018</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>0.00020</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mn>0.04110</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.00072</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>0.00083</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mn>0.999118</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.000036</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>0.000031</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}|V_{ud}|&|V_{us}|&|V_{ub}|\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\|V_{td}|&|V_{ts}|&|V_{tb}|\end{bmatrix}}={\begin{bmatrix}.97435\pm 0.00016&0.22500\pm 0.00067&0.00369\pm 0.00011\\0.22486\pm 0.00067&0.97349\pm 0.00016&0.04182_{-0.00074}^{+0.00085}\\0.00857_{-0.00018}^{+0.00020}&0.04110_{-0.00072}^{+0.00083}&0.999118_{-0.000036}^{+0.000031}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65608ff974ba6f24905c4536d64e502561ecda15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:88.791ex; height:10.509ex;" alt="{\displaystyle {\begin{bmatrix}|V_{ud}|&|V_{us}|&|V_{ub}|\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\|V_{td}|&|V_{ts}|&|V_{tb}|\end{bmatrix}}={\begin{bmatrix}.97435\pm 0.00016&0.22500\pm 0.00067&0.00369\pm 0.00011\\0.22486\pm 0.00067&0.97349\pm 0.00016&0.04182_{-0.00074}^{+0.00085}\\0.00857_{-0.00018}^{+0.00020}&0.04110_{-0.00072}^{+0.00083}&0.999118_{-0.000036}^{+0.000031}\end{bmatrix}}.}"></span></dd></dl> <p>Using those values, one can check the unitarity of the CKM matrix. In particular, we find that the first-row matrix elements give: <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_{\mathrm {ud} }|^{2}+|V_{\mathrm {us} }|^{2}+|V_{\mathrm {ub} }|^{2}=.999997\pm .0007~;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>.999997</mn> <mo>±</mo> <mn>.0007</mn> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V_{\mathrm {ud} }|^{2}+|V_{\mathrm {us} }|^{2}+|V_{\mathrm {ub} }|^{2}=.999997\pm .0007~;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba92472e05d0b82025daea877eab89520fb7e69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.789ex; height:3.343ex;" alt="{\displaystyle |V_{\mathrm {ud} }|^{2}+|V_{\mathrm {us} }|^{2}+|V_{\mathrm {ub} }|^{2}=.999997\pm .0007~;}"></span> </p><p>Making the experimental results in line with the theoretical value of 1. </p><p>The choice of usage of down-type quarks in the definition is a convention, and does not represent a physically preferred asymmetry between up-type and down-type quarks. Other conventions are equally valid: The mass eigenstates <span class="texhtml">u</span>, <span class="texhtml">c</span>, and <span class="texhtml">t</span> of the up-type quarks can equivalently define the matrix in terms of <i>their</i> weak interaction partners <span class="texhtml">u′</span>, <span class="texhtml">c′</span>, and <span class="texhtml">t′</span>. Since the CKM matrix is unitary, its inverse is the same as its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>, which the alternate choices use; it appears as the same matrix, in a slightly altered form. </p> <div class="mw-heading mw-heading2"><h2 id="General_case_construction">General case construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=4" title="Edit section: General case construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To generalize the matrix, count the number of physically important parameters in this matrix <span class="texhtml mvar" style="font-style:italic;">V</span> which appear in experiments. If there are <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> generations of quarks (2<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavours</a>) then </p> <ul><li>An <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> × <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> unitary matrix (that is, a matrix <span class="texhtml mvar" style="font-style:italic;">V</span> such that <span class="texhtml mvar" style="font-style:italic;">V<sup>†</sup>V = I</span>, where <span class="texhtml mvar" style="font-style:italic;">V<sup>†</sup></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <span class="texhtml mvar" style="font-style:italic;">V</span> and <span class="texhtml mvar" style="font-style:italic;">I</span> is the identity matrix) requires <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span><sup>2</sup> real parameters to be specified.</li> <li>2<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> − 1 of these parameters are not physically significant, because one phase can be absorbed into each quark field (both of the mass eigenstates, and of the weak eigenstates), but the matrix is independent of a common phase. Hence, the total number of free variables independent of the choice of the phases of basis vectors is <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span><sup>2</sup> − (2<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> − 1) = (<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> − 1)<sup>2</sup>. <ul><li>Of these, <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><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span>(<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> − 1) are rotation angles called <i>quark mixing angles</i>.</li> <li>The remaining <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="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> − 1)(<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> − 2) are complex phases, which cause <a href="/wiki/CP_violation" title="CP violation">CP violation</a>.</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="N_=_2"><span id="N_.3D_2"></span><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> = 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=5" title="Edit section: N = 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the case <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> = 2, there is only one parameter, which is a mixing angle between two generations of quarks. Historically, this was the first version of CKM matrix when only two generations were known. It is called the <b>Cabibbo angle</b> after its inventor <a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Nicola Cabibbo</a>. </p> <div class="mw-heading mw-heading3"><h3 id="N_=_3"><span id="N_.3D_3"></span><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> = 3</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=6" title="Edit section: N = 3"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> case (<span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;"><span class="TEXHTML MVAR" style="FONT-STYLE:ITALIC;">N</span></span></span> = 3), there are three mixing angles and one CP-violating complex phase.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Observations_and_predictions">Observations and predictions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=7" title="Edit section: Observations and predictions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cabibbo's idea originated from a need to explain two observed phenomena: </p> <ol><li>the transitions <span class="nowrap"> <span class="texhtml">u ↔ d</span>, </span> <span class="nowrap"> <span class="texhtml">e ↔ ν<sub>e</sub></span> ,</span> and <span class="nowrap"> <span class="texhtml">μ ↔ ν<sub>μ</sub></span> </span> had similar amplitudes.</li> <li>the transitions with change in strangeness <span class="nowrap"> <span class="texhtml">ΔS = 1</span> </span> had amplitudes equal to <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">4</span></span>⁠</span> of those with <span class="nowrap"> <span class="texhtml">ΔS = 0</span> .