Chirality (physics)

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<span>Example: u and d quarks in QCD</span> </div> </a> <ul id="toc-Example:_u_and_d_quarks_in_QCD-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-More_flavors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#More_flavors"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>More flavors</span> </div> </a> <ul id="toc-More_flavors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-An_application_in_particle_physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#An_application_in_particle_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>An application in particle physics</span> </div> </a> <ul id="toc-An_application_in_particle_physics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span 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Available in 17 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quiralitat_(f%C3%ADsica)" title="Quiralitat (física) – Catalan" lang="ca" hreflang="ca" data-title="Quiralitat (física)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kiralitet_(fysik)" title="Kiralitet (fysik) – Danish" lang="da" hreflang="da" data-title="Kiralitet (fysik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Chiralit%C3%A4t_(Physik)" title="Chiralität (Physik) – German" lang="de" hreflang="de" data-title="Chiralität (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Quiralidad_(f%C3%ADsica)" title="Quiralidad (física) – Spanish" lang="es" hreflang="es" data-title="Quiralidad (física)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kiralitate_(fisika)" title="Kiralitate (fisika) – Basque" lang="eu" hreflang="eu" data-title="Kiralitate (fisika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%E2%80%8C%D8%B3%D8%A7%D9%86%DB%8C_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="دست‌سانی (فیزیک) – Persian" lang="fa" hreflang="fa" data-title="دست‌سانی (فیزیک)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Chiralit%C3%A0_(fisica)" title="Chiralità (fisica) – Italian" lang="it" hreflang="it" data-title="Chiralità (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Хиралност – Macedonian" lang="mk" hreflang="mk" data-title="Хиралност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%82%A4%E3%83%A9%E3%83%AA%E3%83%86%E3%82%A3" title="カイラリティ – Japanese" lang="ja" hreflang="ja" data-title="カイラリティ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%80%E0%A8%B0%E0%A9%88%E0%A8%B2%E0%A8%BF%E0%A8%9F%E0%A9%80_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਚੀਰੈਲਿਟੀ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – Punjabi" lang="pa" hreflang="pa" data-title="ਚੀਰੈਲਿਟੀ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Quiralidade_(f%C3%ADsica)" title="Quiralidade (física) – Portuguese" lang="pt" hreflang="pt" data-title="Quiralidade (física)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Хиральность (физика) – Russian" lang="ru" hreflang="ru" data-title="Хиральность (физика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kiralnost_(fizika)" title="Kiralnost (fizika) – Slovenian" lang="sl" hreflang="sl" data-title="Kiralnost (fizika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D1%80%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Хиралност (физика) – Serbian" lang="sr" hreflang="sr" data-title="Хиралност (физика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kiralite_(fizik)" title="Kiralite (fizik) – Turkish" lang="tr" hreflang="tr" data-title="Kiralite (fizik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A5%D1%96%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%96%D1%81%D1%82%D1%8C_(%D1%84%D1%96%D0%B7%D0%B8%D0%BA%D0%B0)" title="Хіральність (фізика) – Ukrainian" lang="uk" hreflang="uk" data-title="Хіральність (фізика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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<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">Property of particles related to spin</div> <p>A <b>chiral</b> phenomenon is one that is not identical to its <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a> (see the article on <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">mathematical chirality</a>). The <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> of a <a href="/wiki/Elementary_particle" title="Elementary particle">particle</a> may be used to define a <b>handedness</b>, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)"> symmetry transformation</a> between the two is called <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> transformation. Invariance under parity transformation by a <a href="/wiki/Dirac_fermion" title="Dirac fermion">Dirac fermion</a> is called <b>chiral symmetry</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Chirality_and_helicity">Chirality and helicity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=1" title="Edit section: Chirality and helicity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">See also: <a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">Helicity (particle physics)</a></div> <p>The helicity of a particle is positive ("right-handed") if the direction of its <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> is the same as the direction of its motion. It is negative ("left-handed") if the directions of spin and motion are opposite. So a standard <a href="/wiki/Clock" title="Clock">clock</a>, with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards. </p><p>Mathematically, <i>helicity</i> is the sign of the projection of the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> onto the <a href="/wiki/Momentum" title="Momentum">momentum</a> <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a>: "left" is negative, "right" is positive. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Right_left_helicity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Right_left_helicity.svg/380px-Right_left_helicity.svg.png" decoding="async" width="380" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Right_left_helicity.svg/570px-Right_left_helicity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Right_left_helicity.svg/760px-Right_left_helicity.svg.png 2x" data-file-width="380" data-file-height="120" /></a><figcaption></figcaption></figure> <p>The <b>chirality</b> of a particle is more abstract: It is determined by whether the particle transforms in a right- or left-handed <a href="/wiki/Group_representation" title="Group representation">representation</a> of the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p><p>For massless particles – <a href="/wiki/Photon" title="Photon">photons</a>, <a href="/wiki/Gluon" title="Gluon">gluons</a>, and (hypothetical) <a href="/wiki/Graviton" title="Graviton">gravitons</a> – chirality is the same as <a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a>; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer. </p><p>For massive particles – such as <a href="/wiki/Electron" title="Electron">electrons</a>, <a href="/wiki/Quark" title="Quark">quarks</a>, and <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a> – chirality and helicity must be distinguished: In the case of these particles, it is possible for an observer to change to a <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frame</a> moving faster than the spinning particle, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as "apparent chirality") will be reversed. That is, helicity is a <a href="/wiki/Constant_of_motion" title="Constant of motion">constant of motion</a>, but it is not <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariant</a>. Chirality is Lorentz invariant, but is not a constant of motion: a massive left-handed spinor, when propagating, will evolve into a right handed spinor over time, and vice versa. </p><p>A <i>massless</i> particle moves with the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, so no real observer (who must always travel at less than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>) can be in any reference frame where the particle appears to reverse its relative direction of spin, meaning that all real observers see the same helicity. Because of this, the direction of spin of massless particles is not affected by a change of inertial reference frame (a <a href="/wiki/Lorentz_boost" class="mw-redirect" title="Lorentz boost">Lorentz boost</a>) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: The helicity of massless particles is a <i>relativistic invariant</i> (a quantity whose value is the same in all inertial reference frames) which always matches the massless particle's chirality. </p><p>The discovery of <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillation</a> implies that <a href="/wiki/Neutrino#Mass" title="Neutrino">neutrinos have mass</a>, so the <a href="/wiki/Photon" title="Photon">photon</a> is the only confirmed massless particle; <a href="/wiki/Gluon" title="Gluon">gluons</a> are expected to also be massless, although this has not been conclusively tested.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> Hence, these are the only two particles now known for which helicity could be identical to chirality, and only the <a href="/wiki/Photon" title="Photon">photon</a> has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Chiral_theories">Chiral theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=2" title="Edit section: Chiral theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Particle physicists have only observed or inferred left-chiral <a href="/wiki/Fermion" title="Fermion">fermions</a> and right-chiral antifermions engaging in the <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">charged weak interaction</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In the case of the weak interaction, which can in principle engage with both left- and right-chiral fermions, only two left-handed <a href="/wiki/Fermion" title="Fermion">fermions</a> interact. Interactions involving right-handed or opposite-handed fermions have not been shown to occur, implying that the universe has a preference for left-handed chirality. This preferential treatment of one chiral realization over another violates parity, as first noted by <a href="/wiki/Chien_Shiung_Wu" class="mw-redirect" title="Chien Shiung Wu">Chien Shiung Wu</a> in her famous experiment known as the <a href="/wiki/Wu_experiment" title="Wu experiment">Wu experiment</a>. This is a striking observation, since parity is a symmetry that holds for all other <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">fundamental interactions</a>. </p><p>Chirality for a <a href="/wiki/Fermionic_field#Dirac_fields" title="Fermionic field">Dirac fermion</a> <span class="texhtml mvar" style="font-style:italic;">ψ</span> is defined through the <a href="/wiki/Gamma_matrices#The_fifth_"gamma"_matrix,_γ5" title="Gamma matrices">operator <span class="texhtml"><i>γ</i><sup>5</sup></span></a>, which has <a href="/wiki/Eigenvalue,_eigenvector,_and_eigenspace" class="mw-redirect" title="Eigenvalue, eigenvector, and eigenspace">eigenvalues</a> ±1; the eigenvalue's sign is equal to the particle's chirality: +1 for right-handed, −1 for left-handed. Any Dirac field can thus be projected into its left- or right-handed component by acting with the <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection operators</a> <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(1 − <i>γ</i><sup>5</sup>)</span> or <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(1 + <i>γ</i><sup>5</sup>)</span> on <span class="texhtml mvar" style="font-style:italic;">ψ</span>. </p><p>The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction's <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity symmetry</a> violation. </p><p>A common source of confusion is due to conflating the <span class="texhtml"><i>γ</i><sup>5</sup></span>, chirality operator with the <a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a> operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this paradox is that <em>the chirality operator is equivalent to helicity for massless fields only</em>, for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, or, alternatively, helicity is not Lorentz invariant, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame. </p><p>A theory that is asymmetric with respect to chiralities is called a <b>chiral theory</b>, while a non-chiral (i.e., parity-symmetric) theory is sometimes called a <b>vector theory</b>. Many pieces of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of physics are non-chiral, which is traceable to <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomaly cancellation</a> in chiral theories. <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a> is an example of a vector theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way. </p><p>The <a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theory</a>, developed in the mid 20th century, is an example of a chiral theory. Originally, it assumed that <a href="/wiki/Neutrino#Mass" title="Neutrino">neutrinos were massless</a>, and assumed the existence of only left-handed <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a> and right-handed antineutrinos. After the observation of <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillations</a>, which imply that <a href="/wiki/Neutrino#Mass" title="Neutrino">neutrinos are massive</a> (like all other <a href="/wiki/Fermion" title="Fermion">fermions</a>) the revised <a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">theories of the electroweak interaction</a> now include both right- and left-handed <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a>. However, it is still a chiral theory, as it does not respect parity symmetry. </p><p>The exact nature of the <a href="/wiki/Neutrino" title="Neutrino">neutrino</a> is still unsettled and so the <a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theories</a> that have been proposed are somewhat different, but most accommodate the chirality of <a href="/wiki/Neutrino" title="Neutrino">neutrinos</a> in the same way as was already done for all other <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Chiral_symmetry">Chiral symmetry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=3" title="Edit section: Chiral symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vector <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theories</a> with massless Dirac fermion fields <span class="texhtml mvar" style="font-style:italic;">ψ</span> exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}\rightarrow e^{i\theta _{\rm {L}}}\psi _{\rm {L}}}"> <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">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> </mrow> </msup> <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 \psi _{\rm {L}}\rightarrow e^{i\theta _{\rm {L}}}\psi _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156209bdf6812ccfe54c2330d780e0ff0dd3f061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.812ex; height:3.009ex;" alt="{\displaystyle \psi _{\rm {L}}\rightarrow e^{i\theta _{\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 \psi _{\rm {R}}\rightarrow \psi _{\rm {R}}}"> <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">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <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 \psi _{\rm {R}}\rightarrow \psi _{\rm {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182779744ed6d1c7c8fd242da0e7e4e60d489b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.524ex; height:2.509ex;" alt="{\displaystyle \psi _{\rm {R}}\rightarrow \psi _{\rm {R}}}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\rm {L}}\rightarrow \psi _{\rm {L}}}"> <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">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <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 \psi _{\rm {L}}\rightarrow \psi _{\rm {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ade9acfe587cbeaa29d136bb18f4b2e92ead8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.159ex; height:2.509ex;" alt="{\displaystyle \psi _{\rm {L}}\rightarrow \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 \psi _{\rm {R}}\rightarrow e^{i\theta _{\rm {R}}}\psi _{\rm {R}}.}"> <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">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\rm {R}}\rightarrow e^{i\theta _{\rm {R}}}\psi _{\rm {R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5051d49715cd9d542e7422d2bec368332cc06962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.971ex; height:3.009ex;" alt="{\displaystyle \psi _{\rm {R}}\rightarrow e^{i\theta _{\rm {R}}}\psi _{\rm {R}}.}"></span></dd></dl> <p>With <span class="texhtml mvar" style="font-style:italic;">N</span> <a href="/wiki/Flavor_(particle_physics)" class="mw-redirect" title="Flavor (particle physics)">flavors</a>, we have unitary rotations instead: <span class="texhtml">U(<i>N</i>)<sub>L</sub> × U(<i>N</i>)<sub>R</sub></span>. </p><p>More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {R}}={\frac {1+\gamma ^{5}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {R}}={\frac {1+\gamma ^{5}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe65d8acc42cffccfd3341309cbfd3022f53d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.206ex; height:5.676ex;" alt="{\displaystyle P_{\rm {R}}={\frac {1+\gamma ^{5}}{2}}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {L}}={\frac {1-\gamma ^{5}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {L}}={\frac {1-\gamma ^{5}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45feeb5317cde1b4568d6570c95d22edd9be592d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.023ex; height:5.676ex;" alt="{\displaystyle P_{\rm {L}}={\frac {1-\gamma ^{5}}{2}}}"></span></dd></dl> <p>Massive fermions do not exhibit chiral symmetry, as the mass term in the <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a>, <span class="texhtml"> <i>m</i><span style="text-decoration:overline;"><i>ψ</i></span><i>ψ</i></span>, breaks chiral symmetry explicitly. </p><p><a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">Spontaneous chiral symmetry breaking</a> may also occur in some theories, as it most notably does in <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>. </p><p>The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as <b>vector symmetry</b>, and a component that actually treats them differently, known as <b>axial symmetry</b>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (cf. <i><a href="/wiki/Current_algebra" title="Current algebra">Current algebra</a></i>.) A scalar field model encoding chiral symmetry and its <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">breaking</a> is the <a href="/wiki/Chiral_model" title="Chiral model">chiral model</a>. </p><p>The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference. </p><p>The general principle is often referred to by the name <b>chiral symmetry</b>. The rule is absolutely valid in the <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>, but results from <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> experiments show a difference in the behavior of left-chiral versus right-chiral <a href="/wiki/Subatomic_particles" class="mw-redirect" title="Subatomic particles">subatomic particles</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_u_and_d_quarks_in_QCD">Example: u and d quarks in QCD</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=4" title="Edit section: Example: u and d quarks in QCD"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> (QCD) with two <i>massless</i> <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a> <span class="texhtml">u</span> and <span class="texhtml">d</span> (massive fermions do not exhibit chiral symmetry). The Lagrangian reads </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\overline {u}}\,i\displaystyle {\not }D\,u+{\overline {d}}\,i\displaystyle {\not }D\,d+{\mathcal {L}}_{\mathrm {gluons} }~.