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meson in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></strong></p> <p><strong>and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></strong>:</p> <p><strong><a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a></strong> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>)</p> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the</strong> <br /> <strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</strong></th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a></strong> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <br /> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons</strong>: <br /> <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <br /> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <br /> <br /> <strong><a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <p>(also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>)</p> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a></strong> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meson">meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pion">pion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/kaon">kaon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proton">proton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/neutron">neutron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/carbon">carbon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/caloron">caloron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> </ul> <p><strong>in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a></strong></p> <p>bosinos:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluino">gluino</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axion">axion</a></p> </li> </ul> <p><strong>Exotica</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/preon">preon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monopole">monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antifield">antifield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#FirstGenerationMesons'>First-generation mesons</a></li> <li><a href='#secondgeneration_mesons'>Second-generation mesons</a></li> <li><a href='#thirdgeneration_mesons'>Third-generation mesons</a></li> <li><a href='#see_also'>See also</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#ConceptualizationAndComputationInAdSQCD'>Conceptualization and computation in AdS/QCD</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_chiral_perturbation_theory'>In chiral perturbation theory</a></li> <li><a href='#QuantumHadrodynamicsReferences'>Walecka hadrodynamics with nucleon fields</a></li> <li><a href='#hadrodynamics_via_lightfront_qcd'>Hadrodynamics via Light-front QCD</a></li> <li><a href='#ReferencesSkyrmeHadrodynamicsWithVectorMesons'>Skyrme hadrodynamics with vector mesons (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-model)</a></li> <ul> <li><a href='#SkyrmeModelWithOmegaMesonsReferences'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-meson</a></li> <li><a href='#inclusion_of_the_meson_2'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</a></li> <li><a href='#inclusion_of_the__and_meson'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</a></li> <li><a href='#inclusion_of_the_meson_3'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-meson</a></li> </ul> <li><a href='#SkyrmeHadrodynamicsWithHeavyMesonsReferences'>Skyrme hadrodynamics with heavy quarks/mesons</a></li> <ul> <li><a href='#inclusion_of_strange_quarkskaons'>Inclusion of strange quarks/kaons</a></li> <li><a href='#inclusion_of_charm_quarksdmesons'>Inclusion of charm quarks/D-mesons</a></li> <li><a href='#inclusion_of_bottom_quarksbmesons'>Inclusion of bottom quarks/B-mesons</a></li> </ul> <li><a href='#WZWTermOfChiralPerturbationTheoryReferences'>The WZW term of QCD chiral perturbation theory</a></li> <ul> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesGeneral'>General</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons'>Including light vector mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons'>Including heavy scalar mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons'>Including heavy vector mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions'>Including electroweak interactions</a></li> </ul> <li><a href='#ReferencesHadronsAsKKModesOf5dYangMillsTheory'>Hadrons as KK-modes of 5d Yang-Mills theory</a></li> <li><a href='#in_the_large__limit'>In the large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> limit</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> a <em>meson</em> is a <a class="existingWikiWord" href="/nlab/show/bound+state">bound state</a> of two <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a> via the <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a>.</p> <p><br /></p> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the</strong> <br /> <strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</strong></th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a></strong> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <br /> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons</strong>: <br /> <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <br /> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <br /> <br /> <strong><a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="examples">Examples</h2> <h3 id="FirstGenerationMesons">First-generation mesons</h3> <p>The mesons in the first <a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a>, being <a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of <a class="existingWikiWord" href="/nlab/show/up+quarks">up quarks</a> and <a class="existingWikiWord" href="/nlab/show/down+quarks">down quarks</a> are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%80-meson">π-meson</a> (pion)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a></p> </li> </ul> <p id="WignerClassification"> As <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> (or rather of the<a class="existingWikiWord" href="/nlab/show/Pin+group">Pin group</a>) and <a class="existingWikiWord" href="/nlab/show/isospin">isospin</a> <a class="existingWikiWord" href="/nlab/show/flavour+%28particle+physics%29">flavour</a> group (under <a class="existingWikiWord" href="/nlab/show/Wigner+classification">Wigner classification</a>) these first-generation meson <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> are given as in the first of the following two tables:</p> <div style="margin: 0px 10px 0px 20px"> <img src="/nlab/files/LightAndChiralPartnerMesonFieldsPinIII.jpg" width="600px" /> </div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>n</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{n}</annotation></semantics></math> denotes an <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, and the subscript <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi mathvariant="normal">sgn</mi></msub></mrow><annotation encoding="application/x-tex">{}_{\mathrm{sgn}}</annotation></semantics></math> indicates the one of the two irreps of that dimension which pick up an extra sign under <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>-reversal (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mn>4</mn></mstyle> <mrow><msub><mrow></mrow> <mi>sgn</mi></msub></mrow></msub><mo>=</mo><mstyle mathvariant="bold"><mn>4</mn></mstyle><mo>⊗</mo><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mrow></mrow> <mi>sgn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{4}_{{}_{sgn}} = \mathbf{4} \otimes \mathbf{1}_{{}_{sgn}}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mrow></mrow> <mi>sgn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{1}_{{}_{sgn}}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/sign+representation">sign representation</a>).</p> <p>So the <a class="existingWikiWord" href="/nlab/show/pion">pion</a> is a <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz</a>-<a class="existingWikiWord" href="/nlab/show/pseudoscalar">pseudoscalar</a>, and the <a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a> and <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a> are <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz</a> <a class="existingWikiWord" href="/nlab/show/pseudovector">pseudovector</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>.