</span></li></ol> <p>Cabibbo's solution consisted of postulating <i>weak universality</i> (see below) to resolve the first issue, along with a mixing angle <span class="texhtml"><i>θ</i><sub>c</sub></span>, now called the <i>Cabibbo angle</i>, between the <span class="texhtml">d</span> and <span class="texhtml">s</span> quarks to resolve the second. </p><p>For two generations of quarks, there can be no CP violating phases, as shown by the counting of the previous section. Since CP violations <i>had</i> already been seen in 1964, in neutral <a href="/wiki/Kaon" title="Kaon">kaon</a> decays, the <a href="/wiki/Standard_model_(basic_details)" class="mw-redirect" title="Standard model (basic details)">Standard Model</a> that emerged soon after clearly indicated the existence of a third generation of quarks, as Kobayashi and Maskawa pointed out in 1973. The discovery of the <a href="/wiki/Bottom_quark" title="Bottom quark">bottom quark</a> at <a href="/wiki/Fermilab" title="Fermilab">Fermilab</a> (by <a href="/wiki/Leon_Lederman" class="mw-redirect" title="Leon Lederman">Leon Lederman</a>'s group) in 1976 therefore immediately started off the search for the <a href="/wiki/Top_quark" title="Top quark">top quark</a>, the missing third-generation quark. </p><p>Note, however, that the specific values that the angles take on are <i>not</i> a prediction of the standard model: They are <a href="/wiki/Free_parameter" title="Free parameter">free parameters</a>. At present, there is no generally-accepted theory that explains why the angles should have the values that are measured in experiments. </p> <div class="mw-heading mw-heading2"><h2 id="Weak_universality">Weak universality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=8" title="Edit section: Weak universality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The constraints of unitarity of the CKM-matrix on the diagonal terms can be written as </p> <dl><dd><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 \sum _{k}|V_{jk}|^{2}=\sum _{k}|V_{kj}|^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k}|V_{jk}|^{2}=\sum _{k}|V_{kj}|^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c963eede839b95a5e2bad59c1bc69afdfb56b91b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.782ex; height:5.509ex;" alt="{\displaystyle \sum _{k}|V_{jk}|^{2}=\sum _{k}|V_{kj}|^{2}=1}"></span></dd></dl></dd></dl> <p>separately for each generation <span class="texhtml mvar" style="font-style:italic;">j</span>. This implies that the sum of all couplings of any <i>one</i> of the up-type quarks to <i>all</i> the down-type quarks is the same for all generations. This relation is called <i>weak universality</i> and was first pointed out by <a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Nicola Cabibbo</a> in 1967. Theoretically it is a consequence of the fact that all <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a> doublets couple with the same strength to the <a href="/wiki/Vector_boson" title="Vector boson">vector bosons</a> of weak interactions. It has been subjected to continuing experimental tests. </p> <div class="mw-heading mw-heading2"><h2 id="The_unitarity_triangles">The unitarity triangles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=9" title="Edit section: The unitarity triangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The remaining constraints of unitarity of the CKM-matrix can be written in the form </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 \sum _{k}V_{ik}V_{jk}^{*}=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k}V_{ik}V_{jk}^{*}=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e960236786278baaa41a9a4af8677ff620d27bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.363ex; height:5.509ex;" alt="{\displaystyle \sum _{k}V_{ik}V_{jk}^{*}=0~.}"></span></dd></dl> <p>For any fixed and different <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml mvar" style="font-style:italic;">j</span>, this is a constraint on three complex numbers, one for each <span class="texhtml mvar" style="font-style:italic;">k</span>, which says that these numbers form the sides of a triangle in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. There are six choices of <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml mvar" style="font-style:italic;">j</span> (three independent), and hence six such triangles, each of which is called a <i>unitary triangle</i>. Their shapes can be very different, but they all have the same area, which can be related to the <a href="/wiki/CP_violation" title="CP violation">CP violating</a> phase. The area vanishes for the specific parameters in the Standard Model for which there would be no <a href="/wiki/CP_violation" title="CP violation">CP violation</a>. The orientation of the triangles depend on the phases of the quark fields. </p><p>A popular quantity amounting to twice the area of the unitarity triangle is the <b>Jarlskog invariant</b> (introduced by <a href="/wiki/Cecilia_Jarlskog" title="Cecilia Jarlskog">Cecilia Jarlskog</a> in 1985), </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 J=c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\sin \delta \approx 3\cdot 10^{-5}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mo>≈</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\sin \delta \approx 3\cdot 10^{-5}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b24a4745bf641eb2ca5e089d292fcb42d4a251d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.623ex; height:3.343ex;" alt="{\displaystyle J=c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\sin \delta \approx 3\cdot 10^{-5}~.}"></span></dd></dl> <p>For Greek indices denoting up quarks and Latin ones down quarks, the 4-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 \;(\alpha ,\beta ;i,j)\equiv \operatorname {Im} (V_{\alpha i}V_{\beta j}V_{\alpha j}^{*}V_{\beta i}^{*})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;(\alpha ,\beta ;i,j)\equiv \operatorname {Im} (V_{\alpha i}V_{\beta j}V_{\alpha j}^{*}V_{\beta i}^{*})\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363f816fb32d0ca927e706080262fa127979c29f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.292ex; height:3.343ex;" alt="{\displaystyle \;(\alpha ,\beta ;i,j)\equiv \operatorname {Im} (V_{\alpha i}V_{\beta j}V_{\alpha j}^{*}V_{\beta i}^{*})\;}"></span> is doubly antisymmetric, </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 (\beta ,\alpha ;i,j)=-(\alpha ,\beta ;i,j)=(\alpha ,\beta ;j,i)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mi>α</mi> <mo>;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>;</mo> <mi>j</mi> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\beta ,\alpha ;i,j)=-(\alpha ,\beta ;i,j)=(\alpha ,\beta ;j,i)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23cfcbe4d6c3a62104677e499319cba447692b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.706ex; height:2.843ex;" alt="{\displaystyle (\beta ,\alpha ;i,j)=-(\alpha ,\beta ;i,j)=(\alpha ,\beta ;j,i)~.