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <mi>u</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <mi>d</mi> <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">g</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\overline {u}}\,i\displaystyle {\not }D\,u+{\overline {d}}\,i\displaystyle {\not }D\,d+{\mathcal {L}}_{\mathrm {gluons} }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cdaac110ca949a56d5ac1a234c34acf2c781584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.351ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}={\overline {u}}\,i\displaystyle {\not }D\,u+{\overline {d}}\,i\displaystyle {\not }D\,d+{\mathcal {L}}_{\mathrm {gluons} }~.}"></span></dd></dl> <p>In terms of left-handed and right-handed spinors, it reads </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\overline {u}}_{\rm {L}}\,i\displaystyle {\not }D\,u_{\rm {L}}+{\overline {u}}_{\rm {R}}\,i\displaystyle {\not }D\,u_{\rm {R}}+{\overline {d}}_{\rm {L}}\,i\displaystyle {\not }D\,d_{\rm {L}}+{\overline {d}}_{\rm {R}}\,i\displaystyle {\not }D\,d_{\rm {R}}+{\mathcal {L}}_{\mathrm {gluons} }~.}"> <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"> <mover> <mi>u</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <msub> <mi>u</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"> <mover> <mi>d</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <msub> <mi>d</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"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\overline {u}}_{\rm {L}}\,i\displaystyle {\not }D\,u_{\rm {L}}+{\overline {u}}_{\rm {R}}\,i\displaystyle {\not }D\,u_{\rm {R}}+{\overline {d}}_{\rm {L}}\,i\displaystyle {\not }D\,d_{\rm {L}}+{\overline {d}}_{\rm {R}}\,i\displaystyle {\not }D\,d_{\rm {R}}+{\mathcal {L}}_{\mathrm {gluons} }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb603f8b46b8b49c28b7dcee643a16b64046f37c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.151ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}={\overline {u}}_{\rm {L}}\,i\displaystyle {\not }D\,u_{\rm {L}}+{\overline {u}}_{\rm {R}}\,i\displaystyle {\not }D\,u_{\rm {R}}+{\overline {d}}_{\rm {L}}\,i\displaystyle {\not }D\,d_{\rm {L}}+{\overline {d}}_{\rm {R}}\,i\displaystyle {\not }D\,d_{\rm {R}}+{\mathcal {L}}_{\mathrm {gluons} }~.}"></span></dd></dl> <p>(Here, <span class="texhtml"><i>i</i></span> is the imaginary unit 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 \displaystyle {\not }D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle {\not }D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa8bf256229d235914279d963a2aeaa2868bf42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle \displaystyle {\not }D}"></span> the <a href="/wiki/Dirac_operator" title="Dirac operator">Dirac operator</a>.) </p><p>Defining </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\begin{bmatrix}u\\d\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\begin{bmatrix}u\\d\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ead1b8dbaa7b5b619077ac33cd5f14fb80ec78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.351ex; height:6.176ex;" alt="{\displaystyle q={\begin{bmatrix}u\\d\end{bmatrix}},}"></span></dd></dl> <p>it can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\overline {q}}_{\rm {L}}\,i\displaystyle {\not }D\,q_{\rm {L}}+{\overline {q}}_{\rm {R}}\,i\displaystyle {\not }D\,q_{\rm {R}}+{\mathcal {L}}_{\mathrm {gluons} }~.}"> <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"> <mover> <mi>q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>i</mi> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mpadded width="0"> <mtext>⧸</mtext> </mpadded> </mrow> </mrow> <mi>D</mi> <mspace width="thinmathspace" /> <msub> <mi>q</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"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\overline {q}}_{\rm {L}}\,i\displaystyle {\not }D\,q_{\rm {L}}+{\overline {q}}_{\rm {R}}\,i\displaystyle {\not }D\,q_{\rm {R}}+{\mathcal {L}}_{\mathrm {gluons} }~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bded23eb31251cc3ea90e46984727ddb1ca9532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.914ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}={\overline {q}}_{\rm {L}}\,i\displaystyle {\not }D\,q_{\rm {L}}+{\overline {q}}_{\rm {R}}\,i\displaystyle {\not }D\,q_{\rm {R}}+{\mathcal {L}}_{\mathrm {gluons} }~.}"></span></dd></dl> <p>The Lagrangian is unchanged under a rotation of <i>q</i><sub>L</sub> by any 2×2 unitary matrix <span class="texhtml mvar" style="font-style:italic;">L</span>, and <i>q</i><sub>R</sub> by any 2×2 unitary matrix <span class="texhtml mvar" style="font-style:italic;">R</span>. </p><p>This symmetry of the Lagrangian is called <i>flavor chiral symmetry</i>, and denoted as <span class="texhtml">U(2)<sub>L</sub> × U(2)<sub>R</sub></span>. It decomposes into </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{V}\times \mathrm {U} (1)_{A}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{V}\times \mathrm {U} (1)_{A}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43991cc742473df58360eb231f75772424d5e471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.855ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{V}\times \mathrm {U} (1)_{A}~.}"></span></dd></dl> <p>The singlet vector symmetry, <span class="texhtml">U(1)<sub><i>V</i></sub></span>, acts as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\text{L}}\rightarrow e^{i\theta (x)}q_{\text{L}}\qquad q_{\text{R}}\rightarrow e^{i\theta (x)}q_{\text{R}}~,}"> <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> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mspace width="2em" /> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\text{L}}\rightarrow e^{i\theta (x)}q_{\text{L}}\qquad q_{\text{R}}\rightarrow e^{i\theta (x)}q_{\text{R}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545c3fea185cf45606721267d07edec2597ff2fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.399ex; height:3.176ex;" alt="{\displaystyle q_{\text{L}}\rightarrow e^{i\theta (x)}q_{\text{L}}\qquad q_{\text{R}}\rightarrow e^{i\theta (x)}q_{\text{R}}~,}"></span></dd></dl> <p>and thus invariant under <span class="texhtml">U(1)</span> gauge symmetry. This corresponds to <a href="/wiki/Baryon_number" title="Baryon number">baryon number</a> conservation. </p><p>The singlet axial group <span class="texhtml">U(1)<sub><i>A</i></sub></span> transforms as the following global transformation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\text{L}}\rightarrow e^{i\theta }q_{\text{L}}\qquad q_{\text{R}}\rightarrow e^{-i\theta }q_{\text{R}}~.