In theoretical physics, theμ problem is a problem ofsupersymmetric theories, concerned with understanding the parameters of the theory.

The supersymmetricHiggs mass parameterμ appears as the following term in thesuperpotential:μH_uH_d. It is necessary to provide a mass for the fermionicsuperpartners of the Higgs bosons, i.e. thehiggsinos, and it enters as well the scalar potential of the Higgs bosons.

To ensure thatH_u andH_d get a non-zerovacuum expectation value afterelectroweak symmetry breaking,μ should be of the order of magnitude of theelectroweak scale, many orders of magnitude smaller than thePlanck scale (M_pl), which is the naturalcutoff scale. This brings about a problem of naturalness: Why is that scale so much smaller than the cutoff scale? And why, if theμ term in the superpotential has different physical origins, do the corresponding scale happen to fall so close to each other?

BeforeLHC, it was thought that thesoft 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 theGiudice–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 viaspontaneous breaking of this symmetry. This is proposed to happen together withF-termsupersymmetry breaking, with a spurious fieldX that parameterizes the hidden supersymmetry-breaking sector of the theory (meaning thatF_X is the non-zeroF-term).

Let us assume that theKahler potential includes a term of the form${\displaystyle \frac{X}{M_{\mathsf{p}\mathsf{l}}}{H}_{\mathsf{u}}{H}_{\mathsf{d}}}$ times some dimensionless coefficient, which is naturally of order one, and where M_pl isPlanck mass. Then as supersymmetry breaks,F_X gets a non-zero vacuum expectation value ⟨F_X⟩ and the following effective term is added to the superpotential:${\displaystyle \frac{⟨F_{\mathsf{X}}⟩}{M_{\mathsf{p}\mathsf{l}}}{H}_{\mathsf{u}}{H}_{\mathsf{d}},}$ which gives a measured${\displaystyle \mu =\frac{⟨F_{\mathsf{X}}⟩}{M_{\mathsf{p}\mathsf{l}}}.}$ On the other hand, soft supersymmetry breaking terms are similarly created and also have a natural scale of${\displaystyle \frac{⟨F_{\mathsf{X}}⟩}{M_{\mathsf{p}\mathsf{l}}}.}$

• NMSSM (Next-to-Minimal Supersymmetric Standard Model)
• Minimal Supersymmetric Standard Model
• Doublet–triplet splitting problem
• Hierarchy problem
• Little hierarchy problem

• Supersymmetric Models with extra singlets: a review; DJ Miller, University of Glasgow

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