Mathematical formulation of the Standard Model

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<span class="vector-toc-numb">2</span> <span>Alternative presentations of the fields</span> </div> </a> <button aria-controls="toc-Alternative_presentations_of_the_fields-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 Alternative presentations of the fields subsection</span> </button> <ul id="toc-Alternative_presentations_of_the_fields-sublist" class="vector-toc-list"> <li id="toc-Fermions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fermions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Fermions</span> </div> </a> <ul id="toc-Fermions-sublist" class="vector-toc-list"> <li id="toc-A_chiral_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#A_chiral_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>A chiral theory</span> </div> </a> <ul id="toc-A_chiral_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mass_and_interaction_eigenstates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mass_and_interaction_eigenstates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Mass and interaction eigenstates</span> </div> </a> <ul id="toc-Mass_and_interaction_eigenstates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Positive_and_negative_energies" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Positive_and_negative_energies"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Positive and negative energies</span> </div> </a> <ul id="toc-Positive_and_negative_energies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bosons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bosons"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Bosons</span> </div> </a> <ul id="toc-Bosons-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Perturbative_QFT_and_the_interaction_picture" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Perturbative_QFT_and_the_interaction_picture"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Perturbative QFT and the interaction picture</span> </div> </a> <button aria-controls="toc-Perturbative_QFT_and_the_interaction_picture-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 Perturbative QFT and the interaction picture subsection</span> </button> <ul id="toc-Perturbative_QFT_and_the_interaction_picture-sublist" class="vector-toc-list"> <li id="toc-Free_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Free fields</span> </div> </a> <ul id="toc-Free_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interaction_terms_and_the_path_integral_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interaction_terms_and_the_path_integral_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Interaction terms and the path integral approach</span> </div> </a> <ul id="toc-Interaction_terms_and_the_path_integral_approach-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lagrangian_formalism" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lagrangian_formalism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lagrangian formalism</span> </div> </a> <button aria-controls="toc-Lagrangian_formalism-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 Lagrangian formalism subsection</span> </button> <ul id="toc-Lagrangian_formalism-sublist" class="vector-toc-list"> <li id="toc-Kinetic_terms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinetic_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Kinetic terms</span> </div> </a> <ul id="toc-Kinetic_terms-sublist" class="vector-toc-list"> <li id="toc-Fermion_fields" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fermion_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Fermion fields</span> </div> </a> <ul id="toc-Fermion_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gauge_fields" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gauge_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Gauge fields</span> </div> </a> <ul id="toc-Gauge_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Coupling_terms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coupling_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Coupling terms</span> </div> </a> <ul id="toc-Coupling_terms-sublist" class="vector-toc-list"> <li id="toc-Electroweak_sector" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Electroweak_sector"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Electroweak sector</span> </div> </a> <ul id="toc-Electroweak_sector-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_chromodynamics_sector" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantum_chromodynamics_sector"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Quantum chromodynamics sector</span> </div> </a> <ul id="toc-Quantum_chromodynamics_sector-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mass_terms_and_the_Higgs_mechanism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mass_terms_and_the_Higgs_mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Mass terms and the Higgs mechanism</span> </div> </a> <ul id="toc-Mass_terms_and_the_Higgs_mechanism-sublist" class="vector-toc-list"> <li id="toc-Mass_terms" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mass_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Mass terms</span> </div> </a> <ul id="toc-Mass_terms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higgs_mechanism" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Higgs_mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>Higgs mechanism</span> </div> </a> <ul id="toc-Higgs_mechanism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Yukawa_interaction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Yukawa_interaction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.3</span> <span>Yukawa interaction</span> </div> </a> <ul id="toc-Yukawa_interaction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Neutrino_masses" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Neutrino_masses"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.4</span> <span>Neutrino masses</span> </div> </a> <ul id="toc-Neutrino_masses-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Detailed_information" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Detailed_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Detailed information</span> </div> </a> <button aria-controls="toc-Detailed_information-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 Detailed information subsection</span> </button> <ul id="toc-Detailed_information-sublist" class="vector-toc-list"> <li id="toc-Field_content_in_detail" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Field_content_in_detail"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Field content in detail</span> </div> </a> <ul id="toc-Field_content_in_detail-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fermion_content" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fermion_content"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Fermion content</span> </div> </a> <ul id="toc-Fermion_content-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Free parameters</span> </div> </a> <ul id="toc-Free_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_symmetries_of_the_Standard_Model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additional_symmetries_of_the_Standard_Model"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Additional symmetries of the Standard Model</span> </div> </a> <ul id="toc-Additional_symmetries_of_the_Standard_Model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-U(1)_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#U(1)_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>U(1) symmetry</span> </div> </a> <ul id="toc-U(1)_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Charged_and_neutral_current_couplings_and_Fermi_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charged_and_neutral_current_couplings_and_Fermi_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Charged and neutral current couplings and Fermi theory</span> </div> </a> <ul id="toc-Charged_and_neutral_current_couplings_and_Fermi_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physics_beyond_the_Standard_Model" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Physics_beyond_the_Standard_Model"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Physics beyond the Standard Model</span> </div> </a> <ul id="toc-Physics_beyond_the_Standard_Model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References_and_external_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References_and_external_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References and external links</span> </div> </a> <ul id="toc-References_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 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <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">Mathematics of a particle physics model</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For a less mathematical description and overview, see <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png/500px-Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png" decoding="async" width="500" height="359" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png/750px-Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png/1000px-Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png 2x" data-file-width="6600" data-file-height="4735" /></a><figcaption>Standard Model of Particle Physics. The diagram shows the <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> of the Standard Model (the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>, the three <a href="/wiki/Generation_(particle_physics)" title="Generation (particle physics)">generations</a> of <a href="/wiki/Quark" title="Quark">quarks</a> and <a href="/wiki/Lepton" title="Lepton">leptons</a>, and the <a href="/wiki/Gauge_bosons" class="mw-redirect" title="Gauge bosons">gauge bosons</a>), including their names, masses, spins, charges, chiralities, and interactions with the <a href="/wiki/Strong_interaction" title="Strong interaction">strong</a>, <a href="/wiki/Weak_interaction" title="Weak interaction">weak</a> and <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetic</a> forces. It also depicts the crucial role of the Higgs boson in <a href="/wiki/Electroweak_symmetry_breaking" class="mw-redirect" title="Electroweak symmetry breaking">electroweak symmetry breaking</a>, and shows how the properties of the various particles differ in the (high-energy) symmetric phase (top) and the (low-energy) broken-symmetry phase (bottom).</figcaption></figure> <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 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a></th></tr><tr><td class="sidebar-image"><figure class="skin-invert-image noresize mw-ext-imagemap-desc-bottom-right" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/240px-Standard_Model_of_Elementary_Particles.svg.png" decoding="async" width="240" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/360px-Standard_Model_of_Elementary_Particles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/480px-Standard_Model_of_Elementary_Particles.svg.png 2x" data-file-width="1390" data-file-height="1330" usemap="#ImageMap_af1b7d82cba73703" resource="/wiki/File:Standard_Model_of_Elementary_Particles.svg" /></span><map name="ImageMap_af1b7d82cba73703"><area href="/wiki/Up_quark" shape="rect" coords="17,48,59,89" alt="Up quark" title="Up quark" /><area href="/wiki/Charm_quark" shape="rect" coords="60,48,102,89" alt="Charm quark" title="Charm quark" /><area href="/wiki/Top_quark" shape="rect" coords="104,48,145,89" alt="Top quark" title="Top quark" /><area href="/wiki/Gluon" shape="rect" coords="149,48,190,89" alt="Gluon" title="Gluon" /><area href="/wiki/Higgs_boson" shape="rect" coords="194,48,235,89" alt="Higgs boson" title="Higgs boson" /><area href="/wiki/Down_quark" shape="rect" coords="17,91,59,132" alt="Down quark" title="Down quark" /><area href="/wiki/Strange_quark" shape="rect" coords="60,91,102,132" alt="Strange quark" title="Strange quark" /><area href="/wiki/Bottom_quark" shape="rect" coords="104,91,145,132" alt="Bottom quark" title="Bottom quark" /><area href="/wiki/Photon" shape="rect" coords="149,91,190,132" alt="Photon" title="Photon" /><area href="/wiki/Electron" shape="rect" coords="17,137,59,178" alt="Electron" title="Electron" /><area href="/wiki/Muon" shape="rect" coords="60,137,102,178" alt="Muon" title="Muon" /><area href="/wiki/Tau_(particle)" shape="rect" coords="104,137,145,178" alt="Tau (particle)" title="Tau (particle)" /><area href="/wiki/W_and_Z_bosons#Z_bosons}Z_boson" shape="rect" coords="149,137,190,178" alt="W and Z bosons#Z bosons}Z boson" title="W and Z bosons#Z bosons}Z boson" /><area href="/wiki/Electron_neutrino" shape="rect" coords="17,180,59,221" alt="Electron neutrino" title="Electron neutrino" /><area href="/wiki/Muon_neutrino" shape="rect" coords="60,180,102,221" alt="Muon neutrino" title="Muon neutrino" /><area href="/wiki/Tau_neutrino" shape="rect" coords="104,180,145,221" alt="Tau neutrino" title="Tau neutrino" /><area href="/wiki/W_and_Z_bosons" shape="rect" coords="149,180,190,221" alt="W and Z bosons" title="W and Z bosons" /><area href="/wiki/Standard_Model" shape="rect" coords="16,8,224,19" alt="Standard Model" title="Standard Model" /><area href="/wiki/Fermion" shape="rect" coords="17,24,145,38" alt="Fermion" title="Fermion" /><area href="/wiki/Boson" shape="rect" coords="149,24,235,38" alt="Boson" title="Boson" /><area href="/wiki/Quark" shape="rect" coords="6,93,14,133" alt="Quark" title="Quark" /><area href="/wiki/Lepton" shape="rect" coords="6,176,12,219" alt="Lepton" title="Lepton" /><area href="/wiki/Scalar_boson" shape="rect" coords="227,92,233,172" alt="Scalar boson" title="Scalar boson" /><area href="/wiki/Gauge_boson" shape="rect" coords="196,143,202,219" alt="Gauge boson" title="Gauge boson" /><area href="/wiki/Vector_boson" shape="rect" coords="205,165,209,219" alt="Vector boson" title="Vector boson" /></map><figcaption></figcaption></figure><div class="sidebar-caption"><a href="/wiki/Elementary_particle" title="Elementary particle">Elementary particles</a> of the Standard Model</div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a><br /><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a><br /><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <br /> <a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a> <br /> <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a><br /> <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Constituents</div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a><br /> <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a><br /> <a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">CKM matrix</a><br /><a class="mw-selflink selflink">Standard Model mathematics</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Limitations</div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Strong_CP_problem" title="Strong CP problem">Strong CP problem</a><br /><a href="/wiki/Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a><br /><a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillations</a><br /><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Physics beyond the Standard Model</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Rutherford</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/James_Chadwick" title="James Chadwick">Chadwick</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/E._C._George_Sudarshan" title="E. C. George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Raymond_Davis_Jr." title="Raymond Davis Jr.">Davis Jr</a></li> <li><a href="/wiki/Carl_David_Anderson" title="Carl David Anderson">Anderson</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Carlo_Rubbia" title="Carlo Rubbia">Rubbia</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Richard_E._Taylor" title="Richard E. Taylor">Taylor</a></li> <li><a href="/wiki/Jerome_Isaac_Friedman" title="Jerome Isaac Friedman">Friedman</a></li> <li><a href="/wiki/C._F._Powell" title="C. F. Powell">Powell</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Leon_M._Lederman" title="Leon M. Lederman">Lederman</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Simon_van_der_Meer" title="Simon van der Meer">Meer</a></li> <li><a href="/wiki/Clyde_Cowan" title="Clyde Cowan">Cowan</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Owen_Chamberlain" title="Owen Chamberlain">Chamberlain</a></li> <li><a href="/wiki/Nicola_Cabibbo" title="Nicola Cabibbo">Cabibbo</a></li> <li><a href="/wiki/Melvin_Schwartz" title="Melvin Schwartz">Schwartz</a></li> <li><a href="/wiki/Martin_Lewis_Perl" title="Martin Lewis Perl">Perl</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Makoto_Kobayashi_(physicist)" class="mw-redirect" title="Makoto Kobayashi (physicist)">Kobayashi</a></li> <li><a href="/wiki/Toshihide_Maskawa" title="Toshihide Maskawa">Maskawa</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/Martinus_J._G._Veltman" title="Martinus J. G. Veltman">Veltman</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Abraham_Pais" title="Abraham Pais">Pais</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Hugh_David_Politzer" title="Hugh David Politzer">Politzer</a></li> <li><a href="/wiki/Frederick_Reines" title="Frederick Reines">Reines</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/James_Cronin" title="James Cronin">Cronin</a></li> <li><a href="/wiki/Val_Logsdon_Fitch" title="Val Logsdon Fitch">Fitch</a></li> <li><a href="/wiki/John_Hasbrouck_Van_Vleck" title="John Hasbrouck Van Vleck">Vleck</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Tom_Kibble" title="Tom Kibble">Kibble</a></li> <li><a href="/wiki/Santiago_Ant%C3%BAnez_de_Mayolo" title="Santiago Antúnez de Mayolo">de Mayolo</a></li> <li><a href="/wiki/C%C3%A9sar_Lattes" title="César Lattes">Lattes</a></li> <li><a href="/wiki/George_Zweig" title="George Zweig">Zweig</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output 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class="nv-talk"><a href="/wiki/Template_talk:Standard_model_of_particle_physics" title="Template talk:Standard model of particle physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Standard_model_of_particle_physics" title="Special:EditPage/Template:Standard model of particle physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/Feynman_diagram" title="Feynman diagram"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/211px-Feynmann_Diagram_Gluon_Radiation.svg.png" decoding="async" width="211" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/317px-Feynmann_Diagram_Gluon_Radiation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/422px-Feynmann_Diagram_Gluon_Radiation.svg.png 2x" data-file-width="400" data-file-height="250" /></a></span><div class="sidebar-caption"><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div></td></tr><tr><td class="sidebar-above"> <a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Field_(physics)" title="Field (physics)">Field theory</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">Weak force</a></li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">Symmetries</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">C-symmetry</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/Lorentz_symmetry" class="mw-redirect" title="Lorentz symmetry">Lorentz symmetry</a></li> <li><a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a></li> <li><a href="/wiki/Gauge_symmetry_(mathematics)" title="Gauge symmetry (mathematics)">Gauge symmetry</a></li> <li><a href="/wiki/Explicit_symmetry_breaking" title="Explicit symmetry breaking">Explicit symmetry breaking</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Noether_charge" class="mw-redirect" title="Noether charge">Noether charge</a></li> <li><a href="/wiki/Topological_charge" class="mw-redirect" title="Topological charge">Topological charge</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Tools</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomaly</a></li> <li><a href="/wiki/Background_field_method" title="Background field method">Background field method</a></li> <li><a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a></li> <li><a href="/wiki/Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Correlation function</a></li> <li><a href="/wiki/Crossing_(physics)" title="Crossing (physics)">Crossing</a></li> <li><a href="/wiki/Effective_action" title="Effective action">Effective action</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective field theory</a></li> <li><a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">Expectation value</a></li> <li><a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a></li> <li><a href="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">Partition function</a></li> <li><a href="/wiki/Path_Integral_Formulation" class="mw-redirect" title="Path Integral Formulation">Path Integral Formulation</a></li> <li><a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">Propagator</a></li> <li><a href="/wiki/Quantization_(physics)" title="Quantization (physics)">Quantization</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">Vacuum state</a></li> <li><a href="/wiki/Wick%27s_theorem" title="Wick's theorem">Wick's theorem</a></li> <li><a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca equations</a></li> <li><a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a></li> <li><a href="/wiki/Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Bargmann–Wigner equations</a></li> <li><a href="/wiki/Schwinger-Dyson_equation" class="mw-redirect" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Incomplete theories</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Stephen_Louis_Adler" class="mw-redirect" title="Stephen Louis Adler">Adler</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li> <li><a href="/wiki/Alexey_Andreevich_Anselm" class="mw-redirect" title="Alexey Andreevich Anselm">Anselm</a></li> <li><a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Bargmann</a></li> <li><a href="/wiki/Carlo_Becchi" title="Carlo Becchi">Becchi</a></li> <li><a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Belavin</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/James_Bjorken" title="James Bjorken">Bjorken</a></li> <li><a href="/wiki/Konrad_Bleuler" title="Konrad Bleuler">Bleuer</a></li> <li><a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Bogoliubov</a></li> <li><a href="/wiki/Stanley_Brodsky" title="Stanley Brodsky">Brodsky</a></li> <li><a href="/wiki/Robert_Brout" title="Robert Brout">Brout</a></li> <li><a href="/wiki/Detlev_Buchholz" title="Detlev Buchholz">Buchholz</a></li> <li><a href="/wiki/Freddy_Cachazo" title="Freddy Cachazo">Cachazo</a></li> <li><a href="/wiki/Curtis_Callan" title="Curtis Callan">Callan</a></li> <li><a href="/wiki/John_Cardy" title="John Cardy">Cardy</a></li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Coleman</a></li> <li><a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a></li> <li><a href="/wiki/Roger_Dashen" title="Roger Dashen">Dashen</a></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">DeWitt</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Doplicher</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Fran%C3%A7ois_Englert" title="François Englert">Englert</a></li> <li><a href="/wiki/Ludvig_Faddeev" title="Ludvig Faddeev">Faddeev</a></li> <li><a href="/wiki/Victor_Sergeevich_Fadin" class="mw-redirect" title="Victor Sergeevich Fadin">Fadin</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Markus_Fierz" title="Markus Fierz">Fierz</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Paul_Frampton" title="Paul Frampton">Frampton</a></li> <li><a href="/wiki/Harald_Fritzsch" title="Harald Fritzsch">Fritzsch</a></li> <li><a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Fröhlich</a></li> <li><a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Fredenhagen</a></li> <li><a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Furry</a></li> <li><a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Glashow</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/James_Glimm" title="James Glimm">Glimm</a></li> <li><a href="/wiki/Jeffrey_Goldstone" title="Jeffrey Goldstone">Goldstone</a></li> <li><a href="/wiki/Vladimir_Gribov" title="Vladimir Gribov">Gribov</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Suraj_N._Gupta" title="Suraj N. Gupta">Gupta</a></li> <li><a href="/wiki/Gerald_Guralnik" title="Gerald Guralnik">Guralnik</a></li> <li><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Haag</a></li> <li><a href="/wiki/C._R._Hagen" title="C. R. Hagen">Hagen</a></li> <li><a href="/wiki/Moo-Young_Han" title="Moo-Young Han">Han</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Klaus_Hepp" title="Klaus Hepp">Hepp</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a></li> <li><a href="/wiki/Roman_Jackiw" title="Roman Jackiw">Jackiw</a></li> <li><a href="/wiki/Arthur_Jaffe" title="Arthur Jaffe">Jaffe</a></li> <li><a href="/wiki/Giovanni_Jona-Lasinio" title="Giovanni Jona-Lasinio">Jona-Lasinio</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Res_Jost" title="Res Jost">Jost</a></li> <li><a href="/wiki/Gunnar_K%C3%A4ll%C3%A9n" title="Gunnar Källén">Källén</a></li> <li><a href="/wiki/Henry_Way_Kendall" title="Henry Way Kendall">Kendall</a></li> <li><a href="/wiki/Toichiro_Kinoshita" title="Toichiro Kinoshita">Kinoshita</a></li> <li><a href="/wiki/Kim_Jihn-eui" title="Kim Jihn-eui">Kim</a></li> <li><a href="/wiki/Igor_R._Klebanov" class="mw-redirect" title="Igor R. Klebanov">Klebanov</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Dirk_Kreimer" title="Dirk Kreimer">Kreimer</a></li> <li><a href="/wiki/Eduard_A._Kuraev" title="Eduard A. Kuraev">Kuraev</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Benjamin_W._Lee" title="Benjamin W. Lee">Lee</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Harry_Lehmann" title="Harry Lehmann">Lehmann</a></li> <li><a href="/wiki/Heinrich_Leutwyler" title="Heinrich Leutwyler">Leutwyler</a></li> <li><a href="/wiki/Lev_Lipatov" title="Lev