}"> <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> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mspace width="2em" /> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\text{L}}\rightarrow e^{i\theta }q_{\text{L}}\qquad q_{\text{R}}\rightarrow e^{-i\theta }q_{\text{R}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3db86ec251b0411719e1f2d663cf9d133477f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.239ex; height:3.009ex;" alt="{\displaystyle q_{\text{L}}\rightarrow e^{i\theta }q_{\text{L}}\qquad q_{\text{R}}\rightarrow e^{-i\theta }q_{\text{R}}~.}"></span></dd></dl> <p>However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by a <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">quantum anomaly</a>. </p><p>The remaining chiral symmetry <span class="texhtml">SU(2)<sub>L</sub> × SU(2)<sub>R</sub></span> turns out to be <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneously broken</a> by a <a href="/wiki/Quark_condensate" class="mw-redirect" title="Quark condensate">quark condensate</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 \textstyle \langle {\bar {q}}_{\text{R}}^{a}q_{\text{L}}^{b}\rangle =v\delta ^{ab}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <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>R</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>v</mi> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \langle {\bar {q}}_{\text{R}}^{a}q_{\text{L}}^{b}\rangle =v\delta ^{ab}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01b9f9edd81643a2d224179b63b26c351fa254e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.012ex; height:3.176ex;" alt="{\displaystyle \textstyle \langle {\bar {q}}_{\text{R}}^{a}q_{\text{L}}^{b}\rangle =v\delta ^{ab}}"></span> formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup <span class="texhtml">SU(2)<sub><i>V</i></sub></span> known as <a href="/wiki/Isospin" title="Isospin">isospin</a>. The <a href="/wiki/Goldstone_bosons" class="mw-redirect" title="Goldstone bosons">Goldstone bosons</a> corresponding to the three broken generators are the three <a href="/wiki/Pions" class="mw-redirect" title="Pions">pions</a>. As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">chiral symmetry breaking</a> induces the bulk of hadron masses, such as those for the <a href="/wiki/Nucleon" title="Nucleon">nucleons</a> — in effect, the bulk of the mass of all visible matter. </p><p>In the real world, because of the nonvanishing and differing masses of the quarks, <span class="texhtml">SU(2)<sub>L</sub> × SU(2)<sub>R</sub></span> is only an approximate symmetry<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> to begin with, and therefore the pions are not massless, but have small masses: they are <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">pseudo-Goldstone bosons</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="More_flavors">More flavors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=5" title="Edit section: More flavors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For more "light" quark species, <span class="texhtml mvar" style="font-style:italic;">N</span> <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavors</a> in general, the corresponding chiral symmetries are <span class="texhtml">U(<i>N</i>)<sub>L</sub> × U(<i>N</i>)<sub>R′</sub></span>, decomposing into </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (N)_{\text{L}}\times \mathrm {SU} (N)_{\text{R}}\times \mathrm {U} (1)_{V}\times \mathrm {U} (1)_{A}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (N)_{\text{L}}\times \mathrm {SU} (N)_{\text{R}}\times \mathrm {U} (1)_{V}\times \mathrm {U} (1)_{A}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c08f14294629c410c9f8989b2e0fd0eb0b7fe7cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.658ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (N)_{\text{L}}\times \mathrm {SU} (N)_{\text{R}}\times \mathrm {U} (1)_{V}\times \mathrm {U} (1)_{A}~,}"></span></dd></dl> <p>and exhibiting a very analogous <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">chiral symmetry breaking</a> pattern. </p><p>Most usually, <span class="texhtml"><i>N</i> = 3</span> is taken, the u, d, and s quarks taken to be light (the <a href="/wiki/Eightfold_way_(physics)" title="Eightfold way (physics)">eightfold way</a>), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes. </p> <div class="mw-heading mw-heading3"><h3 id="An_application_in_particle_physics">An application in particle physics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=6" title="Edit section: An application in particle physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, the <a href="/wiki/Electroweak" class="mw-redirect" title="Electroweak">electroweak</a> model breaks <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> maximally. All its <a href="/wiki/Fermion" title="Fermion">fermions</a> are chiral <a href="/wiki/Weyl_fermion" class="mw-redirect" title="Weyl fermion">Weyl fermions</a>, which means that the charged <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">weak gauge bosons W<sup>+</sup> and W<sup>−</sup></a> only couple to left-handed quarks and leptons.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </p><p>Some theorists found this objectionable, and so conjectured a <a href="/wiki/Grand_unification_theory" class="mw-redirect" title="Grand unification theory">GUT</a> extension of the <a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak force</a> which has new, high energy <a href="/wiki/W%E2%80%B2_and_Z%E2%80%B2_bosons" title="W′ and Z′ bosons">W′ and Z′ bosons</a>, which <i>do</i> couple with right handed quarks and leptons: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {SU} (2)_{\text{W}}\times \mathrm {U} (1)_{Y}}{\mathbb {Z} _{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>W</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {SU} (2)_{\text{W}}\times \mathrm {U} (1)_{Y}}{\mathbb {Z} _{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d477b2b6b717b6fd0801908692f28c53c87aa129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.806ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {SU} (2)_{\text{W}}\times \mathrm {U} (1)_{Y}}{\mathbb {Z} _{2}}}}"></span></dd></dl> <p>to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>−</mo> <mi>L</mi> </mrow> </msub> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b425f9cf629a4a33c90df5a99d686e7c142a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.472ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{2}}}.