}"></span></dd></dl> <p>Up to antisymmetry, it only has <span class="nowrap"> 9 = 3 × 3 </span> non-vanishing components, which, remarkably, from the unitarity of <span class="texhtml mvar" style="font-style:italic;">V</span>, can be shown to be <i>all identical in magnitude</i>, that 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 (\alpha ,\beta ;i,j)=J~{\begin{bmatrix}\;~~0&\;~~1&-1\\-1&\;~~0&\;~~1\\\;~~1&-1&\;~~0\end{bmatrix}}_{\alpha \beta }\otimes {\begin{bmatrix}\;~~0&\;~~1&-1\\-1&\;~~0&\;~~1\\\;~~1&-1&\;~~0\end{bmatrix}}_{ij}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>;</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>J</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha ,\beta ;i,j)=J~{\begin{bmatrix}\;~~0&\;~~1&-1\\-1&\;~~0&\;~~1\\\;~~1&-1&\;~~0\end{bmatrix}}_{\alpha \beta }\otimes {\begin{bmatrix}\;~~0&\;~~1&-1\\-1&\;~~0&\;~~1\\\;~~1&-1&\;~~0\end{bmatrix}}_{ij}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227808e8df56bb92d9401e6df1842f9f6c4a4cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:57.295ex; height:9.676ex;" alt="{\displaystyle (\alpha ,\beta ;i,j)=J~{\begin{bmatrix}\;~~0&\;~~1&-1\\-1&\;~~0&\;~~1\\\;~~1&-1&\;~~0\end{bmatrix}}_{\alpha \beta }\otimes {\begin{bmatrix}\;~~0&\;~~1&-1\\-1&\;~~0&\;~~1\\\;~~1&-1&\;~~0\end{bmatrix}}_{ij}\;,}"></span></dd></dl> <p>so that </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 J=(u,c;s,b)=(u,c;d,s)=(u,c;b,d)=(c,t;s,b)=(c,t;d,s)=(c,t;b,d)=(t,u;s,b)=(t,u;b,d)=(t,u;d,s)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>c</mi> <mo>;</mo> <mi>s</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>c</mi> <mo>;</mo> <mi>d</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>c</mi> <mo>;</mo> <mi>b</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>;</mo> <mi>s</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>;</mo> <mi>d</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo>;</mo> <mi>b</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>;</mo> <mi>s</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>;</mo> <mi>b</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>;</mo> <mi>d</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=(u,c;s,b)=(u,c;d,s)=(u,c;b,d)=(c,t;s,b)=(c,t;d,s)=(c,t;b,d)=(t,u;s,b)=(t,u;b,d)=(t,u;d,s)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434c822af573fff89baaa0261e26b74c7585d619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:113.665ex; height:2.843ex;" alt="{\displaystyle J=(u,c;s,b)=(u,c;d,s)=(u,c;b,d)=(c,t;s,b)=(c,t;d,s)=(c,t;b,d)=(t,u;s,b)=(t,u;b,d)=(t,u;d,s)~.}"></span></dd></dl> <p>Since the three sides of the triangles are open to direct experiment, as are the three angles, a class of tests of the Standard Model is to check that the triangle closes. This is the purpose of a modern series of experiments under way at the Japanese <a href="/wiki/Belle_experiment" title="Belle experiment">BELLE</a> and the American <a href="/wiki/BaBar_experiment" title="BaBar experiment">BaBar</a> experiments, as well as at <a href="/wiki/LHCb" class="mw-redirect" title="LHCb">LHCb</a> in CERN, Switzerland. </p> <div class="mw-heading mw-heading2"><h2 id="Parameterizations">Parameterizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=10" title="Edit section: Parameterizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Four independent parameters are required to fully define the CKM matrix. Many parameterizations have been proposed, and three of the most common ones are shown below. </p> <div class="mw-heading mw-heading3"><h3 id="KM_parameters">KM parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=11" title="Edit section: KM parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The original parameterization of Kobayashi and Maskawa used three angles (<span style="white-space: nowrap;"> </span><span class="texhtml mvar" style="font-style:italic;">θ</span><sub>1</sub>, <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>2</sub>, <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>3</sub><span style="white-space: nowrap;"> </span>) and a CP-violating phase angle (<span style="white-space: nowrap;"> </span><span class="texhtml mvar" style="font-style:italic;">δ</span><span style="white-space: nowrap;"> </span>).<sup id="cite_ref-KM_5-1" class="reference"><a href="#cite_note-KM-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>1</sub> is the Cabibbo angle. For brevity, the cosines and sines of the angles <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>k</sub> are denoted <span class="texhtml mvar" style="font-style:italic;">c</span><sub>k</sub> and <span class="texhtml mvar" style="font-style:italic;">s</span><sub>k</sub>, for <span class="nowrap">k = 1,<span style="white-space: nowrap;"> </span>2,<span style="white-space: nowrap;"> </span>3</span> respectively. </p> <dl><dd><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{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta }&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{i\delta }\\s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta }&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{i\delta }\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>δ</mi> </mrow> </msup> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>δ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>δ</mi> </mrow> </msup> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>δ</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta }&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{i\delta }\\s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta }&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{i\delta }\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ace675aa5807ebd6c99a4d0593ac1246b608b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:45.309ex; height:9.509ex;" alt="{\displaystyle {\begin{bmatrix}c_{1}&-s_{1}c_{3}&-s_{1}s_{3}\\s_{1}c_{2}&c_{1}c_{2}c_{3}-s_{2}s_{3}e^{i\delta }&c_{1}c_{2}s_{3}+s_{2}c_{3}e^{i\delta }\\s_{1}s_{2}&c_{1}s_{2}c_{3}+c_{2}s_{3}e^{i\delta }&c_{1}s_{2}s_{3}-c_{2}c_{3}e^{i\delta }\end{bmatrix}}.}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id=""Standard"_parameters"><span id=".22Standard.22_parameters"></span>"Standard" parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=12" title="Edit section: "Standard" parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A "standard" parameterization of the CKM matrix uses three <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a> (<span style="white-space: nowrap;"> </span><span class="texhtml mvar" style="font-style:italic;">θ</span><sub>12</sub>, <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>23</sub>, <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>13</sub><span style="white-space: nowrap;"> </span>) and one CP-violating phase (<span style="white-space: nowrap;"> </span><span class="texhtml mvar" style="font-style:italic;">δ</span><sub>13</sub><span style="white-space: nowrap;"> </span>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>12</sub> is the Cabibbo angle. Couplings between quark generations <span class="texhtml">j</span> and <span class="texhtml">k</span> vanish if <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">θ</span><sub>jk</sub> = 0 </span>. Cosines and sines of the angles are denoted <span class="texhtml mvar" style="font-style:italic;">c</span><sub>jk</sub> and <span class="texhtml mvar" style="font-style:italic;">s</span><sub>jk</sub>, respectively. </p> <dl><dd><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}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{13}}\\0&1&0\\-s_{13}e^{i\delta _{13}}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{13}}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{13}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{13}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{13}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{13}}&c_{23}c_{13}\end{bmatrix}}.\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 /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{13}}\\0&1&0\\-s_{13}e^{i\delta _{13}}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{13}}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{13}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{13}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{13}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{13}}&c_{23}c_{13}\end{bmatrix}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d54f435419500a87b06d0ea7d85a9586ccea25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:68.205ex; height:19.676ex;" alt="{\displaystyle {\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{13}}\\0&1&0\\-s_{13}e^{i\delta _{13}}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{13}}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{13}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{13}}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{13}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{13}}&c_{23}c_{13}\end{bmatrix}}.