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GUT 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> <ul> <li><a href='#general_idea'>General idea</a></li> <li><a href='#IdeaofSU5GUT'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-GUT (Georgi-Glashow)</a></li> <li><a href='#dseries_guts'>D-Series GUTs</a></li> <ul> <li><a href='#TheGUTKnownAsSO10'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(10)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math>”)</a></li> <li><a href='#TheGUTKnownAsSO11'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(11)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(11)</annotation></semantics></math>”)</a></li> <li><a href='#TheGUTKnownAsSO12'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(12)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(12)</annotation></semantics></math>”)</a></li> <li><a href='#TheGUTKnownAsSO16'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>18</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(16), Spin(18)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>18</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16), SO(18)</annotation></semantics></math>”)</a></li> </ul> <li><a href='#eseries_guts'>E-series GUTs</a></li> <ul> <li><a href='#the_gut'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">E_6</annotation></semantics></math>-GUT</a></li> <li><a href='#the_gut_'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">E_7</annotation></semantics></math>-GUT (?)</a></li> <li><a href='#the_gut__2'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-GUT (?)</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToProtonDecay'>Relation to proton decay</a></li> <li><a href='#RelationToNeutrinoMasses'>Relation to neutrino masses</a></li> <li><a href='#RelationToLeptoquarks'>Relation to leptoquarks</a></li> <li><a href='#InStringPhenomenology'>Occurrence in string phenomenology</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='#ReferencesProtonNonDecay'>Proton (non-)decay</a></li> <li><a href='#ReferencesRealisticModels'>Realistic models and phenomenology</a></li> <ul> <li><a href='#gut'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-GUT</a></li> <li><a href='#gut_2'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math>-GUT</a></li> <li><a href='#ReferencesSpin11'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(11)</annotation></semantics></math>-GUT</a></li> <li><a href='#ReferencesSpin12'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(12)</annotation></semantics></math>-GUT</a></li> <li><a href='#ReferencesSpin16'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>18</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(18)</annotation></semantics></math>-GUT</a></li> <li><a href='#gut_5'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-GUT</a></li> </ul> <li><a href='#HeteroticStringPhenomenologyReferences'>Heterotic string phenomenology</a></li> <ul> <li><a href='#HeteroticStringPhenomenologyReferencesForE8'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8 \times E_8</annotation></semantics></math>-heterotic string</a></li> <li><a href='#HeteroticStringPhenomenologyReferencesForSemiSpin32'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SemiSpin</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SemiSpin(32)</annotation></semantics></math>-heterotic string</a></li> <li><a href='#HeteroticStringPhenomenologyReferencesForSO16'>The “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math>”-heterotic string</a></li> </ul> <li><a href='#in_type_ii_string_theory'>In type II string theory</a></li> <li><a href='#in_conneslott_models'>In Connes-Lott models</a></li> <li><a href='#exotica_leptoquarks_bosons_etc'>Exotica: Leptoquarks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Z'</annotation></semantics></math>-bosons, etc.</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general_idea">General idea</h3> <p>The <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> asserts that the fundamental quantum <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">physical fields</a> and <a class="existingWikiWord" href="/nlab/show/particles">particles</a> are modeled as <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of and <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> on a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> that is <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, where the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> – called the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> – is the <a class="existingWikiWord" href="/nlab/show/product">product</a> of (<a class="existingWikiWord" href="/nlab/show/special+unitary+group">special</a>) <a class="existingWikiWord" href="/nlab/show/unitary+groups">unitary groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = SU(3) \times SU(2) \times U(1)</annotation></semantics></math> (or rather a <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of this by the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">/</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">Z/6</annotation></semantics></math>, see <a href="standard+model+of+particle+physics#GaugeGroup">there</a>) and where the <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> used to form the <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> looks fairly ad hoc on first sight.</p> <p>A <strong>grand unified theory</strong> (“GUT” for short) in this context is an attempt to realize the standard model as sitting inside a conceptually simpler <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+physics%29">model</a>, in particular one for which the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> is a bigger but <em>simpler</em> group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{G}</annotation></semantics></math>, preferably a <em><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a></em> in the technical sense, which contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>. Such a grand unified theory would be <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a> viable if a process of <a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous symmetry breaking</a> at some high <a class="existingWikiWord" href="/nlab/show/energy">energy</a> scale – the “GUT scale” – would reduce the model back to the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> without adding spurious extra effects that would not be in agreement with existing observations in <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>.</p> <p>The terminology “grand unified” here refers to the fact that such a single simple group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{G}</annotation></semantics></math> would unify the fundamental <a class="existingWikiWord" href="/nlab/show/forces">forces</a> of <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a>, the <a class="existingWikiWord" href="/nlab/show/weak+nuclear+force">weak nuclear force</a> and the <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> in a way that generalizes the way in which the <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> already unifies the <a class="existingWikiWord" href="/nlab/show/weak+nuclear+force">weak nuclear force</a> and <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a>, and electromagnetism already unifies, as the word says, electricity and magnetism.</p> <p>Since no GUT model has been fully validated yet (but see for instance <a href="#FongMeloni14">Fong-Meloni 14</a>), GUTs are <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">models</a> “beyond the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model</a>”. Often quantum physics “beyond the standard model” is expected to also say something sensible about <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> and hence unify not just the three gauge forces but also the fourth known fundamental force, which is <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>. Models that aim to do all of this would then “unify” the entire content of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> plus the <a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">standard model of cosmology</a>, hence “everything that is known about fundamental physics” to date. Therefore such theories are then sometimes called a <em><a class="existingWikiWord" href="/nlab/show/theory+of+everything">theory of everything</a></em>.</p> <p>(Here it is important to remember the context, both “grand unified” and “of everything” refers to aspects of presently available models of fundamental physics, and not to deeper philosophical questions of <a class="existingWikiWord" href="/nlab/show/ontology">ontology</a>.)</p> <p><br /></p> <p><a class="existingWikiWord" href="/nlab/show/length+scales">length scales</a> in the <a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a> (from <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmic</a> scales, over <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>-<a class="existingWikiWord" href="/nlab/show/masses">masses</a> around the <a class="existingWikiWord" href="/nlab/show/electroweak+symmetry+breaking">electroweak symmetry breaking</a> to <a class="existingWikiWord" href="/nlab/show/GUT+scale">GUT scale</a> and <a class="existingWikiWord" href="/nlab/show/Planck+scale">Planck scale</a>):</p> <center> <img src="https://ncatlab.org/nlab/files/ScalesInFundamentalPhysics.jpg" width="700" /> </center> <blockquote> <p>graphics grabbed from <a href="flavour+in+particle+physics#Zupan19">Zupan 19</a></p> </blockquote> <h3 id="IdeaofSU5GUT">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-GUT (Georgi-Glashow)</h3> <p>The argument for the hypothesis of “grand unification” is fairly compelling if one asks for <em><a class="existingWikiWord" href="/nlab/show/simple+objects">simple algebraic structures</a></em> in the technical sense (<a class="existingWikiWord" href="/nlab/show/simple+Lie+groups">simple Lie groups</a> and their <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a>):</p> <p>The <em>exact</em> gauge group of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> is really a <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>SM</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>6</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G_{SM} \;=\; \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6 \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_6</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/free+action">acts freely</a>, hence exhibiting a subtle global twist in the gauge structure. This seemingly ad hoc group turns out to be <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><munder><mrow><mi>S</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>≃</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>6</mn></msub></mrow></munder><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{ \simeq \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6 }{ \underbrace{ S \big( U(2) \times U(3)\big) }} \;\subset\; SU(5) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(5)</a> (see <a href="#BaezHuerta09">Baez-Huerta 09, p. 33-36</a>). The latter happens to be a <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, thus exhibiting the exact standard model Lie group as being “simply” a “(2+3)-breaking” of a <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>.