Minimal Supersymmetric Standard Model

In the minimal supersymmetric standard model, the Higgs sector has two unknown parameters, usually taken to be tanβ ≡ v2/v1 and mA, the mass of its one physical pseudoscalar particle. By minimizing the minimum of the Higgs potential along a certain direction in parameter space, it is shown that mA = MZ + radiative correction, and if one further plausible assumption is made, tanβ > √ 3. If there is physics beyond the standard SU(2)× U(1) electroweak gauge model, supersymmetry is generally considered to be an excellent candidate. The minimal supersymmetric standard model[1] must then have two scalar doublets Φ1 = (φ + 1 , φ 0 1) and Φ2 = (φ + 2 , φ 0 2), with Yukawa interactions (u, d)LdRΦ1 and (u, d)LuRΦ̃2 respectively, where Φ̃2 = iσ2Φ ∗ 2 = (φ2,−φ2 ). Hence the mass matrix for quarks of charge −1/3 (i.e. d, s, and b) is proportional to the vacuum expectation value 〈φ1〉 ≡ v1, and that for quarks of charge 2/3 (i.e. u, c, and t) is proportional to 〈φ2〉 ≡ v2. There are two unknown parameters in the Higgs sector of this model. They are usually taken to be tanβ ≡ v2/v1 and mA, the mass of its one physical pseudoscalar particle. Numerous phenomenological studies[2] have been made using these as continuous variables. However, there is a plausible theoretical argument for mA = MZ at tree level and perhaps also tanβ > √ 3, as shown below. The Higgs potential of the minimal supersymmetric standard model is given by[3] V = m21Φ † 1Φ1 +m 2 2Φ † 2Φ2 −m12(Φ†1Φ2 + Φ†2Φ1) + 1 8 (g 1 + g 2 2) [ (Φ†1Φ1) 2 + (Φ†2Φ2) 2 ] − 1 4 (g 1 − g 2)(Φ†1Φ1)(Φ†2Φ2)− 1 2 g 2(Φ † 1Φ2)(Φ † 2Φ1), (1) where g1 and g2 are the U(1) and SU(2) gauge couplings respectively. Let V be broken spontaneously, then m21 −m12v2/v1 + 1 4 (g 1 + g 2 2)(v 2 1 − v 2) = 0, (2) and m22 −m212v1/v2 + 1 4 (g 1 + g 2 2)(v 2 2 − v 1) = 0. (3) Hence m21 +m 2 2 = m 2 12(tan β + cot β), (4) and v 1 + v 2 2 = −4m1 cos β + 4m22 sin β (g 1 + g 2 2)(cos 2 β − sin β) . (5)