\end{aligned}}}"></span></dd></dl></dd></dl> <p>The 2008 values for the standard parameters were:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">θ</span><sub>12</sub> = <span class="nowrap"><span data-sort-value="6999227590934460061♠"></span>13.04°<span style="margin-left:0.3em;margin-right:0.15em;">±</span>0.05°</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>13</sub> = <span class="nowrap"><span data-sort-value="6997350811179650861♠"></span>0.201°<span style="margin-left:0.3em;margin-right:0.15em;">±</span>0.011°</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span><sub>23</sub> = <span class="nowrap"><span data-sort-value="6998415388361974651♠"></span>2.38°<span style="margin-left:0.3em;margin-right:0.15em;">±</span>0.06°</span></dd></dl> <p>and </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">δ</span><sub>13</sub> = <span class="nowrap"><span data-sort-value="7000120000000000000♠"></span>1.20<span style="margin-left:0.3em;margin-right:0.15em;">±</span>0.08</span> radians = <span class="nowrap"><span data-sort-value="7000120078652537210♠"></span>68.8°<span style="margin-left:0.3em;margin-right:0.15em;">±</span>4.5°</span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Wolfenstein_parameters">Wolfenstein parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=13" title="Edit section: Wolfenstein parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A third parameterization of the CKM matrix was introduced by <a href="/wiki/Lincoln_Wolfenstein" title="Lincoln Wolfenstein">Lincoln Wolfenstein</a> with the four real parameters <span class="texhtml mvar" style="font-style:italic;">λ</span>, <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">ρ</span>, and <span class="texhtml mvar" style="font-style:italic;">η</span>, which would all 'vanish' (would be zero) if there were no coupling.<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> The four Wolfenstein parameters have the property that all are of order 1 and are related to the 'standard' parameterization: </p> <dl><dd><table> <tbody><tr> <td><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 \lambda =s_{12}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =s_{12}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4936909c113d3352abde74352887f555fdd725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.648ex; height:2.509ex;" alt="{\displaystyle \lambda =s_{12}~,}"></span> </td> <td><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 \lambda =s_{12}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =s_{12}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4936909c113d3352abde74352887f555fdd725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.648ex; height:2.509ex;" alt="{\displaystyle \lambda =s_{12}~,}"></span> </td></tr> <tr> <td><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\lambda ^{2}=s_{23}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lambda ^{2}=s_{23}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb9a35496128550cbbd6cf23efa4a77efee8740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.445ex; height:3.009ex;" alt="{\displaystyle A\lambda ^{2}=s_{23}~,}"></span> </td> <td><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={\frac {s_{23}}{\;s_{12}^{2}\;}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mrow> <mspace width="thickmathspace" /> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {s_{23}}{\;s_{12}^{2}\;}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768c4ffbc0305dc9b193aec3e958fd00a7a07617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.162ex; height:5.676ex;" alt="{\displaystyle A={\frac {s_{23}}{\;s_{12}^{2}\;}}~,}"></span> </td></tr> <tr> <td><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\lambda ^{3}(\rho -i\eta )=s_{13}e^{-i\delta }~,\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>−</mo> <mi>i</mi> <mi>η</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>δ</mi> </mrow> </msup> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lambda ^{3}(\rho -i\eta )=s_{13}e^{-i\delta }~,\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709b04e5ddf3babf33459577350e66d6c9a4811a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.495ex; height:3.176ex;" alt="{\displaystyle A\lambda ^{3}(\rho -i\eta )=s_{13}e^{-i\delta }~,\quad }"></span> </td> <td><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 \rho =\operatorname {\mathcal {R_{e}}} \left\{{\frac {\;s_{13}\,e^{-i\delta }\;}{s_{12}\,s_{23}}}\right\}~,\quad \eta =-\operatorname {\mathcal {I_{m}}} \left\{{\frac {\;s_{13}\,e^{-i\delta }\;}{s_{12}\,s_{23}}}\right\}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">e</mi> </mrow> </msub> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>δ</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </msub> </mrow> </mrow> <mo>⁡</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>δ</mi> </mrow> </msup> <mspace width="thickmathspace" /> </mrow> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\operatorname {\mathcal {R_{e}}} \left\{{\frac {\;s_{13}\,e^{-i\delta }\;}{s_{12}\,s_{23}}}\right\}~,\quad \eta =-\operatorname {\mathcal {I_{m}}} \left\{{\frac {\;s_{13}\,e^{-i\delta }\;}{s_{12}\,s_{23}}}\right\}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e78d3260c1d0632e679f9423079427f42108aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.579ex; height:6.343ex;" alt="{\displaystyle \rho =\operatorname {\mathcal {R_{e}}} \left\{{\frac {\;s_{13}\,e^{-i\delta }\;}{s_{12}\,s_{23}}}\right\}~,\quad \eta =-\operatorname {\mathcal {I_{m}}} \left\{{\frac {\;s_{13}\,e^{-i\delta }\;}{s_{12}\,s_{23}}}\right\}~.}"></span> </td></tr></tbody></table></dd></dl> <p>Although the Wolfenstein parameterization of the CKM matrix can be as exact as desired when carried to high order, it is mainly used for generating convenient approximations to the standard parameterization. The approximation to order <span class="texhtml mvar" style="font-style:italic;">λ</span><sup>3</sup>, good to better than 0.3% accuracy, is: </p> <dl><dd><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{bmatrix}1-{\tfrac {1}{2}}\lambda ^{2}&\lambda &A\lambda ^{3}(\rho -i\eta )\\-\lambda &1-{\tfrac {1}{2}}\lambda ^{2}&A\lambda ^{2}\\A\lambda ^{3}(1-\rho -i\eta )&-A\lambda ^{2}&1\end{bmatrix}}+O(\lambda ^{4})~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>λ</mi> </mtd> <mtd> <mi>A</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>−</mo> <mi>i</mi> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>A</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ρ</mi> <mo>−</mo> <mi>i</mi> <mi>η</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>A</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1-{\tfrac {1}{2}}\lambda ^{2}&\lambda &A\lambda ^{3}(\rho -i\eta )\\-\lambda &1-{\tfrac {1}{2}}\lambda ^{2}&A\lambda ^{2}\\A\lambda ^{3}(1-\rho -i\eta )&-A\lambda ^{2}&1\end{bmatrix}}+O(\lambda ^{4})~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd821696d5a2968d3b2657c0b2168d88b0e57ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:54.583ex; height:11.176ex;" alt="{\displaystyle {\begin{bmatrix}1-{\tfrac {1}{2}}\lambda ^{2}&\lambda &A\lambda ^{3}(\rho -i\eta )\\-\lambda &1-{\tfrac {1}{2}}\lambda ^{2}&A\lambda ^{2}\\A\lambda ^{3}(1-\rho -i\eta )&-A\lambda ^{2}&1\end{bmatrix}}+O(\lambda ^{4})~.}"></span></dd></dl></dd></dl> <p>Rates of <a href="/wiki/CP_violation" title="CP violation">CP violation</a> correspond to the parameters <span class="texhtml mvar" style="font-style:italic;">ρ</span> and <span class="texhtml mvar" style="font-style:italic;">η</span>. </p><p>Using the values of the previous section for the CKM matrix, as of 2008 the best determination of the Wolfenstein parameter values is:<sup id="cite_ref-PDG2023_6-1" class="reference"><a href="#cite_note-PDG2023-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">λ</span> =.22500 ± 0.0067,   <span class="texhtml mvar" style="font-style:italic;">A</span> = <span class="nowrap"><span data-sort-value="6999826000000000000♠"></span>0.826<span style="margin-left:0.3em;"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:85%;text-align:right;">+0.018<br />−0.015</span></span></span>,   <span class="texhtml mvar" style="font-style:italic;">ρ</span> = 0.159±0.010,   and   <span class="texhtml mvar" style="font-style:italic;">η</span> = 0.348±0.010.