</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>-<a class="existingWikiWord" href="/nlab/show/representation">representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>SM</mi></msub></mrow><annotation encoding="application/x-tex">V_{SM}</annotation></semantics></math> that captures one <a class="existingWikiWord" href="/nlab/show/generation+of+fundamental+particles">generation of fundamental particles</a> of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>, which looks fairly ad hoc as a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>SM</mi></msub></mrow><annotation encoding="application/x-tex">G_{SM}</annotation></semantics></math> (e.g. <a href="#BaezHuerta09">Baez-Huerta 09, table 1</a>), but as a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> it is simply</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>SM</mi></msub><mo>≃</mo><mi>Λ</mi><msup><mi>ℂ</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex"> V_{SM} \simeq \Lambda \mathbb{C}^5 </annotation></semantics></math></div> <p>(see <a href="#BaezHuerta09">Baez-Huerta 09, p. 36-41</a>).</p> <p>This leads to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-GUT due to <a href="#GeorgiGlashow74">Georgi-Glashow 74</a></p> <h3 id="dseries_guts">D-Series GUTs</h3> <h4 id="TheGUTKnownAsSO10">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(10)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math>”)</h4> <p>There is a further inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>↪</mo></mrow><annotation encoding="application/x-tex">SU(5) \hookrightarrow </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a> into the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> in 10 (Euclidean) dimensions (still a <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>), and one <a class="existingWikiWord" href="/nlab/show/generation+of+fundamental+particles">generation of fundamental particles</a> regarded as 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>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/representation">representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><msup><mi>ℂ</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\Lambda \mathbb{C}^5</annotation></semantics></math> as <a href="#IdeaofSU5GUT">above</a> extends to a <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a> (see <a href="#BaezHuerta09">Baez-Huerta 09, theorem 2</a>). This has the aesthetically pleasing effect that under this identification the 1-<a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation</a> rep <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>SM</mi></msub></mrow><annotation encoding="application/x-tex">V_{SM}</annotation></semantics></math> is now identified as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>SM</mi></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mn>16</mn></mstyle><mo>⊕</mo><mover><mstyle mathvariant="bold"><mn>16</mn></mstyle><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex"> V_{SM} \;\simeq\; \mathbf{16} \oplus \overline{\mathbf{16}} </annotation></semantics></math></div> <p>namely as the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> two (complex) <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> of <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a>, together being the <a class="existingWikiWord" href="/nlab/show/Dirac+representation">Dirac representation</a> of <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a>.</p> <p>The exact <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> (see <a href="standard+model+of+particle+physics#GaugeGroup">there</a>) is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a> which respects a splitting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi><mo>⊕</mo><mi>ℍ</mi><msub><mo>≃</mo> <mi>ℝ</mi></msub><mi>ℂ</mi><mo>⊕</mo><msup><mi>ℂ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3</annotation></semantics></math> (<a href="Spin9#Krasnov19">Krasnov 19</a>).</p> <p>Again, this means that under the embedding of the gauge group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>SM</mi></msub></mrow><annotation encoding="application/x-tex">G_{SM}</annotation></semantics></math> all the way into the <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a> <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a>, its ingredients become <em>simpler</em>, not just in a naive sense, but in the technical mathematical sense of <em><a class="existingWikiWord" href="/nlab/show/simple+objects">simple algebraic objects</a></em>.</p> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a> (i.e. <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a>) GUT-models with realistic <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> is in <a href="#BLM09">BLM 09</a> <a href="#Malinsky09">Malinský 09</a>, <a href="#LavouraWolfenstein10">Lavoura-Wolfenstein 10</a> <a href="#AltarelliMeloni13">Altarelli-Meloni 13</a> <a href="#DueckRodejohann13">Dueck-Rodejohann 13</a> <a href="#OhlssonPernow19">Ohlsson-Pernow 19</a> <a href="#CPS19">CPS 19</a>.</p> <p><img src="/nlab/files/MinimalNonSusySO10GUTMalinsky.jpg" width="500px" /></p> <blockquote> <p>slide grabbed from <a href="#Malinsky09">Malinský 09</a></p> </blockquote> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/leptoquarks">leptoquarks</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math>-models possibly explaining the <a class="existingWikiWord" href="/nlab/show/flavour+anomalies">flavour anomalies</a>: <a href="#AMM19">AMM 19</a></p> <p><br /></p> <h4 id="TheGUTKnownAsSO11">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(11)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(11)</annotation></semantics></math>”)</h4> <p>Models with <a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a> (“<a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a>”) GUT group.</p> <p>Specifically with <a class="existingWikiWord" href="/nlab/show/gauge-Higgs+unification">gauge-Higgs unification</a> in a <a class="existingWikiWord" href="/nlab/show/Randall-Sundrum+model">Randall-Sundrum model</a>-like 6d spacetime: <a href="#HosotaniYamatsu15">Hosotani-Yamatsu 15</a>, <a href="#FuruiHosotaniYamatsu16">Furui-Hosotani-Yamatsu 16</a>, <a href="#Hosotani17">Hosotani 17</a>, <a href="#HosotaniYamatsu17">Hosotani-Yamatsu 17</a></p> <p>See the references <a href="#ReferencesSpin11">below</a>.</p> <p><br /></p> <h4 id="TheGUTKnownAsSO12">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(12)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(12)</annotation></semantics></math>”)</h4> <p>Models with <a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a> (“<a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a>”) GUT group.</p> <p>Specifically with <a class="existingWikiWord" href="/nlab/show/gauge-Higgs+unification">gauge-Higgs unification</a> in a <a class="existingWikiWord" href="/nlab/show/Randall-Sundrum+model">Randall-Sundrum model</a>-like 6d spacetime: <a href="#NomuraSato08">Nomura-Sato 08</a>, <a href="#Nomura09">Nomura 09</a>, <a href="#ChiangNomuraSato11">Chiang-Nomura-Sato 11</a>)</p> <p>See the references <a href="#ReferencesSpin12">below</a>.</p> <p><br /></p> <h4 id="TheGUTKnownAsSO16">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>18</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(16), Spin(18)</annotation></semantics></math>-GUT (known as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>18</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16), SO(18)</annotation></semantics></math>”)</h4> <p>Models with <a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a> (“<a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a>”) GUT group.</p> <p><a href="#WilczekZee82">Wilczek-Zee 82</a>, <a href="#SenjanovicWilczekZee84">Senjanovic-Wilczek-Zee 84</a>, <a href="#MartínezMelfoNestiSenjanovic11">Martínez-Melfo-Nesti-Senjanovic 11</a></p> <p>See also <a href="#diLucio11">di Lucio 11, p. 44</a> and see the references <a href="#ReferencesSpin16">below</a>.</p> <p>Predicts <a class="existingWikiWord" href="/nlab/show/fourth+generation+of+fermions">fourth generation of fermions</a>…</p> <h3 id="eseries_guts">E-series GUTs</h3> <p>The most studied choices of GUT-groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(5)</a>, <a class="existingWikiWord" href="/nlab/show/spin+group">Spin(10)</a> (in the physics literature often referred to as <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">SO(10)</a>) and <a class="existingWikiWord" href="/nlab/show/E6">E6</a> (review includes <a href="#Witten86">Witten 86, sections 1 and 2</a>).