Lipatov">Lipatov</a></li> <li><a href="/wiki/Jan_%C5%81opusza%C5%84ski_(physicist)" title="Jan Łopuszański (physicist)">Łopuszański</a></li> <li><a href="/wiki/Francis_E._Low" title="Francis E. Low">Low</a></li> <li><a href="/wiki/Gerhart_L%C3%BCders" title="Gerhart Lüders">Lüders</a></li> <li><a href="/wiki/Luciano_Maiani" title="Luciano Maiani">Maiani</a></li> <li><a href="/wiki/Ettore_Majorana" title="Ettore Majorana">Majorana</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Takeo_Matsubara" title="Takeo Matsubara">Matsubara</a></li> <li><a href="/wiki/Alexander_Arkadyevich_Migdal" title="Alexander Arkadyevich Migdal">Migdal</a></li> <li><a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Mills</a></li> <li><a href="/wiki/Christian_M%C3%B8ller" title="Christian Møller">Møller</a></li> <li><a href="/wiki/Mark_Naimark" title="Mark Naimark">Naimark</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Kazuhiko_Nishijima" title="Kazuhiko Nishijima">Nishijima</a></li> <li><a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Oehme</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Hugh_Osborn" title="Hugh Osborn">Osborn</a></li> <li><a href="/wiki/Konrad_Osterwalder" title="Konrad Osterwalder">Osterwalder</a></li> <li><a href="/wiki/Giorgio_Parisi" title="Giorgio Parisi">Parisi</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Roberto_Peccei" title="Roberto Peccei">Peccei</a></li> <li><a href="/wiki/Michael_Peskin" title="Michael Peskin">Peskin</a></li> <li><a href="/wiki/Jan_Christoph_Plefka" title="Jan Christoph Plefka">Plefka</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Pomeranchuk</a></li> <li><a href="/wiki/Victor_Popov" title="Victor Popov">Popov</a></li> <li><a href="/wiki/Alexandru_Proca" title="Alexandru Proca">Proca</a></li> <li><a href="/wiki/Helen_Quinn" title="Helen Quinn">Quinn</a></li> <li><a href="/wiki/Alain_Rouet" title="Alain Rouet">Rouet</a></li> <li><a href="/wiki/Valery_Rubakov" title="Valery Rubakov">Rubakov</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">Ruelle</a></li> <li><a href="/wiki/Jun_John_Sakurai" class="mw-redirect" title="Jun John Sakurai">Sakurai</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Robert_Schrader" title="Robert Schrader">Schrader</a></li> <li><a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Schwarz</a></li> <li><a href="/wiki/Julian_Schwinger" title="Julian Schwinger">Schwinger</a></li> <li><a href="/wiki/Irving_Segal" title="Irving Segal">Segal</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Gordon_Walter_Semenoff" title="Gordon Walter Semenoff">Semenoff</a></li> <li><a href="/wiki/Mikhail_Shifman" title="Mikhail Shifman">Shifman</a></li> <li><a href="/wiki/Dmitry_Shirkov" title="Dmitry Shirkov">Shirkov</a></li> <li><a href="/wiki/Tony_Skyrme" title="Tony Skyrme">Skyrme</a></li> <li><a href="/wiki/Charles_M._Sommerfield" title="Charles M. Sommerfield">Sommerfield</a></li> <li><a href="/wiki/Raymond_Stora" title="Raymond Stora">Stora</a></li> <li><a href="/wiki/Ernst_Stueckelberg" title="Ernst Stueckelberg">Stueckelberg</a></li> <li><a href="/wiki/George_Sudarshan" class="mw-redirect" title="George Sudarshan">Sudarshan</a></li> <li><a href="/wiki/Kurt_Symanzik" title="Kurt Symanzik">Symanzik</a></li> <li><a href="/wiki/Yasushi_Takahashi" title="Yasushi Takahashi">Takahashi</a></li> <li><a href="/wiki/Walter_Thirring" title="Walter Thirring">Thirring</a></li> <li><a href="/wiki/Shin%27ichir%C5%8D_Tomonaga" title="Shin'ichirō Tomonaga">Tomonaga</a></li> <li><a href="/wiki/Igor_Tyutin" title="Igor Tyutin">Tyutin</a></li> <li><a href="/wiki/Arkady_Vainshtein" title="Arkady Vainshtein">Vainshtein</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li> <li><a href="/wiki/Miguel_%C3%81ngel_Virasoro_(physicist)" title="Miguel Ángel Virasoro (physicist)">Virasoro</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward</a></li> <li><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg</a></li> <li><a href="/wiki/Victor_Weisskopf" title="Victor Weisskopf">Weisskopf</a></li> <li><a href="/wiki/Gregor_Wentzel" title="Gregor Wentzel">Wentzel</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Christof_Wetterich" title="Christof Wetterich">Wetterich</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Gian_Carlo_Wick" title="Gian Carlo Wick">Wick</a></li> <li><a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Hideki_Yukawa" title="Hideki Yukawa">Yukawa</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee</a></li> <li><a href="/wiki/Wolfhart_Zimmermann" title="Wolfhart Zimmermann">Zimmermann</a></li> <li><a href="/wiki/Jean_Zinn-Justin" title="Jean Zinn-Justin">Zinn-Justin</a></li> <li><a href="/wiki/Jean-Bernard_Zuber" title="Jean-Bernard Zuber">Zuber</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> <p><br /> </p> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_field_theory" title="Template:Quantum field theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_field_theory" title="Template talk:Quantum field theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theory" title="Special:EditPage/Template:Quantum field theory"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>This article describes the mathematics of the <b>Standard Model</b> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, a <a href="/wiki/Gauge_theory" title="Gauge theory">gauge</a> <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> containing the <a href="/wiki/Field_(physics)#Symmetries_of_fields" title="Field (physics)">internal symmetries</a> of the <a href="/wiki/Unitary_group" title="Unitary group">unitary</a> <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">product group</a> <span class="texhtml"><a href="/wiki/Special_unitary_group" title="Special unitary group">SU</a>(3) × SU(2) × <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a></span>. The theory is commonly viewed as describing the fundamental set of particles – the <a href="/wiki/Lepton" title="Lepton">leptons</a>, <a href="/wiki/Quark" title="Quark">quarks</a>, <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a> and the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a>. </p><p>The Standard Model is <a href="/wiki/Renormalizable" class="mw-redirect" title="Renormalizable">renormalizable</a> and mathematically self-consistent;<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> however, despite having huge and continued successes in providing experimental predictions, it does leave some <a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">unexplained phenomena</a>.<sup id="cite_ref-NYT-20230911_2-0" class="reference"><a href="#cite_note-NYT-20230911-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In particular, although the physics of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> is incorporated, <a href="/wiki/General_relativity" title="General relativity">general relativity</a> is not, and the Standard Model will fail at energies or distances where the <a href="/wiki/Graviton" title="Graviton">graviton</a> is expected to emerge. Therefore, in a modern field theory context, it is seen as an <a href="/wiki/Effective_field_theory" title="Effective field theory">effective field theory</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Quantum_field_theory">Quantum field theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=1" title="Edit section: Quantum field theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Standard_Model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Standard_Model.svg/300px-Standard_Model.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Standard_Model.svg/450px-Standard_Model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Standard_Model.svg/600px-Standard_Model.svg.png 2x" data-file-width="700" data-file-height="700" /></a><figcaption>The pattern of <a href="/wiki/Weak_isospin" title="Weak isospin">weak isospin</a> <span class="texhtml"><i>T</i><sub>3</sub></span>, <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharge</a> <span class="texhtml mvar" style="font-style:italic;">Y<sub>W</sub></span>, and <a href="/wiki/Color_charge" title="Color charge">color charge</a> of all known elementary particles, rotated by the <a href="/wiki/Weak_mixing_angle" class="mw-redirect" title="Weak mixing angle">weak mixing angle</a> to show electric charge <span class="texhtml mvar" style="font-style:italic;">Q</span>, roughly along the vertical. The neutral <a href="/wiki/Higgs_field" class="mw-redirect" title="Higgs field">Higgs field</a> (gray square) breaks the <a href="/wiki/Electroweak_symmetry" class="mw-redirect" title="Electroweak symmetry">electroweak symmetry</a> and interacts with other particles to give them mass.</figcaption></figure> <p>The standard model is a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, meaning its fundamental objects are <i>quantum fields</i>, which are defined at all points in spacetime. QFT treats particles as <a href="/wiki/Excited_state" title="Excited state">excited states</a> (also called <a href="/wiki/Quantum" title="Quantum">quanta</a>) of their underlying quantum <a href="/wiki/Field_(physics)" title="Field (physics)">fields</a>, which are more fundamental than the particles. These fields are </p> <ul><li>the <a href="/wiki/Fermion" title="Fermion">fermion</a> fields, <span class="texhtml mvar" style="font-style:italic;">ψ</span>, which account for "matter particles";</li> <li>the <a href="/wiki/Electroweak_interaction" title="Electroweak interaction">electroweak boson fields</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ab879909bd9762251f679bbb2fa738100baa45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{1}}"></span>, <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 W_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421467fcab2af38ddf3977b9adf66de8ef3abd57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{2}}"></span>, <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 W_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87b911493067a5414c6bc6c4433ff3840062864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{3}}"></span>, and <span class="texhtml mvar" style="font-style:italic;">B</span>;</li> <li>the <a href="/wiki/Gluon_field" title="Gluon field">gluon field</a>, <span class="texhtml mvar" style="font-style:italic;">G<sub>a</sub></span>; and</li> <li>the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs field</a>, <span class="texhtml mvar" style="font-style:italic;">φ</span>.</li></ul> <p>That these are <i>quantum</i> rather than <i>classical</i> fields has the mathematical consequence that they are <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a>-valued. In particular, values of the fields generally do not commute. As operators, they act upon a quantum state (<a href="/wiki/Ket_vector" class="mw-redirect" title="Ket vector">ket vector</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_presentations_of_the_fields">Alternative presentations of the fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=2" title="Edit section: Alternative presentations of the fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As is common in quantum theory, there is more than one way to look at things. At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations that, in particular contexts, may be more appropriate than those that are given above. </p> <div class="mw-heading mw-heading3"><h3 id="Fermions">Fermions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=3" title="Edit section: Fermions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rather than having one fermion field <span class="texhtml mvar" style="font-style:italic;">ψ</span>, it can be split up into separate components for each type of particle. This mirrors the historical evolution of quantum field theory, since the electron component <span class="texhtml"><i>ψ</i><sub>e</sub></span> (describing the <a href="/wiki/Electron" title="Electron">electron</a> and its antiparticle the <a href="/wiki/Positron" title="Positron">positron</a>) is then the original <span class="texhtml mvar" style="font-style:italic;">ψ</span> field of <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>, which was later accompanied by <span class="texhtml mvar" style="font-style:italic;">ψ<sub>μ</sub></span> and <span class="texhtml mvar" style="font-style:italic;">ψ<sub>τ</sub></span> fields for the <a href="/wiki/Muon" title="Muon">muon</a> and <a href="/wiki/Tauon" class="mw-redirect" title="Tauon">tauon</a> respectively (and their antiparticles). Electroweak theory added <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\nu _{\mathrm {e} }},\psi _{\nu _{\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mrow> </msub> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\nu _{\mathrm {e} }},\psi _{\nu _{\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c509be56d58c41737a13f04a8c373e8dd670b901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.875ex; height:3.009ex;" alt="{\displaystyle \psi _{\nu _{\mathrm {e} }},\psi _{\nu _{\mu }}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\nu _{\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\nu _{\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f079b57fe370dd63535aad5043a3bae467a60bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.412ex; height:2.676ex;" alt="{\displaystyle \psi _{\nu _{\tau }}}"></span> for the corresponding <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a>. The <a href="/wiki/Quark" title="Quark">quarks</a> add still further components. In order to be <a href="/wiki/Dirac_spinor#Four-spinor_for_particles" title="Dirac spinor">four-spinors</a> like the electron and other <a href="/wiki/Lepton" title="Lepton">lepton</a> components, there must be one quark component for every combination of <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavor</a> and <a href="/wiki/Color_charge" title="Color charge">color</a>, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four component <a href="/wiki/Bispinor" title="Bispinor">bispinor</a>, for a total of 96 complex-valued components for the fermion field. </p><p>An important definition is the <a href="/wiki/Dirac_adjoint" title="Dirac adjoint">barred</a> fermion field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890adebe2730294079a81b7bf08b2fe0f2c59909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:2.843ex;" alt="{\displaystyle {\bar {\psi }}}"></span>, which is defined to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\dagger }\gamma ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\dagger }\gamma ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3063e2357cbd7fed16b5dec1d773dcfa43177609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.81ex; height:3.176ex;" alt="{\displaystyle \psi ^{\dagger }\gamma ^{0}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dagger }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>†</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dagger }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbce70d5be6fec538cd30d8bc7b7bb2d3ed2d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.032ex; height:2.676ex;" alt="{\displaystyle \dagger }"></span> denotes the <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian adjoint</a> of <span class="texhtml mvar" style="font-style:italic;">ψ</span>, and <span class="texhtml"><i>γ</i><sup>0</sup></span> is the zeroth <a href="/wiki/Gamma_matrix" class="mw-redirect" title="Gamma matrix">gamma matrix</a>. If <span class="texhtml mvar" style="font-style:italic;">ψ</span> is thought of as an <span class="texhtml"><i>n</i> × 1</span> matrix then <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 {\bar {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890adebe2730294079a81b7bf08b2fe0f2c59909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:2.843ex;" alt="{\displaystyle {\bar {\psi }}}"></span> should be thought of as a <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector"><span class="texhtml">1 × <i>n</i></span> matrix</a>. </p> <div class="mw-heading mw-heading4"><h4 id="A_chiral_theory">A chiral theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=4" title="Edit section: A chiral theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An independent decomposition of <span class="texhtml mvar" style="font-style:italic;">ψ</span> is that into <a href="/wiki/Chirality_(physics)" title="Chirality (physics)">chirality</a> components: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="padding-left: 1.5em"><ul><li>"Left" chirality:  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\rm {L}}={\frac {1}{2}}(1-\gamma _{5})\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\rm {L}}={\frac {1}{2}}(1-\gamma _{5})\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e830c5b2c394ab60e017a8fbefb8e4d2c64830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.454ex; height:5.176ex;" alt="{\displaystyle \psi ^{\rm {L}}={\frac {1}{2}}(1-\gamma _{5})\psi }"></span></li><li>"Right" chirality:  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{\rm {R}}={\frac {1}{2}}(1+\gamma _{5})\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{\rm {R}}={\frac {1}{2}}(1+\gamma _{5})\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafbf6ec09a9a1b02b092ca427a942de7583813c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.636ex; height:5.176ex;" alt="{\displaystyle \psi ^{\rm {R}}={\frac {1}{2}}(1+\gamma _{5})\psi }"></span></li></ul></div> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94fc10f503ccc16a12b94646d8fff849fab673d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{5}}"></span> is <a href="/wiki/Gamma_matrices#The_fifth_"gamma"_matrix,_γ5" title="Gamma matrices">the fifth gamma matrix</a>. This is very important in the Standard Model because <i>left and right chirality components are treated differently by the gauge interactions</i>. </p><p>In particular, under <a href="/wiki/Weak_isospin" title="Weak isospin">weak isospin</a> <a href="/wiki/Special_unitary_group" title="Special unitary group">SU(2)</a> transformations the left-handed particles are weak-isospin doublets, whereas the right-handed are singlets – i.e. the weak isospin of <span class="texhtml"><i>ψ</i><sub>R</sub></span> is zero. Put more simply, the weak interaction could rotate e.g. a left-handed electron into a left-handed neutrino (with emission of a <span class="texhtml">W<sup>−</sup></span>), but could not do so with the same right-handed particles. As an aside, the right-handed neutrino originally did not exist in the standard model – but the discovery of <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillation</a> implies that <a href="/wiki/Neutrino#Mass" title="Neutrino">neutrinos must have mass</a>, and since chirality can change during the propagation of a massive particle, right-handed neutrinos must exist in reality. This does not however change the (experimentally proven) chiral nature of the weak interaction. </p><p>Furthermore, <span class="texhtml">U(1)</span> acts differently on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {e} }^{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {e} }^{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0e159932802e86349a54eeb7de27436f0085af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:2.843ex;" alt="{\displaystyle \psi _{\mathrm {e} }^{\rm {L}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {e} }^{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {e} }^{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b392a23968b3fd59cfb72e7dc6fd6d407a125f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.843ex;" alt="{\displaystyle \psi _{\mathrm {e} }^{\rm {R}}}"></span> (because they have different <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharges</a>). </p> <div class="mw-heading mw-heading4"><h4 id="Mass_and_interaction_eigenstates">Mass and interaction eigenstates</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=5" title="Edit section: Mass and interaction eigenstates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A distinction can thus be made between, for example, the mass and interaction <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates</a> of the neutrino. The former is the state that propagates in free space, whereas the latter is the <i>different</i> state that participates in interactions. Which is the "fundamental" particle? For the neutrino, it is conventional to define the "flavor" (<span style="white-space:nowrap;"><a href="/wiki/Electron_neutrino" title="Electron neutrino"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>ν<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">e</sub></span></span></a></span>, <span style="white-space:nowrap;"><a href="/wiki/Muon_neutrino" title="Muon neutrino"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>ν<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">μ</sub></span></span></a></span>, or <span style="white-space:nowrap;"><a href="/wiki/Tau_neutrino" title="Tau neutrino"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>ν<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">τ</sub></span></span></a></span>) by the interaction eigenstate, whereas for the quarks we define the flavor (up, down, etc.) by the mass state. We can switch between these states using the <a href="/wiki/CKM_matrix" class="mw-redirect" title="CKM matrix">CKM matrix</a> for the quarks, or the <a href="/wiki/PMNS_matrix" class="mw-redirect" title="PMNS matrix">PMNS matrix</a> for the neutrinos (the charged leptons on the other hand are eigenstates of both mass and flavor). </p><p>As an aside, if a complex phase term exists within either of these matrices, it will give rise to direct <a href="/wiki/CP_violation" title="CP violation">CP violation</a>, which could explain the dominance of matter over antimatter in our current universe. This has been proven for the CKM matrix, and is expected for the PMNS matrix. </p> <div class="mw-heading mw-heading4"><h4 id="Positive_and_negative_energies">Positive and negative energies</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=6" title="Edit section: Positive and negative energies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Finally, the quantum fields are sometimes decomposed into "positive" and "negative" energy parts: <span class="texhtml"><i>ψ</i> = <i>ψ</i><sup>+</sup> + <i>ψ</i><sup>−</sup></span>. This is not so common when a quantum field theory has been set up, but often features prominently in the process of quantizing a field theory. </p> <div class="mw-heading mw-heading3"><h3 id="Bosons">Bosons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=7" title="Edit section: Bosons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Weinberg_angle_(relation_between_coupling_constants).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Weinberg_angle_%28relation_between_coupling_constants%29.svg/300px-Weinberg_angle_%28relation_between_coupling_constants%29.svg.png" decoding="async" width="300" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Weinberg_angle_%28relation_between_coupling_constants%29.svg/450px-Weinberg_angle_%28relation_between_coupling_constants%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Weinberg_angle_%28relation_between_coupling_constants%29.svg/600px-Weinberg_angle_%28relation_between_coupling_constants%29.svg.png 2x" data-file-width="580" data-file-height="364" /></a><figcaption>Weinberg angle <span class="texhtml"><i>θ</i><sub>W</sub></span>, and relation between coupling constants <i>g</i>, <i>g</i>′, and <i>e</i>. Adapted from T D Lee's book <i>Particle Physics and Introduction to Field Theory</i> (1981).