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Nobel_Prize">Nobel Prize</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=14" title="Edit section: Nobel Prize"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2008, Kobayashi and Maskawa shared one half of the <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature".<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> Some physicists were reported to harbor bitter feelings about the fact that the Nobel Prize committee failed to reward the work of <a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Cabibbo</a>, whose prior work was closely related to that of Kobayashi and Maskawa.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Asked for a reaction on the prize, Cabibbo preferred to give no comment.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </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=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&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/Standard_Model_(mathematical_formulation)" class="mw-redirect" title="Standard Model (mathematical formulation)">Formulation of the Standard Model</a> and <a href="/wiki/CP_violation" title="CP violation">CP violations</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a>, <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavour</a> and <a href="/wiki/CP-violation#Strong_CP_problem" class="mw-redirect" title="CP-violation">strong CP problem</a></li> <li><a href="/wiki/Weinberg_angle" title="Weinberg angle">Weinberg angle</a>, a similar angle for Z and photon mixing</li> <li><a href="/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">Pontecorvo–Maki–Nakagawa–Sakata matrix</a>, the equivalent mixing matrix for <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a></li> <li><a href="/wiki/Koide_formula" title="Koide formula">Koide formula</a></li></ul> <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=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=16" title="Edit section: References"><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-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-Cabibbo-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cabibbo_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="CITEREFCabibbo1963" class="citation journal cs1">Cabibbo, N. (1963). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.10.531">"Unitary Symmetry and Leptonic Decays"</a>. <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>10</b> (12): 531–533. <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/1963PhRvL..10..531C">1963PhRvL..10..531C</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.10.531">10.1103/PhysRevLett.10.531</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=Unitary+Symmetry+and+Leptonic+Decays&rft.volume=10&rft.issue=12&rft.pages=531-533&rft.date=1963&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.10.531&rft_id=info%3Abibcode%2F1963PhRvL..10..531C&rft.aulast=Cabibbo&rft.aufirst=N.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.10.531&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGell-MannLévy1960" class="citation journal cs1"><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann, M.</a>; Lévy, M. (1960). "The Axial Vector Current in Beta Decay". <i><a href="/wiki/Il_Nuovo_Cimento" class="mw-redirect" title="Il Nuovo Cimento">Il Nuovo Cimento</a></i>. <b>16</b> (4): 705–726. <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/1960NCim...16..705G">1960NCim...16..705G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02859738">10.1007/BF02859738</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122945049">122945049</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Il+Nuovo+Cimento&rft.atitle=The+Axial+Vector+Current+in+Beta+Decay&rft.volume=16&rft.issue=4&rft.pages=705-726&rft.date=1960&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122945049%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02859738&rft_id=info%3Abibcode%2F1960NCim...16..705G&rft.aulast=Gell-Mann&rft.aufirst=M.&rft.au=L%C3%A9vy%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaiani2009" class="citation journal cs1">Maiani, L. (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110722053046/http://prometeo.sif.it:8080/papers/online/sag/025/01-02/pdf/78_opinioni.pdf">"Sul premio Nobel per la fisica 2008"</a> [On the Nobel prize in Physics for 2008] <span class="cs1-format">(PDF)</span>. <i>Il Nuovo Saggiatore</i>. <b>25</b> (1–2): 78. Archived from <a rel="nofollow" class="external text" href="http://prometeo.sif.it:8080/papers/online/sag/025/01-02/pdf/78_opinioni.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 22 July 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">30 November</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Il+Nuovo+Saggiatore&rft.atitle=Sul+premio+Nobel+per+la+fisica+2008&rft.volume=25&rft.issue=1%E2%80%932&rft.pages=78&rft.date=2009&rft.aulast=Maiani&rft.aufirst=L.&rft_id=http%3A%2F%2Fprometeo.sif.it%3A8080%2Fpapers%2Fonline%2Fsag%2F025%2F01-02%2Fpdf%2F78_opinioni.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-Hughes-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hughes_4-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughes1991" class="citation book cs1">Hughes, I.S. (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JN6qlZlGUG4C&q=cabbibo+angle&pg=PA242">"Chapter 11.1 – Cabibbo Mixing"</a>. <i>Elementary Particles</i> (3rd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 242–243. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-40402-0" title="Special:BookSources/978-0-521-40402-0"><bdi>978-0-521-40402-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+11.1+%E2%80%93+Cabibbo+Mixing&rft.btitle=Elementary+Particles&rft.pages=242-243&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=1991&rft.isbn=978-0-521-40402-0&rft.aulast=Hughes&rft.aufirst=I.S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJN6qlZlGUG4C%26q%3Dcabbibo%2Bangle%26pg%3DPA242&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-KM-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-KM_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-KM_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKobayashiMaskawa1973" class="citation journal cs1">Kobayashi, M.; Maskawa, T. (1973). <a rel="nofollow" class="external text" href="https://doi.org/10.1143%2FPTP.49.652">"CP-violation in the renormalizable theory of weak interaction"</a>. <i><a href="/wiki/Progress_of_Theoretical_Physics" class="mw-redirect" title="Progress of Theoretical Physics">Progress of Theoretical Physics</a></i>. <b>49</b> (2): 652–657. <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/1973PThPh..49..652K">1973PThPh..49..652K</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/10.1143%2FPTP.49.652">10.1143/PTP.49.652</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/2433%2F66179">2433/66179</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Progress+of+Theoretical+Physics&rft.atitle=CP-violation+in+the+renormalizable+theory+of+weak+interaction&rft.volume=49&rft.issue=2&rft.pages=652-657&rft.date=1973&rft_id=info%3Ahdl%2F2433%2F66179&rft_id=info%3Adoi%2F10.1143%2FPTP.49.652&rft_id=info%3Abibcode%2F1973PThPh..49..652K&rft.aulast=Kobayashi&rft.aufirst=M.