}"></span></dd></dl> <p>Here, <span class="texhtml">SU(2)<sub>L</sub></span> (pronounced "<span class="texhtml">SU(2)</span> left") is <span class="texhtml">SU(2)<sub>W</sub></span> from above, while <span class="texhtml"><i><a href="/wiki/B%E2%88%92L" class="mw-redirect" title="B−L">B−L</a></i></span> is the <a href="/wiki/Baryon_number" title="Baryon number">baryon number</a> minus the <a href="/wiki/Lepton_number" title="Lepton number">lepton number</a>. The electric charge formula in this model is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=T_{\rm {3L}}+T_{\rm {3R}}+{\frac {B-L}{2}}\,;}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo>−</mo> <mi>L</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=T_{\rm {3L}}+T_{\rm {3R}}+{\frac {B-L}{2}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7b00287476f3be3721e4c5aa83749dc8c08cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.735ex; height:5.176ex;" alt="{\displaystyle Q=T_{\rm {3L}}+T_{\rm {3R}}+{\frac {B-L}{2}}\,;}"></span></dd></dl> <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 \ T_{\rm {3L}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ T_{\rm {3L}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a10fb904ca48f05d9036c6581c23ccbb02e2c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.6ex; height:2.509ex;" alt="{\displaystyle \ T_{\rm {3L}}\ }"></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 \ T_{\rm {3R}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ T_{\rm {3R}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e14f83c21924918d8292fa64a9fca70c762f3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.783ex; height:2.509ex;" alt="{\displaystyle \ T_{\rm {3R}}\ }"></span> are the left and right <a href="/wiki/Weak_isospin" title="Weak isospin">weak isospin</a> values of the fields in the theory. </p><p>There is also the <a href="/wiki/Chromodynamic" class="mw-redirect" title="Chromodynamic">chromodynamic</a> <span class="texhtml">SU(3)<sub>C</sub></span>. The idea was to restore parity by introducing a <b>left-right symmetry</b>. This is a <a href="/wiki/Group_extension" title="Group extension">group extension</a> 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 \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}"></span> (the left-right symmetry) by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {SU} (3)_{\text{C}}\times \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{6}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>−</mo> <mi>L</mi> </mrow> </msub> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {SU} (3)_{\text{C}}\times \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{6}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c7b76317292197edb08087baba6dc09f60282c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.092ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {SU} (3)_{\text{C}}\times \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{6}}}}"></span></dd></dl> <p>to the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {SU} (3)_{\text{C}}\times \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{6}}}\rtimes \mathbb {Z} _{2}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>−</mo> <mi>L</mi> </mrow> </msub> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mfrac> </mrow> <mo>⋊</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {SU} (3)_{\text{C}}\times \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{6}}}\rtimes \mathbb {Z} _{2}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c484e1134c34eafe22ad7fa78ab0e14d3810256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.764ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {SU} (3)_{\text{C}}\times \mathrm {SU} (2)_{\text{L}}\times \mathrm {SU} (2)_{\text{R}}\times \mathrm {U} (1)_{B-L}}{\mathbb {Z} _{6}}}\rtimes \mathbb {Z} _{2}\ .}"></span></dd></dl> <p>This has two <a href="/wiki/Connected_space" title="Connected space">connected components</a> 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 \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}"></span> acts as an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>, which is the composition of an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutive</a> <a href="/wiki/Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer automorphism</a> of <span class="texhtml">SU(3)<sub>C</sub></span> with the interchange of the left and right copies of <span class="texhtml">SU(2)</span> with the reversal of <span class="texhtml">U(1)<sub><i>B−L</i></sub></span>. It was shown by <a href="/wiki/Rabindra_Mohapatra" title="Rabindra Mohapatra">Mohapatra</a> & <a href="/wiki/Goran_Senjanovic" class="mw-redirect" title="Goran Senjanovic">Senjanovic</a> (1975)<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> that <a href="/wiki/Left-right_symmetry" class="mw-redirect" title="Left-right symmetry">left-right symmetry</a> can be <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneously broken</a> to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg, and Salam, and also connects the small observed neutrino masses to the breaking of left-right symmetry via the <a href="/wiki/Seesaw_mechanism" title="Seesaw mechanism">seesaw mechanism</a>. </p><p>In this setting, the chiral <a href="/wiki/Quark" title="Quark">quarks</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,2,1)_{+{1 \over 3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,2,1)_{+{1 \over 3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d72170e009621da8f6bb455a2c1a3260fa64d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.379ex; height:4.176ex;" alt="{\displaystyle (3,2,1)_{+{1 \over 3}}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\bar {3}},1,2\right)_{-{1 \over 3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\bar {3}},1,2\right)_{-{1 \over 3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27726a51c32e612206ec87042204c3b1297d4380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.699ex; height:4.343ex;" alt="{\displaystyle \left({\bar {3}},1,2\right)_{-{1 \over 3}}}"></span></dd></dl> <p>are unified into an <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible representation</a> ("irrep") </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,2,1)_{+{1 \over 3}}\oplus \left({\bar {3}},1,2\right)_{-{1 \over 3}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> </msub> <mo>⊕</mo> <msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,2,1)_{+{1 \over 3}}\oplus \left({\bar {3}},1,2\right)_{-{1 \over 3}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd295f3d2e6351075a6ee24bb833a0ef6fbc131d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.146ex; height:4.343ex;" alt="{\displaystyle (3,2,1)_{+{1 \over 3}}\oplus \left({\bar {3}},1,2\right)_{-{1 \over 3}}\ .