</p> <p>As bilinears in the <a class="existingWikiWord" href="/nlab/show/up-quark">up-quark</a> <a class="existingWikiWord" href="/nlab/show/Weyl+spinor">Weyl spinor</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/down-quark">down-quark</a> <a class="existingWikiWord" href="/nlab/show/Weyl+spinor">Weyl spinor</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> these <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> are given as follows (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>σ</mi> <mn>0</mn></msup><mo>,</mo><msup><mi>σ</mi> <mn>1</mn></msup><mo>,</mo><msup><mi>σ</mi> <mn>2</mn></msup><mo>,</mo><msup><mi>σ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sigma^0, \sigma^1, \sigma^2, \sigma^3)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Pauli+matrices">Pauli matrices</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>σ</mi> <mn>0</mn></msup><munder><munder><mrow><msup><mi>ω</mi> <mi>μ</mi></msup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mstyle mathcolor="blue"><mi>omega</mi></mstyle></mrow><mrow><mstyle mathcolor="blue"><mi>meson</mi></mstyle></mrow></mfrac></munder><mo>+</mo><msup><mi>σ</mi> <mi>i</mi></msup><munder><munder><mrow><msubsup><mi>ρ</mi> <mi>i</mi> <mi>μ</mi></msubsup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mstyle mathcolor="blue"><mi>rho</mi></mstyle></mrow><mrow><mstyle mathcolor="blue"><mi>meson</mi></mstyle></mrow></mfrac></munder><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mover><mi>u</mi><mo stretchy="false">¯</mo></mover><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mi>γ</mi> <mn>5</mn></msup><mi>u</mi></mtd> <mtd><mover><mi>u</mi><mo stretchy="false">¯</mo></mover><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mi>γ</mi> <mn>5</mn></msup><mi>d</mi></mtd></mtr> <mtr><mtd><mover><mi>d</mi><mo stretchy="false">¯</mo></mover><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mi>γ</mi> <mn>5</mn></msup><mi>u</mi></mtd> <mtd><mover><mi>d</mi><mo stretchy="false">¯</mo></mover><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mi>γ</mi> <mn>5</mn></msup><mi>d</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \sigma^0 \underset{ {\color{blue}omega} \atop {\color{blue}meson} }{ \underbrace{ \omega^\mu } } + \sigma^i \underset{ {\color{blue}rho} \atop {\color{blue}meson} }{ \underbrace{ \rho^\mu_i } } \;=\; \left( \array{ \bar u \gamma^\mu \gamma^5 u & \bar u \gamma^\mu \gamma^5 d \\ \bar d \gamma^\mu \gamma^5 u & \bar d \gamma^\mu \gamma^5 d } \right) </annotation></semantics></math></div> <h3 id="secondgeneration_mesons">Second-generation mesons</h3> <p>Mesons containing <a class="existingWikiWord" href="/nlab/show/strange+quarks">strange quarks</a> (the first <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a> with respect to 2-flavor <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a>) from the second <a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phi-meson">phi-meson</a></p> </li> </ul> <h3 id="thirdgeneration_mesons">Third-generation mesons</h3> <p><a class="existingWikiWord" href="/nlab/show/heavy+meson">Heavy mesons</a> containing <a class="existingWikiWord" href="/nlab/show/bottom+quarks">bottom quarks</a> from the second <a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B+meson">B meson</a></li> </ul> <h3 id="see_also">See also</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a></li> </ul> <h2 id="properties">Properties</h2> <h3 id="ConceptualizationAndComputationInAdSQCD">Conceptualization and computation in AdS/QCD</h3> <p>In the <a class="existingWikiWord" href="/nlab/show/Witten-Sakai-Sugimoto+model">Witten-Sakai-Sugimoto model</a> for <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">strongly coupled</a> <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> via an <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+model">intersecting D-brane model</a>, the <a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a> in <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> correspond to <a class="existingWikiWord" href="/nlab/show/string+theory">string-theoretic</a>-phenomena in an ambient <a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk field theory</a> on an approximately <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a>:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of 2 <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) correspond to <a class="existingWikiWord" href="/nlab/show/open+strings">open strings</a> in the bulk, whose two endpoints on the <a class="existingWikiWord" href="/nlab/show/asymptotic+boundary">asymptotic boundary</a> correspond to the two <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">N_c</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) appear in two different but equivalent (<a href="#Sugimoto16">Sugimoto 16, 15.4.1</a>) guises:</p> <ol> <li> <p>as <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/D4-branes">D4-branes</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">N_c</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/open+strings">open strings</a> connecting them to the <a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a></p> <p>(<a href="#Witten98b">Witten 98b</a>, <a href="#GrossOoguri98">Gross-Ooguri 98</a>)</p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/skyrmions">skyrmions</a></p> <p>(<a href="#SakaiSugimoto04">Sakai-Sugimoto 04, section 5.2</a>, <a href="#SakaiSugimoto05">Sakai-Sugimoto 05, section 3.3</a>, see <a href="#Bartolini17">Bartolini 17</a>).</p> </li> </ol> </li> </ol> <p>For review see <a href="#Sugimoto16">Sugimoto 16</a>, also <a href="#Rebhan14">Rebhan 14, around (18)</a>.</p> <center> <img src="https://ncatlab.org/nlab/files/BaryonsAsD4Branes.jpg" width="700" /> </center> <blockquote> <p>graphics grabbed from <a href="#Sugimoto16">Sugimoto 16</a></p> </blockquote> <p>This produces <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <a class="existingWikiWord" href="/nlab/show/mass">mass</a> spectra with moderate quantitative agreement with <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> (<a href="#HSSY07">HSSY 07</a>):</p> <center> <img src="https://ncatlab.org/nlab/files/BaryonSpectrumInSakaiSugimoto.jpg" width="700" /> </center> <blockquote> <p>graphics grabbed from <a href="#Sugimoto16">Sugimoto 16</a></p> </blockquote> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a>, <a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a>, <a class="existingWikiWord" href="/nlab/show/tensor+mesons">tensor mesons</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/XYZ+particle">XYZ particle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hadron">hadron</a>, <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hadron+supersymmetry">hadron supersymmetry</a>, <a class="existingWikiWord" href="/nlab/show/hadron+Kaluza-Klein+theory">hadron Kaluza-Klein theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/flavor+brane">flavor brane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/confinement">confinement</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quark+bag+model">quark bag model</a>, <a class="existingWikiWord" href="/nlab/show/Cheshire+cat+principle">Cheshire cat principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson+dominance">vector meson dominance</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li>Eef van Beveren, George Rupp, <em>Scalar and axial-vector mesons</em>, Eur. Phys. J. A31:468-473, 2007 (<a href="https://arxiv.org/abs/hep-ph/0610199">arXiv:hep-ph/0610199</a>)</li> </ul> <p>Origin of <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> of <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> in <a class="existingWikiWord" href="/nlab/show/nuclear+physics">nuclear physics</a>, via <a class="existingWikiWord" href="/nlab/show/hidden+local+gauge+symmetry">hidden local gauge symmetry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jun+John+Sakurai">Jun John Sakurai</a>, <em>Theory of strong interactions</em>, Annals Phys. 11 (1960) 1-48 (<a href="http://inspirehep.net/record/3218">spire:3218</a>, <a href="https://doi.org/10.1016/0003-4916(60)90126-3">doi:10.1016/0003-4916(60)90126-3</a>)</li> </ul> <p>Introduction and survey:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jun+John+Sakurai">Jun John Sakurai</a>, <em>Currents and Mesons</em>, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) (<a href="https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html">ISBN: 9780226733838</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rupert+Machleidt">Rupert Machleidt</a>, <em>The Meson Theory of Nuclear Forces and Nuclear Structure</em>, In: Negele J.W., Vogt E. (eds.) <em>Advances in Nuclear Physics</em> Advances in Nuclear Physics, vol 19. Springer 1989 (<a href="https://doi.org/10.1007/978-1-4613-9907-0_2">doi:10.1007/978-1-4613-9907-0_2</a>)</p> </li> <li id="MachleidtEntem11"> <p><a class="existingWikiWord" href="/nlab/show/Rupert+Machleidt">Rupert Machleidt</a>, <a class="existingWikiWord" href="/nlab/show/David+Rodr%C3%ADguez+Entem">David Rodríguez Entem</a>, <em>Chiral effective field theory and nuclear forces</em>, Phys. Rept. 503:1-75, 2011 (<a href="https://arxiv.org/abs/1105.2919">arXiv:1105.2919</a>, <a href="https://doi.org/10.1016/j.physrep.2011.02.001">doi:10.1016/j.physrep.2011.02.001</a>)</p> </li> <li> <p>Francesco Giacosa, <em>Conventional mesons below 2 GeV</em> [<a href="https://arxiv.org/abs/2211.10092">arXiv:2211.10092</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Franz+Gross">Franz Gross</a>, <a class="existingWikiWord" href="/nlab/show/Eberhard+Klempt">Eberhard Klempt</a> et al., Section 8 of: <em>50 Years of Quantum Chromodynamics</em>, EJPC [<a href="https://arxiv.org/abs/2212.11107">arXiv:2212.11107</a>]</p> </li> </ul> <p>See also</p> <ul> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Meson">Meson</a></em></p> </li> <li> <p>Yi Zhang, <em><a href="http://snst-hu.lzu.edu.cn/zhangyi/ndata/Meson.html">Mesons</a></em></p> </li> </ul> <p>The <span class="newWikiWord">pole approximation<a href="/nlab/new/pole+approximation">?