Mu problem

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

The supersymmetric Higgs mass parameter μ appears as the following term in the superpotential: μH_uH_d. It is necessary to provide a mass for the fermionic superpartners of the Higgs bosons, i.e. the higgsinos, and it enters as well the scalar potential of the Higgs bosons. To ensure that H_u and H_d get a non-zero vacuum expectation value after electroweak symmetry breaking, μ should be of the order of magnitude of the electroweak scale, many orders of magnitude smaller than the Planck scale, which is the natural cutoff scale.

The soft supersymmetry breaking terms should also be of the same order of magnitude as the electroweak scale. This brings about a problem of naturalness: why are these scales so much smaller than the cutoff scale? And why, if the soft supersymmetry breaking terms and the μ term in the superpotential have different physical origins, do the corresponding scales happen to fall so close to each other?

One proposed solution, known as the Giudice-Masiero mechanism, 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 of this symmetry. This is proposed to happen together with F-term supersymmetry breaking, with a spurious field X that parameterizes the hidden supersymmetry-breaking sector of the theory (meaning that F_X is the non-zero F-term). Let us assume that the Kahler potential includes a term of the form ${\displaystyle {X \over M_{pl}}H_{u}H_{d}}$ times some dimensionless coefficient which is naturally of order one where M_pl is Planck 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: ${\displaystyle {\langle F_{X}\rangle \over M_{pl}}H_{u}H_{d}}$, which gives a measured ${\displaystyle \mu ={\langle F_{X}\rangle \over M_{pl}}}$. On the other hand, soft supersymmetry breaking terms are similarly created and also have a natural scale of ${\displaystyle {\langle F_{X}\rangle \over M_{pl}}}$.

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