</figcaption></figure> <p>Due to the <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a>, the electroweak boson fields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ab879909bd9762251f679bbb2fa738100baa45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{1}}"></span>, <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 W_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421467fcab2af38ddf3977b9adf66de8ef3abd57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{2}}"></span>, <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 W_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87b911493067a5414c6bc6c4433ff3840062864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{3}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> "mix" to create the states that are physically observable. To retain gauge invariance, the underlying fields must be massless, but the observable states can <i>gain masses</i> in the process. These states are: </p><p>The massive neutral <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">(Z) boson</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\cos \theta _{\rm {W}}W_{3}-\sin \theta _{\rm {W}}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\cos \theta _{\rm {W}}W_{3}-\sin \theta _{\rm {W}}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a1dd4fe9cf32cacfcb3b4654c92777100ab068" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.396ex; height:2.509ex;" alt="{\displaystyle Z=\cos \theta _{\rm {W}}W_{3}-\sin \theta _{\rm {W}}B}"></span> The massless neutral boson: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\sin \theta _{\rm {W}}W_{3}+\cos \theta _{\rm {W}}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\sin \theta _{\rm {W}}W_{3}+\cos \theta _{\rm {W}}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f19891f46ce48ae160045b6f26ea78e474d623" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.458ex; height:2.509ex;" alt="{\displaystyle A=\sin \theta _{\rm {W}}W_{3}+\cos \theta _{\rm {W}}B}"></span> The massive charged <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W bosons</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{\pm }={\frac {1}{\sqrt {2}}}\left(W_{1}\mp iW_{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∓</mo> <mi>i</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{\pm }={\frac {1}{\sqrt {2}}}\left(W_{1}\mp iW_{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54eae85aa0e3197881c33a5d7bf3e7e51433b92a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.387ex; height:6.176ex;" alt="{\displaystyle W^{\pm }={\frac {1}{\sqrt {2}}}\left(W_{1}\mp iW_{2}\right)}"></span> where <span class="texhtml"><i>θ</i><sub>W</sub></span> is the <a href="/wiki/Weinberg_angle" title="Weinberg angle">Weinberg angle</a>. </p><p>The <span class="texhtml mvar" style="font-style:italic;">A</span> field is the <a href="/wiki/Photon" title="Photon">photon</a>, which corresponds classically to the well-known <a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic four-potential</a> – i.e. the electric and magnetic fields. The <span class="texhtml mvar" style="font-style:italic;">Z</span> field actually contributes in every process the photon does, but due to its large mass, the contribution is usually negligible. </p> <div class="mw-heading mw-heading2"><h2 id="Perturbative_QFT_and_the_interaction_picture">Perturbative QFT and the interaction picture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=8" title="Edit section: Perturbative QFT and the interaction picture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Much of the qualitative descriptions of the standard model in terms of "particles" and "forces" comes from the perturbative <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> view of the model. In this, the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> is decomposed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L}}_{\mathrm {I} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L}}_{\mathrm {I} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f03d0b51a69fc96c659934f42ee5330fda8dd83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.63ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L}}_{\mathrm {I} }}"></span> into separate <i>free field</i> and <i>interaction</i> Lagrangians. The free fields care for particles in isolation, whereas processes involving several particles arise through interactions. The idea is that the state vector should only change when particles interact, meaning a free particle is one whose quantum state is constant. This corresponds to the <a href="/wiki/Interaction_picture" title="Interaction picture">interaction picture</a> in quantum mechanics. </p><p>In the more common <a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a>, even the states of free particles change over time: typically the phase changes at a rate that depends on their energy. In the alternative <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a>, state vectors are kept constant, at the price of having the operators (in particular the <a href="/wiki/Observable" title="Observable">observables</a>) be time-dependent. The interaction picture constitutes an intermediate between the two, where some time dependence is placed in the operators (the quantum fields) and some in the state vector. In QFT, the former is called the free field part of the model, and the latter is called the interaction part. The free field model can be solved exactly, and then the solutions to the full model can be expressed as perturbations of the free field solutions, for example using the <a href="/wiki/Dyson_series" title="Dyson series">Dyson series</a>. </p><p>It should be observed that the decomposition into free fields and interactions is in principle arbitrary. For example, <a href="/wiki/Renormalization#Renormalization_in_QED" title="Renormalization">renormalization</a> in <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">QED</a> modifies the mass of the free field electron to match that of a physical electron (with an electromagnetic field), and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the <a href="/wiki/Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">Feynman diagrams</a>. This is also how the Higgs field is thought to give particles <a href="/wiki/Invariant_mass" title="Invariant mass">mass</a>: the part of the interaction term that corresponds to the nonzero vacuum expectation value of the Higgs field is moved from the interaction to the free field Lagrangian, where it looks just like a mass term having nothing to do with the Higgs field. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagram</a></div> <div class="mw-heading mw-heading3"><h3 id="Free_fields">Free fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=9" title="Edit section: Free fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under the usual free/interaction decomposition, which is suitable for low energies, the free fields obey the following equations: </p> <ul><li>The fermion field <span class="texhtml mvar" style="font-style:italic;">ψ</span> satisfies the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-m_{\rm {f}}c)\psi _{\rm {f}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mi>c</mi> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-m_{\rm {f}}c)\psi _{\rm {f}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179429e573f63bae545355e68ef3e4afc870e7f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.229ex; height:3.009ex;" alt="{\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-m_{\rm {f}}c)\psi _{\rm {f}}=0}"></span> for each type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> of fermion.</li> <li>The photon field <span class="texhtml mvar" style="font-style:italic;">A</span> satisfies the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2841855c2bb7d5fb59926379ed661846f9017c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.132ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }=0}"></span>.</li> <li>The Higgs field <span class="texhtml mvar" style="font-style:italic;">φ</span> satisfies the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a>.</li> <li>The weak interaction fields <span class="texhtml"><i>Z</i>, <i>W</i><sup>±</sup></span> satisfy the <a href="/wiki/Proca_equation" class="mw-redirect" title="Proca equation">Proca equation</a>.</li></ul> <p>These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period <span class="texhtml mvar" style="font-style:italic;">L</span> along each spatial axis; later taking the limit: <span class="texhtml"><i>L</i> → ∞</span> will lift this periodicity restriction. </p><p>In the periodic case, the solution for a field <span class="texhtml mvar" style="font-style:italic;">F</span> (any of the above) can be expressed as a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=\beta \sum _{\mathbf {p} }\sum _{r}E_{\mathbf {p} }^{-{\frac {1}{2}}}\left(a_{r}(\mathbf {p} )u_{r}(\mathbf {p} )e^{-{\frac {ipx}{\hbar }}}+b_{r}^{\dagger }(\mathbf {p} )v_{r}(\mathbf {p} )e^{\frac {ipx}{\hbar }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>β</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>p</mi> <mi>x</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mo>+</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>p</mi> <mi>x</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\beta \sum _{\mathbf {p} }\sum _{r}E_{\mathbf {p} }^{-{\frac {1}{2}}}\left(a_{r}(\mathbf {p} )u_{r}(\mathbf {p} )e^{-{\frac {ipx}{\hbar }}}+b_{r}^{\dagger }(\mathbf {p} )v_{r}(\mathbf {p} )e^{\frac {ipx}{\hbar }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf84429af6b78c38b2d1ae373e0a0b460bc7603" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.769ex; height:6.843ex;" alt="{\displaystyle F(x)=\beta \sum _{\mathbf {p} }\sum _{r}E_{\mathbf {p} }^{-{\frac {1}{2}}}\left(a_{r}(\mathbf {p} )u_{r}(\mathbf {p} )e^{-{\frac {ipx}{\hbar }}}+b_{r}^{\dagger }(\mathbf {p} )v_{r}(\mathbf {p} )e^{\frac {ipx}{\hbar }}\right)}"></span> where: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">β</span> is a normalization factor; for the fermion field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f67f5cb3976f91998c607c4b96b52f389949fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.357ex; height:2.509ex;" alt="{\displaystyle \psi _{\rm {f}}}"></span> it is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {m_{\rm {f}}c^{2}/V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>V</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {m_{\rm {f}}c^{2}/V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f405642e3e0f73195f9b88f6c09e39f4b96d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.219ex; height:3.343ex;" alt="{\textstyle {\sqrt {m_{\rm {f}}c^{2}/V}}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=L^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=L^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d55fea53a6dfdd0cc594af3766b5f2243a87b2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.523ex; height:2.676ex;" alt="{\displaystyle V=L^{3}}"></span> is the volume of the fundamental cell considered; for the photon field <span class="texhtml mvar" style="font-style:italic;">A<sup>μ</sup></span> it is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar c/{\sqrt {2V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>V</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar c/{\sqrt {2V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5107c6aa20c98a8af71111bf97be505ccee1f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.361ex; height:3.176ex;" alt="{\displaystyle \hbar c/{\sqrt {2V}}}"></span>.</li> <li>The sum over <span class="texhtml"><b>p</b></span> is over all momenta consistent with the period <span class="texhtml mvar" style="font-style:italic;">L</span>, i.e., over all vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi \hbar }{L}}(n_{1},n_{2},n_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi \hbar }{L}}(n_{1},n_{2},n_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da495006dc522c29df0f8bf5e8b2d504e5686a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.861ex; height:5.343ex;" alt="{\displaystyle {\frac {2\pi \hbar }{L}}(n_{1},n_{2},n_{3})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1},n_{2},n_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1},n_{2},n_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ad7be042695833cfb38db88f2293c75cfcccd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.415ex; height:2.009ex;" alt="{\displaystyle n_{1},n_{2},n_{3}}"></span> are integers.</li> <li>The sum over <span class="texhtml mvar" style="font-style:italic;">r</span> covers other degrees of freedom specific for the field, such as polarization or spin; it usually comes out as a sum from <span class="texhtml">1</span> to <span class="texhtml">2</span> or from <span class="texhtml">1</span> to <span class="texhtml">3</span>.</li> <li><span class="texhtml"><i>E</i><sub><b>p</b></sub></span> is the relativistic energy for a momentum <span class="texhtml"><b>p</b></span> quantum of the field, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle ={\sqrt {m^{2}c^{4}+c^{2}\mathbf {p} ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle ={\sqrt {m^{2}c^{4}+c^{2}\mathbf {p} ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2846abf1bd17749af9c20277e0cf93cd242b2005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.374ex; height:3.509ex;" alt="{\textstyle ={\sqrt {m^{2}c^{4}+c^{2}\mathbf {p} ^{2}}}}"></span> when the rest mass is <span class="texhtml mvar" style="font-style:italic;">m</span>.</li> <li><span class="texhtml"><i>a<sub>r</sub></i>(<b>p</b>)</span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{r}^{\dagger }(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{r}^{\dagger }(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39277116349803c5bd5d1ba1af20785788bea022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.266ex; height:3.343ex;" alt="{\displaystyle b_{r}^{\dagger }(\mathbf {p} )}"></span> are <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">annihilation and creation</a> operators respectively for "a-particles" and "b-particles" respectively of momentum <span class="texhtml"><b>p</b></span>; "b-particles" are the <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a> of "a-particles". Different fields have different "a-" and "b-particles". For some fields, <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are the same.</li> <li><span class="texhtml"><i>u<sub>r</sub></i>(<b>p</b>)</span> and <span class="texhtml"><i>v<sub>r</sub></i>(<b>p</b>)</span> are non-operators that carry the vector or spinor aspects of the field (where relevant).</li> <li><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 p=(E_{\mathbf {p} }/c,\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(E_{\mathbf {p} }/c,\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4914d9562587afd49e75b8bd3e6b61632ddc82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:13.853ex; height:3.009ex;" alt="{\displaystyle p=(E_{\mathbf {p} }/c,\mathbf {p} )}"></span> is the <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> for a quantum with momentum <span class="texhtml"><b>p</b></span>. <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 px=p_{\mu }x^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle px=p_{\mu }x^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e32c46dd81b349672195264f31c5c691e1163e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:10.633ex; height:3.009ex;" alt="{\displaystyle px=p_{\mu }x^{\mu }}"></span> denotes an inner product of <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a>.</li></ul> <p>In the limit <span class="texhtml"><i>L</i> → ∞</span>, the sum would turn into an integral with help from the <span class="texhtml mvar" style="font-style:italic;">V</span> hidden inside <span class="texhtml mvar" style="font-style:italic;">β</span>. The numeric value of <span class="texhtml mvar" style="font-style:italic;">β</span> also depends on the normalization chosen for <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 u_{r}(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{r}(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c080be01f62184a881f76a40c3e49b326c4504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.598ex; height:2.843ex;" alt="{\displaystyle u_{r}(\mathbf {p} )}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{r}(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{r}(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93f3d96ffd1e11c539eb04616cbf2a83626c547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.396ex; height:2.843ex;" alt="{\displaystyle v_{r}(\mathbf {p} )}"></span>. </p><p>Technically, <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_{r}^{\dagger }(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38b0413c7845de8329f5ffa81b4cf3593b65d14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.498ex; height:3.343ex;" alt="{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}"></span> is the <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian adjoint</a> of the operator <span class="texhtml"><i>a<sub>r</sub></i>(<b>p</b>)</span> in the <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> of <a href="/wiki/Ket_vector" class="mw-redirect" title="Ket vector">ket vectors</a>. The identification of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38b0413c7845de8329f5ffa81b4cf3593b65d14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.498ex; height:3.343ex;" alt="{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}"></span> and <span class="texhtml"><i>a<sub>r</sub></i>(<b>p</b>)</span> as <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a> comes from comparing conserved quantities for a state before and after one of these have acted upon it. <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_{r}^{\dagger }(\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38b0413c7845de8329f5ffa81b4cf3593b65d14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.498ex; height:3.343ex;" alt="{\displaystyle a_{r}^{\dagger }(\mathbf {p} )}"></span> can for example be seen to add one particle, because it will add <span class="texhtml">1</span> to the eigenvalue of the a-particle <a href="/wiki/Number_operator" class="mw-redirect" title="Number operator">number operator</a>, and the momentum of that particle ought to be <span class="texhtml"><b>p</b></span> since the eigenvalue of the vector-valued <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a> increases by that much. For these derivations, one starts out with expressions for the operators in terms of the quantum fields. That the operators with <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 \dagger }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>†</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dagger }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbce70d5be6fec538cd30d8bc7b7bb2d3ed2d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.032ex; height:2.676ex;" alt="{\displaystyle \dagger }"></span> are creation operators and the one without annihilation operators is a convention, imposed by the sign of the commutation relations postulated for them. </p><p>An important step in preparation for calculating in perturbative quantum field theory is to separate the "operator" factors <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> above from their corresponding vector or spinor factors <span class="texhtml mvar" style="font-style:italic;">u</span> and <span class="texhtml mvar" style="font-style:italic;">v</span>. The vertices of <a href="/wiki/Feynman_graph" class="mw-redirect" title="Feynman graph">Feynman graphs</a> come from the way that <span class="texhtml mvar" style="font-style:italic;">u</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> from different factors in the interaction Lagrangian fit together, whereas the edges come from the way that the <span class="texhtml mvar" style="font-style:italic;">a</span>s and <span class="texhtml mvar" style="font-style:italic;">b</span>s must be moved around in order to put terms in the Dyson series on normal form. </p> <div class="mw-heading mw-heading3"><h3 id="Interaction_terms_and_the_path_integral_approach">Interaction terms and the path integral approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=10" title="Edit section: Interaction terms and the path integral approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Lagrangian can also be derived without using creation and annihilation operators (the "canonical" formalism) by using a <a href="/wiki/Path_integral_formulation#Quantum_field_theory" title="Path integral formulation">path integral formulation</a>, pioneered by Feynman building on the earlier work of Dirac. <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagrams</a> are pictorial representations of interaction terms. A quick derivation is indeed presented at the article on <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagrams</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Lagrangian_formalism">Lagrangian formalism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=11" title="Edit section: Lagrangian formalism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg/330px-Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg.png" decoding="async" width="330" height="389" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg/495px-Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg/660px-Standard_Model_%E2%80%93_All_Feynman_diagram_vertices.svg.png 2x" data-file-width="899" data-file-height="1059" /></a><figcaption>Interactions in the Standard Model. All Feynman diagrams in the model are built from combinations of these vertices. <i>q</i> is any quark, <i>g</i> is a gluon, <i>X</i> is any charged particle, γ is a photon, <i>f</i> is any fermion, <i>m</i> is any particle with mass (with the possible exception of the neutrinos), <i>m</i><sub>B</sub> is any boson with mass. In diagrams with multiple particle labels separated by / one particle label is chosen. In diagrams with particle labels separated by | the labels must be chosen in the same order. For example, in the four boson electroweak case the valid diagrams are WWWW, WWZZ, WWγγ, WWZγ. The conjugate of each listed vertex (reversing the direction of arrows) is also allowed.<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></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg/500px-Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg" decoding="async" width="500" height="740" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg/750px-Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg/1000px-Formula_of_the_Standard_Model_of_particle_physics_-_Lagrangian_L_SM.jpg 2x" data-file-width="1842" data-file-height="2727" /></a><figcaption>The full expanded form of the Standard Model Lagrangian</figcaption></figure> <p>We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model <a href="/wiki/Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</a>. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an observer. Therefore, the <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">global</a> <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré symmetry</a>, consisting of <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a>, <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> and the inertial reference frame invariance central to the theory of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> must apply. The <a href="/wiki/Local_symmetry" class="mw-redirect" title="Local symmetry">local</a> <span class="texhtml">SU(3) × SU(2) × U(1)</span> gauge symmetry is the <a href="/wiki/Internal_symmetries" class="mw-redirect" title="Internal symmetries">internal symmetry</a>. The three factors of the gauge symmetry together give rise to the three fundamental interactions, after some appropriate relations have been defined, as we shall see. </p> <div class="mw-heading mw-heading3"><h3 id="Kinetic_terms">Kinetic terms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=12" title="Edit section: Kinetic terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A free particle can be represented by a mass term, and a <i>kinetic</i> term that relates to the "motion" of the fields. </p> <div class="mw-heading mw-heading4"><h4 id="Fermion_fields">Fermion fields</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=13" title="Edit section: Fermion fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The kinetic term for a Dirac fermion is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68cb2b4058d570ef0336f4ad57a7f827c82ab80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.872ex; height:3.176ex;" alt="{\displaystyle i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi }"></span> where the notations are carried from earlier in the article. <span class="texhtml mvar" style="font-style:italic;">ψ</span> can represent any, or all, Dirac fermions in the standard model. Generally, as below, this term is included within the couplings (creating an overall "dynamical" term). </p> <div class="mw-heading mw-heading4"><h4 id="Gauge_fields">Gauge fields</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=14" title="Edit section: Gauge fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the spin-1 fields, first define the field strength <a href="/wiki/Tensor" title="Tensor">tensor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mo>+</mo> <mi>g</mi> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee1f0e6ee46c6e31e5ef520891bf5c2f0d2076e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.746ex; height:3.343ex;" alt="{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}"></span> for a given gauge field (here we use <span class="texhtml mvar" style="font-style:italic;">A</span>), with gauge <a href="/wiki/Coupling_constant" title="Coupling constant">coupling constant</a> <span class="texhtml mvar" style="font-style:italic;">g</span>. The quantity <span class="texhtml"><span style="padding-right:0.15em;"><i>f</i></span><sup><i>abc</i></sup></span> is the <a href="/wiki/Algebra_over_a_field#Structure_coefficients" title="Algebra over a field">structure constant</a> of the particular gauge group, defined by the commutator <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [t_{a},t_{b}]=if^{abc}t_{c},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [t_{a},t_{b}]=if^{abc}t_{c},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af631fdfab4983933a13ca7c98b2076a32546743" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.218ex; height:3.176ex;" alt="{\displaystyle [t_{a},t_{b}]=if^{abc}t_{c},}"></span> where <span class="texhtml mvar" style="font-style:italic;">t<sub>i</sub></span> are the <a href="/wiki/Lie_algebra#Generators_and_dimension" title="Lie algebra">generators</a> of the group. In an <a href="/wiki/Abelian_group" title="Abelian group">abelian (commutative) group</a> (such as the <span class="texhtml">U(1)</span> we use here) the structure constants vanish, since the generators <span class="texhtml mvar" style="font-style:italic;">t<sub>a</sub></span> all commute with each other. Of course, this is not the case in general – the standard model includes the non-Abelian <span class="texhtml">SU(2)</span> and <span class="texhtml">SU(3)</span> groups (such groups lead to what is called a <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills gauge theory</a>). </p><p>We need to introduce three gauge fields corresponding to each of the subgroups <span class="texhtml">SU(3) × SU(2) × <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a></span>. </p> <ul><li>The gluon field tensor will be denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242f3b18426d21e3bfc89678604c9d90fd51987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.921ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }^{a}}"></span>, where the index <span class="texhtml mvar" style="font-style:italic;">a</span> labels elements of the <span class="texhtml"><b>8</b></span> representation of color <a href="/wiki/Special_unitary_group" title="Special unitary group">SU(3)</a>. The strong coupling constant is conventionally labelled <span class="texhtml"><i>g</i><sub>s</sub></span> (or simply <span class="texhtml mvar" style="font-style:italic;">g</span> where there is no ambiguity). <i>The observations leading to the discovery of this part of the Standard Model are discussed in the article in <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>.