</a></span>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a> and Fredrik Zachariasen, <em>Form Factors and Vector Mesons</em>, Phys. Rev. 124, 953 (1961) (<a href="https://doi.org/10.1103/PhysRev.124.953">doi:10.1103/PhysRev.124.953</a>)</li> </ul> <h3 id="in_chiral_perturbation_theory">In chiral perturbation theory</h3> <p>See the references at <em><a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a></em>.</p> <div> <h3 id="QuantumHadrodynamicsReferences">Walecka hadrodynamics with nucleon fields</h3> <p>On <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> (<a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativivist</a> <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> of <a class="existingWikiWord" href="/nlab/show/nuclear+physics">nuclear physics</a>, <a class="existingWikiWord" href="/nlab/show/interaction">coupling</a> <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> and <a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>) in the sense of the <a class="existingWikiWord" href="/nlab/show/Walecka+model">Walecka model</a>, hence with <a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a> appearing as explicit <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> (as opposed to being <a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a> <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a> in the <a class="existingWikiWord" href="/nlab/show/pion+field">pion field</a> as in <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a>).</p> <p>Precursor:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hans-Peter+Duerr">Hans-Peter Duerr</a>, <em>Relativistic Effects in Nuclear Forces</em>, Phys. Rev. 103, 469 (1956) (<a href="https://doi.org/10.1103/PhysRev.103.469">doi:10.1103/PhysRev.103.469</a>)</li> </ul> <p>The original <a class="existingWikiWord" href="/nlab/show/Walecka+model">Walecka model</a> (<em>QHD-I model</em>), with <a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a> coupled to <a class="existingWikiWord" href="/nlab/show/sigma-mesons">sigma-mesons</a> and <a class="existingWikiWord" href="/nlab/show/omega-mesons">omega-mesons</a>:</p> <ul> <li id="Walecka74"><a class="existingWikiWord" href="/nlab/show/John+Dirk+Walecka">John Dirk Walecka</a>, <em>A Theory of highly condensed matter</em>, Annals Phys. 83 (1974) 491 (<a href="http://inspirehep.net/record/91609?ln=en">spire:91609</a>, <a href="https://doi.org/10.1016/0003-4916(74)90208-5">dpi:10.1016/0003-4916(74)90208-5</a>)</li> </ul> <p>Inclusion into the <a class="existingWikiWord" href="/nlab/show/Walecka+model">Walecka model</a> also of the <a class="existingWikiWord" href="/nlab/show/pion">pion</a> and the <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a> (the <em>QHD-II model</em>):</p> <ul> <li id="Serot79"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Serot">Brian Serot</a>, <em>A relativistic nuclear field theory with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> mesons</em>, Physics Letters B Volume 86, Issue 2, 24 (1979), Pages 146-150 (<a href="https://doi.org/10.1016/0370-2693(79)90804-9">doi:10.1016/0370-2693(79)90804-9</a>)</p> </li> <li> <p>T Matsui, <a class="existingWikiWord" href="/nlab/show/Brian+Serot">Brian Serot</a>, <em>The pion propagator in relativistic quantum field theories of the nuclear many-body problem</em>, Annals of Physics Volume 144, Issue 1, November 1982, Pages 107-167 (<a href="https://doi.org/10.1016/0003-4916(82)90106-3">doi:10.1016/0003-4916(82)90106-3</a>)</p> </li> </ul> <p>Further discussion of these models:</p> <ul> <li> <p>S. A. Chin, <em>A relativistic many-body theory of high density matter</em>, Annals of Physics Volume 108, Issue 2, October 1977, Pages 301-367 (<a href="https://doi.org/10.1016/0003-4916(77)90016-1">doi:10.1016/0003-4916(77)90016-1</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brian+Serot">Brian Serot</a>, <a class="existingWikiWord" href="/nlab/show/John+Dirk+Walecka">John Dirk Walecka</a>, <em>The Relativistic Nuclear Many Body Problem</em>, Adv. Nucl. Phys. 16 (1986) 1-327 (<a href="https://inspirehep.net/literature/207866">spire:207866</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brian+Serot">Brian Serot</a>, <em>Quantum hadrodynamics</em>, Reports on Progress in Physics, Volume 55, Number 11 (1992) (<a href="https://doi.org/10.1088/0034-4885/55/11/001">doi:10.1088/0034-4885/55/11/001</a>)</p> </li> <li id="SerotWalecka92"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Serot">Brian Serot</a>, <a class="existingWikiWord" href="/nlab/show/John+Dirk+Walecka">John Dirk Walecka</a>, <em>Chiral QHD with vector mesons</em>, Acta Phys. Polon. B 23 (1992) 655-679 (<a href="https://inspirehep.net/literature/343513">spire:343513</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maciej+Nowak">Maciej Nowak</a>, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, <em><a class="existingWikiWord" href="/nlab/show/Chiral+Nuclear+Dynamics">Chiral Nuclear Dynamics</a></em>, World Scientific 1996 (<a href="https://doi.org/10.1142/1681">doi:10.1142/1681</a>)</p> </li> <li id="SerotWalecka97"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Serot">Brian Serot</a>, <a class="existingWikiWord" href="/nlab/show/John+Dirk+Walecka">John Dirk Walecka</a>, <em>Recent Progress in Quantum Hadrodynamics</em>, Int. J. Mod. Phys. E6:515-631, 1997 (<a href="http://arxiv.org/abs/nucl-th/9701058">arXiv:nucl-th/9701058</a>)</p> </li> <li> <p>R. V. Poberezhnyuk, V. Vovchenko, D. V. Anchishkin, M. I. Gorenstein, <em>Quantum van der Waals and Walecka models of nuclear matter</em> (<a href="https://arxiv.org/abs/1708.05605">arXiv:1708.05605</a>)</p> </li> </ul> <p>Further inclusion of <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a> (<a class="existingWikiWord" href="/nlab/show/photon">photon</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a>):</p> <ul> <li>A. Yu. Korchin, D. Van Neck, M. Waroquier, <em>Electromagnetic interaction in chiral quantum hadrodynamics and decay of vector and axial-vector mesons</em>, Phys. Rev. C67 (2003) 015207 (<a href="https://arxiv.org/abs/nucl-th/0302042">arXiv:nucl-th/0302042</a>)</li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/quark-meson+coupling+model">quark-meson coupling model</a>:</p> <ul> <li>Koichi Saito, <em>Relationship between Quark-Meson Coupling Model and Quantum Hadrodynamics</em>, Prog. Theor. Phys. 108 (2002) 609-614 (<a href="https://arxiv.org/abs/nucl-th/0207053">arXiv:nucl-th/0207053</a>)</li> </ul> <h3 id="hadrodynamics_via_lightfront_qcd">Hadrodynamics via Light-front QCD</h3> <p>On <a class="existingWikiWord" href="/nlab/show/light-front+QCD">light-front QCD</a> for <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <ul> <li> <p>International Light Cone Advisory Committee, <em>Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics</em>, Nuclear Physics B - Proceedings Supplements Volumes 251–252, June–July 2014, Pages 165-174 (<a href="https://arxiv.org/abs/1309.6333">arXiv:1309.6333</a>, <a href="https://doi.org/10.1016/j.nuclphysbps.2014.05.004">doi:10.1016/j.nuclphysbps.2014.05.004</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Shuryak">Edward Shuryak</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, <em>Hadronic structure on the light-front I: Instanton effects and quark-antiquark effective potentials</em> (<a href="https://arxiv.org/abs/2110.15927">arXiv:2110.15927</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Shuryak">Edward Shuryak</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, <em>Hadronic structure on the light-front II: QCD strings, Wilson lines and potentials</em> (<a href="https://arxiv.org/abs/2111.01775">arXiv:2111.01775</a>)</p> </li> </ul> </div><div> <h3 id="ReferencesSkyrmeHadrodynamicsWithVectorMesons">Skyrme hadrodynamics with vector mesons (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-model)</h3> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> (<a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a> and <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a>/<a