&rft.au=Maskawa%2C+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1143%252FPTP.49.652&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-PDG2023-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-PDG2023_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-PDG2023_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="PDG2023" class="citation journal cs1">R.L. Workman et al. (Particle Data Group) (August 2022). <a rel="nofollow" class="external text" href="https://pdg.lbl.gov/">"Review of Particle Physics (and 2023 update)"</a>. <i>Progress of Theoretical and Experimental Physics</i>. <b>2022</b> (8): 083C01. <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/10.1093%2Fptep%2Fptac097">10.1093/ptep/ptac097</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/20.500.11850%2F571164">20.500.11850/571164</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">12 September</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Progress+of+Theoretical+and+Experimental+Physics&rft.atitle=Review+of+Particle+Physics+%28and+2023+update%29&rft.volume=2022&rft.issue=8&rft.pages=083C01&rft.date=2022-08&rft_id=info%3Ahdl%2F20.500.11850%2F571164&rft_id=info%3Adoi%2F10.1093%2Fptep%2Fptac097&rft.au=R.L.+Workman+et+al.+%28Particle+Data+Group%29&rft_id=https%3A%2F%2Fpdg.lbl.gov%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaez2011" class="citation web cs1">Baez, J.C. (4 April 2011). <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/neutrinos.html">"Neutrinos and the mysterious Pontecorvo-Maki-Nakagawa-Sakata matrix"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">13 February</span> 2016</span>. <q>In fact, the <a href="/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">Pontecorvo–Maki–Nakagawa–Sakata matrix</a> actually affects the behavior of all leptons, not just neutrinos. Furthermore, a similar trick works for quarks – but then the matrix <i>U</i> is called the Cabibbo–Kobayashi–Maskawa matrix.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Neutrinos+and+the+mysterious+Pontecorvo-Maki-Nakagawa-Sakata+matrix&rft.date=2011-04-04&rft.aulast=Baez&rft.aufirst=J.C.&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fneutrinos.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChauKeung1984" class="citation journal cs1">Chau, L.L.; Keung, W.-Y. (1984). "Comments on the Parametrization of the Kobayashi-Maskawa Matrix". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>53</b> (19): 1802–1805. <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/1984PhRvL..53.1802C">1984PhRvL..53.1802C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.53.1802">10.1103/PhysRevLett.53.1802</a>.</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=Comments+on+the+Parametrization+of+the+Kobayashi-Maskawa+Matrix&rft.volume=53&rft.issue=19&rft.pages=1802-1805&rft.date=1984&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.53.1802&rft_id=info%3Abibcode%2F1984PhRvL..53.1802C&rft.aulast=Chau&rft.aufirst=L.L.&rft.au=Keung%2C+W.-Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Values obtained from values of Wolfenstein parameters in the 2008 <i><a href="/wiki/Review_of_Particle_Physics" class="mw-redirect" title="Review of Particle Physics">Review of Particle Physics</a></i>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfenstein1983" class="citation journal cs1"><a href="/wiki/Lincoln_Wolfenstein" title="Lincoln Wolfenstein">Wolfenstein, L.</a> (1983). "Parametrization of the Kobayashi-Maskawa Matrix". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>51</b> (21): 1945–1947. <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/1983PhRvL..51.1945W">1983PhRvL..51.1945W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.51.1945">10.1103/PhysRevLett.51.1945</a>.</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=Parametrization+of+the+Kobayashi-Maskawa+Matrix&rft.volume=51&rft.issue=21&rft.pages=1945-1947&rft.date=1983&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.51.1945&rft_id=info%3Abibcode%2F1983PhRvL..51.1945W&rft.aulast=Wolfenstein&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation pressrelease cs1"><a rel="nofollow" class="external text" href="http://nobelprize.org/nobel_prizes/physics/laureates/2008/press.html">"The Nobel Prize in Physics 2008"</a> (Press release). <a href="/wiki/The_Nobel_Foundation" class="mw-redirect" title="The Nobel Foundation">The Nobel Foundation</a>. 7 October 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">24 November</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Nobel+Prize+in+Physics+2008&rft.pub=The+Nobel+Foundation&rft.date=2008-10-07&rft_id=http%3A%2F%2Fnobelprize.org%2Fnobel_prizes%2Fphysics%2Flaureates%2F2008%2Fpress.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamieson2008" class="citation web cs1">Jamieson, V. (7 October 2008). <a rel="nofollow" class="external text" href="https://www.newscientist.com/article/dn14885-physics-nobel-snubs-key-researcher.html?DCMP=ILC-hmts&nsref=news8_head_dn14885">"Physics Nobel Snubs key Researcher"</a>. <i><a href="/wiki/New_Scientist" title="New Scientist">New Scientist</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">24 November</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=New+Scientist&rft.atitle=Physics+Nobel+Snubs+key+Researcher&rft.date=2008-10-07&rft.aulast=Jamieson&rft.aufirst=V.&rft_id=https%3A%2F%2Fwww.newscientist.com%2Farticle%2Fdn14885-physics-nobel-snubs-key-researcher.html%3FDCMP%3DILC-hmts%26nsref%3Dnews8_head_dn14885&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="http://www.corriere.it/scienze_e_tecnologie/08_ottobre_07/nobel_fisica_italiani_traditi_d9993120-946d-11dd-a0d8-00144f02aabc.shtml">"Nobel, l'amarezza dei fisici italiani"</a>. <i><a href="/wiki/Corriere_della_Sera" title="Corriere della Sera">Corriere della Sera</a></i> (in Italian). 7 October 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">24 November</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Corriere+della+Sera&rft.atitle=Nobel%2C+l%27amarezza+dei+fisici+italiani&rft.date=2008-10-07&rft_id=http%3A%2F%2Fwww.corriere.it%2Fscienze_e_tecnologie%2F08_ottobre_07%2Fnobel_fisica_italiani_traditi_d9993120-946d-11dd-a0d8-00144f02aabc.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading_and_external_links">Further reading and external links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix&action=edit&section=17" title="Edit section: Further reading and external links"><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="CITEREFD.J._Griffiths2008" class="citation book cs1">D.J. Griffiths (2008). <i>Introduction to Elementary Particles</i> (2nd ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-40601-2" title="Special:BookSources/978-3-527-40601-2"><bdi>978-3-527-40601-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Elementary+Particles&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=2008&rft.isbn=978-3-527-40601-2&rft.au=D.J.+Griffiths&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._Povh1995" class="citation book cs1">B. Povh; et al. (1995). <i>Particles and Nuclei: An Introduction to the Physical Concepts</i>. <a href="/wiki/Springer_(publisher)" class="mw-redirect" title="Springer (publisher)">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-20168-7" title="Special:BookSources/978-3-540-20168-7"><bdi>978-3-540-20168-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particles+and+Nuclei%3A+An+Introduction+to+the+Physical+Concepts&rft.pub=Springer&rft.date=1995&rft.isbn=978-3-540-20168-7&rft.au=B.+Povh&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFI.I._Bigi,_A.I._Sanda2000" class="citation book cs1">I.I. Bigi, A.I. Sanda (2000). <i>CP violation</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-44349-4" title="Special:BookSources/978-0-521-44349-4"><bdi>978-0-521-44349-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CP+violation&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-44349-4&rft.au=I.I.+Bigi%2C+A.I.