</i></li> <li>The notation <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 W_{\mu \nu }^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\mu \nu }^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63103d3a55400f3d19497e29634009c9d05a6ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.288ex; height:2.843ex;" alt="{\displaystyle W_{\mu \nu }^{a}}"></span> will be used for the gauge field tensor of <span class="texhtml"><a href="/wiki/Special_unitary_group" title="Special unitary group">SU(2)</a></span> where <span class="texhtml mvar" style="font-style:italic;">a</span> runs over the <span class="texhtml">3</span> generators of this group. The coupling can be denoted <span class="texhtml"><i>g</i><sub>w</sub></span> or again simply <span class="texhtml mvar" style="font-style:italic;">g</span>. The gauge field will be denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\mu }^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\mu }^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900a6ee8c880be1e7089aa8bef2aa29b02de241e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.61ex; height:2.843ex;" alt="{\displaystyle W_{\mu }^{a}}"></span>.</li> <li>The gauge field tensor for the <span class="texhtml">U(1)</span> of weak hypercharge will be denoted by <span class="texhtml"><i>B</i><sub><i>μν</i></sub></span>, the coupling by <span class="texhtml"><i>g</i>′</span>, and the gauge field by <span class="texhtml mvar" style="font-style:italic;">B<sub>μ</sub></span>.</li></ul> <p>The kinetic term can now be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\rm {kin}}=-{1 \over 4}B_{\mu \nu }B^{\mu \nu }-{1 \over 2}\mathrm {tr} W_{\mu \nu }W^{\mu \nu }-{1 \over 2}\mathrm {tr} G_{\mu \nu }G^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\rm {kin}}=-{1 \over 4}B_{\mu \nu }B^{\mu \nu }-{1 \over 2}\mathrm {tr} W_{\mu \nu }W^{\mu \nu }-{1 \over 2}\mathrm {tr} G_{\mu \nu }G^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfe5dc666edb3959ea3f7399c60185b6aedeb2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.742ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{\rm {kin}}=-{1 \over 4}B_{\mu \nu }B^{\mu \nu }-{1 \over 2}\mathrm {tr} W_{\mu \nu }W^{\mu \nu }-{1 \over 2}\mathrm {tr} G_{\mu \nu }G^{\mu \nu }}"></span> where the traces are over the <span class="texhtml">SU(2)</span> and <span class="texhtml">SU(3)</span> indices hidden in <span class="texhtml mvar" style="font-style:italic;">W</span> and <span class="texhtml mvar" style="font-style:italic;">G</span> respectively. The two-index objects are the field strengths derived from <span class="texhtml mvar" style="font-style:italic;">W</span> and <span class="texhtml mvar" style="font-style:italic;">G</span> the vector fields. There are also two extra hidden parameters: the theta angles for <span class="texhtml">SU(2)</span> and <span class="texhtml">SU(3)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Coupling_terms">Coupling terms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=15" title="Edit section: Coupling terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The next step is to "couple" the gauge fields to the fermions, allowing for interactions. </p> <div class="mw-heading mw-heading4"><h4 id="Electroweak_sector">Electroweak sector</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=16" title="Edit section: Electroweak sector"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></div> <p>The electroweak sector interacts with the symmetry group <span class="texhtml">U(1) × SU(2)<sub>L</sub></span>, where the subscript L indicates coupling only to left-handed fermions. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\mathrm {EW} }=\sum _{\psi }{\bar {\psi }}\gamma ^{\mu }\left(i\partial _{\mu }-g^{\prime }{1 \over 2}Y_{\mathrm {W} }B_{\mu }-g{1 \over 2}{\boldsymbol {\tau }}\mathbf {W} _{\mu }\right)\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\mathrm {EW} }=\sum _{\psi }{\bar {\psi }}\gamma ^{\mu }\left(i\partial _{\mu }-g^{\prime }{1 \over 2}Y_{\mathrm {W} }B_{\mu }-g{1 \over 2}{\boldsymbol {\tau }}\mathbf {W} _{\mu }\right)\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b39a819006f2c9b1f71871a0d24b9eef783874b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.813ex; height:7.009ex;" alt="{\displaystyle {\mathcal {L}}_{\mathrm {EW} }=\sum _{\psi }{\bar {\psi }}\gamma ^{\mu }\left(i\partial _{\mu }-g^{\prime }{1 \over 2}Y_{\mathrm {W} }B_{\mu }-g{1 \over 2}{\boldsymbol {\tau }}\mathbf {W} _{\mu }\right)\psi }"></span> where <span class="texhtml mvar" style="font-style:italic;">B<sub>μ</sub></span> is the <span class="texhtml">U(1)</span> gauge field; <span class="texhtml"><i>Y</i><sub>W</sub></span> is the <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharge</a> (the generator of the <span class="texhtml">U(1)</span> group); <span class="texhtml"><b>W</b><sub><i>μ</i></sub></span> is the three-component <span class="texhtml">SU(2)</span> gauge field; and the components of <span class="texhtml"><b>τ</b></span> are the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> (infinitesimal generators of the <span class="texhtml">SU(2)</span> group) whose eigenvalues give the weak isospin. Note that we have to redefine a new <span class="texhtml"><a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a></span> symmetry of <i>weak hypercharge</i>, different from QED, in order to achieve the unification with the weak force. The <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> <span class="texhtml mvar" style="font-style:italic;">Q</span>, third component of <a href="/wiki/Weak_isospin" title="Weak isospin">weak isospin</a> <span class="texhtml"><i>T</i><sub>3</sub></span> (also called <span class="texhtml"><i>T<sub>z</sub></i>, <i>I</i><sub>3</sub></span> or <span class="texhtml mvar" style="font-style:italic;">I<sub>z</sub></span>) and weak hypercharge <span class="texhtml"><i>Y</i><sub>W</sub></span> are related by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\rm {W}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\rm {W}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d73ffcf2927cf5490b30a30b6f84ac43892a210" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.766ex; height:3.509ex;" alt="{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\rm {W}},}"></span> (or by the <i>alternative convention</i> <span class="texhtml"><i>Q</i> = <i>T</i><sub>3</sub> + <i>Y</i><sub>W</sub></span>). The first convention, used in this article, is equivalent to the earlier <a href="/wiki/Gell-Mann%E2%80%93Nishijima_formula" title="Gell-Mann–Nishijima formula">Gell-Mann–Nishijima formula</a>. It makes the hypercharge be twice the average charge of a given isomultiplet. </p><p>One may then define the <a href="/wiki/Conserved_current" title="Conserved current">conserved current</a> for weak isospin as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} _{\mu }={1 \over 2}{\bar {\psi }}_{\rm {L}}\gamma _{\mu }{\boldsymbol {\tau }}\psi _{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} _{\mu }={1 \over 2}{\bar {\psi }}_{\rm {L}}\gamma _{\mu }{\boldsymbol {\tau }}\psi _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bfa6b0eb8e59171ffe915c4776fbefa122acb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.164ex; width:16.774ex; height:5.176ex;" alt="{\displaystyle \mathbf {j} _{\mu }={1 \over 2}{\bar {\psi }}_{\rm {L}}\gamma _{\mu }{\boldsymbol {\tau }}\psi _{\rm {L}}}"></span> and for weak hypercharge as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }^{Y}=2(j_{\mu }^{\rm {em}}-j_{\mu }^{3})~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{\mu }^{Y}=2(j_{\mu }^{\rm {em}}-j_{\mu }^{3})~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108b1879dc578c82079460b7ec356a96ca99ee22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:18.08ex; height:3.176ex;" alt="{\displaystyle j_{\mu }^{Y}=2(j_{\mu }^{\rm {em}}-j_{\mu }^{3})~,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }^{\rm {em}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{\mu }^{\rm {em}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5507276fd8729ab0b9f735f33c188a120c2ac32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:3.316ex; height:2.843ex;" alt="{\displaystyle j_{\mu }^{\rm {em}}}"></span> is the electric current and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{\mu }^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1865af81a88e527b47e1949ad1234592907421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:2.208ex; height:3.176ex;" alt="{\displaystyle j_{\mu }^{3}}"></span> the third weak isospin current. As explained <a href="#Bosons">above</a>, <i>these currents mix</i> to create the physically observed bosons, which also leads to testable relations between the coupling constants. </p><p>To explain this in a simpler way, we can see the effect of the electroweak interaction by picking out terms from the Lagrangian. We see that the SU(2) symmetry acts on each (left-handed) fermion doublet contained in <span class="texhtml mvar" style="font-style:italic;">ψ</span>, for example <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{g \over 2}({\bar {\nu }}_{e}\;{\bar {e}})\tau ^{+}\gamma _{\mu }(W^{+})^{\mu }{\begin{pmatrix}{\nu _{e}}\\e\end{pmatrix}}=-{g \over 2}{\bar {\nu }}_{e}\gamma _{\mu }(W^{+})^{\mu }e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>e</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{g \over 2}({\bar {\nu }}_{e}\;{\bar {e}})\tau ^{+}\gamma _{\mu }(W^{+})^{\mu }{\begin{pmatrix}{\nu _{e}}\\e\end{pmatrix}}=-{g \over 2}{\bar {\nu }}_{e}\gamma _{\mu }(W^{+})^{\mu }e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990fe23b5d5a45c16f7600e2609605ccd81f3fed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.217ex; height:6.176ex;" alt="{\displaystyle -{g \over 2}({\bar {\nu }}_{e}\;{\bar {e}})\tau ^{+}\gamma _{\mu }(W^{+})^{\mu }{\begin{pmatrix}{\nu _{e}}\\e\end{pmatrix}}=-{g \over 2}{\bar {\nu }}_{e}\gamma _{\mu }(W^{+})^{\mu }e}"></span> where the particles are understood to be left-handed, and where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ^{+}\equiv {1 \over 2}(\tau ^{1}{+}i\tau ^{2})={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mi>i</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ^{+}\equiv {1 \over 2}(\tau ^{1}{+}i\tau ^{2})={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a61324392a83c5ea36eed270dd71889c4594ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.828ex; height:6.176ex;" alt="{\displaystyle \tau ^{+}\equiv {1 \over 2}(\tau ^{1}{+}i\tau ^{2})={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}"></span> </p><p>This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a transformation between <span class="texhtml">e<sub>L</sub></span> and <span class="texhtml">ν<sub>eL</sub></span> via emission of a <span class="texhtml"><i>W</i><sup>−</sup></span> boson. The <span class="texhtml">U(1)</span> symmetry, on the other hand, is similar to electromagnetism, but acts on all "<i>weak hypercharged</i>" fermions (both left- and right-handed) via the neutral <span class="texhtml"><i>Z</i><sup>0</sup></span>, as well as the <i>charged</i> fermions via the photon. </p> <div class="mw-heading mw-heading4"><h4 id="Quantum_chromodynamics_sector">Quantum chromodynamics sector</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=17" title="Edit section: Quantum chromodynamics sector"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></div> <p>The quantum chromodynamics (QCD) sector defines the interactions between <a href="/wiki/Quark" title="Quark">quarks</a> and <a href="/wiki/Gluon" title="Gluon">gluons</a>, with <span class="texhtml">SU(3)</span> symmetry, generated by <span class="texhtml mvar" style="font-style:italic;">T<sub>a</sub></span>. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=i{\overline {U}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }U+i{\overline {D}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }D.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>U</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>U</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=i{\overline {U}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }U+i{\overline {D}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }D.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd79064f32b46806dc803efd275a8b9d40eacd3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.861ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=i{\overline {U}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }U+i{\overline {D}}\left(\partial _{\mu }-ig_{s}G_{\mu }^{a}T^{a}\right)\gamma ^{\mu }D.}"></span> where <span class="texhtml mvar" style="font-style:italic;">U</span> and <span class="texhtml mvar" style="font-style:italic;">D</span> are the Dirac spinors associated with up and down-type quarks, and other notations are continued from the previous section. </p> <div class="mw-heading mw-heading3"><h3 id="Mass_terms_and_the_Higgs_mechanism">Mass terms and the Higgs mechanism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=18" title="Edit section: Mass terms and the Higgs mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Mass_terms">Mass terms</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=19" title="Edit section: Mass terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mass term arising from the Dirac Lagrangian (for any fermion <span class="texhtml mvar" style="font-style:italic;">ψ</span>) is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m{\bar {\psi }}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -m{\bar {\psi }}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072b70f2c59b501059640d0d33f53469cd5534df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.957ex; height:2.843ex;" alt="{\displaystyle -m{\bar {\psi }}\psi }"></span>, which is <i>not</i> invariant under the electroweak symmetry. This can be seen by writing <span class="texhtml mvar" style="font-style:italic;">ψ</span> in terms of left and right-handed components (skipping the actual calculation): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m{\bar {\psi }}\psi =-m({\bar {\psi }}_{\rm {L}}\psi _{\rm {R}}+{\bar {\psi }}_{\rm {R}}\psi _{\rm {L}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -m{\bar {\psi }}\psi =-m({\bar {\psi }}_{\rm {L}}\psi _{\rm {R}}+{\bar {\psi }}_{\rm {R}}\psi _{\rm {L}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfdc43d8c7bd5bf8b0291cd7484fe3f5d9f9bfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.174ex; height:3.009ex;" alt="{\displaystyle -m{\bar {\psi }}\psi =-m({\bar {\psi }}_{\rm {L}}\psi _{\rm {R}}+{\bar {\psi }}_{\rm {R}}\psi _{\rm {L}})}"></span> i.e. contribution from <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 {\bar {\psi }}_{\rm {L}}\psi _{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}_{\rm {L}}\psi _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfcb6c389b54b0d823316ea13f5e4598e9efbf8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.628ex; height:3.009ex;" alt="{\displaystyle {\bar {\psi }}_{\rm {L}}\psi _{\rm {L}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\psi }}_{\rm {R}}\psi _{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\psi }}_{\rm {R}}\psi _{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e8162533006c98f3ecc1bb10c201ba075889ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.992ex; height:3.009ex;" alt="{\displaystyle {\bar {\psi }}_{\rm {R}}\psi _{\rm {R}}}"></span> terms do not appear. We see that the mass-generating interaction is achieved by constant flipping of particle chirality. The spin-half particles have no right/left chirality pair with the same <span class="texhtml">SU(2)</span> representations and equal and opposite weak hypercharges, so assuming these gauge charges are conserved in the vacuum, none of the spin-half particles could ever swap chirality, and must remain massless. Additionally, we know experimentally that the W and Z bosons are massive, but a boson mass term contains the combination e.g. <span class="texhtml mvar" style="font-style:italic;">A<sup>μ</sup>A<sub>μ</sub></span>, which clearly depends on the choice of gauge. Therefore, none of the standard model fermions <i>or</i> bosons can "begin" with mass, but must acquire it by some other mechanism. </p> <div class="mw-heading mw-heading4"><h4 id="Higgs_mechanism">Higgs mechanism</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=20" title="Edit section: Higgs mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></div> <p>The solution to both these problems comes from the <a href="/wiki/Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a>, which involves scalar fields (the number of which depend on the exact form of Higgs mechanism) which (to give the briefest possible description) are "absorbed" by the massive bosons as degrees of freedom, and which couple to the fermions via Yukawa coupling to create what looks like mass terms. </p><p>In the Standard Model, the <a href="/wiki/Higgs_field" class="mw-redirect" title="Higgs field">Higgs field</a> is a complex scalar field of the group <span class="texhtml"><a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a><sub>L</sub></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\phi ^{+}\\\phi ^{0}\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\phi ^{+}\\\phi ^{0}\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7e234d193354fd4dcd0c16d24a0636ad696429" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.134ex; height:6.509ex;" alt="{\displaystyle \phi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\phi ^{+}\\\phi ^{0}\end{pmatrix}},}"></span> where the superscripts <span class="texhtml">+</span> and <span class="texhtml">0</span> indicate the electric charge (<span class="texhtml mvar" style="font-style:italic;">Q</span>) of the components. The weak hypercharge (<span class="texhtml"><i>Y</i><sub>W</sub></span>) of both components is <span class="texhtml">1</span>. </p><p>The Higgs part of the Lagrangian is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\rm {H}}=\left[\left(\partial _{\mu }-igW_{\mu }^{a}t^{a}-ig'Y_{\phi }B_{\mu }\right)\phi \right]^{2}+\mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi>g</mi> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>−</mo> <mi>i</mi> <msup> <mi>g</mi> <mo>′</mo> </msup> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>ϕ</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>ϕ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\rm {H}}=\left[\left(\partial _{\mu }-igW_{\mu }^{a}t^{a}-ig'Y_{\phi }B_{\mu }\right)\phi \right]^{2}+\mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfc646435cfe64e21ef791f7a52d07f11b3ae10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.329ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{\rm {H}}=\left[\left(\partial _{\mu }-igW_{\mu }^{a}t^{a}-ig'Y_{\phi }B_{\mu }\right)\phi \right]^{2}+\mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2},}"></span> where <span class="texhtml"><i>λ</i> > 0</span> and <span class="texhtml"><i>μ</i><sup>2</sup> > 0</span>, so that the mechanism of <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a> can be used. There is a parameter here, at first hidden within the shape of the potential, that is very important. In a <a href="/wiki/Unitarity_gauge" title="Unitarity gauge">unitarity gauge</a> one can set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{+}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{+}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264b5eaba9047758c83908386f441b9e288753d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.157ex; height:2.843ex;" alt="{\displaystyle \phi ^{+}=0}"></span> and make <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391d8cde37179d147230a1f7c6db6722c5b580e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.44ex; height:3.009ex;" alt="{\displaystyle \phi ^{0}}"></span> real. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi ^{0}\rangle =v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi ^{0}\rangle =v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b110368a2da1190524f89d7413a48f3b556395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.475ex; height:3.176ex;" alt="{\displaystyle \langle \phi ^{0}\rangle =v}"></span> is the non-vanishing <a href="/wiki/Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a> of the Higgs field. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> has units of mass, and it is the only parameter in the Standard Model that is not dimensionless. It is also much smaller than the Planck scale and about twice the Higgs mass, setting the scale for the mass of all other particles in the Standard Model. This is the only real fine-tuning to a small nonzero value in the Standard Model. Quadratic terms in <span class="texhtml mvar" style="font-style:italic;">W<sub>μ</sub></span> and <span class="texhtml mvar" style="font-style:italic;">B<sub>μ</sub></span> arise, which give masses to the W and Z bosons: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}M_{\rm {W}}&={\tfrac {1}{2}}vg\\M_{\rm {Z}}&={\tfrac {1}{2}}v{\sqrt {g^{2}+{g'}^{2}}}\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> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>v</mi> <mi>g</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>g</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{\rm {W}}&={\tfrac {1}{2}}vg\\M_{\rm {Z}}&={\tfrac {1}{2}}v{\sqrt {g^{2}+{g'}^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df619db5f1fa0fe6d7cef0431b1926c159d497c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:21.005ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}M_{\rm {W}}&={\tfrac {1}{2}}vg\\M_{\rm {Z}}&={\tfrac {1}{2}}v{\sqrt {g^{2}+{g'}^{2}}}\end{aligned}}}"></span> </p><p>The mass of the Higgs boson itself is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M_{\rm {H}}={\sqrt {2\mu ^{2}}}\equiv {\sqrt {2\lambda v^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>λ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M_{\rm {H}}={\sqrt {2\mu ^{2}}}\equiv {\sqrt {2\lambda v^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7cec2d04eef32635d5b207f0029cc8b6554a37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.14ex; height:3.509ex;" alt="{\textstyle M_{\rm {H}}={\sqrt {2\mu ^{2}}}\equiv {\sqrt {2\lambda v^{2}}}.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Yukawa_interaction">Yukawa interaction</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=21" title="Edit section: Yukawa interaction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Yukawa_interaction" title="Yukawa interaction">Yukawa interaction</a> terms are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\text{Yukawa}}=(Y_{\text{u}})_{mn}({\bar {q}}_{\text{L}})_{m}{\tilde {\varphi }}(u_{\text{R}})_{n}+(Y_{\text{d}})_{mn}({\bar {q}}_{\text{L}})_{m}\varphi (d_{\text{R}})_{n}+(Y_{\text{e}})_{mn}({\bar {L}}_{\text{L}})_{m}{\tilde {\varphi }}(e_{\text{R}})_{n}+\mathrm {h.c.} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Yukawa</mtext> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>u</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{Yukawa}}=(Y_{\text{u}})_{mn}({\bar {q}}_{\text{L}})_{m}{\tilde {\varphi }}(u_{\text{R}})_{n}+(Y_{\text{d}})_{mn}({\bar {q}}_{\text{L}})_{m}\varphi (d_{\text{R}})_{n}+(Y_{\text{e}})_{mn}({\bar {L}}_{\text{L}})_{m}{\tilde {\varphi }}(e_{\text{R}})_{n}+\mathrm {h.c.} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7a0b6d7ea61073cbf6940463529197a78028dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:84.214ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}_{\text{Yukawa}}=(Y_{\text{u}})_{mn}({\bar {q}}_{\text{L}})_{m}{\tilde {\varphi }}(u_{\text{R}})_{n}+(Y_{\text{d}})_{mn}({\bar {q}}_{\text{L}})_{m}\varphi (d_{\text{R}})_{n}+(Y_{\text{e}})_{mn}({\bar {L}}_{\text{L}})_{m}{\tilde {\varphi }}(e_{\text{R}})_{n}+\mathrm {h.c.} }"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\text{u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>u</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\text{u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/981c3edfdac996c9d21da2440a206e692692cf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.497ex; height:2.509ex;" alt="{\displaystyle Y_{\text{u}}}"></span>, <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 Y_{\text{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\text{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e39cac0d67e06243c7e65ffc0a78b748eb0455a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.497ex; height:2.509ex;" alt="{\displaystyle Y_{\text{d}}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\text{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\text{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edaf2398a78c9c4ad6924a848d3438fcd430ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.313ex; height:2.509ex;" alt="{\displaystyle Y_{\text{e}}}"></span> are <span class="texhtml">3 × 3</span> matrices of Yukawa couplings, with the <span class="texhtml mvar" style="font-style:italic;">mn</span> term giving the coupling of the generations <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span>, and h.c. means Hermitian conjugate of preceding terms. The fields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1a082c5a2c186b958595955e56d2a601453a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.297ex; height:2.009ex;" alt="{\displaystyle q_{\text{L}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b1139c23d9ede1a53953e8609cd9f619e3d8fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.842ex; height:2.509ex;" alt="{\displaystyle L_{\text{L}}}"></span> are left-handed quark and lepton doublets. Likewise, <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 u_{\text{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\text{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957b25b7d09c4a553a9824301f0732df1607e868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.772ex; height:2.009ex;" alt="{\displaystyle u_{\text{R}}}"></span>, <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_{\text{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\text{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd94931576e4b4fa23236106ad2e5e6d74c029c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.651ex; height:2.509ex;" alt="{\displaystyle d_{\text{R}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\text{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\text{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2898f8fed9bc6d9cde28f60f7af1883eede490bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.525ex; height:2.009ex;" alt="{\displaystyle e_{\text{R}}}"></span> are right-handed up-type quark, down-type quark, and lepton singlets. Finally <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is the Higgs doublet and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\varphi }}=i\tau _{2}\varphi ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>i</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\varphi }}=i\tau _{2}\varphi ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70704c70119d005759715c450a1f1d6813def007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.08ex; height:2.843ex;" alt="{\displaystyle {\tilde {\varphi }}=i\tau _{2}\varphi ^{*}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Neutrino_masses">Neutrino masses</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=22" title="Edit section: Neutrino masses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As previously mentioned, evidence shows neutrinos must have mass. But within the standard model, the right-handed neutrino does not exist, so even with a Yukawa coupling neutrinos remain massless. An obvious solution<sup id="cite_ref-fas.org_4-0" class="reference"><a href="#cite_note-fas.org-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> is to simply <i>add a right-handed neutrino</i> <span class="texhtml"><i>ν</i><sub>R</sub></span>, which requires the addition of a new <b>Dirac mass</b> term in the Yukawa sector: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\nu }^{\text{Dir}}=(Y_{\nu })_{mn}({\bar {L}}_{L})_{m}\varphi (\nu _{R})_{n}+\mathrm {h.c.} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Dir</mtext> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\nu }^{\text{Dir}}=(Y_{\nu })_{mn}({\bar {L}}_{L})_{m}\varphi (\nu _{R})_{n}+\mathrm {h.c.} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/642c73835214aeb962409a3ff62fe28271744a85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.657ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}_{\nu }^{\text{Dir}}=(Y_{\nu })_{mn}({\bar {L}}_{L})_{m}\varphi (\nu _{R})_{n}+\mathrm {h.c.} }"></span> </p><p>This field however must be a <a href="/wiki/Sterile_neutrino" title="Sterile neutrino">sterile neutrino</a>, since being right-handed it experimentally belongs to an isospin singlet (<span class="texhtml"><i>T</i><sub>3</sub> = 0</span>) and also has charge <span class="texhtml"><i>Q</i> = 0</span>, implying <span class="texhtml"><i>Y</i><sub>W</sub> = 0</span> (see <a href="#Electroweak_sector">above</a>) i.e. it does not even participate in the weak interaction. The experimental evidence for sterile neutrinos is currently inconclusive.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Another possibility to consider is that the neutrino satisfies the <b><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana equation</a></b>, which at first seems possible due to its zero electric charge. In this case a new <b>Majorana mass</b> term is added to the Yukawa sector: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\nu }^{\text{Maj}}=-{\frac {1}{2}}m\left({\overline {\nu }}^{C}\nu +{\overline {\nu }}\nu ^{C}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Maj</mtext> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> <mi>ν</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\nu }^{\text{Maj}}=-{\frac {1}{2}}m\left({\overline {\nu }}^{C}\nu +{\overline {\nu }}\nu ^{C}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e29d7deb9469846c862c8de0a00574e8fda9517" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.872ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{\nu }^{\text{Maj}}=-{\frac {1}{2}}m\left({\overline {\nu }}^{C}\nu +{\overline {\nu }}\nu ^{C}\right)}"></span> where <span class="texhtml mvar" style="font-style:italic;">C</span> denotes a charge conjugated (i.e. anti-) particle, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> terms are consistently all left (or all right) chirality (note that a left-chirality projection of an antiparticle is a right-handed field; care must be taken here due to different notations sometimes used). Here we are essentially flipping between left-handed neutrinos and right-handed anti-neutrinos (it is furthermore possible but <i>not</i> necessary that neutrinos are their own antiparticle, so these particles are the same). However, for left-chirality neutrinos, this term changes weak hypercharge by 2 units – not possible with the standard Higgs interaction, requiring the Higgs field to be extended to include an extra triplet with weak hypercharge = 2<sup id="cite_ref-fas.org_4-1" class="reference"><a href="#cite_note-fas.org-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> – whereas for right-chirality neutrinos, no Higgs extensions are necessary. For both left and right chirality cases, Majorana terms violate <a href="/wiki/Lepton_number" title="Lepton number">lepton number</a>, but possibly at a level beyond the current sensitivity of experiments to detect such violations. </p><p>It is possible to include <b>both</b> Dirac and Majorana mass terms in the same theory, which (in contrast to the Dirac-mass-only approach) can provide a “natural” explanation for the smallness of the observed neutrino masses, by linking the right-handed neutrinos to yet-unknown physics around the GUT scale<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (see <a href="/wiki/Seesaw_mechanism" title="Seesaw mechanism">seesaw mechanism</a>). </p><p>Since in any case new fields must be postulated to explain the experimental results, neutrinos are an obvious gateway to searching physics <a href="/wiki/Beyond_the_Standard_Model" class="mw-redirect" title="Beyond the Standard Model">beyond the Standard Model</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Detailed_information">Detailed information</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=23" title="Edit section: Detailed information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section provides more detail on some aspects, and some reference material. Explicit Lagrangian terms are also provided <a href="/wiki/Electroweak_interaction#After_electroweak_symmetry_breaking" title="Electroweak interaction">here</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Field_content_in_detail">Field content in detail</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=24" title="Edit section: Field content in detail"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Standard Model has the following fields. These describe one <i>generation</i> of leptons and quarks, and there are three generations, so there are three copies of each fermionic field. By CPT symmetry, there is a set of fermions and antifermions with opposite parity and charges. If a left-handed fermion spans some representation its antiparticle (right-handed antifermion) spans the <a href="/wiki/Dual_representation" title="Dual representation">dual representation</a><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> (note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbf {2} }}={\mathbf {2} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {2} }}={\mathbf {2} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7116953efd1441e0846d2688bc417074ba090a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.772ex; height:2.509ex;" alt="{\displaystyle {\bar {\mathbf {2} }}={\mathbf {2} }}"></span> for SU(2), because it is <a href="/wiki/Pseudo-real" class="mw-redirect" title="Pseudo-real">pseudo-real</a>). The column "<b>representation</b>" indicates under which <a href="/wiki/Representation_theory" title="Representation theory">representations</a> of the <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge groups</a> that each field transforms, in the order (SU(3), SU(2), U(1)) and for the U(1) group, the value of the <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharge</a> is listed. There are twice as many left-handed lepton field components as right-handed lepton field components in each generation, but an equal number of left-handed quark and right-handed quark field components. </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th colspan="5">Field content of the standard model </th></tr> <tr> <th colspan="5" style="background:#ffdead">Spin 1 – the gauge fields </th></tr> <tr> <th>Symbol</th> <th>Associated charge</th> <th>Group</th> <th>Coupling</th> <th>Representation<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> </th></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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></td> <td><a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a></td> <td><span class="texhtml">U(1)<sub><i>Y</i></sub></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 g'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a53c0df5d85b36e3fd327c74db998f679f4f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.803ex; height:2.843ex;" alt="{\displaystyle g'}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755e3e04ec295992b2b5331655ef83a500a05c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}"></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 (\mathbf {1} ,\mathbf {1} ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {1} ,\mathbf {1} ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a359fd93b783d4ea95a4a7db11ab0aef628c1d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.713ex; height:2.843ex;" alt="{\displaystyle (\mathbf {1} ,\mathbf {1} ,0)}"></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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span></td> <td><a href="/wiki/Weak_isospin" title="Weak isospin">Weak isospin</a></td> <td><span class="texhtml">SU(2)<sub>L</sub></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 g_{w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{w}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5cb25bc5be5673bbec228d4a5c6e1287e4cfd90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.518ex; height:2.009ex;" alt="{\displaystyle g_{w}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"></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 (\mathbf {1} ,\mathbf {3} ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {1} ,\mathbf {3} ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d409ebebfaaefc595ef0fb49c2bd0f214c9d2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.713ex; height:2.843ex;" alt="{\displaystyle (\mathbf {1} ,\mathbf {3} ,0)}"></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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span></td> <td><a href="/wiki/Color_charge" title="Color charge">color</a></td> <td><span class="texhtml">SU(3)<sub><i>C</i></sub></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 g_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddd218a9b5b346d0dd85e51359c7cab5adf8697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.009ex;" alt="{\displaystyle g_{s}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207852a9b2705498c65b929b79fdda27b257ec87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}"></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 (\mathbf {8} ,\mathbf {1} ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">8</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {8} ,\mathbf {1} ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220b21322f7784660d79eb404d9fc66e42352ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.713ex; height:2.843ex;" alt="{\displaystyle (\mathbf {8} ,\mathbf {1} ,0)}"></span> </td></tr> <tr> <th colspan="5" style="background:#ffdead">Spin <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">2</span></span> – the fermions </th></tr> <tr> <th>Symbol</th> <th>Name</th> <th>Baryon number</th> <th>Lepton number</th> <th>Representation </th></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 q_{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d23d2731a0222d45d5d415886fd99f59b5ed0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.297ex; height:2.009ex;" alt="{\displaystyle q_{\rm {L}}}"></span></td> <td>Left-handed <a href="/wiki/Quark" title="Quark">quark</a></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 \textstyle {\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd0a7660677cda599a22c28518d7cea18e4e302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{3}}}"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 (\mathbf {3} ,\mathbf {2} ,\textstyle {\frac {1}{3}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> <mo>,</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {3} ,\mathbf {2} ,\textstyle {\frac {1}{3}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084e9555f5f280b5b679423a07da5bd17c682d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.209ex; height:3.676ex;" alt="{\displaystyle (\mathbf {3} ,\mathbf {2} ,\textstyle {\frac {1}{3}})}"></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 u_{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49533e4d687f9b4f1ba7ed87eac2e959b3ecba1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.772ex; height:2.009ex;" alt="{\displaystyle u_{\rm {R}}}"></span></td> <td>Right-handed quark (up)</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 \textstyle {\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd0a7660677cda599a22c28518d7cea18e4e302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{3}}}"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 ({\mathbf {3} },\mathbf {1} ,\textstyle {\frac {4}{3}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbf {3} },\mathbf {1} ,\textstyle {\frac {4}{3}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038194b3aaa1e0dc3b01f2adbe9aabfb324bf32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.209ex; height:3.676ex;" alt="{\displaystyle ({\mathbf {3} },\mathbf {1} ,\textstyle {\frac {4}{3}})}"></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 d_{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a349e59b77d5d20a291c2edd634da9d1817d37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.651ex; height:2.509ex;" alt="{\displaystyle d_{\rm {R}}}"></span></td> <td>Right-handed quark (down)</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 \textstyle {\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd0a7660677cda599a22c28518d7cea18e4e302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{3}}}"></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 ({\mathbf {3} },\mathbf {1} ,-\textstyle {\frac {2}{3}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbf {3} },\mathbf {1} ,-\textstyle {\frac {2}{3}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c32c4a68dcf39bd727803dc4908b7ad4304c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.017ex; height:3.676ex;" alt="{\displaystyle ({\mathbf {3} },\mathbf {1} ,-\textstyle {\frac {2}{3}})}"></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 \ell _{\rm {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82acd12a01ec1bf1994dcdca266a022d08cf2800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.229ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {L}}}"></span></td> <td>Left-handed <a href="/wiki/Lepton" title="Lepton">lepton</a></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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></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 (\mathbf {1} ,\mathbf {2} ,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {1} ,\mathbf {2} ,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7700b856902a01702629f9ae68d128e04e2c14b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.521ex; height:2.843ex;" alt="{\displaystyle (\mathbf {1} ,\mathbf {2} ,-1)}"></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 \ell _{\rm {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbe43f028177fab036f11e92dbba44f73db5c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle \ell _{\rm {R}}}"></span></td> <td>Right-handed lepton</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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></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 (\mathbf {1} ,\mathbf {1} ,-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {1} ,\mathbf {1} ,-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2b4703617cc6b5780328b4605a01919ce3e400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.521ex; height:2.843ex;" alt="{\displaystyle (\mathbf {1} ,\mathbf {1} ,-2)}"></span> </td></tr> <tr> <th colspan="5" style="background:#ffdead">Spin 0 – the scalar boson </th></tr> <tr> <th>Symbol</th> <th>Name</th> <th colspan="3">Representation </th></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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span></td> <td><a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a></td> <td colspan="3"><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 (\mathbf {1} ,\mathbf {2} ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {1} ,\mathbf {2} ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5d02fd11853aeacb8f4a3575e924176b4a601b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.713ex; height:2.843ex;" alt="{\displaystyle (\mathbf {1} ,\mathbf {2} ,1)}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Fermion_content">Fermion content</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=25" title="Edit section: Fermion content"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This table is based in part on data gathered by the <a href="/wiki/Particle_Data_Group" title="Particle Data Group">Particle Data Group</a>.<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> <table class="wikitable collapsible collapsed"> <tbody><tr> <th colspan="8">Left-handed fermions in the Standard Model </th></tr> <tr> <th colspan="8" style="background:#ffdead">Generation 1 </th></tr> <tr style="background:#fdd;"> <th>Fermion<br />(left-handed) </th> <th>Symbol </th> <th><a href="/wiki/Electric_charge" title="Electric charge">Electric<br />charge</a> </th> <th><a href="/wiki/Weak_isospin" title="Weak isospin">Weak<br />isospin</a> </th> <th><a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak<br />hypercharge</a> </th> <th><a href="/wiki/Color_charge" title="Color charge">Color<br />charge</a><br /><sup id="cite_ref-s1_10-0" class="reference"><a href="#cite_note-s1-10"><span class="cite-bracket">[</span>lhf 1<span class="cite-bracket">]</span></a></sup> </th> <th><a href="/wiki/Mass" title="Mass">Mass</a><sup id="cite_ref-s2_11-0" class="reference"><a href="#cite_note-s2-11"><span class="cite-bracket">[</span>lhf 2<span class="cite-bracket">]</span></a></sup> </th></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Electron" title="Electron">Electron</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>e<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>511 keV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Positron" title="Positron">Positron</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>e<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425979b37896d1911b58956146264cc1fb29aec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +2}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>511 keV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Electron_neutrino" title="Electron neutrino">Electron neutrino</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>ν<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">e</sub></span></span></span> </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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588217aab824f29bbd55e94f999fe5e700e2fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>< 0.28 eV<sup id="cite_ref-s4_12-0" class="reference"><a href="#cite_note-s4-12"><span class="cite-bracket">[</span>lhf 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s4b_13-0" class="reference"><a href="#cite_note-s4b-13"><span class="cite-bracket">[</span>lhf 4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Electron_antineutrino" class="mw-redirect" title="Electron antineutrino">Electron antineutrino</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">ν</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">e</sub></span></span></span> </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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>< 0.28 eV<sup id="cite_ref-s4_12-1" class="reference"><a href="#cite_note-s4-12"><span class="cite-bracket">[</span>lhf 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s4b_13-1" class="reference"><a href="#cite_note-s4b-13"><span class="cite-bracket">[</span>lhf 4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Up_quark" title="Up quark">Up quark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>u<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fdbfd39640b32923ac8d500266b717dfb25588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {2}{3}}}"></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 +{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588217aab824f29bbd55e94f999fe5e700e2fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}}"></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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }"></span> </td> <td>~ 3 MeV<sup id="cite_ref-s3_14-0" class="reference"><a href="#cite_note-s3-14"><span class="cite-bracket">[</span>lhf 5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Up_antiquark" class="mw-redirect" title="Up antiquark">Up antiquark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">u</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c863378259b3b309d6fb284b2698b0bea3e296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {2}{3}}}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 -{\tfrac {4}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {4}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af51191589a19a154f1008d6ca3d6ee8b7baeac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {4}{3}}}"></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 \mathbf {\bar {3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\bar {3}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2b4b8ffd1620c4bdbcbfef78b4555894a2d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.509ex;" alt="{\displaystyle \mathbf {\bar {3}} }"></span> </td> <td>~ 3 MeV<sup id="cite_ref-s3_14-1" class="reference"><a href="#cite_note-s3-14"><span class="cite-bracket">[</span>lhf 5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Down_quark" title="Down quark">Down quark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>d<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb3e88f738057777c96ebd0a7a7c7c1d1934696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {1}{3}}}"></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 -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"></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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }"></span> </td> <td>~ 6 MeV<sup id="cite_ref-s3_14-2" class="reference"><a href="#cite_note-s3-14"><span class="cite-bracket">[</span>lhf 5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Down_antiquark" class="mw-redirect" title="Down antiquark">Down antiquark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">d</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fdbfd39640b32923ac8d500266b717dfb25588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {2}{3}}}"></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 \mathbf {\bar {3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\bar {3}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2b4b8ffd1620c4bdbcbfef78b4555894a2d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.509ex;" alt="{\displaystyle \mathbf {\bar {3}} }"></span> </td> <td>~ 6 MeV<sup id="cite_ref-s3_14-3" class="reference"><a href="#cite_note-s3-14"><span class="cite-bracket">[</span>lhf 5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <th colspan="8">  </th></tr> <tr> <th colspan="8" style="background:#ffdead">Generation 2 </th></tr> <tr style="background:#fdd;"> <th>Fermion<br />(left-handed) </th> <th>Symbol </th> <th>Electric<br />charge </th> <th>Weak<br />isospin </th> <th>Weak<br />hypercharge </th> <th>Color<br />charge<br /><sup id="cite_ref-s1_10-1" class="reference"><a href="#cite_note-s1-10"><span class="cite-bracket">[</span>lhf 1<span class="cite-bracket">]</span></a></sup> </th> <th>Mass <sup id="cite_ref-s2_11-1" class="reference"><a href="#cite_note-s2-11"><span class="cite-bracket">[</span>lhf 2<span class="cite-bracket">]</span></a></sup> </th></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Muon" title="Muon">Muon</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>μ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>106 MeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Antimuon" class="mw-redirect" title="Antimuon">Antimuon</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>μ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425979b37896d1911b58956146264cc1fb29aec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +2}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>106 MeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Muon_neutrino" title="Muon neutrino">Muon neutrino</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>ν<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">μ</sub></span></span></span> </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 ~0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4d8f6b255cdb78e8d22b95546cc62caa13fa8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle ~0}"></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 +{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588217aab824f29bbd55e94f999fe5e700e2fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>< 0.28 eV<sup id="cite_ref-s4_12-2" class="reference"><a href="#cite_note-s4-12"><span class="cite-bracket">[</span>lhf 