class="existingWikiWord" href="/nlab/show/A1-meson">A1-meson</a>) into the <a class="existingWikiWord" href="/nlab/show/Skyrmion+model">Skyrmion model</a> of <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>, in addition to the <a class="existingWikiWord" href="/nlab/show/pion">pion</a>:</p> <p>First, on the equivalence between <a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a>- and <a class="existingWikiWord" href="/nlab/show/massive+Yang-Mills+theory">massive Yang-Mills theory</a>-description of <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, H. Toki, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, <em>Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 (<a href="http://inspirehep.net/record/24079">spire:24079</a>)</em></li> </ul> <p>See also</p> <ul> <li>Marcelo Ipinza, Patricio Salgado-Rebolledo, <em>Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model</em> (<a href="https://arxiv.org/abs/2005.04920">arXiv:2005.04920</a>)</li> </ul> <h4 id="SkyrmeModelWithOmegaMesonsReferences">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-meson</h4> <p>Original proposal for inclusion of the <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> in the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li id="AdkinsNappi84"><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <a class="existingWikiWord" href="/nlab/show/Chiara+Nappi">Chiara Nappi</a>, <em>Stabilization of Chiral Solitons via Vector Mesons</em>, Phys. Lett. 137B (1984) 251-256 (<a href="http://inspirehep.net/record/194727">spire:194727</a>, <a href="https://doi.org/10.1016/0370-2693(84)90239-9">doi:10.1016/0370-2693(84)90239-9</a>)</li> </ul> <p>Relating to <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a>-<a class="existingWikiWord" href="/nlab/show/scattering">scattering</a>:</p> <ul> <li>J. M. Eisenberg, A. Erell, R. R. Silbar, <em>Nucleon-nucleon force in a skyrmion model stabilized by omega exchange</em>, Phys. Rev. C 33, 1531 (1986) (<a href="https://doi.org/10.1103/PhysRevC.33.1531">doi:10.1103/PhysRevC.33.1531</a>)</li> </ul> <p>Combination of the <a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a>-stabilized <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a> with the <a class="existingWikiWord" href="/nlab/show/bag+model+for+quark+confinement">bag model</a> for <a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, <em>Omega stabilized chiral bag model with a surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mi>q</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">\omega q q</annotation></semantics></math> coupling</em>, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (<a href="https://doi.org/10.1016/0375-9474(92)90544-T">doi:10.1016/0375-9474(92)90544-T</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-stabilized Skyrme model:</p> <ul> <li id="GudnasonSpeight20"> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/James+Martin+Speight">James Martin Speight</a>, <em>Realistic classical binding energies in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-Skyrme model</em> (<a href="https://arxiv.org/abs/2004.12862">arXiv:2004.12862</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Leask">Paul Leask</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Speight">Martin Speight</a>, <em>Skyrmion crystals stabilized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-mesons</em> &lbrack;<a href="https://arxiv.org/abs/2404.11287">arXiv:2404.11287</a>&rbrack;</p> </li> </ul> <h4 id="inclusion_of_the_meson_2">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</h4> <p>Original proposal for inclusion of the <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a>:</p> <ul> <li> <p>Y. Igarashi, M. Johmura, A. Kobayashi, H. Otsu, T. Sato, S. Sawada, <em>Stabilization of Skyrmions via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-Mesons</em>, Nucl.Phys. B259 (1985) 721-729 (<a href="http://inspirehep.net/record/213451">spire:213451</a>, <a href="https://doi.org/10.1016/0550-3213(85)90010-0">doi:10.1016/0550-3213(85)90010-0</a>)</p> </li> <li id="Adkins86"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <em>Rho mesons in the Skyrme model</em>, Phys. Rev. D 33, 193 (1986) (<a href="http://inspirehep.net/record/16895">spire:16895</a>, <a href="https://doi.org/10.1103/PhysRevD.33.193">doi:10.1103/PhysRevD.33.193</a>)</p> </li> </ul> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> of light <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a>:</p> <ul> <li id="NayaSutcliffe18"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions and clustering in light nuclei</em>, Phys. Rev. Lett. 121, 232002 (2018) (<a href="https://arxiv.org/abs/1811.02064">arXiv:1811.02064</a>)</p> </li> <li id="NayaSutcliffe18b"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions in models with pions and rho</em>, JHEP 05 (2018) 174 (<a href="https://arxiv.org/abs/1803.06098">arXiv:1803.06098</a>)</p> <p>APS Synopsis: <em><a href="https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002">Revamping the Skyrmion Model</a></em>, 2018</p> </li> </ul> <p>See also:</p> <ul> <li>Miguel Huidobro, <a class="existingWikiWord" href="/nlab/show/Paul+Leask">Paul Leask</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, Andrzej Wereszczynski, <em>Compressibility of dense nuclear matter in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson variant of the Skyrme model</em> [<a href="https://arxiv.org/abs/2405.20757">arXiv:2405.20757</a>]</li> </ul> <h4 id="inclusion_of_the__and_meson">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</h4> <p>The resulting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> model:</p> <ul> <li id="MeissnerZahed86"> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, <em>Skyrmions in the Presence of Vector Mesons</em>, Phys. Rev. Lett. 56, 1035 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.56.1035">doi:10.1103/PhysRevLett.56.1035</a>)</p> <p>(includes also the <a class="existingWikiWord" href="/nlab/show/A1-meson">A1-meson</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, <em>Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties</em>, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, Andreas Wirzba, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, <em>Skyrmions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> Mesons as Dynamical Gauge Bosons</em>, Phys. Rev. Lett. 57, 1676 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.57.1676">doi:10.1103/PhysRevLett.57.1676</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <em>Low-energy hadron physics from effective chiral Lagrangians with vector mesons</em>, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (<a href="https://doi.org/10.1016/0370-1573(88)90090-7">doi:10.1016/0370-1573(88)90090-7</a>)</p> </li> <li> <p>L. Zhang, Nimai C. Mukhopadhyay, <em>Baryon physics from mesons: Leading order properties of the nucleon and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mn>1232</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(1232)</annotation></semantics></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mi>ρ</mi><mi>ω</mi><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi \rho\omega a_1(f_1)</annotation></semantics></math> chiral soliton model</em>, Phys. Rev. D 50, 4668 (1994) (<a href="https://doi.org/10.1103/PhysRevD.50.4668">doi:10.1103/PhysRevD.50.4668</a>, <a href="http://inspirehep.net/record/384906">spire:384906</a>)</p> </li> <li> <p>Yong-Liang Ma, Ghil-Seok Yang, Yongseok Oh, Masayasu Harada, <em>Skyrmions with vector mesons in the hidden local symmetry approach</em>, Phys. Rev. D87:034023, 2013 (<a href="https://arxiv.org/abs/1209.3554">arXiv:1209.3554</a>)</p> <p>(<a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a>)</p> </li> <li> <p>Ju-Hyun Jung, Ulugbek T. Yakhshiev, Hyun-Chul Kim, <em>In-medium modified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> mesonic Lagrangian and properties of nuclear matter</em>, Physics Letters B Volume 723, Issues 4–5, 25 June 2013, Pages 442-447 (<a href="https://arxiv.org/abs/1212.4616">arXiv:1212.4616</a>, <a href="https://doi.org/10.1016/j.physletb.2013.05.042">doi:10.1016/j.physletb.2013.05.042</a>)</p> </li> <li> <p>Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim, Peter Schweitzerm, <em>In-medium modified energy-momentum tensor form factors of the nucleon within the framework of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">omgea</mo></mrow><annotation encoding="application/x-tex">\omgea</annotation></semantics></math> soliton model</em>, Phys. Rev. D 89, 114021 (2014) (<a href="https://arxiv.org/abs/1402.0161">arXiv:1402.0161</a>)</p> </li> <li> <p>Yongseok Oh, <em>Skyrmions with vector mesons revisited</em> (<a href="https://arxiv.org/abs/1402.2821">arXiv:1402.2821</a>)</p> </li> </ul> <p>See also</p> <ul> <li>Ki-Hoon Hong, Ulugbek Yakhshiev, Hyun-Chul Kim, <em>Modification of hyperon masses in nuclear matter</em>, Phys. Rev. C 99, 035212 (2019) (<a href="https://arxiv.org/abs/1806.06504">arXiv:1806.06504</a>)</li> </ul> <p>Review:</p> <ul> <li id="Kaiser00"> <p>Roland Kaiser, <em>Anomalies and WZW-term of two-flavour QCD</em>, Phys. Rev. D63:076010, 2001 (<a href="https://arxiv.org/abs/hep-ph/0011377">arXiv:hep-ph/0011377</a>, <a href="https://inspirehep.net/literature/537600">spire:537600</a>)</p> </li> <li id="Holzwarth05"> <p>Gottfried Holzwarth, Section 2.3 of: <em>Electromagnetic Form Factors of the Nucleon in Chiral Soliton Models</em> (<a href="https://arxiv.org/abs/hep-ph/0511194">arXiv:hep-ph/0511194</a>), Chapter 2 in: <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> <li> <p>Yongseok Oh, <em>Skyrmions with vector mesons: Single Skyrmion and baryonic matter</em>, 2013 (<a class="existingWikiWord" href="/nlab/files/OhSkyrmionsWithVectorMesons.