</p> <p>It so happens that, mathematically, the sequence <a class="existingWikiWord" href="/nlab/show/special+unitary+groups">SU(5)</a>, <a class="existingWikiWord" href="/nlab/show/spin+group">Spin(10)</a>, <a class="existingWikiWord" href="/nlab/show/E6">E6</a> naturally continues (each step by consecutively adding a node to the <a class="existingWikiWord" href="/nlab/show/Dynkin+diagrams">Dynkin diagrams</a>) with the <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+groups">exceptional Lie groups</a> <a class="existingWikiWord" href="/nlab/show/E7">E7</a>, <a class="existingWikiWord" href="/nlab/show/E8">E8</a> that naturally appear in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic</a> <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> (exposition is in <a href="#Witten02a">Witten 02a</a>) and conjecturally further via the <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> <a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a> <a class="existingWikiWord" href="/nlab/show/E9">E9</a>, <a class="existingWikiWord" href="/nlab/show/E10">E10</a>, <a class="existingWikiWord" href="/nlab/show/E11">E11</a> that are being argued to underly <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>. In the context of <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> model building, also properties of the observes <a class="existingWikiWord" href="/nlab/show/Yukawa+couplings">Yukawa couplings</a> may point to exceptional GUT groups (<a href="#Zoccarato14">Zoccarato 14, slide 11</a>, <a href="#Vafa15">Vafa 15, slide 11</a>).</p> <h4 id="the_gut">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">E_6</annotation></semantics></math>-GUT</h4> <p>(…)</p> <p>review in <a href="#Britto17">Britto 17</a></p> <p>(…)</p> <h4 id="the_gut_">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">E_7</annotation></semantics></math>-GUT (?)</h4> <p>(…)</p> <h4 id="the_gut__2">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-GUT (?)</h4> <p>(…)</p> <h2 id="properties">Properties</h2> <h3 id="RelationToProtonDecay">Relation to proton decay</h3> <p>Many GUT models imply that the <a class="existingWikiWord" href="/nlab/show/proton">proton</a> – which in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> is a stable <a class="existingWikiWord" href="/nlab/show/bound+state">bound state</a> (of <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) – is in fact unstable, albeit with an extremely long mean liftetime, and hence may decay (e.g. <a href="#KM14">KM 14</a>). <a class="existingWikiWord" href="/nlab/show/experiment">Experimental</a> searches for such <em><a class="existingWikiWord" href="/nlab/show/proton+decay">proton decay</a></em> (see there for more) put strong bounds on this effect and hence heavily constrain or rule out many <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">models</a>.</p> <p>But in recent years it is claimed that there are in fact realistic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">models</a> that do not imply any <a class="existingWikiWord" href="/nlab/show/proton+decay">proton decay</a>, quite generically so for <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a>-models (<a href="#MuetterRatzVaudrvange16">Mütter-Ratz-Vaudrvange 16</a>), but also for non-supersymmetric models ( <a href="#FornalGrinstein17">Fornal-Grinstein 17</a>, <a href="#FornalGrinstein18">Fornal-Grinstein 18</a>, in particular in <a class="existingWikiWord" href="/nlab/show/gauge-Higgs+grand+unification">gauge-Higgs grand unification</a> such as <a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a>- (“<a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a>”-) and <a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a>- (“<a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a>”-) models: (<a href="#HosotaniYamatsu15">Hosotani-Yamatsu 15</a>, <a href="#FuruiHosotaniYamatsu16">Furui-Hosotani-Yamatsu 16, Sec. 2.6</a> <a href="#Hosotani17">Hosotani 17, Section 6</a>).</p> <h3 id="RelationToNeutrinoMasses">Relation to neutrino masses</h3> <p>The high energy scale required by a <a class="existingWikiWord" href="/nlab/show/seesaw+mechanism">seesaw mechanism</a> to produce the experimentally observer <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> masses happens to be about the conventional <a class="existingWikiWord" href="/nlab/show/GUT+scale">GUT scale</a>, for review see for instance (<a href="#Mohapatra06">Mohapatra 06</a>).</p> <blockquote> <p>I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">M^{-1}</annotation></semantics></math>, and that therefore with a reasonable estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mn>16</mn></msup><mi>GeV</mi></mrow><annotation encoding="application/x-tex">10^{16} GeV</annotation></semantics></math>. (<a href="neutrino#Weinberg09">Weinberg 09, p. 15</a>)</p> </blockquote> <p>Detailed matching of parameters of non-supersymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(10)</annotation></semantics></math>-GUT to <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> <a class="existingWikiWord" href="/nlab/show/masses">masses</a> is discussed in <a href="#OhlssonPernow19">Ohlsson-Pernow 19</a></p> <h3 id="RelationToLeptoquarks">Relation to leptoquarks</h3> <p>Generically, GUT-theories predict the existence of <a class="existingWikiWord" href="/nlab/show/leptoquarks">leptoquarks</a> (<a href="#MurayamaYanagida92">Murayama-Yanagida 92</a>), possibly related to the apparently observed</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/flavour+anomalies">flavour anomalies</a> (<a href="#BDFKFS18">BDFKFS 18</a>, <a href="#AMM19">AMM 19</a>, <a href="#HeekTeresi18">Heek-Teresi 18</a>, <a href="#HeekTeresi19">Heek-Teresi 19</a>)</p> </li> <li> <p><a href="anomalous+magnetic+moment#Anomalies">anomalies</a> in <a class="existingWikiWord" href="/nlab/show/anomalous+magnetic+moment">anomalous magnetic moment</a> of <a class="existingWikiWord" href="/nlab/show/muon">muon</a> and <a class="existingWikiWord" href="/nlab/show/electron">electron</a></p> </li> </ul> <h3 id="InStringPhenomenology">Occurrence in string phenomenology</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> of <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactified</a> with non-geometric <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> such that the would-be string <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a> with these <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> are in fact <a class="existingWikiWord" href="/nlab/show/Gepner+models">Gepner models</a> (in the sense of <em><a href="https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/">Spectral Standard Model and String Compactifications</a></em>) is in (<a href="#DijkstraHuiszoonSchellekens04a">Dijkstra-Huiszoon-Schellekens 04a</a>, <a href="#DijkstraHuiszoonSchellekens04b">Dijkstra-Huiszoon-Schellekens 04b</a>):</p> <center> <img src="https://ncatlab.org/nlab/files/SchellekensSuperGepnerModelScanII.jpg" width="600" /> </center> <blockquote> <p>A plot of <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model</a>-like <a class="existingWikiWord" href="/nlab/show/coupling+constants">coupling constants</a> in a computer scan of <a class="existingWikiWord" href="/nlab/show/Gepner+model">Gepner model</a>-<a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a> according to <a href="#DijkstraHuiszoonSchellekens04b">Dijkstra-Huiszoon-Schellekens 04b</a>.</p> <p>The blue dot indicates the couplings in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.</p> </blockquote> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory+of+everything">theory of everything</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+coupling+unification">gauge coupling unification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Connes-Lott-Chamseddine-Barrett+model">Connes-Lott-Chamseddine-Barrett model</a></p> </li> </ul> <div> <p><strong>fundamental <a class="existingWikiWord" href="/nlab/show/scales">scales</a></strong> (fundamental/<a class="existingWikiWord" href="/nlab/show/natural+unit">natural</a> <a class="existingWikiWord" href="/nlab/show/physical+units">physical units</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitational+constant">gravitational constant</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>N</mi></msub><mo>=</mo><msup><mi>κ</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mn>8</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">G_N = \kappa^2/8\pi</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck+scale">Planck scale</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck+length">Planck length</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>p</mi></msub><mo>=</mo><msqrt><mrow><mi>ℏ</mi><mi>G</mi><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>3</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\ell_p = \sqrt{ \hbar G / c^3 } </annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck+mass">Planck mass</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>p</mi></msub><mo>=</mo><msqrt><mrow><mi>ℏ</mi><mi>c</mi><mo stretchy="false">/</mo><mi>G</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">m_p = \sqrt{\hbar c / G}</annotation></semantics></math></p> </li> </ul> </li> <li> <p>depending on a given <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Compton+wavelength">Compton wavelength</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>m</mi></msub><mo>=</mo><mi>ℏ</mi><mo stretchy="false">/</mo><mi>m</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">\lambda_m = \hbar / m c</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwarzschild+radius">Schwarzschild radius</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>m</mi><mi>G</mi><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">2 m G / c^2</annotation></semantics></math></p> </li> <li> <p>depending also on a given <a class="existingWikiWord" href="/nlab/show/charge">charge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Schwinger+limit">Schwinger limit</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>crit</mi></msub><mo>=</mo><msup><mi>m</mi> <mn>2</mn></msup><msup><mi>c</mi> <mn>3</mn></msup><mo stretchy="false">/</mo><mi>e</mi><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">E_{crit} = m^2 c^3 / e \hbar</annotation></semantics></math></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT+scale">GUT scale</a></p> </li> <li> <p><a href="string+theory#Scales">string scale</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+tension">string tension</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mi>α</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T = 1/(2\pi \alpha^\prime)</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+length+scale">string length scale</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>s</mi></msub><mo>=</mo><msqrt><mrow><mi>α</mi><mo>′</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\ell_s = \sqrt{\alpha'}</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+coupling+constant">string coupling constant</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>s</mi></msub><mo>=</mo><msup><mi>e</mi> <mi>λ</mi></msup></mrow><annotation encoding="application/x-tex">g_s = e^\lambda</annotation></semantics></math></p> </li> </ul> </li> </ul> </div> <h2 id="References">References</h2> <h3 id="general">General</h3> <p>Original articles include</p> <ul> <li id="PatiSalam74"> <p><a class="existingWikiWord" href="/nlab/show/Jogesh+Pati">Jogesh Pati</a>, <a class="existingWikiWord" href="/nlab/show/Abdus+Salam">Abdus Salam</a>, <em>Lepton number as the fourth “color”</em>, Phys. Rev. D 10, 275 – Published 1 July 1974 (<a href="https://doi.org/10.1103/PhysRevD.10.275">doi:10.1103/PhysRevD.10.275</a>)</p> </li> <li id="GeorgiGlashow74"> <p><a class="existingWikiWord" href="/nlab/show/Howard+Georgi">Howard Georgi</a>, <a class="existingWikiWord" href="/nlab/show/Sheldon+Glashow">Sheldon Glashow</a>, <em>Unity of All Elementary-Particle Forces</em>, Phys. Rev. Lett. <strong>32</strong> (1974) 438 [<a href="https://doi.org/10.1103/PhysRevLett.32.438">doi:10.1103/PhysRevLett.32.438</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+E.