Can the Supersymmetric Μ Parameter Be Generated Dynamically without a Light Singlet?

It is generally assumed that the dynamical generation of the Higgs mass parameter of the superpotential, μ, implies the existence of a light singlet at or below the supersymmetry breaking scale, MSUSY. We present a counter-example in which the singlet field can receive an arbitrarily heavy mass (e.g., of the order of the Planck scale, MP ≈ 1019 GeV). In this example, a non-zero value of μ is generated through soft supersymmetry breaking parameters and is thus naturally of the order of MSUSY. The cancellation of quadratic divergences in the unrenormalized Green functions is one of the main motivations of supersymmetry (SUSY). It stabilizes any mass scale under radiative corrections and thus allows the existence of different mass scales such as the electroweak scale, given by the Z boson mass, mz, and the Planck scale, MP. The minimal supersymmetric standard model (MSSM) is the most popular model of this kind due to its minimal particle content [1]. In this model, the SU(2)L⊗U(1)Y symmetry breaking is driven by soft SUSY breaking parameters. Thus, the SUSY breaking scale, MSUSY, has to be at or slightly above mz. For this mechanism to work it is also necessary that the SUSY Higgs mass parameter, |μ| < ∼ MSUSY. This parameter also determines the chargino and neutralino mass spectrum. From here one can deduce a experimental lower bound from LEP experiments of |μ| >∼ mz/4 independent of tanβ [2]. The fact that in the MSSM the μ-parameter, which is a priori arbitrary, has to lie within the narrow range 1 4mz < ∼ |μ| < ∼ MSUSY , (1) has been considered a problem of fine-tuning. Possible attempts to try and solve this problem are the inclusion of gravitational couplings [3] or the introduction of additional fields [4]. The introduction of a singlet, N1, under the SU(3)c⊗SU(2)L⊗U(1)Y standard model (SM) gauge group is the most economical extension of the MSSM in which eq. (1) is natural [4]. Here, the μ parameter is generated dynamically: μ = λ 〈N1〉 6 = 0, where 〈N1〉 is the vacuum expectation value (VEV) of N1. In this model, one can achieve that the potential vanishes in the direction of N1 in the SUSY limit by imposing a discrete symmetry. If one includes soft SUSY breaking terms then N1 acquires a VEV and a mass of order of MSUSY. Thus, one inevitable consequence of this mechanism is the presence of a singlet field under the SM gauge group in the low energy theory. This leads to a severe loss of predictability of the SUSY Higgs sector. In particular, the MSSM prediction mh0 ≤ mz + radiative corrections , (2)