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s4b_13-2" class="reference"><a href="#cite_note-s4b-13"><span class="cite-bracket">[</span>lhf 4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Muon_antineutrino" class="mw-redirect" title="Muon antineutrino">Muon antineutrino</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">ν</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">μ</sub></span></span></span> </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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>< 0.28 eV<sup id="cite_ref-s4_12-3" class="reference"><a href="#cite_note-s4-12"><span class="cite-bracket">[</span>lhf 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s4b_13-3" class="reference"><a href="#cite_note-s4b-13"><span class="cite-bracket">[</span>lhf 4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Charm_quark" title="Charm quark">Charm quark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>c<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fdbfd39640b32923ac8d500266b717dfb25588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {2}{3}}}"></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 +{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588217aab824f29bbd55e94f999fe5e700e2fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}}"></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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }"></span> </td> <td>~ 1.3 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Charm_antiquark" class="mw-redirect" title="Charm antiquark">Charm antiquark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">c</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c863378259b3b309d6fb284b2698b0bea3e296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {2}{3}}}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 -{\tfrac {4}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {4}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af51191589a19a154f1008d6ca3d6ee8b7baeac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {4}{3}}}"></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 \mathbf {\bar {3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\bar {3}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2b4b8ffd1620c4bdbcbfef78b4555894a2d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.509ex;" alt="{\displaystyle \mathbf {\bar {3}} }"></span> </td> <td>~ 1.3 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Strange_quark" title="Strange quark">Strange quark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>s<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb3e88f738057777c96ebd0a7a7c7c1d1934696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {1}{3}}}"></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 -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"></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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }"></span> </td> <td>~ 100 MeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Strange_antiquark" class="mw-redirect" title="Strange antiquark">Strange antiquark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">s</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fdbfd39640b32923ac8d500266b717dfb25588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {2}{3}}}"></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 \mathbf {\bar {3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\bar {3}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2b4b8ffd1620c4bdbcbfef78b4555894a2d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.509ex;" alt="{\displaystyle \mathbf {\bar {3}} }"></span> </td> <td>~ 100 MeV </td></tr> <tr> <th colspan="8">  </th></tr> <tr> <th colspan="8" style="background:#ffdead">Generation 3 </th></tr> <tr style="background:#fdd;"> <th>Fermion<br />(left-handed) </th> <th>Symbol </th> <th>Electric<br />charge </th> <th>Weak<br />isospin </th> <th>Weak<br />hypercharge </th> <th>Color<br />charge<br /><sup id="cite_ref-s1_10-2" class="reference"><a href="#cite_note-s1-10"><span class="cite-bracket">[</span>lhf 1<span class="cite-bracket">]</span></a></sup> </th> <th>Mass<sup id="cite_ref-s2_11-2" class="reference"><a href="#cite_note-s2-11"><span class="cite-bracket">[</span>lhf 2<span class="cite-bracket">]</span></a></sup> </th></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Tau_(particle)" title="Tau (particle)">Tau</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>τ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>1.78 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Antitau" class="mw-redirect" title="Antitau">Antitau</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>τ<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425979b37896d1911b58956146264cc1fb29aec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +2}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>1.78 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Tau_neutrino" title="Tau neutrino">Tau neutrino</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>ν<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">τ</sub></span></span></span> </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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588217aab824f29bbd55e94f999fe5e700e2fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}}"></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 -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>< 0.28 eV<sup id="cite_ref-s4_12-4" class="reference"><a href="#cite_note-s4-12"><span class="cite-bracket">[</span>lhf 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s4b_13-4" class="reference"><a href="#cite_note-s4b-13"><span class="cite-bracket">[</span>lhf 4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Tau_antineutrino" class="mw-redirect" title="Tau antineutrino">Tau antineutrino</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">ν</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">τ</sub></span></span></span> </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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }"></span> </td> <td>< 0.28 eV<sup id="cite_ref-s4_12-5" class="reference"><a href="#cite_note-s4-12"><span class="cite-bracket">[</span>lhf 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s4b_13-5" class="reference"><a href="#cite_note-s4b-13"><span class="cite-bracket">[</span>lhf 4<span class="cite-bracket">]</span></a></sup> </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Top_quark" title="Top quark">Top quark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>t<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fdbfd39640b32923ac8d500266b717dfb25588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {2}{3}}}"></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 +{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7588217aab824f29bbd55e94f999fe5e700e2fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}}"></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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }"></span> </td> <td>171 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Top_antiquark" class="mw-redirect" title="Top antiquark">Top antiquark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">t</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c863378259b3b309d6fb284b2698b0bea3e296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {2}{3}}}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 -{\tfrac {4}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {4}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af51191589a19a154f1008d6ca3d6ee8b7baeac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {4}{3}}}"></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 \mathbf {\bar {3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\bar {3}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2b4b8ffd1620c4bdbcbfef78b4555894a2d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.509ex;" alt="{\displaystyle \mathbf {\bar {3}} }"></span> </td> <td>171 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Bottom_quark" title="Bottom quark">Bottom quark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>b<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 -{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb3e88f738057777c96ebd0a7a7c7c1d1934696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {1}{3}}}"></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 -{\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f99130274f09ef6b857419bf5c2529a6423597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}"></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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 \mathbf {3} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2210f71f47c2aef89cebab1b74ee6b14a05fa4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }"></span> </td> <td>~ 4.2 GeV </td></tr> <tr style="text-align:center;"> <td style="background:#efefef"><a href="/wiki/Bottom_antiquark" class="mw-redirect" title="Bottom antiquark">Bottom antiquark</a> </td> <td><span class="texhtml"> <span style="white-space:nowrap;"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span><span style="text-decoration:overline;">b</span><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></span> </span> </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 +{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178c7ff027440be4798036dec82f7787ca0ed319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {1}{3}}}"></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 ~\ 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\ 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3ab79b7d8d2c4721a4cfc4bd4a7d88ce6659c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle ~\ 0}"></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 +{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fdbfd39640b32923ac8d500266b717dfb25588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle +{\tfrac {2}{3}}}"></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 \mathbf {\bar {3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\bar {3}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2b4b8ffd1620c4bdbcbfef78b4555894a2d7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.509ex;" alt="{\displaystyle \mathbf {\bar {3}} }"></span> </td> <td>~ 4.2 GeV </td></tr> <tr> <th colspan="8">  </th></tr> <tr> <td colspan="8"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-s1-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-s1_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-s1_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-s1_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">These are not ordinary <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Charge_(physics)" title="Charge (physics)">charges</a>, which can be added together, but are labels of <a href="/wiki/Group_representation" title="Group representation">group representations</a> of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>.</span> </li> <li id="cite_note-s2-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-s2_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-s2_11-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-s2_11-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Mass is really a coupling between a left-handed fermion and a right-handed fermion. For example, the mass of an electron is really a coupling between a left-handed electron and a right-handed electron, which is the <a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</a> of a left-handed <a href="/wiki/Positron" title="Positron">positron</a>. Also neutrinos show large mixings in their mass coupling, so it's not accurate to talk about neutrino masses in the <a href="/wiki/Flavor_(particle_physics)" class="mw-redirect" title="Flavor (particle physics)">flavor</a> basis or to suggest a left-handed electron antineutrino.</span> </li> <li id="cite_note-s4-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-s4_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-s4_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-s4_12-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-s4_12-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-s4_12-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-s4_12-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text">The Standard Model assumes that neutrinos are massless. However, many contemporary experiments prove that <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrinos oscillate</a> between their <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavor</a> states, which could not happen if all were massless. It is straightforward to extend the model to fit these data but there are many possibilities, so the mass <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates</a> are still <a href="/wiki/Open_problem" title="Open problem">open</a>. See <a href="/wiki/Neutrino_mass" class="mw-redirect" title="Neutrino mass">neutrino mass</a>.</span> </li> <li id="cite_note-s4b-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-s4b_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-s4b_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-s4b_13-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-s4b_13-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-s4b_13-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-s4b_13-5"><sup><i><b>f</b></i></sup></a></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="CITEREFYao,_W.-M.2006" class="citation journal cs1">Yao, W.-M.; et al. (<a href="/wiki/Particle_Data_Group" title="Particle Data Group">Particle Data Group</a>) (2006). <a rel="nofollow" class="external text" href="http://pdg.lbl.gov/2007/reviews/numixrpp.pdf">"Review of Particle Physics: Neutrino mass, mixing, and flavor change"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_of_Physics_G" title="Journal of Physics G">Journal of Physics G</a></i>. <b>33</b> (1): 1. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0601168">astro-ph/0601168</a></span>. <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/2006JPhG...33....1Y">2006JPhG...33....1Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0954-3899%2F33%2F1%2F001">10.1088/0954-3899/33/1/001</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:117958297">117958297</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+G&rft.atitle=Review+of+Particle+Physics%3A+Neutrino+mass%2C+mixing%2C+and+flavor+change&rft.volume=33&rft.issue=1&rft.pages=1&rft.date=2006&rft_id=info%3Aarxiv%2Fastro-ph%2F0601168&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117958297%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0954-3899%2F33%2F1%2F001&rft_id=info%3Abibcode%2F2006JPhG...33....1Y&rft.au=Yao%2C+W.-M.&rft_id=http%3A%2F%2Fpdg.lbl.gov%2F2007%2Freviews%2Fnumixrpp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+the+Standard+Model" class="Z3988"></span> </span> </li> <li id="cite_note-s3-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-s3_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-s3_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-s3_14-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-s3_14-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">The <a href="/wiki/Mass" title="Mass">masses</a> of <a href="/wiki/Baryon" title="Baryon">baryons</a> and <a href="/wiki/Hadron" title="Hadron">hadrons</a> and various cross-sections are the experimentally measured quantities. Since quarks can't be isolated because of <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">QCD</a> <a href="/wiki/Color_confinement" title="Color confinement">confinement</a>, the quantity here is supposed to be the mass of the quark at the <a href="/wiki/Renormalization" title="Renormalization">renormalization</a> scale of the QCD scale.</span> </li> </ol></div> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Free_parameters">Free parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=26" title="Edit section: Free parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Upon writing the most general Lagrangian with massless neutrinos, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. Straightforward extensions of the Standard Model with massive neutrinos need 7 more parameters (3 masses and 4 PMNS matrix parameters) for a total of 26 parameters.<sup id="cite_ref-Thomson499_15-0" class="reference"><a href="#cite_note-Thomson499-15"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The neutrino parameter values are still uncertain. The 19 certain parameters are summarized here. </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th colspan="5">Parameters of the Standard Model </th></tr> <tr> <th>Symbol </th> <th>Description </th> <th>Renormalization<br /> scheme (point) </th> <th>Value </th></tr> <tr> <td><i>m</i><sub>e</sub> </td> <td>electron mass </td> <td> </td> <td><span class="nowrap"><span data-sort-value="6986818710503677190♠"></span>0.510<span style="margin-left:.25em;">998</span><span style="margin-left:.25em;">950</span><span style="margin-left:.25em;">69</span>(16) MeV</span>/<i>c</i><sup>2</sup> </td></tr> <tr> <td><i>m</i><sub>μ</sub> </td> <td>muon mass </td> <td> </td> <td><span class="nowrap"><span data-sort-value="6989169283364880716♠"></span>105.658<span style="margin-left:.25em;">3755</span>(23) MeV</span>/<i>c</i><sup>2</sup> </td></tr> <tr> <td><i>m</i><sub>τ</sub> </td> <td>tau mass </td> <td> </td> <td><span class="nowrap"><span data-sort-value="6990284684331269082♠"></span>1<span style="margin-left:.25em;">776</span>.86(12) MeV</span>/<i>c</i><sup>2</sup> </td></tr> <tr> <td><i>m</i><sub>u</sub> </td> <td>up quark mass </td> <td><i>μ</i><sub><a href="/wiki/MSbar_scheme" class="mw-redirect" title="MSbar scheme"><span style="text-decoration:overline;">MS</span></a></sub> = 2 GeV </td> <td><span class="nowrap"><span data-sort-value="7000216000000000000♠"></span>2.16<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;">+0.49<br />−0.26</span></span> MeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>m</i><sub>d</sub> </td> <td>down quark mass </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = 2 GeV </td> <td><span class="nowrap"><span data-sort-value="7000467000000000000♠"></span>4.67<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;">+0.48<br />−0.17</span></span> MeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>m</i><sub>s</sub> </td> <td>strange quark mass </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = 2 GeV </td> <td><span class="nowrap"><span data-sort-value="7001934000000000000♠"></span>93.4<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;">+8.6<br />−3.4</span></span> MeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>m</i><sub>c</sub> </td> <td>charm quark mass </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = <i>m</i><sub>c</sub> </td> <td><span class="nowrap"><span data-sort-value="7000127000000000000♠"></span>1.27(2) GeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>m</i><sub>b</sub> </td> <td>bottom quark mass </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = <i>m</i><sub>b</sub> </td> <td><span class="nowrap"><span data-sort-value="7000418000000000000♠"></span>4.18<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;">+0.03<br />−0.02</span></span> GeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>m</i><sub>t</sub> </td> <td>top quark mass </td> <td><a href="/wiki/On-shell_scheme" class="mw-redirect" title="On-shell scheme">on-shell scheme</a> </td> <td><span class="nowrap"><span data-sort-value="7002172690000000000♠"></span>172.69(30) GeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>θ</i><sub>12</sub> </td> <td>CKM 12-mixing angle </td> <td> </td> <td>13.1° </td></tr> <tr> <td><i>θ</i><sub>23</sub> </td> <td>CKM 23-mixing angle </td> <td> </td> <td>2.4° </td></tr> <tr> <td><i>θ</i><sub>13</sub> </td> <td>CKM 13-mixing angle </td> <td> </td> <td>0.2° </td></tr> <tr> <td><i>δ</i> </td> <td>CKM <a href="/wiki/CP_violation" title="CP violation">CP-violating</a> Phase </td> <td> </td> <td>0.995 </td></tr> <tr> <td><i>g</i><sub>1</sub> or <i>g</i>′ </td> <td>U(1) gauge coupling </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = <i>m</i><sub>Z</sub> </td> <td>0.357 </td></tr> <tr> <td><i>g</i><sub>2</sub> or <i>g</i> </td> <td>SU(2) gauge coupling </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = <i>m</i><sub>Z</sub> </td> <td>0.652 </td></tr> <tr> <td><i>g</i><sub>3</sub> or <i>g</i><sub>s</sub> </td> <td>SU(3) gauge coupling </td> <td><i>μ</i><sub><span style="text-decoration:overline;">MS</span></sub> = <i>m</i><sub>Z</sub> </td> <td>1.221 </td></tr> <tr> <td><i>θ</i><sub>QCD</sub> </td> <td>QCD <a href="/wiki/Theta_vacuum" title="Theta vacuum">vacuum angle</a> </td> <td> </td> <td>~ 0 </td></tr> <tr> <td><i>v</i> </td> <td>Higgs vacuum expectation value </td> <td> </td> <td><span class="nowrap"><span data-sort-value="7002246219600000000♠"></span>246.2196(2) GeV/<i>c</i><sup>2</sup></span> </td></tr> <tr> <td><i>m</i><sub>H</sub> </td> <td>Higgs mass </td> <td> </td> <td><span class="nowrap"><span data-sort-value="7002125180000000000♠"></span>125.18(16) GeV/<i>c</i><sup>2</sup></span> </td></tr></tbody></table> <p>The choice of free parameters is somewhat arbitrary. In the table above, gauge couplings are listed as free parameters, therefore with this choice the Weinberg angle is not a free parameter – it is defined as <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 _{\rm {W}}={g_{1}}/{g_{2}}}"> <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">W</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta _{\rm {W}}={g_{1}}/{g_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e9f9a3cb35f592cc114a2b9d54714ea9009c06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.346ex; height:2.843ex;" alt="{\displaystyle \tan \theta _{\rm {W}}={g_{1}}/{g_{2}}}"></span>. Likewise, the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a> of QED is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {1}{4\pi }}{\frac {(g_{1}g_{2})^{2}}{g_{1}^{2}+g_{2}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {1}{4\pi }}{\frac {(g_{1}g_{2})^{2}}{g_{1}^{2}+g_{2}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160099bca4d0b3e0a6a2d43fc5194bc73373b990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.943ex; height:6.843ex;" alt="{\displaystyle \alpha ={\frac {1}{4\pi }}{\frac {(g_{1}g_{2})^{2}}{g_{1}^{2}+g_{2}^{2}}}}"></span>. Instead of fermion masses, dimensionless Yukawa couplings can be chosen as free parameters. For example, the electron mass depends on the Yukawa coupling of the electron to the Higgs field, and its value is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\rm {e}}=y_{\rm {e}}v/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\rm {e}}=y_{\rm {e}}v/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a37dc97580c9c5b5ec724fc34eebcacb38377a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.591ex; height:3.176ex;" alt="{\displaystyle m_{\rm {e}}=y_{\rm {e}}v/{\sqrt {2}}}"></span>. Instead of the Higgs mass, the Higgs self-coupling strength <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {m_{\rm {H}}^{2}}{2v^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {m_{\rm {H}}^{2}}{2v^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ffd0a29d9ee141ef0e5ba4f194c8ca69515863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.795ex; height:6.343ex;" alt="{\displaystyle \lambda ={\frac {m_{\rm {H}}^{2}}{2v^{2}}}}"></span>, which is approximately 0.129, can be chosen as a free parameter. Instead of the Higgs vacuum expectation value, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155c5b5630673f7c1e819577a3ddf1f45cd6cfc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:3.176ex;" alt="{\displaystyle \mu ^{2}}"></span> parameter directly from the Higgs self-interaction term <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 \mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>ϕ</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>ϕ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20f50045dcbc6fc5eb877364091ecae93191f08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.981ex; height:3.176ex;" alt="{\displaystyle \mu ^{2}\phi ^{\dagger }\phi -\lambda (\phi ^{\dagger }\phi )^{2}}"></span> can be chosen. Its value is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{2}=\lambda v^{2}={m_{\rm {H}}^{2}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>λ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{2}=\lambda v^{2}={m_{\rm {H}}^{2}}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee8e3b7ac52d47515ac7c3b37330ca385833436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.02ex; height:3.343ex;" alt="{\displaystyle \mu ^{2}=\lambda v^{2}={m_{\rm {H}}^{2}}/2}"></span>, or approximately <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> = <span class="nowrap"><span data-sort-value="6992141712510275150♠"></span>88.45 GeV</span>. </p><p>The value of the <a href="/wiki/Vacuum_energy" title="Vacuum energy">vacuum energy</a> (or more precisely, the <a href="/wiki/Renormalization" title="Renormalization">renormalization</a> scale used to calculate this energy) may also be treated as an additional free parameter. The renormalization scale may be identified with the <a href="/wiki/Planck_scale" class="mw-redirect" title="Planck scale">Planck scale</a> or fine-tuned to match the observed <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a>. However, both options <a href="/wiki/Cosmological_constant_problem" title="Cosmological constant problem">are problematic</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Additional_symmetries_of_the_Standard_Model">Additional symmetries of the Standard Model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=27" title="Edit section: Additional symmetries of the Standard Model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the theoretical point of view, the Standard Model exhibits four additional global symmetries, not postulated at the outset of its construction, collectively denoted <b><a href="/wiki/Accidental_symmetry" title="Accidental