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Several crucial features of dynamical symmetry breakdown (that is, without elementary scalar fields) cannot be fully understood from calculations of anomalous dimensions and linearized symmetry-breaking Dyson equations. In particular, the usual criterion of positive anomalous dimensions for the symmetry-breaking operators allows arbitrary CP violation in the linearized theory. It is also impossible at this level to determine all the parameters of the effective (Ginzburg-Landau) Lagrangian for composite Higgs fields. Instead, one must study the nonlinearities of the effective action in the dressed-loop expansion. We first show that, because the loop expansion of the effective action (sum of connected vacuum graphs) deals with dressed lines and vertices (skeletons with no self-energy insertions), it is possible to order the expansion in powers of the renormalized coupling constant, even though inverse powers of g appear (signaling nonperturbative effects in the symmetry-breaking sector). Symmetric corrections to symmetry-breaking effects are ordered in positive powers of g . These results hold for both Abelian and non-Abelian gauge theories. Then we show that, for an Abelian gauge theory, CP violation does not occur at the level of two dressed loops in the effective action. We compute all masses and coupling constants appearing in he Ginzburg-Landau Lagrangian in terms of the coupling constants and masses of the symmetric theory, again at the two-loop level. A mass hierarchy appears, much like that of Coleman and Weinberg, in which the scalar mass is O ( g ) times the vector mass.
Physical Review D – American Physical Society (APS)
Published: Aug 15, 1978
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