Effective-potential study of symmetry breaking in scalarless SU(2) x U(1)

We derive the effective potential as a functional of composite operators for massless, scalarless SU(2) x U(1); and we obtain conditions under which this effective potential is the vacuum energy density for the case of composite operators which are nonlocal in time. General considerations of the number of Goldstone bosons prevent the construction of a realistic model, but we are able to study dynamical symmetry breaking for various unrealistic spectra of physical particles. We first use the standard linearized approximation (LA) to solve the equations obtained for the propagators from the effective potential, and we compare these results to the most-attractive-channel hypothesis. These LA solutions reproduce the standard vector-boson mixing and in addition yield a (reasonable) relation between the vector masses and those of the fermions. However, the linearized equations are also satisfied by the symmetric (massless) solution. In order to determine which solution corresponds to the true vacuum, we use the effective potential in a variational calculation. The linearized-approximation solutions are used to determine the functional forms of the propagators, and the physical masses are treated as variational parameters in minimizing the effective potential. In an Abelian approximation, in which the effects of the vector self-couplings are absent, the mass relations of the LA survive the inclusion of nonlinear effects. On the other hand, if a Hartree-Fock approximation is used for the vector self-couplings, all the desirable features of the LA are lost: The breaking is no longer I = (1/2), and the relation between vector and fermion masses requires fermions much heavier than the vectors (or very large numbers of lighter fermions). In either approximation, whether spontaneous symmetry breaking occurs depends on the number and quantum numbers of the fermions. Significance of the results and possible future directions are discussed