symmetry">accidental symmetries</a></b>, which are continuous <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">global symmetries</a>. The transformations leaving the Lagrangian invariant are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\text{q}}\to e^{i\alpha /3}\psi _{\text{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>q</mtext> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>q</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\text{q}}\to e^{i\alpha /3}\psi _{\text{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a24a57df7d8d5165bb307c7aad52f8b4cafdd83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.443ex; height:3.509ex;" alt="{\displaystyle \psi _{\text{q}}\to e^{i\alpha /3}\psi _{\text{q}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\rm {L}}\to e^{i\beta }E_{\rm {L}}{\text{ and }}(e_{\rm {R}})^{\text{c}}\to e^{i\beta }(e_{\rm {R}})^{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\rm {L}}\to e^{i\beta }E_{\rm {L}}{\text{ and }}(e_{\rm {R}})^{\text{c}}\to e^{i\beta }(e_{\rm {R}})^{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe9e87e742ae54184c889b2b7ed6a2b4a115130" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.33ex; height:3.176ex;" alt="{\displaystyle E_{\rm {L}}\to e^{i\beta }E_{\rm {L}}{\text{ and }}(e_{\rm {R}})^{\text{c}}\to e^{i\beta }(e_{\rm {R}})^{\text{c}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\rm {L}}\to e^{i\beta }M_{\rm {L}}{\text{ and }}(\mu _{\rm {R}})^{\text{c}}\to e^{i\beta }(\mu _{\rm {R}})^{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\rm {L}}\to e^{i\beta }M_{\rm {L}}{\text{ and }}(\mu _{\rm {R}})^{\text{c}}\to e^{i\beta }(\mu _{\rm {R}})^{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c43d0f01833ac2bf2e30f4c8f5b4d89b06beeb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.044ex; height:3.176ex;" alt="{\displaystyle M_{\rm {L}}\to e^{i\beta }M_{\rm {L}}{\text{ and }}(\mu _{\rm {R}})^{\text{c}}\to e^{i\beta }(\mu _{\rm {R}})^{\text{c}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\rm {L}}\to e^{i\beta }T_{\rm {L}}{\text{ and }}(\tau _{\rm {R}})^{\text{c}}\to e^{i\beta }(\tau _{\rm {R}})^{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {L}}\to e^{i\beta }T_{\rm {L}}{\text{ and }}(\tau _{\rm {R}})^{\text{c}}\to e^{i\beta }(\tau _{\rm {R}})^{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7787363ebb6b699650299c0e0deee4bae3526ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.48ex; height:3.176ex;" alt="{\displaystyle T_{\rm {L}}\to e^{i\beta }T_{\rm {L}}{\text{ and }}(\tau _{\rm {R}})^{\text{c}}\to e^{i\beta }(\tau _{\rm {R}})^{\text{c}}}"></span> </p><p>The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields <span class="texhtml"><i>M</i><sub>L</sub>, <i>T</i><sub>L</sub></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{\rm {R}})^{\text{c}},(\tau _{\rm {R}})^{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{\rm {R}})^{\text{c}},(\tau _{\rm {R}})^{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c51d73df36c08a110fc6b8d8c42675eb5c3784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.879ex; height:2.843ex;" alt="{\displaystyle (\mu _{\rm {R}})^{\text{c}},(\tau _{\rm {R}})^{\text{c}}}"></span> are the 2nd (muon) and 3rd (tau) generation analogs of <span class="texhtml"><i>E</i><sub>L</sub></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{\rm {R}})^{\text{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{\rm {R}})^{\text{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5ea47f948eaca84066b9a6b5ece69ee509fd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.297ex; height:2.843ex;" alt="{\displaystyle (e_{\rm {R}})^{\text{c}}}"></span> fields. </p><p>By <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>, each symmetry above has an associated <a href="/wiki/Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conservation law</a>: the conservation of <a href="/wiki/Baryon_number" title="Baryon number">baryon number</a>,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Lepton_number" title="Lepton number">electron number</a>, <a href="/wiki/Lepton_number" title="Lepton number">muon number</a>, and <a href="/wiki/Lepton_number" title="Lepton number">tau number</a>. Each quark is assigned a baryon number of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}"></span>, while each antiquark is assigned a baryon number of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70def1351375878dbed3e58e789193c219f2347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {1}{3}}}"></span>. Conservation of baryon number implies that the number of quarks minus the number of antiquarks is a constant. Within experimental limits, no violation of this conservation law has been found. </p><p>Similarly, each electron and its associated neutrino is assigned an electron number of +1, while the <a href="/wiki/Positron" title="Positron">anti-electron</a> and the associated anti-neutrino carry a −1 electron number. Similarly, the muons and their neutrinos are assigned a muon number of +1 and the tau leptons are assigned a tau lepton number of +1. The Standard Model predicts that each of these three numbers should be conserved separately in a manner similar to the way baryon number is conserved. These numbers are collectively known as <a href="/wiki/Lepton_family_number" class="mw-redirect" title="Lepton family number">lepton family numbers</a> (LF). (This result depends on the assumption made in Standard Model that neutrinos are massless. Experimentally, neutrino oscillations imply that individual electron, muon and tau numbers are not conserved.)<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>In addition to the accidental (but exact) symmetries described above, the Standard Model exhibits several <b><a href="/wiki/Particle_physics_and_representation_theory#Approximate_symmetries" title="Particle physics and representation theory">approximate symmetries</a></b>. These are the "SU(2) <a href="/wiki/Custodial_symmetry" title="Custodial symmetry">custodial symmetry</a>" and the "SU(2) or SU(3) quark flavor symmetry". </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th colspan="4">Symmetries of the Standard Model and associated conservation laws </th></tr> <tr> <th><a href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">Symmetry</a> </th> <th><a href="/wiki/Lie_group" title="Lie group">Lie group</a> </th> <th>Symmetry Type </th> <th><a href="/wiki/Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">Conservation law</a> </th></tr> <tr> <td><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a> </td> <td style="text-align:center;"><a href="/wiki/Translational_symmetry" title="Translational symmetry">Translations</a><a href="/wiki/Semidirect_product" title="Semidirect product">⋊</a><a href="/wiki/Lorentz_group" title="Lorentz group">SO(3,1)</a> </td> <td style="text-align:center;"><a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">Global symmetry</a> </td> <td><a href="/wiki/Energy" title="Energy">Energy</a>, <a href="/wiki/Momentum" title="Momentum">Momentum</a>, <a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a> </td></tr> <tr> <td><a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">Gauge</a> </td> <td style="text-align:center;"><a href="/wiki/SU(3)" class="mw-redirect" title="SU(3)">SU(3)</a>×<a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a>×<a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> </td> <td style="text-align:center;"><a href="/wiki/Local_symmetry" class="mw-redirect" title="Local symmetry">Local symmetry</a> </td> <td><a href="/wiki/Color_charge" title="Color charge">Color charge</a>, <a href="/wiki/Weak_isospin" title="Weak isospin">Weak isospin</a>, <a href="/wiki/Electric_charge" title="Electric charge">Electric charge</a>, <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a> </td></tr> <tr> <td><a href="/wiki/Baryon" title="Baryon">Baryon</a> phase </td> <td style="text-align:center;"><a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> </td> <td style="text-align:center;">Accidental <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">Global symmetry</a> </td> <td><a href="/wiki/Baryon_number" title="Baryon number">Baryon number</a> </td></tr> <tr> <td><a href="/wiki/Electron" title="Electron">Electron</a> phase </td> <td style="text-align:center;"><a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> </td> <td style="text-align:center;">Accidental <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">Global symmetry</a> </td> <td><a href="/wiki/Lepton_number" title="Lepton number">Electron number</a> </td></tr> <tr> <td><a href="/wiki/Muon" title="Muon">Muon</a> phase </td> <td style="text-align:center;"><a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> </td> <td style="text-align:center;">Accidental <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">Global symmetry</a> </td> <td><a href="/wiki/Lepton_number" title="Lepton number">Muon number</a> </td></tr> <tr> <td><a href="/wiki/Tau_(particle)" title="Tau (particle)">Tau</a> phase </td> <td style="text-align:center;"><a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> </td> <td style="text-align:center;">Accidental <a href="/wiki/Global_symmetry" class="mw-redirect" title="Global symmetry">Global symmetry</a> </td> <td><a href="/wiki/Lepton_number" title="Lepton number">Tau number</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="U(1)_symmetry"><span id="U.281.29_symmetry"></span>U(1) symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=28" title="Edit section: U(1) symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the <a href="/wiki/Lepton" title="Lepton">leptons</a>, the gauge group can be written <span class="texhtml">SU(2)<sub><i>l</i></sub> × U(1)<sub>L</sub> × U(1)<sub>R</sub></span>. The two <span class="texhtml">U(1)</span> factors can be combined into <span class="texhtml">U(1)<sub>Y</sub> × U(1)<sub><i>l</i></sub></span>, where <span class="texhtml"><i>l</i></span> is the <a href="/wiki/Lepton_number" title="Lepton number">lepton number</a>. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group <span class="texhtml">SU(2)<sub>L</sub> × U(1)<sub>Y</sub></span>. A similar argument in the quark sector also gives the same result for the electroweak theory. </p> <div class="mw-heading mw-heading3"><h3 id="Charged_and_neutral_current_couplings_and_Fermi_theory">Charged and neutral current couplings and Fermi theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=29" title="Edit section: Charged and neutral current couplings and Fermi theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The charged currents <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^{\mp }=j^{1}\pm ij^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∓</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>±</mo> <mi>i</mi> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j^{\mp }=j^{1}\pm ij^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5260eac464855e26b7662bf65a41124612b7369c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:13.262ex; height:3.009ex;" alt="{\displaystyle j^{\mp }=j^{1}\pm ij^{2}}"></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }^{-}={\overline {U}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>U</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{\mu }^{-}={\overline {U}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034d8f45bd79ee5f82cffc2ef1fa0cdb5fd918d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:27.288ex; height:3.676ex;" alt="{\displaystyle j_{\mu }^{-}={\overline {U}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }.}"></span> These charged currents are precisely those that entered the <a href="/wiki/Fermi_theory_of_beta_decay" class="mw-redirect" title="Fermi theory of beta decay">Fermi theory of beta decay</a>. The action contains the charge current piece <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\rm {CC}}={\frac {g}{\sqrt {2}}}(j_{\mu }^{+}W^{-\mu }+j_{\mu }^{-}W^{+\mu }).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\rm {CC}}={\frac {g}{\sqrt {2}}}(j_{\mu }^{+}W^{-\mu }+j_{\mu }^{-}W^{+\mu }).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bcd1467e5da58fea1a7b10f11c9619aa423e65c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.496ex; height:5.843ex;" alt="{\displaystyle {\mathcal {L}}_{\rm {CC}}={\frac {g}{\sqrt {2}}}(j_{\mu }^{+}W^{-\mu }+j_{\mu }^{-}W^{+\mu }).}"></span> For energy much less than the mass of the W-boson, the effective theory becomes the current–current contact interaction of the <a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi theory</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {2}}G_{\rm {F}}~~J_{\mu }^{+}J^{\mu ~~-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mtext> </mtext> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mtext> </mtext> <mtext> </mtext> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {2}}G_{\rm {F}}~~J_{\mu }^{+}J^{\mu ~~-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff3481ecdd75509b4bf57fa4fb0adc01ef78cf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.78ex; height:3.343ex;" alt="{\displaystyle 2{\sqrt {2}}G_{\rm {F}}~~J_{\mu }^{+}J^{\mu ~~-}}"></span>. </p><p>However, gauge invariance now requires that the component <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 W^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d421d5feca0131907716978ba5cc6598ce49192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.562ex; height:2.676ex;" alt="{\displaystyle W^{3}}"></span> of the gauge field also be coupled to a current that lies in the triplet of SU(2). However, this mixes with the <span class="texhtml">U(1)</span>, and another current in that sector is needed. These currents must be uncharged in order to conserve charge. So <b>neutral currents</b> are also required, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }^{3}={\frac {1}{2}}\left({\overline {U}}_{i\mathrm {L} }\gamma _{\mu }U_{i\mathrm {L} }-{\overline {D}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }\nu _{i\mathrm {L} }-{\overline {l}}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>U</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{\mu }^{3}={\frac {1}{2}}\left({\overline {U}}_{i\mathrm {L} }\gamma _{\mu }U_{i\mathrm {L} }-{\overline {D}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }\nu _{i\mathrm {L} }-{\overline {l}}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f340b5467f07f0d25fdd822601b2e35c58aa9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.027ex; width:54.942ex; height:5.176ex;" alt="{\displaystyle j_{\mu }^{3}={\frac {1}{2}}\left({\overline {U}}_{i\mathrm {L} }\gamma _{\mu }U_{i\mathrm {L} }-{\overline {D}}_{i\mathrm {L} }\gamma _{\mu }D_{i\mathrm {L} }+{\overline {\nu }}_{i\mathrm {L} }\gamma _{\mu }\nu _{i\mathrm {L} }-{\overline {l}}_{i\mathrm {L} }\gamma _{\mu }l_{i\mathrm {L} }\right)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{\mu }^{\rm {em}}={\frac {2}{3}}{\overline {U}}_{i}\gamma _{\mu }U_{i}-{\frac {1}{3}}{\overline {D}}_{i}\gamma _{\mu }D_{i}-{\overline {l}}_{i}\gamma _{\mu }l_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>U</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{\mu }^{\rm {em}}={\frac {2}{3}}{\overline {U}}_{i}\gamma _{\mu }U_{i}-{\frac {1}{3}}{\overline {D}}_{i}\gamma _{\mu }D_{i}-{\overline {l}}_{i}\gamma _{\mu }l_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921fdf80a7c667ee213b68585901a7790c907dec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.027ex; width:37.897ex; height:5.176ex;" alt="{\displaystyle j_{\mu }^{\rm {em}}={\frac {2}{3}}{\overline {U}}_{i}\gamma _{\mu }U_{i}-{\frac {1}{3}}{\overline {D}}_{i}\gamma _{\mu }D_{i}-{\overline {l}}_{i}\gamma _{\mu }l_{i}.}"></span> The neutral current piece in the Lagrangian is then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\rm {NC}}=ej_{\mu }^{\rm {em}}A^{\mu }+{\frac {g}{\cos \theta _{\rm {W}}}}(J_{\mu }^{3}-\sin ^{2}\theta _{\rm {W}}J_{\mu }^{\rm {em}})Z^{\mu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>e</mi> <msubsup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msubsup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> </mrow> </msub> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">)</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\rm {NC}}=ej_{\mu }^{\rm {em}}A^{\mu }+{\frac {g}{\cos \theta _{\rm {W}}}}(J_{\mu }^{3}-\sin ^{2}\theta _{\rm {W}}J_{\mu }^{\rm {em}})Z^{\mu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff5c565b5c7b1ed81b538042178a4ea0c33c142" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.854ex; height:5.343ex;" alt="{\displaystyle {\mathcal {L}}_{\rm {NC}}=ej_{\mu }^{\rm {em}}A^{\mu }+{\frac {g}{\cos \theta _{\rm {W}}}}(J_{\mu }^{3}-\sin ^{2}\theta _{\rm {W}}J_{\mu }^{\rm {em}})Z^{\mu }.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Physics_beyond_the_Standard_Model">Physics beyond the Standard Model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=30" title="Edit section: Physics beyond the Standard Model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Physics beyond the Standard Model</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Physics_beyond_the_Standard_Model&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the Standard Model</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:CMS_Higgs-event.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CMS_Higgs-event.jpg/220px-CMS_Higgs-event.jpg" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CMS_Higgs-event.jpg/330px-CMS_Higgs-event.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CMS_Higgs-event.jpg/440px-CMS_Higgs-event.jpg 2x" data-file-width="1104" data-file-height="1018" /></a></span><div class="sidebar-caption">Simulated <a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">Large Hadron Collider</a> <a href="/wiki/Compact_Muon_Solenoid" title="Compact Muon Solenoid">CMS</a> particle detector data depicting a <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a> produced by colliding protons decaying into hadron jets and electrons</div></td></tr><tr><th class="sidebar-heading"> <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Evidence</div><div class="sidebar-list-content mw-collapsible-content plainlist"> <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/Dark_energy" title="Dark energy">Dark energy</a></li> <li><a href="/wiki/Quintessence_(physics)" title="Quintessence (physics)">Quintessence</a></li> <li><a href="/wiki/Phantom_energy" title="Phantom energy">Phantom energy</a></li> <li><a href="/wiki/Dark_radiation" title="Dark radiation">Dark radiation</a></li> <li><a href="/wiki/Dark_photon" title="Dark photon">Dark photon</a></li> <li><a href="/wiki/Cosmological_constant_problem" title="Cosmological constant problem">Cosmological constant problem</a></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></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Theories</div><div class="sidebar-list-content mw-collapsible-content plainlist"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship hypothesis</a></li> <li><a href="/wiki/Fifth_force" title="Fifth force">Fifth force</a></li> <li><a href="/wiki/F-theory" title="F-theory">F-theory</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li> <li><a href="/wiki/Unified_field_theory" title="Unified field theory">Unified field theory</a></li> <li><a href="/wiki/Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li> <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/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0) superconformal field theory</a></li> <li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative quantum field theory</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li> <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/M-theory" title="M-theory">M-theory</a></li> <li><a href="/wiki/Mathematical_universe_hypothesis" title="Mathematical universe hypothesis">Mathematical universe hypothesis</a></li> <li><a href="/wiki/Mirror_matter" title="Mirror matter">Mirror matter</a></li> <li><a href="/wiki/Randall%E2%80%93Sundrum_model" title="Randall–Sundrum model">Randall–Sundrum model</a></li> <li><a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 supersymmetric Yang–Mills theory</a></li> <li><a href="/wiki/Twistor_string_theory" title="Twistor string theory">Twistor string theory</a></li> <li><a href="/wiki/Dark_fluid" title="Dark fluid">Dark fluid</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li> <li><a href="/wiki/De_Sitter_invariant_special_relativity" title="De Sitter invariant special relativity">de Sitter invariant special relativity</a></li> <li><a href="/wiki/Causal_fermion_system" class="mw-redirect" title="Causal fermion system">Causal fermion systems</a></li> <li><a href="/wiki/Black_hole_thermodynamics" title="Black hole thermodynamics">Black hole thermodynamics</a></li> <li><a href="/wiki/Unparticle_physics" title="Unparticle physics">Unparticle physics</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviscalar" title="Graviscalar">Graviscalar</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Gravitino" title="Gravitino">Gravitino</a></li> <li><a href="/wiki/Massive_gravity" title="Massive gravity">Massive gravity</a></li> <li><a href="/wiki/Gauge_gravitation_theory" title="Gauge gravitation theory">Gauge gravitation theory</a></li> <li><a href="/wiki/Gauge_theory_gravity" title="Gauge theory gravity">Gauge theory gravity</a></li> <li><a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></div><div class="sidebar-list-content mw-collapsible-content plainlist"> <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/Superstring_theory" title="Superstring theory">Superstring theory</a></li> <li><a href="/wiki/M-theory" title="M-theory">M-theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Supersymmetry_breaking" title="Supersymmetry breaking">Supersymmetry breaking</a></li> <li><a href="/wiki/Extra_dimensions" title="Extra dimensions">Extra dimensions</a></li> <li><a href="/wiki/Large_extra_dimensions" title="Large extra dimensions">Large extra dimensions</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></div><div class="sidebar-list-content mw-collapsible-content plainlist"> <ul><li><a href="/wiki/False_vacuum" title="False vacuum">False vacuum</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Spin_foam" title="Spin foam">Spin foam</a></li> <li><a href="/wiki/Quantum_foam" title="Quantum foam">Quantum foam</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Causal_fermion_systems" title="Causal fermion systems">Causal fermion systems</a></li> <li><a href="/wiki/Causal_sets" title="Causal sets">Causal sets</a></li> <li><a href="/wiki/Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li> <li><a href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li> <li><a href="/wiki/Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content plainlist"> <ul><li><a href="/wiki/Accelerator_Neutrino_Neutron_Interaction_Experiment" title="Accelerator Neutrino Neutron Interaction Experiment">ANNIE</a></li> <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> <li><a href="/wiki/Sudbury_Neutrino_Observatory" title="Sudbury Neutrino Observatory">SNO</a></li> <li><a href="/wiki/Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li> <li><a href="/wiki/Tevatron" title="Tevatron">Tevatron</a></li> <li><a href="/wiki/NOvA" title="NOvA">NOvA</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Beyond_the_Standard_Model" title="Template:Beyond the Standard Model"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Beyond_the_Standard_Model" title="Template talk:Beyond the Standard Model"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Beyond_the_Standard_Model" title="Special:EditPage/Template:Beyond the Standard Model"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Physics beyond the Standard Model</a> (BSM) refers to the theoretical developments needed to explain the deficiencies of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>, such as the inability to explain the fundamental parameters of the standard model, the <a href="/wiki/Strong_CP_problem" title="Strong CP problem">strong CP problem</a>, <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillations</a>, <a href="/wiki/Baryon_asymmetry" title="Baryon asymmetry">matter–antimatter asymmetry</a>, and the nature of <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a> and <a href="/wiki/Dark_energy" title="Dark energy">dark energy</a>.<sup id="cite_ref-Physics_beyond_the_Standard_Model_sym-v2-feb-05_20-0" class="reference"><a href="#cite_note-Physics_beyond_the_Standard_Model_sym-v2-feb-05-20"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Another problem lies within the <a href="/wiki/Quantum_field_theory" title="Quantum field theory">mathematical framework</a> of the Standard Model itself: the Standard Model is inconsistent with that of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, and one or both theories break down under certain conditions, such as <a href="/wiki/Gravitational_singularity" title="Gravitational singularity">spacetime singularities</a> like the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a> and <a href="/wiki/Black_hole" title="Black hole">black hole</a> <a href="/wiki/Event_horizon" title="Event horizon">event horizons</a>. </p> Theories that lie beyond the Standard Model include various extensions of the standard model through <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a>, such as the <a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">Minimal Supersymmetric Standard Model</a> (MSSM) and <a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">Next-to-Minimal Supersymmetric Standard Model</a> (NMSSM), and entirely novel explanations, such as <a href="/wiki/String_theory" title="String theory">string theory</a>, <a href="/wiki/M-theory" title="M-theory">M-theory</a>, and <a href="/wiki/Extra_dimensions" title="Extra dimensions">extra dimensions</a>. As these theories tend to reproduce the entirety of current phenomena, the question of which theory is the right one, or at least the "best step" towards a <a href="/wiki/Theory_of_Everything" class="mw-redirect" title="Theory of Everything">Theory of Everything</a>, can only be settled via experiments, and is one of the most active areas of research in both <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical</a> and <a href="/wiki/Experimental_physics" title="Experimental physics">experimental physics</a>.