+Sanda&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span></li></ul> <ul><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://pdg.lbl.gov/2013/reviews/rpp2012-rev-ckm-matrix.pdf">"Particle Data Group: The CKM quark-mixing matrix"</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=Particle+Data+Group%3A+The+CKM+quark-mixing+matrix&rft_id=http%3A%2F%2Fpdg.lbl.gov%2F2013%2Freviews%2Frpp2012-rev-ckm-matrix.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" 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://pdg.lbl.gov/2013/reviews/rpp2012-rev-cp-violation.pdf">"Particle Data Group: CP violation in meson decays"</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=Particle+Data+Group%3A+CP+violation+in+meson+decays&rft_id=http%3A%2F%2Fpdg.lbl.gov%2F2013%2Freviews%2Frpp2012-rev-cp-violation.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" 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://www-public.slac.stanford.edu/babar/">"The Babar experiment"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Babar+experiment&rft_id=http%3A%2F%2Fwww-public.slac.stanford.edu%2Fbabar%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span> at <a href="/wiki/SLAC" class="mw-redirect" title="SLAC">SLAC</a>, California, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://belle.kek.jp">"the BELLE experiment"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=the+BELLE+experiment&rft_id=http%3A%2F%2Fbelle.kek.jp&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACabibbo%E2%80%93Kobayashi%E2%80%93Maskawa+matrix" class="Z3988"></span> at <a href="/wiki/KEK" title="KEK">KEK</a>, Japan.</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 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Matrix_classes" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Matrix_classes" title="Template:Matrix classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Matrix_classes" title="Template talk:Matrix classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Matrix_classes" title="Special:EditPage/Template:Matrix classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Matrix_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicitly constrained entries</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/Alternant_matrix" title="Alternant matrix">Alternant</a></li> <li><a href="/wiki/Anti-diagonal_matrix" title="Anti-diagonal matrix">Anti-diagonal</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric</a></li> <li><a href="/wiki/Arrowhead_matrix" title="Arrowhead matrix">Arrowhead</a></li> <li><a href="/wiki/Band_matrix" title="Band matrix">Band</a></li> <li><a href="/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li> <li><a href="/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li> <li><a href="/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li> <li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li> <li><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean</a></li> <li><a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li> <li><a href="/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li> <li><a href="/wiki/Conference_matrix" title="Conference matrix">Conference</a></li> <li><a href="/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li> <li><a href="/wiki/Copositive_matrix" title="Copositive matrix">Copositive</a></li> <li><a href="/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li> <li><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li> <li><a href="/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li> <li><a href="/wiki/Elementary_matrix" title="Elementary matrix">Elementary</a></li> <li><a href="/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li> <li><a href="/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li> <li><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li> <li><a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li> <li><a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel</a></li> <li><a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a></li> <li><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li> <li><a href="/wiki/Hollow_matrix" title="Hollow matrix">Hollow</a></li> <li><a href="/wiki/Integer_matrix" title="Integer matrix">Integer</a></li> <li><a href="/wiki/Logical_matrix" title="Logical matrix">Logical</a></li> <li><a href="/wiki/Matrix_unit" title="Matrix unit">Matrix unit</a></li> <li><a href="/wiki/Metzler_matrix" title="Metzler matrix">Metzler</a></li> <li><a href="/wiki/Moore_matrix" title="Moore matrix">Moore</a></li> <li><a href="/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li> <li><a href="/wiki/Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal</a></li> <li><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation</a></li> <li><a href="/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li> <li><a href="/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li> <li><a href="/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li> <li><a href="/wiki/Signature_matrix" title="Signature matrix">Signature</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li> <li><a href="/wiki/Skyline_matrix" title="Skyline matrix">Skyline</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse</a></li> <li><a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li> <li><a href="/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li> <li><a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li> <li><a href="/wiki/Triangular_matrix" title="Triangular matrix">Triangular</a></li> <li><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li> <li><a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li> <li><a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh</a></li> <li><a href="/wiki/Z-matrix_(mathematics)" title="Z-matrix (mathematics)">Z</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constant</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/Exchange_matrix" title="Exchange matrix">Exchange</a></li> <li><a href="/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert</a></li> <li><a href="/wiki/Identity_matrix" title="Identity matrix">Identity</a></li> <li><a href="/wiki/Lehmer_matrix" title="Lehmer matrix">Lehmer</a></li> <li><a href="/wiki/Matrix_of_ones" title="Matrix of ones">Of ones</a></li> <li><a href="/wiki/Pascal_matrix" title="Pascal matrix">Pascal</a></li> <li><a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli</a></li> <li><a href="/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer</a></li> <li><a href="/wiki/Shift_matrix" title="Shift matrix">Shift</a></li> <li><a href="/wiki/Zero_matrix" title="Zero matrix">Zero</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conditions on <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues or eigenvectors</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/Companion_matrix" title="Companion matrix">Companion</a></li> <li><a href="/wiki/Convergent_matrix" title="Convergent matrix">Convergent</a></li> <li><a href="/wiki/Defective_matrix" title="Defective matrix">Defective</a></li> <li><a href="/wiki/Definite_matrix" title="Definite matrix">Definite</a></li> <li><a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable</a></li> <li><a href="/wiki/Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz-stable</a></li> <li><a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite</a></li> <li><a href="/wiki/Stieltjes_matrix" title="Stieltjes matrix">Stieltjes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satisfying conditions on <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">products</a> or <a href="/wiki/Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverses</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/Matrix_congruence" title="Matrix congruence">Congruent</a></li> <li><a href="/wiki/Idempotent_matrix" title="Idempotent matrix">Idempotent</a> or <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Projection</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Involutory_matrix" title="Involutory