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Combination of the <a class="existingWikiWord" href="/nlab/show/omega-meson">omega</a>-<a class="existingWikiWord" href="/nlab/show/rho+meson">rho</a>-<a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a> with the <a class="existingWikiWord" href="/nlab/show/bag+model+of+quark+confinement">bag model of quark confinement</a>:</p> <ul> <li>H. Takashita, S. Yoro, H. Toki, <em>Chiral bag plus skyrmion hybrid model with vector mesons for nucleon</em>, Nuclear Physics A Volume 485, Issues 3–4, August 1988, Pages 589-605 (<a href="https://doi.org/10.1016/0375-9474(88)90555-6">doi:10.1016/0375-9474(88)90555-6</a>)</li> </ul> <h4 id="inclusion_of_the_meson_3">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-meson</h4> <p>Inclusion of the <a class="existingWikiWord" href="/nlab/show/sigma-meson">sigma-meson</a>:</p> <ul> <li>Thomas D. Cohen, <em>Explicit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> meson, topology, and the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> limit of the Skyrmion</em>, Phys. Rev. D 37 (1988) (<a href="https://doi.org/10.1103/PhysRevD.37.3344">doi:10.1103/PhysRevD.37.3344</a>)</li> </ul> <p>For analysis of <a class="existingWikiWord" href="/nlab/show/neutron+star">neutron star</a> <a class="existingWikiWord" href="/nlab/show/equation+of+state">equation of state</a>:</p> <ul> <li>David Alvarez-Castillo, Alexander Ayriyan, Gergely Gábor Barnaföldi, Hovik Grigorian, Péter Pósfay, <em>Studying the parameters of the extended <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> model for neutron star matter</em> (<a href="https://arxiv.org/abs/2006.03676">arXiv:2006.03676</a>)</li> </ul> </div><div> <h3 id="SkyrmeHadrodynamicsWithHeavyMesonsReferences">Skyrme hadrodynamics with heavy quarks/mesons</h3> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/heavy+flavor">heavy</a> <a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a> for <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <h4 id="inclusion_of_strange_quarkskaons">Inclusion of strange quarks/kaons</h4> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/strange+quarks">strange quarks</a>/<a class="existingWikiWord" href="/nlab/show/kaons">kaons</a> into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, <em>Bound-state approach to strangeness in the Skyrme model</em>, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (<a href="https://doi.org/10.1016/0550-3213(85)90292-5">doi10.1016/0550-3213(85)90292-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, K. Hornbostel, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, <em>Baryon masses in the bound state approach to strangeness in the skyrme model</em>, Physics Letters B Volume 202, Issue 2, 3 March 1988, Pages 269-275 (<a href="https://doi.org/10.1016/0370-2693(88)90022-6">doi10.1016/0370-2693(88)90022-6</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, D. P. Min, H. Nadeau, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>The strangeness problem: An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3)</annotation></semantics></math> skyrmion with vector mesons</em>, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (<a href="https://doi.org/10.1016/0375-9474(89)90029-8">doi:10.1016/0375-9474(89)90029-8</a>)</p> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, section 6 of: <em>Strangeness in the Skyrme model</em>, in: D. Vauthrin, F. Lenz, J. W. Negele, <em>Hadrons and Hadronic Matter</em>, Plenum Press 1989 (<a href="https://link.springer.com/book/10.1007/978-1-4684-1336-6">doi:10.1007/978-1-4684-1336-6</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, Section 2.2 of: <em>Cheshire Cat Hadrons</em>, Phys. Rept. 240 (1994) 1-142 (<a href="https://arxiv.org/abs/hep-ph/9310300">arXiv:hep-ph/9310300</a>, <a href="https://doi.org/10.1016/0370-1573(94)90002-7">doi:10.1016/0370-1573(94)90002-7</a>)</p> </li> </ul> <h4 id="inclusion_of_charm_quarksdmesons">Inclusion of charm quarks/D-mesons</h4> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/charm+quarks">charm quarks</a>/<a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a> into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, <a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>Charmed baryons as soliton - D meson bound states</em>, Phys. Lett.B 251 (1990) 597-602 (<a href="https://inspirehep.net/literature/297771">spire:297771</a>, <a href="https://doi.org/10.1016/0370-2693(90)90802-D">doi:10.1016/0370-2693(90)90802-D</a>)</p> </li> <li> <p>Yongseok Oh, Dong-Pil Min, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>Massive-quark baryons as skyrmions: Magnetic moments</em>, Nuclear Physics A Volume 534, Issues 3–4 (1991) Pages 493-512 (<a href="https://doi.org/10.1016/0375-9474(91)90458-I">doi:10.1016/0375-9474(91)90458-I</a>)</p> </li> </ul> <h4 id="inclusion_of_bottom_quarksbmesons">Inclusion of bottom quarks/B-mesons</h4> <p>Inclusion of further <a class="existingWikiWord" href="/nlab/show/heavy+flavor">heavy</a> <a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> beyond <a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a>/<a class="existingWikiWord" href="/nlab/show/kaons">kaons</a>, namely <a class="existingWikiWord" href="/nlab/show/charm+quarks">charm quarks</a>/<a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a> and <a class="existingWikiWord" href="/nlab/show/bottom+quarks">bottom quarks</a>/<a class="existingWikiWord" href="/nlab/show/B-mesons">B-mesons</a>, into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, <a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>The energy levels of the heavy flavour baryons in the topological soliton model</em>, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages 343–352 (1992) (<a href="https://doi.org/10.1007/BF01283544">doi:10.1007/BF01283544</a>)</p> </li> <li> <p>Arshad Momen, Joseph Schechter, Anand Subbaraman, <em>Heavy Quark Solitons: Strangeness and Symmetry Breaking</em>, Phys. Rev. D49:5970-5978, 1994 (<a href="https://arxiv.org/abs/hep-ph/9401209">arXiv:hep-ph/9401209</a>)</p> </li> <li> <p>Yongseok Oh, Byung-Yoon Park, Dong-Pil Min, <em>Heavy Baryons as Skyrmion with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><msub><mi>m</mi> <mi>Q</mi></msub></mrow><annotation encoding="application/x-tex">1/m_Q</annotation></semantics></math> Corrections</em>, Phys. Rev. D49 (1994) 4649-4658 (<a href="https://arxiv.org/abs/hep-ph/9402205">arXiv:hep-ph/9402205</a>)</p> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>Massive-quark baryons as Skyrmions</em>, Modern Physics Letters A, Vol. 06, No. 23 (1991) (<a href="https://doi.org/10.1142/S0217732391002268">doi:10.1142/S0217732391002268</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>Heavy quark skyrmions</em>, (<a href="https://arxiv.org/abs/0905.2722">arXiv:0905.2722</a>, <a href="https://doi.org/10.1142/9789814280709_0004">doi:10.1142/9789814280709_0004</a>), Chapter 4 in: <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> </ul> </div><div> <h3 id="WZWTermOfChiralPerturbationTheoryReferences">The WZW term of QCD chiral perturbation theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/gauged+WZW+model">gauged</a> <a class="existingWikiWord" href="/nlab/show/WZW+term">WZW term</a> of <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> which reproduces the <a class="existingWikiWord" href="/nlab/show/chiral+anomaly">chiral anomaly</a> of <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> in the <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> of <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> and <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a>:</p> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesGeneral">General</h4> <p>The original articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Julius+Wess">Julius Wess</a>, <a class="existingWikiWord" href="/nlab/show/Bruno+Zumino">Bruno Zumino</a>, <em>Consequences of anomalous Ward identities</em>, Phys. Lett. B 37 (1971) 95-97 (<a href="https://inspirehep.net/literature/67330">spire:67330</a>, <a href="https://doi.org/10.1016/0370-2693(71)90582-X">doi:10.1016/0370-2693(71)90582-X</a>)</p> </li> <li id="Witten83a"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Global aspects of current algebra</em>, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (<a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>)</p> </li> </ul> <p>See also:</p> <ul> <li>O. Kaymakcalan, S. Rajeev, J. Schechter, <em>Nonabelian Anomaly and Vector Meson Decays</em>, Phys. Rev. D 30 (1984) 594 (<a href="https://inspirehep.net/literature/194756">spire:194756</a>)</li> </ul> <p>Corrections and streamlining of the computations:</p> <ul> <li> <p>Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, <em>On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action</em>, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (<a href="https://doi.org/10.1016/0370-2693(84)90986-9">doi:10.1016/0370-2693(84)90986-9</a>))</p> </li> <li> <p>H. Kawai, S.