+Marshak">Robert E. Marshak</a>, Chapter 8 of: <em>Conceptual Foundations of Modern Particle Physics</em>, World Scientific 1993 (<a href="https://doi.org/10.1142/1767">doi:10.1142/1767</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Howard+Georgi">Howard Georgi</a>, §18 in: <em>Lie Algebras In Particle Physics</em>, Westview Press (1999), CRC Press (2019) [<a href="https://doi.org/10.1201/9780429499210">doi:10.1201/9780429499210</a>]</p> </li> </ul> <p>See also</p> <ul> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Pati%E2%80%93Salam_model">Pati-Salam model</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Grand_unification_energy">Grand unification energy</a></em></p> </li> </ul> <p>Discussion with an eye towards <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> unification:</p> <ul> <li id="GellMann83"> <p><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a>, introductory talk at <em><a href="https://en.wikipedia.org/wiki/Shelter_Island_Conference">Shelter Island II</a></em>, 1983 (<a class="existingWikiWord" href="/nlab/files/Gell-Mann_ShelterIslandII_1983.pdf" title="pdf">pdf</a>)</p> <p>in: <em>Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics</em>. MIT Press. pp. 301–343. ISBN 0-262-10031-2.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Ramond">Pierre Ramond</a>, Richard Slansky, <em>Complex Spinors and Unified Theories</em>, Supergravity, <a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a> and D.Z. Freedman, eds, North Holland Publishing Co, 1979, (reprinted as <a href="http://arxiv.org/abs/1306.4669">arXiv:1306.4669</a>)</p> </li> </ul> <p>A basic textbook account is in</p> <ul> <li id="IbanezUranga12"><a class="existingWikiWord" href="/nlab/show/Luis+Ib%C3%A1%C3%B1ez">Luis Ibáñez</a>, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, section 1.2 of <em><a class="existingWikiWord" href="/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology">String Theory and Particle Physics – An Introduction to String Phenomenology</a></em>, Cambridge University Press 2012</li> </ul> <p>and a detailed one is in</p> <ul> <li>A. Hebecker, J. Hisano, <em>Grand Unified Theories</em>, chapter 114. in Particle Data Group’s <em><a href="http://pdg.lbl.gov/2018/">The Review of Particle Physics</a></em> 2018 (<a href="http://pdg.lbl.gov/2018/reviews/rpp2018-rev-guts.pdf">pdf</a>)</li> </ul> <p>See also</p> <ul> <li id="diLucio11">Luca di Lucio, <em>Aspects of Symmetry Breaking inGrand Unified Theories</em>, 2011 (<a href="https://www.sissa.it/tpp/phdsection/AlumniThesis/Luca%20Di%20Luzio.pdf">pdf</a>)</li> </ul> <p>Survey of arguments for the hypothesis of grand unification includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Peskin">Michael Peskin</a>, <em>Beyond the Standard Model</em> (<a href="http://arxiv.org/abs/hep-ph/9705479">arXiv:hep-ph/9705479</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jogesh+Pati">Jogesh Pati</a>, <em>Discovery of Proton Decay: A Must for Theory, a Challenge for Experiment</em> (<a href="http://arxiv.org/abs/hep-ph/0005095">arXiv:hep-ph/0005095</a>)</p> </li> <li id="Witten02a"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Quest For Unification</em>, Heinrich Hertz lecture at <a href="http://www.desy.de/susy02/">SUSY 2002</a> at DESY, Hamburg (<a href="http://arxiv.org/abs/hep-ph/0207124">arXiv:hep-ph/0207124</a>)</p> </li> </ul> <p>Introductions to GUTs aimed more at mathematicians:</p> <ul> <li id="Witten86"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 1 and 2 of <em>Physics and geometry</em>, Proceedings of the international congress of mathematicians, 1986 (<a href="http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf">pdf</a>)</p> </li> <li id="BaezHuerta09"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <em>The algebra of grand unified theories</em>, Bull. Am. Math. Soc. 47:483-552,2010 (<a href="http://arxiv.org/abs/0904.1556">arXiv:0904.1556</a>)</p> </li> <li id="Britto17"> <p>Vivian Anthony Britto, <em>A Mathematical Construction of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">E_6</annotation></semantics></math> Grand Unified Theory</em>, Munich 2017 (<a href="https://www.theorie.physik.uni-muenchen.de/TMP/theses/thesisbritto.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BrittoGUT.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/division+algebra">division-algebraic</a> relations in GUT-<a class="existingWikiWord" href="/nlab/show/broken+symmetry">symmetry breaking</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Nichol+Furey">Nichol Furey</a>, <em>An Algebraic Roadmap of Particle Theories, Part I: General construction</em> [<a href="https://arxiv.org/abs/2312.12377">arXiv:2312.12377</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nichol+Furey">Nichol Furey</a>, <em>An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints</em> [<a href="https://arxiv.org/abs/2312.12799">arXiv:2312.12799</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nichol+Furey">Nichol Furey</a>, <em>An Algebraic Roadmap of Particle Theories, Part III: Intersections</em> [<a href="https://arxiv.org/abs/2312.14207">arXiv:2312.14207</a>]</p> </li> </ul> <p>See also</p> <ul> <li>Mario Fernández Navarro, Stephen F. King, Avelino Vicente, <em>Tri-unification: a separate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> for each fermion family</em> [<a href="https://arxiv.org/abs/2311.05683">arXiv:2311.05683</a>]</li> </ul> <h3 id="ReferencesProtonNonDecay">Proton (non-)decay</h3> <p>Discussion of experimental bounds on <a class="existingWikiWord" href="/nlab/show/proton+decay">proton decay</a> in GUTs includes</p> <ul> <li id="KM14">Helena Kolešová, <a class="existingWikiWord" href="/nlab/show/Michal+Malinsk%C3%BD">Michal Malinský</a>, <em>Proton lifetime in the minimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> GUT and its implications for the LHC</em>, Phys. Rev. D 90, 115001 (2014) (<a href="http://arxiv.org/abs/1409.4961">arXiv:1409.4961</a>)</li> </ul> <p>Claim that <a class="existingWikiWord" href="/nlab/show/proton+decay">proton decay</a> may be entirely avoided:</p> <ul> <li id="MuetterRatzVaudrvange16"> <p>Andreas Mütter, Michael Ratz, Patrick K.S. Vaudrevange, <em>Grand Unification without Proton Decay</em> (<a href="https://arxiv.org/abs/1606.02303">arXiv:1606.02303</a>)</p> <p>(claims that many <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> and <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> models have this property)</p> </li> <li id="FornalGrinstein17"> <p>Bartosz Fornal, Benjamin Grinstein, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> Unification without Proton Decay</em>, Physics Review Letters (<a href="https://arxiv.org/abs/1706.08535">arXiv:1706.08535</a>)</p> </li> <li id="FornalGrinstein18"> <p>Bartosz Fornal, Benjamin Grinstein, <em>Grand Unified Theory with a Stable Proton</em>, Int. J. Mod. Phys. A 33 (2018) 1844013 (<a href="https://arxiv.org/abs/1808.00953">arXiv:1808.00953</a>)</p> </li> </ul> <p>Claim that <a class="existingWikiWord" href="/nlab/show/threshold+corrections">threshold corrections</a> can considerably alter (decrease) proton decay rate predictions in non-supersymmetric GUTs:</p> <ul> <li>Joydeep Chakrabortty, Stephen F. King, Rinku Maji, <em>Unification, Proton Decay and Topological Defects in non-SUSY GUTs with Thresholds</em> (<a href="https://arxiv.org/abs/1901.05867">arXiv:1901.05867</a>)</li> </ul> <h3 id="ReferencesRealisticModels">Realistic models and phenomenology</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a> viable GUT-<a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">models</a> (compatible with <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> and the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>):</p> <h4 id="gut"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-GUT</h4> <p>More on the original <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math>-approach:</p> <ul> <li>Ilja Doršner, <em>Towards a minimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> model</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/2206.06036">arXiv:2206.06036</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>On quasi-realistic “flipped” <a class="existingWikiWord" href="/nlab/show/SU%285%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>SU</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">SU(5)</annotation> </semantics> </math></a>-<a class="existingWikiWord" href="/nlab/show/GUT">GUT</a>, modeled in 4d <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> and subsuming realistic <a class="existingWikiWord" href="/nlab/show/Starobinsky+model+of+cosmic+inflation">Starobinsky-type cosmic inflation</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ignatios+Antoniadis">Ignatios Antoniadis</a>, <a class="existingWikiWord" href="/nlab/show/John+Ellis">John Ellis</a>, <a class="existingWikiWord" href="/nlab/show/John+S.