}"></span></dd></dl> <p>The <a href="/wiki/Lepton" title="Lepton">leptons</a> are also unified into an <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible representation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,1)_{-1}\oplus (1,1,2)_{+1}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,2,1)_{-1}\oplus (1,1,2)_{+1}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db51f3727bf8d991a4a9233d8690719a3c60796e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.463ex; height:2.843ex;" alt="{\displaystyle (1,2,1)_{-1}\oplus (1,1,2)_{+1}\ .}"></span></dd></dl> <p>The <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs bosons</a> needed to implement the breaking of left-right symmetry down to the Standard Model are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,3,1)_{2}\oplus (1,1,3)_{2}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊕</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,3,1)_{2}\oplus (1,1,3)_{2}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b211b7ddd44d724d325000342e2bb7f2ff439e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.905ex; height:2.843ex;" alt="{\displaystyle (1,3,1)_{2}\oplus (1,1,3)_{2}\ .}"></span></dd></dl> <p>This then provides three <a href="/wiki/Sterile_neutrino" title="Sterile neutrino">sterile neutrinos</a> which are perfectly consistent with current<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Chirality_(physics)&action=edit">[update]</a></sup> <a href="/wiki/Neutrino_oscillation" title="Neutrino oscillation">neutrino oscillation</a> data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies. </p><p>Because the left–right symmetry is spontaneously broken, left–right models predict <a href="/wiki/Domain_wall_(string_theory)" title="Domain wall (string theory)">domain walls</a>. This left-right symmetry idea first appeared in the <a href="/wiki/Pati%E2%80%93Salam_model" title="Pati–Salam model">Pati–Salam model</a> (1974)<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and Mohapatra–Pati models (1975).<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 12em;"> <ul><li><a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theory</a></li> <li><a href="/wiki/Chirality_(chemistry)" title="Chirality (chemistry)">chirality (chemistry)</a></li> <li><a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chirality (mathematics)</a></li> <li><a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">chiral symmetry breaking</a></li> <li><a href="/wiki/Handedness" title="Handedness">handedness</a></li> <li><a href="/wiki/Spinors" class="mw-redirect" title="Spinors">spinors</a> and <a href="/wiki/Fermionic_field#Dirac_fields" title="Fermionic field">Dirac fields</a></li> <li><a href="/wiki/Sigma_model" title="Sigma model">sigma model</a></li> <li><a href="/wiki/Chiral_model" title="Chiral model">chiral model</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Note, however, that representations such as <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinors</a> and others, necessarily have both right- and left-handed components. In such cases, we can define <a href="/wiki/Projection_operator" class="mw-redirect" title="Projection operator">projection operators</a> that remove (set to zero) either the right- or left-hand components, and discuss the left- or right-handed portions of the representation that remain.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Graviton" title="Graviton">Gravitons</a> are also assumed to be massless, but so far are merely hypothetical.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">It is still possible that as-yet unobserved particles, like the <a href="/wiki/Graviton" title="Graviton">graviton</a>, might be massless, and like the <a href="/wiki/Photon" title="Photon">photon</a>, have invariant helicity that matches their chirality.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Unlike the W<sup>+</sup> and W<sup>−</sup> bosons, the neutral electroweak <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">Z<sup>0</sup> boson</a> couples to both left <i>and</i> right-handed fermions, although not equally.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPovh,_BogdanRith,_KlausScholz,_ChristophZetsche,_Frank2006" class="citation book cs1">Povh, Bogdan; Rith, Klaus; Scholz, Christoph; Zetsche, Frank (2006). <i>Particles and Nuclei: An introduction to the physical concepts</i>. Springer. p. 145. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-36683-6" title="Special:BookSources/978-3-540-36683-6"><bdi>978-3-540-36683-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particles+and+Nuclei%3A+An+introduction+to+the+physical+concepts&rft.pages=145&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-36683-6&rft.au=Povh%2C+Bogdan&rft.au=Rith%2C+Klaus&rft.au=Scholz%2C+Christoph&rft.au=Zetsche%2C+Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Ta-Pei Cheng and <a href="/w/index.php?title=Ling-Fong_Li&action=edit&redlink=1" class="new" title="Ling-Fong Li (page does not exist)">Ling-Fong Li</a>, <i>Gauge Theory of Elementary Particle Physics</i>, (Oxford 1984) <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/978-0198519614" title="Special:BookSources/978-0198519614">978-0198519614</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGell-MannRenner1968" class="citation journal cs1">Gell-Mann, M.; Renner, B. (1968). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/3634/1/GELpr68.pdf">"Behavior of Current Divergences under SU<sub>3</sub>×SU<sub>3</sub>"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review</i>. <b>175</b> (5): 2195. <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/1968PhRv..175.2195G">1968PhRv..175.2195G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.175.2195">10.1103/PhysRev.175.2195</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Behavior+of+Current+Divergences+under+SU%3Csub%3E3%3C%2Fsub%3E%C3%97SU%3Csub%3E3%3C%2Fsub%3E&rft.volume=175&rft.issue=5&rft.pages=2195&rft.date=1968&rft_id=info%3Adoi%2F10.1103%2FPhysRev.175.2195&rft_id=info%3Abibcode%2F1968PhRv..175.2195G&rft.aulast=Gell-Mann&rft.aufirst=M.&rft.au=Renner%2C+B.&rft_id=https%3A%2F%2Fauthors.library.caltech.edu%2F3634%2F1%2FGELpr68.