-H. H. Tye, <em>Chiral anomalies, effective lagrangians and differential geometry</em>, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (<a href="https://doi.org/10.1016/0370-2693(84)90780-9">doi:10.1016/0370-2693(84)90780-9</a>)</p> </li> <li> <p>J. L. Mañes, <em>Differential geometric construction of the gauged Wess-Zumino action</em>, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (<a href="https://doi.org/10.1016/0550-3213(85)90487-0">doi:10.1016/0550-3213(85)90487-0</a>)</p> </li> <li> <p>Tomáš Brauner, Helena Kolešová, <em>Gauged Wess-Zumino terms for a general coset space</em>, Nuclear Physics B Volume 945, August 2019, 114676 (<a href="https://doi.org/10.1016/j.nuclphysb.2019.114676">doi:10.1016/j.nuclphysb.2019.114676</a>)</p> </li> </ul> <p>See also</p> <ul> <li>Yasunori Lee, <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <em>Revisiting Wess-Zumino-Witten terms</em> (<a href="https://arxiv.org/abs/2009.00033">arXiv:2009.00033</a>)</li> </ul> <p>Interpretation as <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a>/<a class="existingWikiWord" href="/nlab/show/baryon+current">baryon current</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jeffrey+Goldstone">Jeffrey Goldstone</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Fractional Quantum Numbers on Solitons</em>, Phys. Rev. Lett. 47, 986 (1981) (<a href="https://doi.org/10.1103/PhysRevLett.47.986">doi:10.1103/PhysRevLett.47.986</a>)</p> </li> <li id="Witten83b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Current algebra, baryons, and quark confinement</em>, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (<a href="https://doi.org/10.1016/0550-3213(83)90064-0">doi:10.1016/0550-3213(83)90064-0</a>)</p> </li> <li id="AdkinsNappi84"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <a class="existingWikiWord" href="/nlab/show/Chiara+Nappi">Chiara Nappi</a>, <em>Stabilization of Chiral Solitons via Vector Mesons</em>, Phys. Lett. 137B (1984) 251-256 (<a href="http://inspirehep.net/record/194727">spire:194727</a>, <a href="https://doi.org/10.1016/0370-2693(84)90239-9">doi:10.1016/0370-2693(84)90239-9</a>)</p> <p>(beware that the two copies of the text at these two sources differ!)</p> </li> <li id="RhoEtAl16"> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a> et al., <em>Introduction</em>, In: <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a> et al. (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> </ul> <p>Concrete form for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> in 2d:</p> <ul> <li>C. R. Lee, H. C. Yen, <em>A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators</em>, Chinese Journal of Physics, Vol 23 No. 1 (1985) (<a href="https://inspirehep.net/literature/16389">spire:16389</a>, <a class="existingWikiWord" href="/nlab/files/LeeYenWZW85.pdf" title="pdf">pdf</a>)</li> </ul> <p>Concrete form for 2 <a class="existingWikiWord" href="/nlab/show/flavor+physics">flavors</a> in 4d:</p> <ul> <li>Masashi Wakamatsu, <em>On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action</em>, (<a href="https://arxiv.org/abs/1108.1236">arXiv:1108.1236</a>, <a href="https://inspirehep.net/literature/922302">spire:922302</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons">Including light vector mesons</h4> <p>Concrete form for 2-<a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> in 4d with <a class="existingWikiWord" href="/nlab/show/light+meson">light</a> <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> included (<a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a> and <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a>):</p> <ul> <li id="MeissnerZahed86"> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, equation (6) in: <em>Skyrmions in the Presence of Vector Mesons</em>, Phys. Rev. Lett. 56, 1035 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.56.1035">doi:10.1103/PhysRevLett.56.1035</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, equation (2.18) in: <em>Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties</em>, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, equation (2.45) in: <em>Low-energy hadron physics from effective chiral Lagrangians with vector mesons</em>, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (<a href="https://doi.org/10.1016/0370-1573(88)90090-7">doi:10.1016/0370-1573(88)90090-7</a>)</p> </li> <li id="Kaiser00"> <p>Roland Kaiser, equation (12) in: <em>Anomalies and WZW-term of two-flavour QCD</em>, Phys. Rev. D63:076010, 2001 (<a href="https://arxiv.org/abs/hep-ph/0011377">arXiv:hep-ph/0011377</a>, <a href="https://inspirehep.net/literature/537600">spire:537600</a>)</p> </li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons">Including heavy scalar mesons</h4> <p>Including <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy</a> <a class="existingWikiWord" href="/nlab/show/scalar+mesons">scalar mesons</a>:</p> <p>specifically <a class="existingWikiWord" href="/nlab/show/kaons">kaons</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, equation (4.1) in: <em>Bound-state approach to strangeness in the Skyrme model</em>, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (<a href="https://doi.org/10.1016/0550-3213(85)90292-5">doi10.1016/0550-3213(85)90292-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, equation (99) of: <em>Strangeness in the Skyrme model</em>, in: D. Vauthrin, F. Lenz, J. W. Negele, <em>Hadrons and Hadronic Matter</em>, Plenum Press 1989 (<a href="https://link.springer.com/book/10.1007/978-1-4684-1336-6">doi:10.1007/978-1-4684-1336-6</a>)</p> </li> <li> <p>N. N. Scoccola, D. P. Min, H. Nadeau, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, equation (2.20) in: <em>The strangeness problem: An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3)</annotation></semantics></math> skyrmion with vector mesons</em>, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (<a href="https://doi.org/10.1016/0375-9474(89)90029-8">doi:10.1016/0375-9474(89)90029-8</a>)</p> </li> </ul> <p>specifically <a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a>:</p> <p>(…)</p> <p>specifically <a class="existingWikiWord" href="/nlab/show/B-mesons">B-mesons</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, N. N. Scoccola, above (2.1) in: <em>The energy levels of the heavy flavour baryons in the topological soliton model</em>, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (<a href="https://doi.org/10.1007/BF01283544">doi:10.1007/BF01283544</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons">Including heavy vector mesons</h4> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy</a> <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a>:</p> <p>specifically <a class="existingWikiWord" href="/nlab/show/K%2A-mesons">K*-mesons</a>:</p> <ul> <li>S. Ozaki, H. Nagahiro, <a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, Equations (3) and (9) in: <em>Magnetic interaction induced by the anomaly in kaon-photoproductions</em>, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 (<a href="https://arxiv.org/abs/0710.5581">arXiv:0710.5581</a>, <a href="https://doi.org/10.1016/j.physletb.2008.06.020">doi:10.1016/j.physletb.2008.06.020</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions">Including electroweak interactions</h4> <p>Including <a class="existingWikiWord" href="/nlab/show/electroweak+fields">electroweak fields</a>:</p> <ul> <li> <p>J. Bijnens, G. Ecker, A. Picha, <em>The chiral anomaly in non-leptonic weak interactions</em>, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (<a href="https://doi.org/10.1016/0370-2693(92)91785-8">doi:10.1016/0370-2693(92)91785-8</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerhard+Ecker">Gerhard