+Hagelin">John S. Hagelin</a>, <a class="existingWikiWord" href="/nlab/show/Dimitri+Nanopoulos">Dimitri Nanopoulos</a>, <em>The flipped <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5) \times U(1)</annotation></semantics></math> string model revamped</em>, Physics Letters B <strong>231</strong> 1–2 (1989) 65-74 [<a href="https://doi.org/10.1016/0370-2693(89)90115-9">doi:10.1016/0370-2693(89)90115-9</a>]</p> </li> <li id="AntoniadisNanopoulosRizos21a"> <p><a class="existingWikiWord" href="/nlab/show/Ignatios+Antoniadis">Ignatios Antoniadis</a>, <a class="existingWikiWord" href="/nlab/show/Dimitri+Nanopoulos">Dimitri Nanopoulos</a>, <a class="existingWikiWord" href="/nlab/show/John+Rizos">John Rizos</a>, <em>Cosmology of the string derived flipped <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math></em>, J. Cosm. Astrop. Phys <strong>03</strong> (2021) 017 [<a href="https://arxiv.org/abs/2011.09396">arXiv:2011.09396</a>, <a href="https://doi.org/10.1088/1475-7516/2021/03/017">doi:10.1088/1475-7516/2021/03/017</a>]</p> </li> <li id="AntoniadisNanopoulosRizos21a"> <p><a class="existingWikiWord" href="/nlab/show/Ignatios+Antoniadis">Ignatios Antoniadis</a>, <a class="existingWikiWord" href="/nlab/show/Dimitri+Nanopoulos">Dimitri Nanopoulos</a>, <a class="existingWikiWord" href="/nlab/show/John+Rizos">John Rizos</a>, <em>Particle physics and cosmology of the string derived no-scale flipped <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math></em>, Eur. Phys. J. C <strong>82</strong> (2022) 377 [<a href="https://arxiv.org/abs/2112.01211">arXiv:2112.01211</a>, <a href="https://doi.org/10.1140/epjc/s10052-022-10353-6">doi:10.1140/epjc/s10052-022-10353-6</a>]</p> </li> </ul> <h4 id="gut_2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math>-GUT</h4> <p>The idea of “SO(10)” (<a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a>) GUT originates with</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Harald+Fritzsch">Harald Fritzsch</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Minkowski">Peter Minkowski</a>, <em>Unified interactions of leptons and hadrons</em>, Annals of Physics Volume 93, Issues 1–2, 5 September 1975, Pages 193-266 (<a href="https://doi.org/10.1016/0003-4916(75)90211-0">doi:10.1016/0003-4916(75)90211-0</a>)</li> </ul> <p>Review:</p> <ul> <li id="Malinsky09"> <p><a class="existingWikiWord" href="/nlab/show/Michal+Malinsk%C3%BD">Michal Malinský</a>, <em>35 years of GUTs - where do we stand?</em>, 2009 (<a href="https://www.mpi-hd.mpg.de/lin/seminar_theory/talks/Malinsky.pdf">pdf</a>)</p> </li> <li id="BLM09"> <p>Stefano Bertolini, Luca Di Luzio, <a class="existingWikiWord" href="/nlab/show/Michal+Malinsk%C3%BD">Michal Malinský</a>, <em>Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal</em>, Phys. Rev. D80:015013, 2009 (<a href="https://arxiv.org/abs/0903.4049">arXiv:0903.4049</a>)</p> </li> </ul> <p>for non-superymmetric <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">models</a>:</p> <ul> <li id="LavouraWolfenstein10"> <p>L. Lavoura and Lincoln Wolfenstein, <em>Resuscitation of minimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> grand unification</em>, Phys. Rev. D 48, 264 (<a href="https://doi.org/10.1103/PhysRevD.48.264">doi:10.1103/PhysRevD.48.264</a>)</p> </li> <li id="AltarelliMeloni13"> <p>Guido Altarelli, Davide Meloni, <em>A non Supersymmetric SO(10) Grand Unified Model for All the Physics below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>GUT</mi></msub></mrow><annotation encoding="application/x-tex">M_{GUT}</annotation></semantics></math></em> (<a href="https://arxiv.org/abs/1305.1001">arXiv:1305.1001</a>)</p> </li> <li id="DueckRodejohann13"> <p>Alexander Dueck, Werner Rodejohann, <em>Fits to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> Grand Unified Models</em> (<a href="http://arxiv.org/abs/1306.4468">arXiv:1306.4468</a>)</p> </li> <li id="FongMeloni14"> <p>Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, <em>Leptogenesis in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math></em> (<a href="http://arxiv.org/abs/1412.4776">arXiv:1412.4776</a>)</p> <p>(in view of <a class="existingWikiWord" href="/nlab/show/leptogenesis">leptogenesis</a>)</p> </li> <li id="OhlssonPernow19"> <p>Tommy Ohlsson, Marcus Pernow, <em>Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution</em> (<a href="https://arxiv.org/abs/1903.08241">arXiv:1903.08241</a>)</p> </li> <li id="CPS19"> <p>Mainak Chakraborty, M.K. Parida, Biswonath Sahoo, <em>Triplet Leptogenesis, Type-II Seesaw Dominance, Intrinsic Dark Matter, Vacuum Stability and Proton Decay in Minimal SO(10) Breakings</em> (<a href="https://arxiv.org/abs/1906.05601">arXiv:1906.05601</a>)</p> <blockquote> <p>Results indicating non-SUSY <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> as self sufficient theory for neutrino masses, baryon asymmetry, dark matter, vacuum stability of SM scalar potential, origin of three gauge forces, and observed proton stability.</p> </blockquote> </li> <li> <p>Nobuchika Okada, Digesh Raut, Qaisar Shafi, <em>Inflation, Proton Decay, and Higgs-Portal Dark Matter in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mo lspace="0em" rspace="thinmathspace">pri</mo></msub></mrow><annotation encoding="application/x-tex">SO(10) \times U(1)_\pri</annotation></semantics></math></em> (<a href="https://arxiv.org/abs/1906.06869">arXiv:1906.06869</a>)</p> </li> <li> <p>V. Suryanarayana Mummidi, Ketan M. Patel, <em>Leptogenesis and fermion mass fit in a renormalizable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> model</em> (<a href="https://arxiv.org/abs/2109.04050">arXiv:2109.04050</a>)</p> </li> <li> <p>George Lazarides, Rinku Maji, Rishav Roshan, Qaisar Shafi, <em>A Predictive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> Model</em> [<a href="https://arxiv.org/abs/2210.03710">arXiv:2210.03710</a>]</p> </li> </ul> <p>with possible relation to <a class="existingWikiWord" href="/nlab/show/flavour+anomalies">flavour anomalies</a>:</p> <ul> <li>Purushottam Sahu, Aishwarya Bhatta, Rukmani Mohanta, Shivaramakrishna Singirala, Sudhanwa Patra, <em>Flavour anomalies and dark matter assisted unification in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> GUT</em> [<a href="https://arxiv.org/abs/2204.06392">arXiv:2204.06392</a>]</li> </ul> <p>for <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">models</a>:</p> <ul> <li> <p>Archana Anandakrishnan, B. Charles Bryant, Stuart Raby, <em>LHC Phenomenology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> Models with Yukawa Unification II</em> (<a href="http://arxiv.org/abs/1404.5628">arXiv:1404.5628</a>)</p> </li> <li> <p>Ila Garg, <em>New minimal supersymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> GUT phenomenology and its cosmological implications</em> (<a href="http://arxiv.org/abs/1506.05204">arXiv:1506.05204</a>)</p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/symmetry+breaking">symmetry breaking</a> from <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a> to the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model</a> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> in terms of <a class="existingWikiWord" href="/nlab/show/real+normed+division+algebras">real normed division algebras</a> (<a class="existingWikiWord" href="/nlab/show/octonions">octonions</a>, <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kirill+Krasnov">Kirill Krasnov</a>, <em>Geometry of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(10)</annotation></semantics></math> Symmetry Breaking</em> [<a href="https://arxiv.org/abs/2209.05088">arXiv:2209.05088</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nichol+Furey">Nichol Furey</a>, M. J. Hughes, <em>Division algebraic symmetry breaking</em> [<a href="https://arxiv.org/abs/2210.10126">arXiv:2210.10126</a>]</p> </li> </ul> <h4 id="ReferencesSpin11"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(11)</annotation></semantics></math>-GUT</h4> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a> GUT group (“<a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a>”):</p> <ul> <li id="HosotaniYamatsu15"> <p><a class="existingWikiWord" href="/nlab/show/Yutaka+Hosotani">Yutaka Hosotani</a>, Naoki Yamatsu, <em>Gauge–Higgs grand unification</em>, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 (<a href="https://doi.org/10.1093/ptep/ptv153">doi:10.1093/ptep/ptv153</a>, <a href="https://doi.org/10.1093/ptep/ptw116">doi:10.1093/ptep/ptw116</a>)</p> </li> <li id="FuruiHosotaniYamatsu16"> <p>Atsushi Furui, <a class="existingWikiWord" href="/nlab/show/Yutaka+Hosotani">Yutaka Hosotani</a>, Naoki Yamatsu, <em>Toward Realistic Gauge-Higgs Grand Unification</em>, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 (<a href="https://arxiv.org/abs/1606.07222">arXiv:1606.07222</a>)</p> </li> <li id="Hosotoni16"> <p><a class="existingWikiWord" href="/nlab/show/Yutaka+Hosotani">Yutaka Hosotani</a>, <em>Gauge-Higgs EW and Grand Unification</em>, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016) (<a href="https://arxiv.org/abs/1606.08108">arXiv:1606.08108</a>)</p> </li> <li id="Hosotani17"> <p><a class="existingWikiWord" href="/nlab/show/Yutaka+Hosotani">Yutaka Hosotani</a>, <em>New dimensions from gauge-Higgs unification</em> (<a href="https://arxiv.org/abs/1702.08161">arXiv:1702.08161</a>)</p> </li> <li id="HosotaniYamatsu17"> <p><a class="existingWikiWord" href="/nlab/show/Yutaka+Hosotani">Yutaka Hosotani</a>, Naoki Yamatsu, <em>Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification</em> (<a href="https://arxiv.org/abs/1710.04811">arXiv:1710.04811</a>)</p> </li> </ul> <h4 id="ReferencesSpin12"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(12)</annotation></semantics></math>-GUT</h4> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a> GUT group (“<a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a>”):</p> <ul> <li> <p>S. Rajpoot and P. Sithikong, <em>Implications of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(12)</annotation></semantics></math> gauge symmetry for grand unification</em>, Phys. Rev. D 