matrix">Involutory</a></li> <li><a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">Nilpotent</a></li> <li><a href="/wiki/Normal_matrix" title="Normal matrix">Normal</a></li> <li><a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal</a></li> <li><a href="/wiki/Unimodular_matrix" title="Unimodular matrix">Unimodular</a></li> <li><a href="/wiki/Unipotent" title="Unipotent">Unipotent</a></li> <li><a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary</a></li> <li><a href="/wiki/Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular</a></li> <li><a href="/wiki/Weighing_matrix" title="Weighing matrix">Weighing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With specific applications</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/Adjugate_matrix" title="Adjugate matrix">Adjugate</a></li> <li><a href="/wiki/Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign</a></li> <li><a href="/wiki/Augmented_matrix" title="Augmented matrix">Augmented</a></li> <li><a href="/wiki/B%C3%A9zout_matrix" title="Bézout matrix">Bézout</a></li> <li><a href="/wiki/Carleman_matrix" title="Carleman matrix">Carleman</a></li> <li><a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan</a></li> <li><a href="/wiki/Circulant_matrix" title="Circulant matrix">Circulant</a></li> <li><a href="/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor</a></li> <li><a href="/wiki/Commutation_matrix" title="Commutation matrix">Commutation</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion</a></li> <li><a href="/wiki/Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter</a></li> <li><a href="/wiki/Distance_matrix" title="Distance matrix">Distance</a></li> <li><a href="/wiki/Duplication_and_elimination_matrices" title="Duplication and elimination matrices">Duplication and elimination</a></li> <li><a href="/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance</a></li> <li><a href="/wiki/Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental (linear differential equation)</a></li> <li><a href="/wiki/Generator_matrix" title="Generator matrix">Generator</a></li> <li><a href="/wiki/Gram_matrix" title="Gram matrix">Gram</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li> <li><a href="/wiki/Householder_transformation" title="Householder transformation">Householder</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Moment_matrix" title="Moment matrix">Moment</a></li> <li><a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff</a></li> <li><a href="/wiki/Pick_matrix" class="mw-redirect" title="Pick matrix">Pick</a></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Random</a></li> <li><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation</a></li> <li><a href="/wiki/Routh%E2%80%93Hurwitz_matrix" title="Routh–Hurwitz matrix">Routh-Hurwitz</a></li> <li><a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert</a></li> <li><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear matrix">Shear</a></li> <li><a href="/wiki/Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity</a></li> <li><a href="/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic</a></li> <li><a href="/wiki/Totally_positive_matrix" title="Totally positive matrix">Totally positive</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Statistics" title="Statistics">statistics</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/Centering_matrix" title="Centering matrix">Centering</a></li> <li><a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation</a></li> <li><a href="/wiki/Covariance_matrix" title="Covariance matrix">Covariance</a></li> <li><a href="/wiki/Design_matrix" title="Design matrix">Design</a></li> <li><a href="/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic</a></li> <li><a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information</a></li> <li><a href="/wiki/Projection_matrix" title="Projection matrix">Hat</a></li> <li><a href="/wiki/Precision_(statistics)" title="Precision (statistics)">Precision</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</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/Adjacency_matrix" title="Adjacency matrix">Adjacency</a></li> <li><a href="/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency</a></li> <li><a href="/wiki/Degree_matrix" title="Degree matrix">Degree</a></li> <li><a href="/wiki/Edmonds_matrix" title="Edmonds matrix">Edmonds</a></li> <li><a href="/wiki/Incidence_matrix" title="Incidence matrix">Incidence</a></li> <li><a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian</a></li> <li><a href="/wiki/Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency</a></li> <li><a href="/wiki/Tutte_matrix" title="Tutte matrix">Tutte</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in science and engineering</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 class="mw-selflink selflink">Cabibbo–Kobayashi–Maskawa</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density</a></li> <li><a href="/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental (computer vision)</a></li> <li><a href="/wiki/Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative</a></li> <li><a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma</a></li> <li><a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann</a></li> <li><a href="/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian</a></li> <li><a href="/wiki/Irregular_matrix" title="Irregular matrix">Irregular</a></li> <li><a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap</a></li> <li><a href="/wiki/S-matrix" title="S-matrix">S</a></li> <li><a href="/wiki/State-transition_matrix" title="State-transition matrix">State transition</a></li> <li><a href="/wiki/Substitution_matrix" title="Substitution matrix">Substitution</a></li> <li><a href="/wiki/Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z (chemistry)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related terms</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/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li> <li><a href="/wiki/Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li> <li><a href="/wiki/Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li> <li><a href="/wiki/Pseudoinverse" class="mw-redirect" title="Pseudoinverse">Pseudoinverse</a></li> <li><a href="/wiki/Row_echelon_form" title="Row echelon form">Row echelon form</a></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><a href="/wiki/List_of_matrices" class="mw-redirect" title="List of matrices">List of matrices</a></li> <li><a href="/wiki/Category:Matrices" title="Category:Matrices">Category:Matrices</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Standard_Model" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" 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" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/225px-Standard_Model_of_Elementary_Particles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/300px-Standard_Model_of_Elementary_Particles.svg.png 2x" data-file-width="1390" data-file-height="1330" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constituents</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 class="mw-selflink selflink">CKM matrix</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li> <li><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical 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 href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li> <li><a href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</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/Laboratori_Nazionali_del_Gran_Sasso" title="Laboratori Nazionali del Gran Sasso">Gran Sasso</a></li> <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> 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Cabibbo–Kobayashi–Maskawa matrix - Wikipedia