Mu problem

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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 responsible for the fact that both H_u and H_d gets a non-zero vacuum expectation value after electroweak symmetry breaking, so that all quarks and leptons can get masses. It should therefore be of the order of magnitude of the electroweak scale, many orders of magnitude smaller than that of 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 yet happen to fall so close to each other?

One proposed solution is that this term does not appear explicitly in the Lagrangian, because it violates some U(1) symmetry, and can therefore be created only via spontaneous symmetry breaking of this symmetry. This is proposed to happen together with F-term supersymmetry breaking, with the spurious field X (meaning that F_X is the non-zero F-term). Let us assume that the Kahler potential includes a term of the form ${X \over M_{pl}} H_u H_d$ 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: ${<F_X> \over M_{pl}} H_u H_d$ , which gives a measured $\mu = {<F_X> \over M_{pl}}$ . On the other hand, soft supersymmetry breaking term are similarly created and also have a natural scale of ${<F_X> \over M_{pl}}$ .