23, 1649 (1981) (<a href="https://doi.org/10.1103/PhysRevD.23.1649">doi:10.1103/PhysRevD.23.1649</a>)</p> </li> <li id="NomuraSato08"> <p><a class="existingWikiWord" href="/nlab/show/Takaaki+Nomura">Takaaki Nomura</a>, Joe Sato, <em>Standard(-like) Model from an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(12)</annotation></semantics></math> Grand Unified Theory in six-dimensions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> extra-space</em>, Nucl.Phys.B811:109-122, 2009 (<a href="https://arxiv.org/abs/0810.0898">arXiv:0810.0898</a>)</p> </li> <li id="Nomura09"> <p><a class="existingWikiWord" href="/nlab/show/Takaaki+Nomura">Takaaki Nomura</a>, <em>Physics beyond the standard model with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> extra-space</em>, 2009 (<a href="http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/NomuraSO12GUT.pdf" title="pdf">pdf</a>)</p> </li> <li id="ChiangNomuraSato11"> <p>Cheng-Wei Chiang, <a class="existingWikiWord" href="/nlab/show/Takaaki+Nomura">Takaaki Nomura</a>, Joe Sato, <em>Gauge-Higgs unification models in six dimensions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S^2/\mathbb{Z}_2</annotation></semantics></math> extra space and GUT gauge symmetry</em> (<a href="https://arxiv.org/abs/1109.5835">arXiv:1109.5835</a>)</p> </li> </ul> <h4 id="ReferencesSpin16"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>18</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(18)</annotation></semantics></math>-GUT</h4> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a> and <span class="newWikiWord">Spin(18)<a href="/nlab/new/Spin%2818%29">?</a></span> GUT group (“<a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a>” and “<span class="newWikiWord">SO(18)<a href="/nlab/new/SO%2818%29">?</a></span>”):</p> <ul> <li id="WilczekZee82"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <a class="existingWikiWord" href="/nlab/show/Anthony+Zee">Anthony Zee</a>, <em>Families from spinors</em>, Phys. Rev. D 25, 553 (1982) (<a href="https://doi.org/10.1103/PhysRevD.25.553">doi:10.1103/PhysRevD.25.55310.1103/PhysRevD.25.553</a>)</p> </li> <li id="SenjanovicWilczekZee84"> <p>Goran Senjanović, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <a class="existingWikiWord" href="/nlab/show/Anthony+Zee">Anthony Zee</a>, <em>Reflections on mirror fermions</em>, Physics Letters B Volume 141, Issues 5–6, 5 July 1984, Pages 389-394 Physics Letters B (<a href="https://doi.org/10.1016/0370-2693(84)90269-7">doi:10.1016/0370-2693(84)90269-7</a>)</p> </li> <li id="MartínezMelfoNestiSenjanovic11"> <p>Homero Martínez, Alejandra Melfo, Fabrizio Nesti, Goran Senjanović, <em>Three Extra Mirror or Sequential Families: a Case for Heavy Higgs and Inert Doublet</em>, Phys. Rev. Lett.106:191802, 2011 (<a href="https://arxiv.org/abs/1101.3796">arXiv:1101.3796</a>)</p> </li> <li id="Guigan19"> <p>Michael McGuigan, <em>Dark Horse, Dark Matter: Revisiting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">SO(16) \times SO(16)'</annotation></semantics></math> Nonsupersymmetric Model in the LHC and Dark Energy Era</em> (<a href="https://arxiv.org/abs/1907.01944">arXiv:1907.01944</a>)</p> </li> </ul> <h4 id="gut_5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math>-GUT</h4> <p>See at <em><a href="heterotic+string+theory#ReferencesPhenomenology">heterotic string – phenomenology</a></em></p> <p>See also:</p> <ul> <li> <p>Alfredo Aranda, Francisco J. de Anda, Stephen F. King, <em>Exceptional Unification of Families and Forces</em> (<a href="https://arxiv.org/abs/2005.03048">arXiv:2005.03048</a>)</p> </li> <li> <p>Alfredo Aranda, Francisco J. de Anda, <em>Complete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Unification in 10 Dimensions</em> (<a href="https://arxiv.org/abs/2007.13248">arXiv:2007.13248</a>)</p> </li> <li> <p>George Manolakos, Gregory Patellis, George Zoupanos, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathcal{N}=1</annotation></semantics></math> trinification from dimensional reduction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math>, 10D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> over <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><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">SU(3)/U(1)\times U(1) \times \mathbb{Z}_3</annotation></semantics></math> and its phenomenological consequences</em> (<a href="https://arxiv.org/abs/2009.07059">arXiv:2009.07059</a>)</p> </li> <li> <p>Francisco J. de Anda, Alfredo Aranda, António P. Morais, Roman Pasechnik, <em>Gauge couplings evolution from the Standard Model, through Pati-Salam theory, into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> unification of families and forces</em> (<a href="https://arxiv.org/abs/2011.13902">arXiv:2011.13902</a>)</p> </li> <li> <p>Alfredo Aranda, Francisco J. de Anda, António P. Morais, Roman Pasechnik, <em>A Different Take on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Unification</em> (<a href="https://arxiv.org/abs/2107.05421">arXiv:2107.05421</a>)</p> </li> <li> <p>Alfredo Aranda, Francisco J. de Anda, António P. Morais, Roman Pasechnik, <em>Sculpting the Standard Model from low-scale Gauge-Higgs-Matter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Grand Unification in ten dimensions</em> (<a href="https://arxiv.org/abs/2107.05495">arXiv:2107.05495</a>)</p> </li> </ul> <div> <h3 id="HeteroticStringPhenomenologyReferences">Heterotic string phenomenology</h3> <p>The historical origin of all <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> is the <a class="existingWikiWord" href="/nlab/show/top-down+model+building">top-down</a> <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a>-model building in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> due to</p> <ul> <li id="CHSW85"><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Gary+Horowitz">Gary Horowitz</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Vacuum configurations for superstrings</em>, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) [<a href="https://doi.org/10.1016/0550-3213(85)90602-9">doi:10.1016/0550-3213(85)90602-9</a>]</li> </ul> <p>Review and exposition:</p> <ul> <li id="Witten02"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Quest For Unification</em>, Heinrich Hertz lecture at <a href="http://www.desy.de/susy02/">SUSY 2002</a> at DESY, Hamburg [<a href="http://arxiv.org/abs/hep-ph/0207124">arXiv:hep-ph/0207124</a>]</p> </li> <li id="Nilles11"> <p><a class="existingWikiWord" href="/nlab/show/Hans-Peter+Nilles">Hans-Peter Nilles</a>, <em>Strings, Exceptional Groups and Grand Unification</em>, talk at <em><a href="https://indico.cern.ch/event/112851/">Planck 2011</a></em> [<a href="http://www.th.physik.uni-bonn.de/nilles/db/HPtalks/114planck.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Nilles11GUT.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Ratz">Michael Ratz</a>, <em>Heterotic Orbifold Models</em>, in <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a></em>, Springer (2024) [<a href="https://doi.org/10.1007/978-981-19-3079-9">doi:10.1007/978-981-19-3079-9</a>,</p> <p><a href="https://arxiv.org/abs/2401.03125">arXiv:2401.03125</a>]</p> </li> </ul> <h4 id="HeteroticStringPhenomenologyReferencesForE8">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8 \times E_8</annotation></semantics></math>-heterotic string</h4> <p>The following articles claim the existence of exact realization of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> and <a class="existingWikiWord" href="/nlab/show/matter">matter</a>-content of the <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> (not yet checking <a class="existingWikiWord" href="/nlab/show/Yukawa+couplings">Yukawa couplings</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, <em>A Heterotic Standard Model</em>, Phys. Lett. B618 : 252-258 2005 (<a href="http://arxiv.org/abs/hep-th/0501070">arXiv:hep-th/0501070</a>)</p> </li> <li> <p>Wilfried Buchmuller, Koichi Hamaguchi, Oleg Lebedev, <a class="existingWikiWord" href="/nlab/show/Michael+Ratz">Michael Ratz</a>, <em>Supersymmetric Standard Model from the Heterotic String</em>, Phys. Rev. Lett. <strong>96</strong> 121602 (2006) ([<a href="https://doi.org/10.1103/PhysRevLett.96.121602">doi:v</a><a href="https://arxiv.org/abs/hep-ph/0511035">arXiv:hep-ph/0511035</a>]</p> </li> <li id="BraunHeOvrutPantev06"> <p><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, <em>The Exact MSSM Spectrum from String Theory</em>, JHEP 0605:043, 2006 (<a href="http://arxiv.org/abs/hep-th/0512177">arXiv:hep-th/0512177</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vincent+Bouchard">Vincent Bouchard</a>, <a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, <em>An SU(5) Heterotic Standard Model</em>, Phys. Lett. B633:783-791,2006 (<a href="http://arxiv.org/abs/hep-th/0512149">arXiv:hep-th/0512149</a>)</p> </li> </ul> <p>A computer search through the “<a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape</a>” of <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+varieties">Calabi-Yau varieties</a> showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as <a class="existingWikiWord" href="/nlab/show/SU%285%29">SU(5)</a>-<a class="existingWikiWord" href="/nlab/show/GUTs">GUTs</a>):</p> <p>general computational theory:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <em>Heterotic Compactification, An Algorithmic Approach</em>, JHEP 0707:049, 2007 (<a href="https://arxiv.org/abs/hep-th/0702210">arXiv:hep-th/0702210</a>)</li> </ul> <p>using <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle+models">heterotic line bundle models</a>:</p> <ul> <li id="AndersonGrayLukasPalti11"> <p><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, <em>Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds</em>, Phys. Rev. D 84, 106005 (2011) (<a href="http://arxiv.org/abs/1106.4804">arXiv:1106.4804</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, <em>Heterotic Line Bundle Standard Models</em> JHEP06(2012)113 (<a href="https://arxiv.org/abs/1202.1757">arXiv:1202.1757</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/Andrei+Constantin">Andrei Constantin</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, <em>A Comprehensive Scan for Heterotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> GUT models</em>, JHEP01(2014)047 (<a href="https://arxiv.org/abs/1307.4787">arXiv:1307.4787</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Seung-Joo Lee, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, Chuang Sun, <em>Heterotic Model Building: 16 Special Manifolds</em>, J. High Energ. Phys. 2014, 77 (2014) (<a href="http://arxiv.org/abs/1309.0223">arXiv:1309.0223</a>)</p> </li> <li> <p>Stefan Groot Nibbelink, Orestis Loukas, Fabian Ruehle, Patrick K.S. Vaudrevange, <em>Infinite number of MSSMs from heterotic line bundles?