A Topological Bound for Electroweak Vortices from Supersymmetry

We study the connection between N = 2 supersymmetry and a topological bound in a two-Higgsdoublet system having an SU(2) × U(1)Y × U(1)Y ′ gauge group. We derive Bogomol’nyi equations from supersymmetry considerations showing that they hold provided certain conditions on the coupling constants, which are a consequence of the huge symmetry of the theory, are satisfied. Supersymmetric Grand Unified Theories (SUSY GUTs) have attracted much attention in connection with the hierarchy problem in possible unified theories of strong and electroweak interactions [1, 2]. In view of the requirement of electroweak symmetry breaking, these models necessitate an enrichment of the Higgs sector [3], posing many interesting questions both from the classical and quantum point of view. In fact, many authors have explored the existence of stable vortex solutions in a variety of multi-Higgs systems [4, 5] which mimic the bosonic sector of SUSY GUTs, as it happens in the abelian Higgs model [6]. Vortices emerging as finite energy solutions of gauge theories can be usually shown to satisfy a topological bound for the energy, the so-called Bogomol’nyi bound [7]. Bogomol’nyi bounds were shown to reflect the presence of an extended supersymmetric structure [8]-[11] this requiring certain conditions on coupling constants where the central charge coincides with the topological charge. Being originated in the supercharge algebra, the bound is expected to be exact quantum mechanically. Since multi-Higgs models can be understood to be motivated by SUSY GUTs, Supersymmetry is a natural framework to investigate Bogomol’nyi bounds. We shall study, then, the supersymmetric generalization of the SU(2)×U(1)Y ×U(1)Y ′ model with two-Higgs first introduced in Ref.[5]. The theory has the same gauge group structure as that of supersymmetric extensions of the Weinberg-Salam Model that arise as low energy limits of E6 based Grand Unified or superstring theories. In spite of being a simplified model (in the sense that its Higgs structure is not so rich as that of Grand Unified theories), it can be seen as the minimal extension of the Standard Model necessary for having Bogomol’nyi equations. We show that the Bogomol’nyi bound of the model, as well as the Bogomol’nyi equations, are straight consequences of the requirement of N = 2 supersymmetry imposed on the theory. We also show explicitely that a necessary condition to achieve the N = 2 model implies certain relations between coupling constants that equal those found in [5] for the existence of a Bogomol’nyi bound. The SU(2)× U(1)Y × U(1)Y ′ gauge theory in 2 + 1, introduced in Ref.[5], is described by the action S = ∫ dx [ − 4 ~ Wμν · ~ W − 1 4 FμνF μν − 1 4 GμνG μν + 1 2 2 ∑ q=1 |D μ Φ(q)| − V (Φ(1),Φ(2)) ] (1) where Φ(1) and Φ(2) are a couple of Higgs doublets under the SU(2) factor of the gauge group, A and B are real scalar fields and ~ W = W τ is a real scalar in the adjoint representation of SU(2). The specific Consejo Nacional de Investigaciones Cient́ıficas y Técnicas. 1 form of the potential will be determined below. The strength fields can be written in terms of the gauge fields Aμ, Bμ and ~ Wμ. The covariant derivative is defined as: D μ Φ(q) = ( ∂μ + i 2 gW a μ τ a + i 2 α(q)Aμ + i 2 β(q)Bμ ) Φ(q), q=1,2 (2) where g is the SU(2) coupling constant while α(q) and β(q) represents the different couplings of Φ(q) with Aμ and Bμ. A minimal N = 1 supersymmetric extension of this model is given by an action which in superspace reads: SN=1 = 1 2 ∫