A natural solution for the μ problem with anomalous U ( 1 ) A gauge symmetry

Recently we proposed an attractive scenario of grand unified theories with anomalous U(1)A gauge symmetry, in which doublet-triplet splitting is naturally realized in SO(10) unification using Dimopoulos-Wilczek mechanism and realistic quark and lepton mass matrices can be obtained in a simple way. In this paper we show that there is a mechanism in which the doublet Higgs obtains the supersymmetric mass which is proportional to the SUSY breaking parameters. This mechanism can be applied easily in the above scenario. The point is that the mass term, which is forbidden by SUSY zero mechanism, can be induced by SUSY breaking. The proportional coefficient is controlled by the anomalous U(1)A charges. e-mail: [email protected] Recently we proposed an attractive scenario of supersymmetric (SUSY) grand unified theories (GUTs) with anomalous U(1)A gauge symmetry, in which doublettriplet splitting is naturally realized in SO(10) unification using DimopoulosWilczek mechanism, and realistic quark and lepton mass matrices can be obtained in a simple way [1]. In the scenario, the mass term of the Higgs field is forbidden by the holomorphy ( SUSY zero mechanism). This is because the anomalous U(1)A charge of the Higgs field is taken to be negative. However, in order to give a mass to higgsino, the SUSY Higgs mass term is required. The SUSY Higgs mass μ must be of oder of the weak scale, namely, the SUSY breaking scale. This is a mystery in the minimal SUSY standard model, because at a glance we have no reason that the SUSY parameter becomes the same order of the SUSY breaking parameters. This is called the μ problem [2]. In the super gravity scenario, there are several natural solutions for the μ problem [3, 4] by using non-renormalization operator in the Kähler potential (Giudice and Masiero [3]) or in the superpotential (Casas and Muños [4]). However, if the Higgs mass term in tree level is forbidden by some symmetry as in our scenario, these mechanisms for the μ problem do not work well, though R-symmetry can be an exceptional one. Since our model has no R-symmetry, it is important to examine other mechanisms to induce the supersymmetric Higgs mass term related with the SUSY breaking scale. There are several other attempts [5, 6, 7, 8, 9] to solve the μ problem. One of them is to introduce a light singlet which couples with the Higgs doublet [5, 7]. The vacuum expectation value (VEV) of the singlet field can become of order of the SUSY breaking scale, so suitable μ parameter is induced. In addition to the problem that the induced μ parameter is unstable under radiative correction of heavy particle and non-renormalizable terms [6], however, in our scenario, it is not easy to introduce a light singlet S with positive charge (the positive charge is required for the singlet to couple to Higgs field), because the mass term of the singlet field with positive charge is not forbidden. In this paper, we examine a mechanism which solves the μ problem. The mechanism can be naturally applied to our scenario. The generated Higgs mass μ is proportional to the SUSY breaking parameters and the coefficient is controlled by anomalous U(1)A charges. The point is that since the Higgs mass term is forbidden by the SUSY zero mechanism, when SUSY is broken, the μ term must be induced. Note that if all the SUSY breaking parameter become zero, the μ term must vanish. Therefore the μ parameter must be proportional to the SUSY breaking parameters. Since all the coefficients are controlled by anomalous U(1)A charges, the proportional coefficient is also determined by the anomalous U(1)A charges. Let us recall the SUSY zero mechanism. First of all, we assume that Dflatness condition of the anomalous U(1)A gauge symmetry leads to 〈Φ〉 = λMP , because DA = gA 2 (λM P − |Φ|2), where Φ has negative anomalous U(1)A charge φ = −1 and λ < 1 (Actually we usually adopt λ ∼ 0.2 for reproducing the