A Simplified Scheme for GUT-inspired Theories with Multiple Abelian Factors

Grand Unified Theories often involve additional Abelian group factors apart from the standard model hypercharge, that generally lead to loop-induced mixing gauge kinetic terms. In this letter, we show that at the one-loop level this effect can be avoided in many cases by a suitable choice of basis in group space and present a general scheme for the construction of this basis. In supersymmetric theories however, a residual mixing in the soft SUSY breaking gaugino mass terms may appear. We generalize the renormalization group equations for the gaugino mass terms to account for this effect. In a further calculation we also present the necessary adjustments in the renormalization group equations of the trilinear soft breaking couplings and the soft breaking scalar mass squares. [email protected] [email protected] The renormalization group equations (RGEs) describe the dependence of the coupling constants on the choice of the renormalization scale μ, which is commonly translated into an energy dependence, as the perturbative series usually converges best if one chooses μ to be of the order of the characteristic energy scale of a given process. In (supersymmetric) grand unified model building [1,2,3,4,5,6] these equations constitute the framework which is employed to derive the potential unification of the gauge interactions into one fundamental force. They also describe the evolution of all other Lagrangian parameters (including the soft supersymmetry breaking parameters mediated to the “visible” sector through some mechanism at high scales) from a high unification scale down to the energy scales accessible to current collider experiments. The RGEs for a (supersymmetric) model with an arbitrary semi-simple gauge group augmented by at most one U(1) gauge group were given in [7,8,9,10] ([11,12,13,13]). We present a way of treating the case with several Abelian gauge groups, including a consistent generalization of the one-loop SUSY RGEs from [13] in the case, where a mixing of gauge kinetic terms at the tree-level does not occur, i.e. κF i μν F μν,j = 0 ∀i 6= j . (1) In this situation we will show that the general concept presented in [14,15] which has recently been applied to the complete set of SUSY two-loop RGEs in [16], where an additional coupling parametrizing the mixing is introduced, can be simplified considerably at the one-loop level by an appropriate choice of basis for the U(1) groups. First, we specify the type of models in which condition (1) holds, i.e. where our formalism is applicable: Consider some potentially multi-scale symmetry breaking scenario: G (0) N Λ → G (1) N−d ∣ ∣ ∣ d≥3 × U(1) → SM, (2) where G (0) N denotes a simple 1 Lie group of rank N and G (i) N ′ an arbitrary semi-simple non-Abelian subgroup of it. This implies, that condition (1) holds at Λ for all U(1) groups in G (1) N−d × U(1) 2, as they all originate from non-Abelian gauge multiplets above Λ. A term as in Eq. (1) would have to necessarily arise from a matching condition at scale Λ κGμν,1 G μν 2 → κ ′Fμν,1 F μν 2 , (3) with Gμν being a non-Abelian field-strengh tensor, which istself is not gauge invariant. So the left-hand side of Eq. (3) is forbidden by the gauge symmetry. Our argumentation holds if there are intermediate symmetry breaking steps above Λ, with arbitrary semi-simple gauge groups, as long as the rank N is preserved and the matter content still fills complete multiplets of G (0) N . At the tree-level, there cannot appear mixing terms in the course of symmetry 1 G (0) N does not necessarily have to be simple as long as the particle content can be assembled into complete multiplets of a simple Lie group and there are no mixing gauge kinetic terms at the tree-level.