# Mu problem

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Problem of supersymmetric theories

Not to be confused with [MU puzzle](/source/MU_puzzle).

In theoretical physics, the **μ problem** is a problem of [supersymmetric](/source/Supersymmetry) theories, concerned with understanding the parameters of the theory.

## Background

The supersymmetric [Higgs](/source/Peter_Higgs) mass parameter μ appears as the following term in the [superpotential](/source/Superpotential): μ Hu Hd. It is necessary to provide a mass for the fermionic [superpartners](/source/Superpartner) of the Higgs bosons, i.e. the [higgsinos](/source/Higgsino), and it enters as well the scalar potential of the Higgs bosons.

To ensure that Hu and Hd get a non-zero [vacuum expectation value](/source/Vacuum_expectation_value) after [electroweak symmetry breaking](/source/Electroweak_symmetry_breaking), μ should be of the order of magnitude of the [electroweak scale](/source/Electroweak_scale), many orders of magnitude smaller than the [Planck scale](/source/Planck_scale) (Mpl), which is the natural [cutoff](/source/Cutoff_(physics)) scale. This brings about a problem of naturalness: Why is that scale so much smaller than the cutoff scale? And why, if the μ term in the superpotential has different physical origins, do the corresponding scale happen to fall so close to each other?

Before [LHC](/source/LHC), it was thought that the [soft supersymmetry breaking](/source/Soft_supersymmetry_breaking) 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.[1]

One proposed solution, known as the [Giudice](/source/Gian_Francesco_Giudice)–Masiero mechanism,[2] is that this term does not appear explicitly in the Lagrangian, because it violates some global symmetry, and can therefore be created only via [spontaneous breaking](/source/Spontaneous_symmetry_breaking) of this symmetry. This is proposed to happen together with [F-term](/source/F-term) [supersymmetry breaking](/source/Supersymmetry_breaking), with a spurious field X that parameterizes the hidden supersymmetry-breaking sector of the theory (meaning that FX is the non-zero F-term).

Let us assume that the [Kahler potential](/source/Kahler_potential) includes a term of the form X M p l H u H d {\displaystyle \ {\frac {X}{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ } times some dimensionless coefficient, which is naturally of order one, and where Mpl is [Planck mass](/source/Planck_mass). Then as supersymmetry breaks, FX gets a non-zero vacuum expectation value ⟨FX⟩ and the following effective term is added to the superpotential: ⟨ F X ⟩ M p l H u H d , {\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ H_{\mathsf {u}}\ H_{\mathsf {d}}\ ,} which gives a measured μ = ⟨ F X ⟩ M p l . {\displaystyle \ \mu ={\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .} On the other hand, soft supersymmetry breaking terms are similarly created and also have a natural scale of ⟨ F X ⟩ M p l . {\displaystyle \ {\frac {\ \langle F_{\mathsf {X}}\rangle \ }{\ M_{\mathsf {pl}}\ }}\ .}

## See also

- [NMSSM](/source/NMSSM) (Next-to-Minimal Supersymmetric Standard Model)

- [Minimal Supersymmetric Standard Model](/source/Minimal_Supersymmetric_Standard_Model)

- [Doublet–triplet splitting problem](/source/Doublet%E2%80%93triplet_splitting_problem)

- [Hierarchy problem](/source/Hierarchy_problem)

- [Little hierarchy problem](/source/Little_hierarchy_problem)

## References

1. **[^](#cite_ref-1)** Fowlie, Andrew (2014). ["Is the CNMSSM more credible than the CMSSM?"](https://doi.org/10.1140%2Fepjc%2Fs10052-014-3105-y). *The European Physical Journal C*. **74** (10): 3105. [arXiv](/source/ArXiv_(identifier)):[1407.7534](https://arxiv.org/abs/1407.7534). [Bibcode](/source/Bibcode_(identifier)):[2014EPJC...74.3105F](https://ui.adsabs.harvard.edu/abs/2014EPJC...74.3105F). [doi](/source/Doi_(identifier)):[10.1140/epjc/s10052-014-3105-y](https://doi.org/10.1140%2Fepjc%2Fs10052-014-3105-y). [S2CID](/source/S2CID_(identifier)) [119304794](https://api.semanticscholar.org/CorpusID:119304794).

1. **[^](#cite_ref-2)** Giudice, G.F.; Masiero, A. (1988). "A natural solution to the mu problem in supergravity theories". *Physics Letters B*. **206** (3): 480–484. [Bibcode](/source/Bibcode_(identifier)):[1988PhLB..206..480G](https://ui.adsabs.harvard.edu/abs/1988PhLB..206..480G). [doi](/source/Doi_(identifier)):[10.1016/0370-2693(88)91613-9](https://doi.org/10.1016%2F0370-2693%2888%2991613-9).

## External links

- [Supersymmetric Models with extra singlets: a review; DJ Miller, University of Glasgow](https://web.archive.org/web/20070927021637/http://whepp9.iopb.res.in/talks/DJ_Miller.ppt)

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