Predictions of the sign of μ from supersymmetry breaking models

The sign of the supersymmetric Higgs mass μ is usually taken as an independent input parameter in analyses of the supersymmetric standard model. I study the role of theories of supersymmetry breaking in determining the sign of μ as an output. Models with vanishing soft scalar couplings at the apparent gauge coupling unification scale are known to predict positive μ. I investigate more general results for the sign of μ as a function of the holomorphic soft scalar couplings, and compare to predictions of models with gaugino mass dominance at higher scales. In a significant region of the B0/m1/2 versus A0/m1/2 plane including A0 = B0 = 0, μ must be positive. In another region, μ is definitely negative. Only in a smaller intermediate region does knowledge of the supersymmetry breaking mechanism not permit a definite prediction of the sign of μ. The last region will shrink considerably as the top quark mass becomes more accurately known. In the minimal supersymmetric standard model (MSSM) [1, 2], the Higgs mass term μ is the only coupling which does not explicitly break supersymmetry that has not already been directly measured by experiment. Nevertheless, in phenomenological treatments of supersymmetric models, it is usual to treat |μ| as an output rather than an input parameter, because it can be fixed in terms of the other parameters from our knowledge of the electroweak scale. However, this condition alone does not address the phase of μ, which is left unfixed by the conditions of electroweak symmetry breaking (EWSB). The lack of observed CP violation in the electric dipole moments of the neutron and electron requires that large relative phases in the MSSM lagrangian must either be absent or aligned to rather particular values. Barring the latter possibility, it follows that all gaugino masses should be (at least nearly) relatively real, and that with appropriately chosen phase conventions μ is real and the phases of scalar cubic couplings are equal to their Yukawa coupling counterparts. The remaining discrete phase freedom sign(μ) is therefore usually regarded as an independent input parameter. However, if the mechanism of supersymmetry breaking is known, the phase of μ including its sign is often determined purely from the theory and knowledge of already-measured dimensionless supersymmetry-preserving couplings. This has been noted before in the contexts of flipped SU(5)×U(1) no-scale supergravity models [4] and in gauge-mediated supersymmetry breaking models[13]-[16]. More generally, a complete model of supersymmetry breaking should predict boundary conditions for all soft parameters in terms of supersymmetric parameters. This implies that, under many (but not all!) circumstances, the sign of μ should properly be regarded as an output prediction rather than an input assumption. Conversely, an experimental determination of the sign of μ will provide a non-trivial test of different models of supersymmetry breaking. In this paper I will study the ability of flavor-preserving high-scale theories of supersymmetry breaking to predict the sign of μ, and consider under what circumstances such a prediction can be made unambiguously. In this paper, it is assumed that the gaugino mass parameters M1, M2, and M3 indeed have the same phase, so that they can be taken real and positive without loss of generality. To fix conventions explicitly, the tree-level neutral Higgs potential is given by V = (|μ| +m2Hu)|H0 u| + (|μ| +m2Hd)|H 0 d | − (bH uH0 d + c.c.) + 1 8 (g + g)(|H u| − |H d |), (1) Here b is the holomorphic soft supersymmetry-breaking Higgs squared mass parameter. (Other common notations in the literature for this term are Bμ and m212 and m 2 3.) Without loss of generality, a suitably renormalized b is taken to be real and positive at a renormalization group (RG) scale near or below 1 TeV, to fulfill the condition that at the minimum This would follow, for example, in GUT models in which all gaugino masses are unified.