<sup id="cite_ref-Physics_beyond_the_Standard_Model_NYT-20230911_21-0" class="reference"><a href="#cite_note-Physics_beyond_the_Standard_Model_NYT-20230911-21"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></div></div> <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=Mathematical_formulation_of_the_Standard_Model&action=edit&section=31" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Overview of <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a></li> <li><a href="/wiki/Fundamental_interaction" title="Fundamental interaction">Fundamental interaction</a></li> <li><a href="/wiki/Noncommutative_standard_model" title="Noncommutative standard model">Noncommutative standard model</a></li> <li>Open questions: <a href="/wiki/CP_violation" title="CP violation">CP violation</a>, <a href="/wiki/Neutrino" title="Neutrino">Neutrino masses</a>, <a href="/wiki/QCD_matter" title="QCD matter">Quark matter</a></li> <li><a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Physics beyond the Standard Model</a></li> <li><a href="/wiki/Strong_interaction" title="Strong interaction">Strong interactions</a> <ul><li><a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">Flavor</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/Weak_interaction" title="Weak interaction">Weak interactions</a> <ul><li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi's interaction</a></li></ul></li> <li><a href="/wiki/Weinberg_angle" title="Weinberg angle">Weinberg angle</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><i><a href="/wiki/Quantum_Field_Theory_in_a_Nutshell" title="Quantum Field Theory in a Nutshell">Quantum Field Theory in a Nutshell</a></i> by A. Zee</li></ul> <div class="mw-heading mw-heading2"><h2 id="References_and_external_links">References and external links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_the_Standard_Model&action=edit&section=32" title="Edit section: References and external links"><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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">In fact, there are mathematical issues regarding quantum field theories still under debate (see e.g. <a href="/wiki/Landau_pole" title="Landau pole">Landau pole</a>), but the predictions extracted from the Standard Model by current methods are all self-consistent. For a further discussion see e.g. R. Mann, chapter 25.</span> </li> <li id="cite_note-NYT-20230911-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-NYT-20230911_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOverbye2023" class="citation news cs1"><a href="/wiki/Dennis_Overbye" title="Dennis Overbye">Overbye, Dennis</a> (11 September 2023). <a rel="nofollow" class="external text" href="https://www.nytimes.com/2023/09/11/science/space/astronomy-universe-simulations.html">"Don't Expect a 'Theory of Everything' to Explain It All - Not even the most advanced physics can reveal everything we want to know about the history and future of the cosmos, or about ourselves"</a>. <i><a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a></i>. <a rel="nofollow" class="external text" href="https://archive.today/20230911043212/https://www.nytimes.com/2023/09/11/science/space/astronomy-universe-simulations.html">Archived</a> from the original on 11 September 2023<span class="reference-accessdate">. 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Cambridge University Press. pp. <span class="nowrap">499–</span>500. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-29254-3" title="Special:BookSources/978-1-107-29254-3"><bdi>978-1-107-29254-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Particle+Physics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E499-%3C%2Fspan%3E500&rft.pub=Cambridge+University+Press&rft.date=2013-09-05&rft.isbn=978-1-107-29254-3&rft.au=Mark+Thomson&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHamyAAAAQBAJ%26pg%3DPA499&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+the+Standard+Model" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin2012" class="citation journal cs1">Martin, Jérôme (July 2012). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S1631070512000497">"Everything you always wanted to know about the cosmological constant problem (but were afraid to ask)"</a>. <i>Comptes Rendus Physique</i>. <b>13</b> (<span class="nowrap">6–</span>7): <span class="nowrap">566–</span>665. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1205.3365">1205.3365</a></span>. <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/2012CRPhy..13..566M">2012CRPhy..13..566M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.crhy.2012.04.008">10.1016/j.crhy.2012.04.008</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:119272967">119272967</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+Rendus+Physique&rft.atitle=Everything+you+always+wanted+to+know+about+the+cosmological+constant+problem+%28but+were+afraid+to+ask%29&rft.volume=13&rft.issue=%3Cspan+class%3D%22nowrap%22%3E6%E2%80%93%3C%2Fspan%3E7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E566-%3C%2Fspan%3E665&rft.date=2012-07&rft_id=info%3Aarxiv%2F1205.3365&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119272967%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.crhy.2012.04.008&rft_id=info%3Abibcode%2F2012CRPhy..13..566M&rft.aulast=Martin&rft.aufirst=J%C3%A9r%C3%B4me&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS1631070512000497&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+the+Standard+Model" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">The baryon number in SM is only conserved at the classical level. There are non-perturbative effects that do not conserve baryon number: <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1311.5285">Baryon Number Violation, report prepared for the Community Planning Study – Snowmass 2013</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">The lepton number in SM is only conserved at the classical level. There are non-perturbative effects that do not conserve lepton number: see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuentes-MartínPortolésRuiz-Femenía2015" class="citation journal cs1">Fuentes-Martín, J.; Portolés, J.; Ruiz-Femenía, P. (January 2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP01%282015%29134">"Instanton-mediated baryon number violation in non-universal gauge extended models"</a>. <i>Journal of High Energy Physics</i>. <b>2015</b> (1): 134. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.2471">1411.2471</a></span>. <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/2015JHEP...01..134F">2015JHEP...01..134F</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/10.1007%2FJHEP01%282015%29134">10.1007/JHEP01(2015)134</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1029-8479">1029-8479</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Instanton-mediated+baryon+number+violation+in+non-universal+gauge+extended+models&rft.volume=2015&rft.issue=1&rft.pages=134&rft.date=2015-01&rft_id=info%3Aarxiv%2F1411.2471&rft.issn=1029-8479&rft_id=info%3Adoi%2F10.1007%2FJHEP01%282015%29134&rft_id=info%3Abibcode%2F2015JHEP...01..134F&rft.aulast=Fuentes-Mart%C3%ADn&rft.aufirst=J.&rft.au=Portol%C3%A9s%2C+J.&rft.au=Ruiz-Femen%C3%ADa%2C+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FJHEP01%25282015%2529134&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+the+Standard+Model" class="Z3988"></span> or <a rel="nofollow" class="external text" href="https://thesis.library.caltech.edu/8211/1/JArnold_THESIS.pdf">Baryon and lepton numbers in particle physics beyond the standard model</a></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">The violation of lepton number and baryon number cancel each other out and in effect <a href="/wiki/B_%E2%88%92_L" title="B − L">B − L</a> is an exact symmetry of the Standard Model. Extension of the Standard Model with massive Majorana neutrinos breaks B-L symmetry, but extension with massive Dirac neutrinos does not: see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaSrivastava2015" class="citation journal cs1">Ma, Ernest; Srivastava, Rahul (2015-08-30). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/doi/abs/10.1142/S0217732315300207">"Dirac or inverse seesaw neutrino masses from gauged B–L symmetry"</a>. <i>Modern Physics Letters A</i>. <b>30</b> (26): 1530020. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1504.00111">1504.00111</a></span>. <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/2015MPLA...3030020M">2015MPLA...3030020M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217732315300207">10.1142/S0217732315300207</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0217-7323">0217-7323</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:119111538">119111538</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Modern+Physics+Letters+A&rft.atitle=Dirac+or+inverse+seesaw+neutrino+masses+from+gauged+B%E2%80%93L+symmetry&rft.volume=30&rft.issue=26&rft.pages=1530020&rft.date=2015-08-30&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119111538%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015MPLA...3030020M&rft_id=info%3Aarxiv%2F1504.00111&rft.issn=0217-7323&rft_id=info%3Adoi%2F10.1142%2FS0217732315300207&rft.aulast=Ma&rft.aufirst=Ernest&rft.au=Srivastava%2C+Rahul&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fdoi%2Fabs%2F10.1142%2FS0217732315300207&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+the+Standard+Model" class="Z3988"></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeeck2014" class="citation journal cs1">Heeck, Julian (December 2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physletb.2014.10.067">"Unbroken B – L symmetry"</a>. <i>Physics Letters B</i>. <b>739</b>: <span class="nowrap">256–</span>262. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1408.6845">1408.6845</a></span>. <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/2014PhLB..739..256H">2014PhLB..739..256H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physletb.2014.10.067">10.1016/j.physletb.2014.10.067</a></span>.</cite><span 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Retrieved <span class="nowrap">11 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=The+New+York+Times&rft.atitle=Don%27t+Expect+a+%27Theory+of+Everything%27+to+Explain+It+All+-+Not+even+the+most+advanced+physics+can+reveal+everything+we+want+to+know+about+the+history+and+future+of+the+cosmos%2C+or+about+ourselves.&rft.date=2023-09-11&rft.aulast=Overbye&rft.aufirst=Dennis&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2023%2F09%2F11%2Fscience%2Fspace%2Fastronomy-universe-simulations.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+the+Standard+Model" class="Z3988"></span></span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><i>An introduction to quantum field theory</i>, by M.E. Peskin and D.V. Schroeder (HarperCollins, 1995) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-50397-2" title="Special:BookSources/0-201-50397-2">0-201-50397-2</a>.</li> <li><i>Gauge theory of elementary particle physics</i>, by T.P. Cheng and L.F. Li (Oxford University Press, 1982) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-851961-3" title="Special:BookSources/0-19-851961-3">0-19-851961-3</a>.</li> <li><a rel="nofollow" class="external text" href="http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html">Standard Model Lagrangian with explicit Higgs terms</a> (T.D. Gutierrez, ca 1999) (PDF, PostScript, and LaTeX version)</li> <li><i>The quantum theory of fields</i> (vol 2), by S. Weinberg (Cambridge University Press, 1996) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-55002-5" title="Special:BookSources/0-521-55002-5">0-521-55002-5</a>.</li> <li><i>Quantum Field Theory in a Nutshell</i> (Second Edition), by A. Zee (Princeton University Press, 2010) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-3532-4" title="Special:BookSources/978-1-4008-3532-4">978-1-4008-3532-4</a>.</li> <li><i>An Introduction to Particle Physics and the Standard Model</i>, by R. Mann (CRC Press, 2010) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1420082982" title="Special:BookSources/978-1420082982">978-1420082982</a></li> <li><i>Physics From Symmetry</i> by J. Schwichtenberg (Springer, 2015) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3319192000" title="Special:BookSources/3319192000">3319192000</a>. Especially <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_vLLCQAAQBAJ&dq=physics%20from%20symmetry&pg=PA86">page 86</a></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .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="Standard_Model377" 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_Model377" 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 href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">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 class="mw-selflink selflink">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> <li><a href="/wiki/Sudbury_Neutrino_Observatory" title="Sudbury Neutrino Observatory">SNO</a></li> <li><a href="/wiki/Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li> <li><a href="/wiki/Tevatron" title="Tevatron">Tevatron</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Standard_Model" title="Category:Standard Model">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" 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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:Particles" title="Template:Particles"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Particles" title="Template talk:Particles"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Particles" title="Special:EditPage/Template:Particles"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Particles_in_physics237" style="font-size:114%;margin:0 4em"><a href="/wiki/Particle_physics" title="Particle physics">Particles in physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Elementary_particle" title="Elementary particle">Elementary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" 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%;text-align: center;"><a href="/wiki/Fermion" title="Fermion">Fermions</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%;font-weight:normal;"><a href="/wiki/Quark" title="Quark">Quarks</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/Up_quark" title="Up quark">Up (quark</a></li> <li><a href="/wiki/Up_antiquark" class="mw-redirect" title="Up antiquark">antiquark)</a></li> <li><a href="/wiki/Down_quark" title="Down quark">Down (quark</a></li> <li><a href="/wiki/Down_antiquark" class="mw-redirect" title="Down antiquark">antiquark)</a></li> <li><a href="/wiki/Charm_quark" title="Charm quark">Charm (quark</a></li> <li><a href="/wiki/Charm_antiquark" class="mw-redirect" title="Charm antiquark">antiquark)</a></li> <li><a href="/wiki/Strange_quark" title="Strange quark">Strange (quark</a></li> <li><a href="/wiki/Strange_antiquark" class="mw-redirect" title="Strange antiquark">antiquark)</a></li> <li><a href="/wiki/Top_quark" title="Top quark">Top (quark</a></li> <li><a href="/wiki/Top_antiquark" class="mw-redirect" title="Top antiquark">antiquark)</a></li> <li><a href="/wiki/Bottom_quark" title="Bottom quark">Bottom (quark</a></li> <li><a href="/wiki/Bottom_antiquark" class="mw-redirect" title="Bottom antiquark">antiquark)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Lepton" title="Lepton">Leptons</a></th><td 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href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Tau antineutrino</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Boson" title="Boson">Bosons</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:4em;font-weight:normal; text-align: center;"><a href="/wiki/Gauge_boson" title="Gauge boson">Gauge</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Photon" title="Photon">Photon</a></li> <li><a href="/wiki/Gluon" title="Gluon">Gluon</a></li> <li><a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:4em;font-weight:normal; text-align: center;"><a href="/wiki/Scalar_boson" title="Scalar boson">Scalar</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson </a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;"><a href="/wiki/Ghost_(physics)" title="Ghost (physics)">Ghost fields</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/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov ghosts</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/List_of_hypothetical_particles" title="List of hypothetical particles">Hypothetical</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%;font-weight:normal; text-align: center;"><a href="/wiki/Superpartner" title="Superpartner">Superpartners</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%;font-weight:normal; text-align: center;"><a href="/wiki/Gaugino" title="Gaugino">Gauginos</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/Gluino" title="Gluino">Gluino</a></li> <li><a href="/wiki/Gravitino" title="Gravitino">Gravitino</a></li> <li><a href="/wiki/Photino" title="Photino">Photino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</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/Axino" title="Axino">Axino</a></li> <li><a href="/wiki/Chargino" title="Chargino">Chargino</a></li> <li><a href="/wiki/Higgsino" title="Higgsino">Higgsino</a></li> <li><a href="/wiki/Neutralino" title="Neutralino">Neutralino</a></li> <li><a href="/wiki/Sfermion" title="Sfermion">Sfermion</a> (<a href="/wiki/Stop_squark" title="Stop squark">Stop squark</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</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/Axion" title="Axion">Axion</a></li> <li><a href="/wiki/Curvaton" title="Curvaton">Curvaton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Inflaton" title="Inflaton">Inflaton</a></li> <li><a href="/wiki/Leptoquark" title="Leptoquark">Leptoquark</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Majoron" title="Majoron">Majoron</a></li> <li><a href="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermion</a></li> <li><a href="/wiki/Dark_photon" title="Dark photon">Dark photon</a></li> <li><a href="/wiki/Preon" title="Preon">Preon</a></li> <li><a href="/wiki/Sterile_neutrino" title="Sterile neutrino">Sterile neutrino</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/W%E2%80%B2_and_Z%E2%80%B2_bosons" title="W′ and Z′ bosons">W′ and Z′ bosons</a></li> <li><a href="/wiki/X_and_Y_bosons" title="X and Y bosons">X and Y bosons</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Bound_state" title="Bound state">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Hadrons64" scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Hadron" title="Hadron">Hadrons</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%;font-weight:normal; text-align: center;"><a href="/wiki/Baryon" title="Baryon">Baryons</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/Nucleon" title="Nucleon">Nucleon</a> <ul><li><a href="/wiki/Proton" title="Proton">Proton</a></li> <li><a href="/wiki/Antiproton" title="Antiproton">Antiproton</a></li> <li><a href="/wiki/Neutron" title="Neutron">Neutron</a></li> <li><a href="/wiki/Antineutron" title="Antineutron">Antineutron</a></li></ul></li> <li><a href="/wiki/Delta_baryon" title="Delta baryon">Delta baryon</a></li> <li><a href="/wiki/Lambda_baryon" title="Lambda baryon">Lambda baryon</a></li> <li><a href="/wiki/Sigma_baryon" title="Sigma baryon">Sigma baryon</a></li> <li><a href="/wiki/Xi_baryon" title="Xi baryon">Xi baryon</a></li> <li><a href="/wiki/Omega_baryon" title="Omega baryon">Omega baryon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Meson" title="Meson">Mesons</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/Pion" title="Pion">Pion</a></li> <li><a href="/wiki/Rho_meson" title="Rho meson">Rho meson</a></li> <li><a href="/wiki/Eta_meson" class="mw-redirect" title="Eta meson">Eta and eta prime mesons</a></li> <li><a href="/wiki/Bottom_eta_meson" title="Bottom eta meson">Bottom eta meson</a></li> <li><a href="/wiki/Phi_meson" title="Phi meson">Phi meson</a></li> <li><a href="/wiki/J/psi_meson" title="J/psi meson">J/psi meson</a></li> <li><a href="/wiki/Omega_meson" title="Omega meson">Omega meson</a></li> <li><a href="/wiki/Upsilon_meson" title="Upsilon meson">Upsilon meson</a></li> <li><a href="/wiki/Kaon" title="Kaon">Kaon</a></li> <li><a href="/wiki/B_meson" title="B meson">B meson</a></li> <li><a href="/wiki/D_meson" title="D meson">D meson</a></li> <li><a href="/wiki/Quarkonium" title="Quarkonium">Quarkonium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Exotic_hadron" title="Exotic hadron">Exotic hadrons</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/Tetraquark" title="Tetraquark">Tetraquark</a> (<a href="/wiki/Double-charm_tetraquark" title="Double-charm tetraquark">Double-charm tetraquark</a>)</li> <li><a href="/wiki/Pentaquark" title="Pentaquark">Pentaquark</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;">Others</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/Atomic_nucleus" title="Atomic nucleus">Atomic nuclei</a></li> <li><a href="/wiki/Atom" title="Atom">Atoms</a></li> <li><a href="/wiki/Exotic_atom" title="Exotic atom">Exotic atoms</a> <ul><li><a href="/wiki/Positronium" title="Positronium">Positronium</a></li> <li><a href="/wiki/Muonium" title="Muonium">Muonium</a></li> <li><a href="/wiki/Tauonium" class="mw-redirect" title="Tauonium">Tauonium</a></li> <li><a href="/wiki/Onium" title="Onium">Onia</a></li> <li><a href="/wiki/Pionium" title="Pionium">Pionium</a></li> <li><a href="/wiki/Protonium" title="Protonium">Protonium</a></li></ul></li> <li><a href="/wiki/Superatom" title="Superatom">Superatoms</a></li> <li><a href="/wiki/Molecule" title="Molecule">Molecules</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/List_of_hypothetical_particles" title="List of hypothetical particles">Hypothetical</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><td colspan="2" class="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%;font-weight:normal;">Baryons</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/Hexaquark" title="Hexaquark">Hexaquark</a></li> <li><a href="/wiki/Heptaquark" title="Heptaquark">Heptaquark</a></li> <li><a href="/wiki/Skyrmion" title="Skyrmion">Skyrmion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mesons</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/Glueball" title="Glueball">Glueball</a></li> <li><a href="/wiki/Theta_meson" title="Theta meson">Theta meson</a></li> <li><a href="/wiki/T_meson" title="T meson">T meson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;;font-weight:normal; text-align: center;">Others</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/Mesonic_molecule" title="Mesonic molecule">Mesonic molecule</a></li> <li><a href="/wiki/Pomeron" title="Pomeron">Pomeron</a></li> <li><a href="/wiki/Diquark" title="Diquark">Diquark</a></li> <li><a href="/wiki/R-hadron" title="R-hadron">R-hadron</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anyon" title="Anyon">Anyon</a></li> <li><a href="/wiki/Davydov_soliton" title="Davydov soliton">Davydov soliton</a></li> <li><a href="/wiki/Dropleton" title="Dropleton">Dropleton</a></li> <li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li> <li><a href="/wiki/Fracton_(subdimensional_particle)" title="Fracton (subdimensional particle)">Fracton</a></li> <li><a href="/wiki/Electron_hole" title="Electron hole">Hole</a></li> <li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li> <li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li> <li><a href="/wiki/Plasmaron" title="Plasmaron">Plasmaron</a></li> <li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a></li> <li><a href="/wiki/Polariton" title="Polariton">Polariton</a></li> <li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li> <li><a href="/wiki/Roton" title="Roton">Roton</a></li> <li><a href="/wiki/Trion_(physics)" title="Trion (physics)">Trion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_baryons" title="List of baryons">Baryons</a></li> <li><a href="/wiki/List_of_mesons" title="List of mesons">Mesons</a></li> <li><a href="/wiki/List_of_particles" title="List of particles">Particles</a></li> <li><a href="/wiki/List_of_quasiparticles" title="List of quasiparticles">Quasiparticles</a></li> <li><a href="/wiki/Timeline_of_particle_discoveries" title="Timeline of particle discoveries">Timeline of particle discoveries</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_subatomic_physics" title="History of subatomic physics">History of subatomic physics</a> <ul><li><a href="/wiki/Timeline_of_atomic_and_subatomic_physics" title="Timeline of atomic and subatomic physics">timeline</a></li></ul></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> <ul><li><a class="mw-selflink selflink">mathematical formulation</a></li></ul></li> <li><a href="/wiki/Subatomic_particle" title="Subatomic particle">Subatomic particles</a></li> <li><a href="/wiki/Particle" title="Particle">Particles</a></li> <li><a href="/wiki/Antiparticle" title="Antiparticle">Antiparticles</a></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Eightfold_way_(physics)" title="Eightfold way (physics)">Eightfold way</a> <ul><li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Massless_particle" title="Massless particle">Massless particle</a></li> <li><a href="/wiki/Relativistic_particle" title="Relativistic particle">Relativistic particle</a></li> <li><a href="/wiki/Virtual_particle" title="Virtual particle">Virtual particle</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/wiki/Particle_chauvinism" title="Particle chauvinism">Particle chauvinism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" 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Mathematical formulation of the Standard Model - Wikipedia