Ecker</a>, <a class="existingWikiWord" href="/nlab/show/Helmut+Neufeld">Helmut Neufeld</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+Pich">Antonio Pich</a>, <em>Non-leptonic kaon decays and the chiral anomaly</em>, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (<a href="https://doi.org/10.1016/0550-3213(94)90623-8">doi:10.1016/0550-3213(94)90623-8</a>)</p> </li> </ul> <p>Discussion for the full <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, Christopher T. Hill, Richard J. Hill, <em>Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions</em>, Phys. Rev. D77:085017, 2008 (<a href="https://arxiv.org/abs/0712.1230">arXiv:0712.1230</a>)</li> </ul> </div><div> <h3 id="ReferencesHadronsAsKKModesOf5dYangMillsTheory">Hadrons as KK-modes of 5d Yang-Mills theory</h3> <p>The suggestion that the tower of observed <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> – when regarded as <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> of <a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetries</a> of <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a> – is reasonably modeled as a <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">Kaluza-Klein tower</a> of <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a>:</p> <ul> <li id="SonStephanov03">D. T. Son, M. A. Stephanov, <em>QCD and dimensional deconstruction</em>, Phys. Rev. D69 (2004) 065020 (<a href="https://arxiv.org/abs/hep-ph/0304182">arXiv:hep-ph/0304182</a>, <a href="https://doi.org/10.1103/PhysRevD.69.065020">doi:10.1103/PhysRevD.69.065020</a>)</li> </ul> <p>That the pure <a class="existingWikiWord" href="/nlab/show/pion">pion</a>-<a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a>-model of <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> is approximately the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">KK-reduction</a> of <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> is already due to:</p> <ul> <li id="AtiyahManton89"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions from instantons</em>, Phys. Lett. B, 222(3):438–442, 1989 (<a href="https://doi.org/10.1016/0370-2693(89)90340-7">doi:10.1016/0370-2693(89)90340-7</a>)</li> </ul> <p>with a <a class="existingWikiWord" href="/nlab/show/hyperbolic+space">hyperbolic space</a>-variant in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions, instantons, mass and curvature</em>, Phys. Lett. B605 (2005) 106-114 (<a href="https://arxiv.org/abs/hep-th/0411052">arXiv:hep-th/0411052</a>)</li> </ul> <p>Further discussion of this approximation:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>ADHM skyrmions</em> (<a href="https://arxiv.org/abs/2110.15190">arXiv:2110.15190</a>)</li> </ul> <p>The observation that the result of <a href="#AtiyahManton89">Atiyah-Manton 89</a> becomes an exact <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">Kaluza-Klein construction</a> of <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a>/<a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> from <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5</a> <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> when the full <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">KK-tower</a> of <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> as in <a href="#SonStephanov03">Son-Stephanov 03</a> is included into the Skyrmion model (see also <a href="skyrmion#ReferencesSkyrmeHadrodynamicsWithVectorMesons">there</a>) is due to:</p> <ul> <li id="Sutcliffe10"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions, instantons and holography</em>, JHEP 1008:019, 2010 (<a href="https://arxiv.org/abs/1003.0023">arXiv:1003.0023</a>)</p> </li> <li id="Sutcliffe15"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Holographic Skyrmions</em>, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 (<a href="http://inspirehep.net/record/1383608">spire:1383608</a>, <a href="https://doi.org/10.1142/S0217984915400515">doi:10.1142/S0217984915400515</a>)</p> </li> </ul> <p>In the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a> of <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a> the <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> of this <a class="existingWikiWord" href="/nlab/show/hadron+Kaluza-Klein+theory">hadron Kaluza-Klein theory</a> is identified with the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>-theory of <a class="existingWikiWord" href="/nlab/show/D8-brane">D8</a>-<a class="existingWikiWord" href="/nlab/show/flavour+branes">flavour branes</a> <a class="existingWikiWord" href="/nlab/show/D4%2FD8-brane+intersection">intersected with D4-branes</a> in an <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+model">intersecting D-brane model</a>:</p> <ul> <li id="SakaiSugimoto04"> <p><a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, section 5.2 of <em>Low energy hadron physics in holographic QCD</em>, Prog.Theor.Phys.113:843-882, 2005 (<a href="https://arxiv.org/abs/hep-th/0412141">arXiv:hep-th/0412141</a>)</p> </li> <li id="SakaiSugimoto05"> <p><a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, section 3.3. of <em>More on a holographic dual of QCD</em>, Prog.Theor.Phys.114:1083-1118, 2005 (<a href="https://arxiv.org/abs/hep-th/0507073">arXiv:hep-th/0507073</a>)</p> </li> <li id="HSSY07"> <p>Hiroyuki Hata, <a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, Shinichiro Yamato, <em>Baryons from instantons in holographic QCD</em>, Prog.Theor.Phys.117:1157, 2007 (<a href="https://arxiv.org/abs/hep-th/0701280">arXiv:hep-th/0701280</a>)</p> </li> <li id="BolognesiSutcliffe13"> <p><a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>The Sakai-Sugimoto soliton</em>, JHEP 1401:078, 2014 (<a href="https://arxiv.org/abs/1309.1396">arXiv:1309.1396</a>)</p> </li> <li id="Bartolini17"> <p><a class="existingWikiWord" href="/nlab/show/Lorenzo+Bartolini">Lorenzo Bartolini</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, Andrea Proto, <em>From the Sakai-Sugimoto Model to the Generalized Skyrme Model</em>, Phys. Rev. D <strong>97</strong> 014024 (2018) [<a href="https://doi.org/10.1103/PhysRevD.97.014024">doi:10.1103/PhysRevD.97.014024</a>, <a href="https://arxiv.org/abs/1711.03873">arXiv:1711.03873</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lorenzo+Bartolini">Lorenzo Bartolini</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, Tommaso Rainaldi, <em>Mass and Isospin Breaking Effects in the Skyrme Model and in Holographic QCD</em> [<a href="https://arxiv.org/abs/2312.15404">arXiv:2312.15404</a>]</p> </li> </ul> <p>Extensive review of this <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic</a>/<a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">KK-theoretic</a>-realization of <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> from <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> is in:</p> <ul> <li id="RhoEtAl16"> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific, Second edition, 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> <ul> <li id="Sugimoto16"><a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, <em>Skyrmion and String theory</em>, chapter 15 in <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</li> </ul> </li> </ul> <p>Via the realization of <a class="existingWikiWord" href="/nlab/show/D4%2FD8+brane+bound+states">D4/D8 brane bound states</a> as <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in the <a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>-theory (see <a href="DpDpPlus4BraneBoundState#AsYangMillsInstantons">there</a> and <a href="D4-D8+brane+bound+state#AsInstantonsOnTheD8Brane">there</a>), this relates also to the model of <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> as <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/D4-branes">D4-branes</a>, originally due to</p> <ul> <li id="Witten98b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Baryons And Branes In Anti de Sitter Space</em>, JHEP 9807:006, 1998 (<a href="https://arxiv.org/abs/hep-th/9805112">arXiv:hep-th/9805112</a>)</p> </li> <li id="GrossOoguri98"> <p><a class="existingWikiWord" href="/nlab/show/David+Gross">David Gross</a>, <a class="existingWikiWord" href="/nlab/show/Hirosi+Ooguri">Hirosi Ooguri</a>, <em>Aspects of Large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> Gauge Theory Dynamics as Seen by String Theory</em>, Phys. Rev. D58:106002, 1998 (<a href="https://arxiv.org/abs/hep-th/9805129">arXiv:hep-th/9805129</a>)</p> </li> </ul> <p>and further developed in the <a class="existingWikiWord" href="/nlab/show/nuclear+matrix+model">nuclear matrix model</a>:</p> <ul> <li id="HashimotoIizukaYi10"> <p><a class="existingWikiWord" href="/nlab/show/Koji+Hashimoto">Koji Hashimoto</a>, Norihiro Iizuka, Piljin Yi, <em>A Matrix Model for