On the High T Phase Transition in the Gauged SU (2) Higgs Model

We study the effective field theory of a weakly coupled 3+1d gauged φ type model at high temperature. Our model has 4N real scalars (N complex Higgs doublets) and a gauge group SU(2) which is spontaneously broken by a nonzero scalar field vev at zero temperature. We find, for sufficiently large N , that the transition from the high temperature symmetry restoring phase to the low temperature phase can be either first order or second order depending on the ratio of the gauge coupling to the scalar self coupling. Recently there has been much interest in the nature of the high temperature electroweak phase transition. It has been known for some time that at high enough temperature the ground state of the electroweak model is symmetry preserving [1], even if the zero temperature ground state is not. What is not well understood is how the transition from the high temperature phase to the low temperature phase proceeds. If the phase transition is first order, it may be possible to generate the baryon asymmetry of the universe (BAU) within the electroweak model itself [2],[3],[4], It is currently believed that the phase transition is first order, but too weakly first order for the purposes of generating the BAU [6],[7]. A one–loop high temperature calculation suggests that the electroweak phase transition is first order [8]. However, the high temperature field theory suffers from infrared divergence problems. A partial resummation of higher loop contributions implies that the strength of the first order transition is reduced relative to the one–loop result [9],[6], [7],[10]. In this letter we would like to point out that higher loop contributions may not only reduce the strength of the first order transition but actually result in a second order phase transition. We study not the full electroweak theory but a weakly coupled gauged φ model. The gauge group is SU(2) and we work with 4N real scalars (N complex Higgs doublets). Each of the SU(2) Higgs doublets is coupled to the same gauge fields. Our analysis follows [11]. The 3+1d model is, at high temperature, formally equivalent to a euclidean field theory with one compact dimension. For physics at scales less than O(T ) it is sufficient to study the effective 3d Lagrangian that results from integrating out the compact dimension. Vacuum polarization effects can be computed exactly in three dimensions, in contrast to four dimensions. Furthermore, the explicit φ term can be removed by introducing a dimension two auxiliary field χ [12], [13],[14]. The auxiliary field allows for a straightforward computation of all the quantum corrections that survive in the limit of arbitrarily large N . The corrections that survive depend on how the gauge coupling and scalar self coupling are held fixed as N increases. We find that if g/λ ≤ O(1), where g is the 4d gauge coupling and λ is the scalar self–coupling, the effective scalar potential Veff admits only a 2nd order phase transition at large N . This is because vacuum polarization effects at nonzero external momentum prevent any Tφ term from appearing in Veff . This is in contrast to the case g /λ ∼ O(N) for which a first order