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeskinSchroeder1995" class="citation book cs1">Peskin, Michael; Schroeder, Daniel (1995). <i>An Introduction to Quantum Field Theory</i>. Westview Press. p. 670. <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"><bdi>0-201-50397-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Quantum+Field+Theory&rft.pages=670&rft.pub=Westview+Press&rft.date=1995&rft.isbn=0-201-50397-2&rft.aulast=Peskin&rft.aufirst=Michael&rft.au=Schroeder%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSenjanovicMohapatra1975" class="citation journal cs1"><a href="/wiki/Goran_Senjanovic" class="mw-redirect" title="Goran Senjanovic">Senjanovic, Goran</a>; <a href="/wiki/Rabindra_Mohapatra" title="Rabindra Mohapatra">Mohapatra, Rabindra N.</a> (1975). "Exact left-right symmetry and spontaneous violation of parity". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>12</b> (5): 1502. <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/1975PhRvD..12.1502S">1975PhRvD..12.1502S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.12.1502">10.1103/PhysRevD.12.1502</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Exact+left-right+symmetry+and+spontaneous+violation+of+parity&rft.volume=12&rft.issue=5&rft.pages=1502&rft.date=1975&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.12.1502&rft_id=info%3Abibcode%2F1975PhRvD..12.1502S&rft.aulast=Senjanovic&rft.aufirst=Goran&rft.au=Mohapatra%2C+Rabindra+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatiSalam1974" class="citation journal cs1">Pati, Jogesh C.; Salam, Abdus (1 June 1974). "Lepton number as the fourth "color"<span class="cs1-kern-right"></span>". <i><a href="/wiki/Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>10</b> (1): 275–289. <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/1974PhRvD..10..275P">1974PhRvD..10..275P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevd.10.275">10.1103/physrevd.10.275</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Lepton+number+as+the+fourth+%22color%22&rft.volume=10&rft.issue=1&rft.pages=275-289&rft.date=1974-06-01&rft_id=info%3Adoi%2F10.1103%2Fphysrevd.10.275&rft_id=info%3Abibcode%2F1974PhRvD..10..275P&rft.aulast=Pati&rft.aufirst=Jogesh+C.&rft.au=Salam%2C+Abdus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMohapatraPati1975" class="citation journal cs1">Mohapatra, R.N.; Pati, J.C. (1975). "<span class="cs1-kern-left"></span>'Natural' left-right symmetry". <i>Physical Review D</i>. <b>11</b> (9): 2558–2561. <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/1975PhRvD..11.2558M">1975PhRvD..11.2558M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.11.2558">10.1103/PhysRevD.11.2558</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=%27Natural%27+left-right+symmetry&rft.volume=11&rft.issue=9&rft.pages=2558-2561&rft.date=1975&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.11.2558&rft_id=info%3Abibcode%2F1975PhRvD..11.2558M&rft.aulast=Mohapatra&rft.aufirst=R.N.&rft.au=Pati%2C+J.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></span> </li> </ol></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalter_GreinerBerndt_Müller2000" class="citation book cs1">Walter Greiner; Berndt Müller (2000). <i>Gauge Theory of Weak Interactions</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-67672-4" title="Special:BookSources/3-540-67672-4"><bdi>3-540-67672-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theory+of+Weak+Interactions&rft.pub=Springer&rft.date=2000&rft.isbn=3-540-67672-4&rft.au=Walter+Greiner&rft.au=Berndt+M%C3%BCller&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGordon_L._Kane1987" class="citation book cs1">Gordon L. Kane (1987). <i>Modern Elementary Particle Physics</i>. Perseus Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-11749-5" title="Special:BookSources/0-201-11749-5"><bdi>0-201-11749-5</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+Elementary+Particle+Physics&rft.pub=Perseus+Books&rft.date=1987&rft.isbn=0-201-11749-5&rft.au=Gordon+L.+Kane&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKondepudiHegstrom1990" class="citation journal cs1">Kondepudi, Dilip K.; Hegstrom, Roger A. (January 1990). "The Handedness of the Universe". <i>Scientific American</i>. <b>262</b> (1): 108–115. <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/1990SciAm.262a.108H">1990SciAm.262a.108H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0190-108">10.1038/scientificamerican0190-108</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=The+Handedness+of+the+Universe&rft.volume=262&rft.issue=1&rft.pages=108-115&rft.date=1990-01&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0190-108&rft_id=info%3Abibcode%2F1990SciAm.262a.108H&rft.aulast=Kondepudi&rft.aufirst=Dilip+K.&rft.au=Hegstrom%2C+Roger+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWinters1995" class="citation journal cs1">Winters, Jeffrey (November 1995). <a rel="nofollow" class="external text" href="http://discovermagazine.com/1995/nov/lookingfortherig591">"Looking for the Right Hand"</a>. <i>Discover</i><span class="reference-accessdate">. Retrieved <span class="nowrap">12 September</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discover&rft.atitle=Looking+for+the+Right+Hand&rft.date=1995-11&rft.aulast=Winters&rft.aufirst=Jeffrey&rft_id=http%3A%2F%2Fdiscovermagazine.com%2F1995%2Fnov%2Flookingfortherig591&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChirality+%28physics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chirality_(physics)&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>To see a summary of the differences and similarities between chirality and helicity (those covered here and more) in chart form, one may go to <a rel="nofollow" class="external text" href="http://www.quantumfieldtheory.info">Pedagogic Aids to Quantum Field Theory</a> and click on the link near the bottom of the page entitled "Chirality and Helicity Summary". To see an in depth discussion of the two with examples, which also shows how chirality and helicity approach the same thing as speed approaches that of light, click the link entitled "Chirality and Helicity in Depth" on the same page.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050403125400/http://ccreweb.org/documents/parity/parity.html">History of science: parity violation</a></li> <li><a rel="nofollow" class="external text" href="http://www.quantumdiaries.org/2011/06/19/helicity-chirality-mass-and-the-higgs/">Helicity, Chirality, Mass, and the Higgs</a> (Quantum Diaries blog)</li> <li><a rel="nofollow" class="external text" href="http://www.quantumfieldtheory.info/Chiralityvshelicitychart.htm">Chirality vs helicity chart</a> (Robert D. Klauber)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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Chirality (physics) - Wikipedia