Baryons and Nuclear Forces</em>, JHEP 1010:003, 2010 (<a href="https://arxiv.org/abs/1003.4988">arXiv:1003.4988</a>)</p> </li> <li> <p>Si-wen Li, Tuo Jia, <em>Matrix model and Holographic Baryons in the D0-D4 background</em>, Phys. Rev. D 92, 046007 (2015) (<a href="https://arxiv.org/abs/1506.00068">arXiv:1506.00068</a>)</p> </li> <li id="HashimotoMatsuoMorita19"> <p><a class="existingWikiWord" href="/nlab/show/Koji+Hashimoto">Koji Hashimoto</a>, <a class="existingWikiWord" href="/nlab/show/Yoshinori+Matsuo">Yoshinori Matsuo</a>, <a class="existingWikiWord" href="/nlab/show/Takeshi+Morita">Takeshi Morita</a>, <em>Nuclear states and spectra in holographic QCD</em>, JHEP12 (2019) 001 (<a href="https://arxiv.org/abs/1902.07444">arXiv:1902.07444</a>)</p> </li> <li> <p>Yasuhiro Hayashi, Takahiro Ogino, <a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, <em>Stringy excited baryons in holographic QCD</em>, Prog Theor Exp Phys (2020) (<a href="https://arxiv.org/abs/2001.01461">arXiv:2001.01461</a>)</p> </li> </ul> <p>In relation to <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <em>Solitons, Large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> and Holography</em>, 2015 (<a class="existingWikiWord" href="/nlab/files/Bolognesi_SolitonsHolography.pdf" title="pdf">pdf</a>)</li> </ul> <p>Discussion, in this context, of <a class="existingWikiWord" href="/nlab/show/D-term">D-term</a> effects affecting <a class="existingWikiWord" href="/nlab/show/hadron">hadron</a> stability:</p> <ul> <li>Mitsutoshi Fujita, Yoshitaka Hatta, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, Takahiro Ueda, <em>Nucleon D-term in holographic QCD</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2206.06578">arXiv:2206.06578</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>More on <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> in the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a> of <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>:</p> <ul> <li>Salvatore Baldino, <a class="existingWikiWord" href="/nlab/show/Lorenzo+Bartolini">Lorenzo Bartolini</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <em>Holographic Nuclear Physics with Massive Quarks</em>, Phys. Rev. D <strong>103</strong> 126015 (2021) [<a href="https://doi.org/10.1103/PhysRevD.103.126015">doi:10.1103/PhysRevD.103.126015</a>, <a href="https://arxiv.org/abs/2102.00680">arXiv:2102.00680</a>]</li> </ul> <p>More on <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> in <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Johanna+Erdmenger">Johanna Erdmenger</a>, <a class="existingWikiWord" href="/nlab/show/Nick+Evans">Nick Evans</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Kirsch">Ingo Kirsch</a>, <a class="existingWikiWord" href="/nlab/show/Ed+Threlfall">Ed Threlfall</a>, <em>Mesons in Gauge/Gravity Duals - A Review</em>, Eur. Phys. J. A <strong>35</strong> (2008) 81-133 &lbrack;<a href="https://arxiv.org/abs/0711.4467">arXiv:0711.4467</a>, <a href="https://doi.org/10.1140/epja/i2007-10540-1">doi:10.1140/epja/i2007-10540-1</a>&rbrack;</p> </li> <li> <p>Daniel Ávila, Leonardo Patiño, <em>Melting holographic mesons by cooling a magnetized quark gluon plasma</em> (<a href="https://arxiv.org/abs/2002.02470">arXiv:2002.02470</a>)</p> </li> <li> <p>Xuanmin Cao, Hui Liu, Danning Li, <em>Pion quasiparticles and QCD phase transitions at finite temperature and isospin density from holography</em>, Phys. Rev. D 102, 126014 (2020) (<a href="https://arxiv.org/abs/2009.00289">arXiv:2009.00289</a>)</p> </li> <li> <p>Xuanmin Cao, Songyu Qiu, Hui Liu, Danning Li, <em>Thermal properties of light mesons from holography</em> (<a href="https://arxiv.org/abs/2102.10946">arXiv:2102.10946</a>)</p> </li> <li> <p>Artur Amorim, Miguel S. Costa, Matti Järvinen, <em>Regge theory in a Holographic dual of QCD in the Veneziano Limit</em> (<a href="https://arxiv.org/abs/2102.11296">arXiv:2102.11296</a>)</p> </li> <li id="daRocha21"> <p><a class="existingWikiWord" href="/nlab/show/Rold%C3%A3o+da+Rocha">Roldão da Rocha</a>, <em>Information in AdS/QCD: mass spectroscopy of isovector mesons</em>, Phys. Rev. D <strong>103</strong> 106027 (2021) &lbrack;<a href="https://arxiv.org/abs/2103.03924">arXiv:2103.03924</a>, <a href="https://doi.org/10.1103/PhysRevD.103.106027">doi:10.1103/PhysRevD.103.106027</a>&rbrack;</p> </li> <li> <p>Shahin Mamedov, Narmin Nasibova, <em>Temperature dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> meson-nucleon coupling constant from the soft-wall model</em> (<a href="https://arxiv.org/abs/2103.10494">arXiv:2103.10494</a>)</p> </li> </ul> <p>An alternative scenario of derivation of 4d <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a> by <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a>, now on a <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a>, motivated by <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> instead of by <a class="existingWikiWord" href="/nlab/show/D4%2FD8-brane+intersections">D4/D8-brane intersections</a> as in the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a>, is discussed in:</p> <ul> <li id="IvanovaLechtenfeldPopov18"><a class="existingWikiWord" href="/nlab/show/Tatiana+Ivanova">Tatiana Ivanova</a>, <a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Popov">Alexander Popov</a>, <em>Skyrme model from 6d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}= (2,0)</annotation></semantics></math> theory</em>, Physics Letters B <strong>783</strong> (2018), 222-226 [<a href="https://doi.org/10.1016/j.physletb.2018.06.052">doi:10.1016/j.physletb.2018.06.052</a>, <a href="https://arxiv.org/abs/1805.07241">arXiv:1805.07241</a>]</li> </ul> <p>with related discussion in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tatiana+Ivanova">Tatiana Ivanova</a>, <a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Popov">Alexander Popov</a>, <em>Non-Abelian sigma models from Yang-Mills theory compactified on a circle</em>, Physics Letters B <strong>781</strong> (2018) 322-326 [<a href="https://doi.org/10.1016/j.physletb.2018.04.013">doi:10.1016/j.physletb.2018.04.013</a><a href="https://arxiv.org/abs/1803.07322">arXiv:1803.07322</a>]</p> </li> <li id="LechtenfeldPopov18"> <p><a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+D.+Popov">Alexander D. Popov</a>, <em>Skyrme-Faddeev model from 5d super-Yang-Mills</em>, Phys. Lett. B <strong>786</strong> (2018) 39-44 [<a href="https://doi.org/10.1016/j.physletb.2018.09.028">doi:10.1016/j.physletb.2018.09.028</a>, <a href="https://arxiv.org/abs/1807.11803">arXiv:1807.11803</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+D.+Popov">Alexander D. Popov</a>, <em>Skyrme and Faddeev models in the low-energy limit of 4d Yang-Mills-Higgs theories</em>, Nuclear Physics B <strong>945</strong> (2019) 114675 [<a href="https://doi.org/10.1016/j.nuclphysb.2019.114675">doi:10.1016/j.nuclphysb.2019.114675</a>, <a href="https://arxiv.org/abs/1808.08972">arXiv:1808.08972</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>Y. H. Ahn, Sin Kyu Kang, Hyun Min Lee, <em>Towards a Model of Quarks and Leptons</em> (<a href="https://arxiv.org/abs/2112.13392">arXiv:2112.13392</a>)</li> </ul> </div> <h3 id="in_the_large__limit">In the large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> limit</h3> <p>In the <a class="existingWikiWord" href="/nlab/show/large+N+limit">large N limit</a>:</p> <ul> <li id="Witten79"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Baryons in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1/n</annotation></semantics></math> Expansion</em>, Nucl. Phys. B160 (1979) 57-115 (<a href="http://inspirehep.net/record/140391">spire:140391</a>, <a href="https://doi.org/10.1016/0550-3213(79)90232-3">doi:10.1016/0550-3213(79)90232-3</a>)</p> </li> <li> <p>Jan Albert, Johan Henriksson, <a class="existingWikiWord" href="/nlab/show/Leonardo+Rastelli">Leonardo Rastelli</a>, Alessandro Vichi, <em>Bootstrapping mesons at large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>: Regge trajectory from spin-two maximization</em> [<a href="https://arxiv.org/abs/2312.15013">arXiv:2312.15013</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 28, 2023 at 11:59:40. See the <a href="/nlab/history/meson" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/meson" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11017/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/meson/45" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/meson" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/meson" accesskey="S" class="navlink" id="history" rel="nofollow">History (45 revisions)</a> <a href="/nlab/show/meson/cite" style="color: black">Cite</a> <a href="/nlab/print/meson" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/meson" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>