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<title>Mu problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Mu problem</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="MU_puzzle" title="MU puzzle">MU puzzle</a>.</div>

<p>In theoretical physics, the <b><span class="texhtml">μ</span> problem</b> is a problem of <a href="Supersymmetry" title="Supersymmetry">supersymmetric</a> theories, concerned with understanding the parameters of the theory.

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<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

<p>The supersymmetric <a href="Peter_Higgs" title="Peter Higgs">Higgs</a> mass parameter <span class="texhtml mvar" style="font-style:italic;">μ</span> appears as the following term in the <a href="Superpotential" title="Superpotential">superpotential</a>: <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">μ</span> <span class="texhtml mvar" style="font-style:italic;">H</span><sub>u</sub> <span class="texhtml mvar" style="font-style:italic;">H</span><sub>d</sub></span>. It is necessary to provide a mass for the fermionic <a href="Superpartner" title="Superpartner">superpartners</a> of the Higgs bosons, i.e. the <a href="Higgsino" title="Higgsino">higgsinos</a>, and it enters as well the scalar potential of the Higgs bosons.

</p><p>To ensure that <span class="texhtml mvar" style="font-style:italic;">H</span><sub>u</sub> and <span class="texhtml mvar" style="font-style:italic;">H</span><sub>d</sub> get a non-zero <a href="Vacuum_expectation_value" title="Vacuum expectation value">vacuum expectation value</a> after <a href="Electroweak_symmetry_breaking" class="mw-redirect" title="Electroweak symmetry breaking">electroweak symmetry breaking</a>, <span class="texhtml mvar" style="font-style:italic;">μ</span> should be of the order of magnitude of the <a href="Electroweak_scale" title="Electroweak scale">electroweak scale</a>, many orders of magnitude smaller than the <a href="Planck_scale" class="mw-redirect" title="Planck scale">Planck scale</a> (<span class="texhtml mvar" style="font-style:italic;">M</span><sub>pl</sub>), which is the natural <a href="Cutoff_(physics)" title="Cutoff (physics)">cutoff</a> scale. This brings about a problem of naturalness: Why is that scale so much smaller than the cutoff scale? And why, if the <span class="texhtml mvar" style="font-style:italic;">μ</span> term in the superpotential has different physical origins, do the corresponding scale happen to fall so close to each other?

</p><p>Before <a href="LHC" class="mw-redirect" title="LHC">LHC</a>, it was thought that the <a href="Soft_supersymmetry_breaking" class="mw-redirect" title="Soft supersymmetry breaking">soft supersymmetry breaking</a> terms should also be of the same order of magnitude as the electroweak scale. This was negated by the Higgs mass measurements and limits on supersymmetry models.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>One proposed solution, known as the <a href="Gian_Francesco_Giudice" title="Gian Francesco Giudice">Giudice</a>–Masiero mechanism,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is that this term does not appear explicitly in the Lagrangian, because it violates some global symmetry, and can therefore be created only via <a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous breaking</a> of this symmetry. This is proposed to happen together with <a href="F-term" title="F-term">F-term</a> <a href="Supersymmetry_breaking" title="Supersymmetry breaking">supersymmetry breaking</a>, with a spurious field <span class="texhtml">X</span> that parameterizes the hidden supersymmetry-breaking sector of the theory (meaning that <span class="texhtml mvar" style="font-style:italic;">F</span><sub>X</sub> is the non-zero <span class="texhtml mvar" style="font-style:italic;">F</span>-term).

</p><p>Let us assume that the <a href="Kahler_potential" class="mw-redirect" title="Kahler potential">Kahler potential</a> includes a term of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {X}{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ }">

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<annotation encoding="application/x-tex">{\displaystyle \ {\frac {X}{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ }</annotation>

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</math></span><img src="./b9b060e684f995682503fcf6389aafb6476556df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.076ex; height:5.843ex;" alt="{\displaystyle \ {\frac {X}{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ }" loading="lazy"></span> times some dimensionless coefficient, which is naturally of order one, and where M<sub>pl</sub> is <a href="Planck_mass" class="mw-redirect" title="Planck mass">Planck mass</a>. Then as supersymmetry breaks, <span class="texhtml mvar" style="font-style:italic;">F</span><sub>X</sub> gets a non-zero vacuum expectation value ⟨<span class="texhtml mvar" style="font-style:italic;">F</span><sub>X</sub>⟩ and the following effective term is added to the superpotential: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ ,}">

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<annotation encoding="application/x-tex">{\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ ,}</annotation>

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</math></span><img src="./61670a2a6fb27c8320f009f497b4635b11cca4ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.626ex; height:6.343ex;" alt="{\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ ,}" loading="lazy"></span> which gives a measured <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu ={\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}">

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<annotation encoding="application/x-tex">{\displaystyle \ \mu ={\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}</annotation>

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</math></span><img src="./c92ccc2459f2c61df58a6f1e1826c62adb9f7f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.938ex; height:6.343ex;" alt="{\displaystyle \ \mu ={\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}" loading="lazy"></span> On the other hand, soft supersymmetry breaking terms are similarly created and also have a natural scale of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}">

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<annotation encoding="application/x-tex">{\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}</annotation>

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</math></span><img src="./1d8ddea65dd747d7b36ca855626bfcf0668bd398.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.438ex; height:6.343ex;" alt="{\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}" loading="lazy"></span>

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<ul><li><a href="NMSSM" class="mw-redirect" title="NMSSM">NMSSM</a> (Next-to-Minimal Supersymmetric Standard Model)</li>

<li><a href="Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">Minimal Supersymmetric Standard Model</a></li>

<li><a href="Doublet%E2%80%93triplet_splitting_problem" title="Doublet–triplet splitting problem">Doublet–triplet splitting problem</a></li>

<li><a href="Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li>

<li><a href="Little_hierarchy_problem" title="Little hierarchy problem">Little hierarchy problem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFFowlie2014" class="citation journal cs1">Fowlie, Andrew (2014). "Is the CNMSSM more credible than the CMSSM?". <i>The European Physical Journal C</i>. <b>74</b> (10): 3105. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1407.7534">1407.7534</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014EPJC...74.3105F">2014EPJC...74.3105F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjc%2Fs10052-014-3105-y">10.1140/epjc/s10052-014-3105-y</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119304794">119304794</a>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFGiudice,_G.F.Masiero,_A.1988" class="citation journal cs1">Giudice, G.F.; Masiero, A. (1988). "A natural solution to the mu problem in supergravity theories". <i>Physics Letters B</i>. <b>206</b> (3): <span class="nowrap">480–</span>484. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988PhLB..206..480G">1988PhLB..206..480G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2888%2991613-9">10.1016/0370-2693(88)91613-9</a>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070927021637/http://whepp9.iopb.res.in/talks/DJ_Miller.ppt">Supersymmetric Models with extra singlets: a review; DJ Miller, University of Glasgow</a></li></ul>

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