Orbifold-induced μ term and electroweak symmetry breaking

It is known that a Higgs μ term can be naturally generated through the Kähler potential in orbifold string models in which one of the three compactified complex planes has order two. In this class of models explicit expressions for both the μ parameter and the soft SUSY-breaking parameters can be obtained under the assumption that the goldstino is an arbitrary linear combination of the fermionic partners of the dilaton S and all the moduli Ti, Ui. We apply this picture to the MSSM and explore the consistency of the obtained boundary conditions with radiative gauge symmetry breaking. We find that consistency with the measured value of the top-quark mass can only be achieved if the goldstino has a negligible dilatino component and relevant components along the T3, U3 moduli associated to the order-two complex plane. CERN-TH/96-183 July 1996 1. The Minimal Supersymmetric extension of the Standard Model (MSSM) contains two kinds of mass terms: a set of soft supersymmetry-breaking terms, including scalar and gaugino mass terms, and a globally supersymmetric Higgs mass term, the so-called μ term. If SUSY breaking originates from a super-Higgs mechanism in an underlying supergravity theory, the gravitino gets a mass m3/2 and soft parameters O(m3/2) are usually generated. In addition, if the supergravity Kähler potential contains appropriate terms bilinear in the Higgs fields, an effective μ parameter O(m3/2) can be generated as well [1]. Although not unique, this way to generate a μ term is particularly attractive because μ becomes directly related to SUSY breaking and thus stands essentially on the same footing as the other mass parameters. In the restricted and motivated class of effective supergravity theories corresponding to 4-D superstring compactifications, specific patterns of soft terms emerge under the assumption that SUSY breaking is due to non-vanishing F -components for the dilaton (S) and moduli (Ti, Ui) fields [2]–[7]. Such an assumption implies that the goldstino is a linear combination of the fermionic partners of the dilaton and the moduli, the coefficients of the combination just measuring the relative contribution of each field to SUSY breaking. We recall that, in the case of orbifold models, the set of Kähler moduli Ti always includes the three diagonal Kähler moduli T1,T2, T3 associated to the three compactified complex planes, whereas the number of complex structure moduli Ui can be at most three. Here we will follow the approach of [4, 7], where the soft parameters are expressed in terms of the gravitino mass m3/2, the angles specifying the (free) goldstino direction and the modular weights of the matter fields. However, we recall that some ambiguity affects the results for the μ parameter and the associated soft B parameter, depending on the source of the μ term itself. In this respect, an interesting and predictive class of models consists of orbifold models in which one of the three compactified complex planes (the third, say) has order two. We will focus on such a class of models and denote by T3 and U3 the Kähler and complex structure moduli associated to that plane. It was found in [8, 9] that the Kähler potential corresponding to T3 and U3, and to charged untwisted fields C1, C2 in conjugate representations associated to the same plane, has the form K = − log [(T3 + T ∗ 3 )(U3 + U∗ 3 )− (C1 + C∗ 2)(C∗ 1 + C2)] . (1) After SUSY breaking, a certain effective μ term [9] is induced for the fields C1 and C2, which we will identify here with the electroweak Higgs fields. Under the assumption that this is the main source of the μ term and that SUSY breaking is dilaton/moduli-dominated, simple expressions for μ and B in terms of goldstino angles can be derived [7]. Therefore a quite predictive scenario is obtained, with explicit and correlated expressions for the full set of mass parameters. The aim of the present letter is to apply such a scenario to the MSSM and check its consistency with radiative electroweak symmetry breaking and the constraints on the top quark mass. 2. In the following we will assume that the MSSM can be obtained from a string model of the kind mentioned above and regard the resulting tree-level expressions for the soft and μ parameters as boundary conditions given at some high scale MX . We will then use one-loop RGEs to evolve the parameters down to the electroweak scale mZ , where we will impose the standard requirement of electroweak symmetry breaking and evaluate the top mass. Such ‘leading-order’ procedure, which neglects e.g. highand low-energy threshold corrections, will be sufficient for our purpose.