</em>, Phys. Rev. D 92, 046002 (2015) (<a href="https://arxiv.org/abs/1506.00879">arXiv:1506.00879</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andreas+Braun">Andreas Braun</a>, Callum R. Brodie, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <em>Heterotic Line Bundle Models on Elliptically Fibered Calabi-Yau Three-folds</em>, JHEP04 (2018) 087 (<a href="https://arxiv.org/abs/1706.07688">arXiv:1706.07688</a>)</p> </li> <li id="CHE18"> <p><a class="existingWikiWord" href="/nlab/show/Andrei+Constantin">Andrei Constantin</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <em>Counting String Theory Standard Models</em>, Physics Letters B</p> <p>Volume 792, 10 May 2019, Pages 258-262 (<a href="https://arxiv.org/abs/1810.00444">arXiv:1810.00444</a>)</p> </li> <li> <p>Alon E. Faraggi, Glyn Harries, Benjamin Percival, John Rizos, <em>Towards machine learning in the classification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \times \mathbb{Z}_2</annotation></semantics></math> orbifold compactifications (<a href="https://arxiv.org/abs/1901.04448">arXiv:1901.04448</a>)</em></p> </li> <li> <p>Magdalena Larfors, Robin Schneider, <em>Explore and Exploit with Heterotic Line Bundle Models</em>, Fortschritte der Physik Vol 86 Nr. 5 (<a href="https://arxiv.org/abs/2003.04817">arXiv:2003.04817</a>)</p> </li> </ul> <p>The resulting database of <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle+models">heterotic line bundle models</a> is here:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Eran+Palti">Eran Palti</a>, <em>Heterotic standard model database</em> (<a href="http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.">web</a>)</li> </ul> <p>Review includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <em>New aspects of heterotic geometry and phenomenology</em>, talk at <a href="http://wwwth.mpp.mpg.de/conf/strings2012/">Strings2012</a>, Munich 2012 (<a href="http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf">pdf</a>)</p> </li> <li id="He18"> <p><a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <em>Deep-learning the landscape</em>, talk at <em><a href="https://ims.nus.edu.sg/events/2018/wstring/wk.php">String and M-Theory: The new geometry of the 21st century</a></em> (<a href="https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf">pdf slides</a>, <a href="https://www.youtube.com/watch?v=x3ThgBgkPlE">video recording</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <em>Calabi-Yau Spaces in the String Landscape</em> (<a href="https://arxiv.org/abs/2006.16623">arXiv:2006.16623</a>)</p> </li> </ul> <p>Computation of <a class="existingWikiWord" href="/nlab/show/metrics">metrics</a> on these Calabi-Yau compactifications (eventually needed for computing their induced <a class="existingWikiWord" href="/nlab/show/Yukawa+couplings">Yukawa couplings</a>) is started in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, Tamaz Brelidze, <a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <em>Calabi-Yau Metrics for Quotients and Complete Intersections</em>, JHEP 0805:080, 2008 (<a href="https://arxiv.org/abs/0712.3563">arXiv:0712.3563</a>)</li> </ul> <p>and via <a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>:</p> <ul> <li> <p>Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <em>Computation of Quark Masses from String Theory</em> [<a href="https://arxiv.org/abs/2402.01615">arXiv:2402.01615</a>]</p> </li> <li> <p>Per Berglund, Giorgi Butbaia, <a class="existingWikiWord" href="/nlab/show/Tristan+H%C3%BCbsch">Tristan Hübsch</a>, Vishnu Jejjala, Damián Mayorga Peña, Challenger Mishra, Justin Tan: <em>Precision String Phenomenology</em> [<a href="https://arxiv.org/abs/2407.13836">arXiv:2407.13836</a>]</p> </li> </ul> <p>This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Volker+Braun">Volker Braun</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <em>Supersymmetric Hidden Sectors for Heterotic Standard Models</em> (<a href="http://arxiv.org/abs/1301.6767">arXiv:1301.6767</a>)</li> </ul> <p>The issue of <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> in these kinds of models is discussed in</p> <ul> <li> <p>Michele Cicoli, Senarath de Alwis, Alexander Westphal, <em>Heterotic Moduli Stabilization</em> (<a href="http://arxiv.org/abs/1304.1809">arXiv:1304.1809</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lara+Anderson">Lara Anderson</a>, <a class="existingWikiWord" href="/nlab/show/James+Gray">James Gray</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Lukas">Andre Lukas</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <em>Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories</em> (<a href="http://arxiv.org/abs/1304.2704">arXiv:1304.2704</a>)</p> </li> </ul> <p>Principles singling out heterotic models with three generations of fundamental particles are discussed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Xenia+de+la+Ossa">Xenia de la Ossa</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Hui+He">Yang-Hui He</a>, Balazs Szendroi, <em>Triadophilia: A Special Corner in the Landscape</em>, Adv.Theor.Math.Phys.12:2,2008 (<a href="http://arxiv.org/abs/0706.3134">arXiv:0706.3134</a>)</li> </ul> <p>Discussion of non-<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a>: <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> models:</p> <ul> <li>Alon E. Faraggi, Viktor G. Matyas, Benjamin Percival, <em>Classification of Non-Supersymmetric Pati-Salam Heterotic String Models</em> (<a href="https://arxiv.org/abs/2011.04113">arXiv:2011.04113</a>)</li> </ul> <p>See also:</p> <ul> <li id="AngelantonjFlorakis18"><a class="existingWikiWord" href="/nlab/show/Carlo+Angelantonj">Carlo Angelantonj</a>, <a class="existingWikiWord" href="/nlab/show/Ioannis+Florakis">Ioannis Florakis</a>, <em>GUT Scale Unification in Heterotic Strings</em>, Physics Letters B <strong>789</strong> (2019) 496-501 [<a href="https://doi.org/10.1016/j.physletb.2018.12.054">doi:10.1016/j.physletb.2018.12.054</a><a href="https://arxiv.org/abs/1812.06915">arXiv:1812.06915</a>]</li> </ul> <h4 id="HeteroticStringPhenomenologyReferencesForSemiSpin32">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SemiSpin</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SemiSpin(32)</annotation></semantics></math>-heterotic string</h4> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> for the <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a>-<a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> (see also at <a href="type+I+string+theory#Phenomenology">type I phenomenology</a>):</p> <ul> <li> <p>Kang-Sin Choi, Stefan Groot Nibbelink, Michele Trapletti, <em>Heterotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> model building in four dimensions</em>, JHEP 0412:063, 2004 (<a href="https://arxiv.org/abs/hep-th/0410232">arXiv:hep-th/0410232</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hans-Peter+Nilles">Hans-Peter Nilles</a>, <a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, Patrick K.S. Vaudrevange, Akin Wingerter, <em>Exploring the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> Heterotic String</em>, JHEP 0604:050 (2006) [<a href="https://doi.org/10.1088/1126-6708/2006/04/050">doi:10.1088/1126-6708/2006/04/050</a><a href="https://arxiv.org/abs/hep-th/0603086">arXiv:hep-th/0603086</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saul+Ramos-Sanchez">Saul Ramos-Sanchez</a>, <em>Towards Low Energy Physics from the Heterotic String</em>, Fortsch. Phys. <strong>10</strong> (2009) 907-1036 [<a href="https://doi.org/10.1002/prop.200900073">doi:10.1002/prop.200900073</a><a href="https://arxiv.org/abs/0812.3560">arXiv:0812.3560</a>]</p> <p>(<a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a> <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a> <a class="existingWikiWord" href="/nlab/show/R-parity">R-parity</a> <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> <a class="existingWikiWord" href="/nlab/show/string+theory+vacua">vacua</a>)</p> </li> <li id="Yamatsu17"> <p>Naoki Yamatsu, <em>String-Inspired Special Grand Unification</em>, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (<a href="https://arxiv.org/abs/1708.02078">arXiv:1708.02078</a>, <a href="https://doi.org/10.1093/ptep/ptx135">doi:10.1093/ptep/ptx135</a>)</p> </li> <li> <p>Jihn E. Kim, <em>Grand unfication models from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> heterotic string</em> (<a href="https://arxiv.org/abs/2008.00367">arXov:2008.00367</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/heterotic+line+bundle+models">heterotic line bundle models</a>:</p> <ul> <li>Hajime Otsuka, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> heterotic line bundle models</em>, (<a href="https://arxiv.org/abs/1801.03684">arXiv:1801.03684</a>)</li> </ul> <h4 id="HeteroticStringPhenomenologyReferencesForSO16">The “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math>”-heterotic string</h4> <p>This non-<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> string theory was first described in:</p> <ul> <li id="AGMV86het"> <p><a class="existingWikiWord" href="/nlab/show/Luis+Alvarez-Gaum%C3%A9">Luis Alvarez-Gaumé</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Ginsparg">Paul Ginsparg</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a> and <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(16)\times O(16)</annotation></semantics></math> heterotic string</em>, Phys. Lett. B <strong>171</strong> (1986) 155 [<a href="https://doi.org/10.1016/0370-2693(86)91524-8">doi:10.1016/0370-2693(86)91524-8</a>]</p> </li> <li id="DH86het"> <p><a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a> and <a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, <em>String Theories in Ten-Dimensions Without Space-Time Supersymmetry</em>, Nucl. Phys. B <strong>274</strong> (1986) 93 [<a href="https://doi.org/10.1016/0550-3213(86)90619-X">doi:10.1016/0550-3213(86)90619-X</a>]</p> </li> </ul> <p>A proposal on what the correct global character of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> is appears in:</p> <ul> <li id="McIn99spin"><a class="existingWikiWord" href="/nlab/show/Brett+McInnes">Brett McInnes</a>, <em>The Semispin Groups in String Theory</em>, J. Math. Phys. <strong>40</strong> (1999) 4699-4712 [<a href="https://arxiv.org/abs/hep-th/9906059">arXiv:hep-th/9906059</a>, <a href="https://doi.org/10.1063/1.532999">doi:10.1063/1.532999</a>]</li> </ul> <p>A suggestion that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(16)\times SO(16)</annotation></semantics></math> heterotic string is a the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8\times E_8</annotation></semantics></math> string “minus” the <a class="existingWikiWord" href="/nlab/show/semispin+group">semispin group</a> <a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mtext>Spin</mtext> <mo stretchy="false">(</mo> <mn>32</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <msub><mi>ℤ</mi> <mn>2</mn></msub> </mrow> <annotation encoding="application/x-tex">\text{Spin}(32)/\mathbb{Z}_2</annotation> </semantics> </math></a>:</p> <ul> <li id="SW87anom"><a class="existingWikiWord" href="/nlab/show/Adrian+Norbert+Schellekens">Adrian Norbert Schellekens</a>, and <a class="existingWikiWord" href="/nlab/show/Nicholas+Warner">Nicholas Warner</a> <em>Anomalies, characters and strings</em>, Nuclear Physics B <strong>287</strong> (1987) 317-361 [<a href="https://doi.org/10.1016/0550-3213(87)90108-8">doi:10.1016/0550-3213(87)90108-8</a>]</li> </ul> </div> <h3 id="in_type_ii_string_theory">In type II string theory</h3> <p>Computer scan of <a class="existingWikiWord" href="/nlab/show/Gepner+model">Gepner model</a>-compactifications in relation to GUT-models is in</p> <ul> <li id="DijkstraHuiszoonSchellekens04a"> <p>T.P.T. Dijkstra, L. R. Huiszoon, <a class="existingWikiWord" href="/nlab/show/Bert+Schellekens">Bert Schellekens</a>, <em>Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models</em>, Phys.Lett. B609 (2005) 408-417 (<a href="https://arxiv.org/abs/hep-th/0403196">arXiv:hep-th/0403196</a>)</p> </li> <li id="DijkstraHuiszoonSchellekens04b"> <p>T.P.T. Dijkstra, L. R. Huiszoon, <a class="existingWikiWord" href="/nlab/show/Bert+Schellekens">Bert Schellekens</a>, <em>Supersymmetric Standard Model Spectra from RCFT orientifolds</em>, Nucl.Phys.B710:3-57,2005 (<a href="https://arxiv.org/abs/hep-th/0411129">arXiv:hep-th/0411129</a>)</p> </li> </ul> <p>Realization of GUTs in the context of <a class="existingWikiWord" href="/nlab/show/M-theory+on+G2-manifolds">M-theory on G2-manifolds</a> and possible resolution of the <a class="existingWikiWord" href="/nlab/show/doublet-triplet+splitting+problem">doublet-triplet splitting problem</a> is discussed in</p> <ul> <li id="Witten02b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Deconstruction, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy, and Doublet-Triplet Splitting</em>, (<a href="http://arxiv.org/abs/hep-ph/0201018">arXiv:hep-ph/0201018</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> Grand Unification in M theory on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> manifold</em> (<a href="http://arxiv.org/abs/1502.01727">arXiv:1502.01727</a>)</p> </li> </ul> <p>Discussion of GUTs in <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Beasley">Chris Beasley</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Heckman">Jonathan Heckman</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>GUTs and Exceptional Branes in F-theory - I</em> (<a href="http://arxiv.org/abs/0802.3391">arxiv:0802.3391</a>), <em>II: Experimental Predictions</em> (<a href="http://arxiv.org/abs/0806.0102">arxiv:0806.0102</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Beasley">Chris Beasley</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Heckman">Jonathan Heckman</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>GUTs and Exceptional Branes in F-theory - I</em>, JHEP 0901:058,2009 (<a href="http://arxiv.org/abs/0802.3391">arXiv:0802.3391</a>)</p> </li> <li id="Zoccarato14"> <p>Gianluca Zoccarato, <em>Yukawa couplings at the point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> in F-theory</em>, 2014 (<a href="http://stringpheno2014.ictp.it/parallels/tuesday/F-theory(B">pdf</a>/zoccarato.pdf))</p> </li> <li id="Vafa15"> <p><a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>Reflections on F-theory</em>, 2015 (<a href="http://f-theory15.mpp.mpg.de/talks/Vafa.pdf">pdf</a>)</p> </li> </ul> <h3 id="in_conneslott_models">In Connes-Lott models</h3> <p>Discussion of GUTs within <a class="existingWikiWord" href="/nlab/show/Connes-Lott+models">Connes-Lott models</a>:</p> <ul> <li id="ChamseddineConnesMukhanov14"> <p><a class="existingWikiWord" href="/nlab/show/Ali+Chamseddine">Ali Chamseddine</a>, <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, Viatcheslav Mukhanov, <em>Quanta of Geometry: Noncommutative Aspects</em>, Phys. Rev. Lett. 114 (2015) 9, 091302 (<a href="https://arxiv.org/abs/1409.2471">arXiv:1409.2471</a>)</p> </li> <li id="ChamseddineConnesMukhanov14"> <p><a class="existingWikiWord" href="/nlab/show/Ali+Chamseddine">Ali Chamseddine</a>, <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, Viatcheslav Mukhanov, <em>Geometry and the Quantum: Basics</em>, JHEP 12 (2014) 098 (<a href="https://arxiv.org/abs/1411.0977">arXiv:1411.0977</a>)</p> </li> <li id="Connes17"> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, section 4 of <em>Geometry and the Quantum</em>, in <em>Foundations of Mathematics and Physics One Century After Hilbert</em>, Springer 2018. 159-196 (<a href="https://arxiv.org/abs/1703.02470">arXiv:1703.02470</a>, <a href="https://www.springer.com/gp/book/9783319648125">doi:10.1007/978-3-319-64813-2</a>)</p> </li> </ul> <h3 id="exotica_leptoquarks_bosons_etc">Exotica: Leptoquarks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Z'</annotation></semantics></math>-bosons, etc.</h3> <p>Topological defects can play considerable role to constrain the non-SUSY and SUSY GUTs:</p> <ul> <li>Joydeep Chakrabortty, Rinku Maji, Sunando Kumar Patra, Tripurari Srivastava, Subhendra Mohanty, <em>Roadmap of left-right models based on GUTs</em> (<a href="https://arxiv.org/abs/1711.11391">arXiv:1711.11391</a>, Phys.Rev. D97 (2018) no.9, 095010.)</li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/Z%27-bosons">Z'-bosons</a>:</p> <ul> <li id="Sahoo06">S. Sahoo, <em>The prediction of mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Z'</annotation></semantics></math>-boson in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math>-based model</em>, Indian J. Phys. 80 (2), 191-194 (2006) (<a href="http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6019/1/The%20Prediction%20of%20Mass%20of%20Z%27-Boson_By%20S%20Sahoo.pdf">pdf</a>)</li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/leptoquarks">leptoquarks</a> and <a class="existingWikiWord" href="/nlab/show/flavour+anomalies">flavour anomalies</a>:</p> <ul> <li id="MurayamaYanagida92"> <p>H. Murayama, T. Yanagida, <em>A viable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(5)</annotation></semantics></math> GUT with light leptoquark bosons</em>, Mod.Phys.Lett. A7 (1992) 147-152 (<a href="inspirehep.net/record/315898">arXiv:315898</a>, <a href="https://doi.org/10.1142/S0217732392000070">doi:10.1142/S0217732392000070</a>)</p> </li> <li id="BDFKFS18"> <p>Damir Bečirević, Ilja Doršner, Svjetlana Fajfer, Nejc Košnik, Darius A. Faroughy, Olcyr Sumensari, <em>Scalar leptoquarks from GUT to accommodate the <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>-physics anomalies</em>, Phys. Rev. D 98, 055003 (2018) (<a href="https://arxiv.org/abs/1806.05689">arXiv:1806.05689</a>)</p> </li> <li id="AMM19"> <p>Ufuk Aydemir, Tanumoy Mandal, Subhadip Mitra, <em>A single TeV-scale scalar leptoquark in SO(10) grand unification and B-decay anomalies</em> (<a href="https://arxiv.org/abs/1902.08108">arXiv:1902.08108</a>)</p> </li> <li id="HeekTeresi18"> <p>Julian Heeck, Daniele Teresi, <em>Pati-Salam explanations of the B-meson anomalies</em>, JHEP 12 (2018) 103 (<a href="https://arxiv.org/abs/1808.07492">arXiv:1808.07492</a>)</p> </li> <li id="HeekTeresi19"> <p>Julian Heeck, Daniele Teresi, <em>Pati-Salam and lepton universality in B decays</em> (<a href="https://arxiv.org/abs/1905.05211">arXiv:1905.05211</a>)</p> </li> <li id="Malinsky19"> <p><a class="existingWikiWord" href="/nlab/show/Michal+Malinsk%C3%BD">Michal Malinský</a>, <em>Lepton non-universality in <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>-decays in the minimal leptoquark gauge model</em> (<a href="https://arxiv.org/abs/1906.09174">arXiv:1906.09174</a>)</p> </li> <li> <p>Shyam Balaji, Michael A. Schmidt, <em>A unified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(4)</annotation></semantics></math> theory for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>R</mi> <mi>D</mi> <mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">R_D^{(\ast)}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>R</mi> <mi>K</mi> <mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">R_K^{(\ast)}</annotation></semantics></math> anomalies</em> (<a href="https://arxiv.org/abs/1911.08873">arXiv:1911.08873</a>)</p> </li> </ul> <p>Modeled <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+model">on branes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Javier+Fuentes-Martin">Javier Fuentes-Martin</a>, <a class="existingWikiWord" href="/nlab/show/Gino+Isidori">Gino Isidori</a>, Julie Pagès, Ben A. Stefanek, <p><em>Flavor Non-universal Pati-Salam Unification and Neutrino Masses</em>, Physics Letters B Volume 820, 10 September 2021, 136484 (<a href="https://arxiv.org/abs/2012.10492">arXiv:2012.10492</a>, <a href="https://www.youtube.com/watch?v=ko4R8S8XYV4&t=2s">talk recording</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 27, 2024 at 04:36:15. 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