Abstract
We derive the one-loop effective action for scalar, pseudoscalar, and electromagnetic fields coupled to a Dirac fermion in an extension of QED with Yukawa couplings. Using the Schwinger proper-time formalism and zeta-function regularization, we calculate the full nonperturbative effective action to one loop in the constant background field approximation. Our result is nonperturbative in the external fields, and goes beyond existing results in the literature which treat only the first nontrivial order involving the pseudoscalar. The result has an even and odd part, which are related to the modulus and phase of the fermion functional determinant. The even contribution to the effective action involves the modulus of the effective Yukawa couplings and is invariant under global chiral transformations while the odd contribution is proportional to the angle between the scalar and pseudoscalar couplings. In different limits the effective action reduces either to the Euler–Heisenberg effective action or the Coleman–Weinberg potential. We also comment on the relationship between the odd part of the effective action and the chiral anomaly in QED.
Highlights
The effective action provides rich insight into the low energy regime of an underlying quantum field theory
We find a much simpler way of calculating the even and odd contributions to the effective action which circumvents the need for computing the phase of the functional determinant directly
We have presented a derivation of the effective action for QED with Yukawa couplings in the one-loop and constant background field approximations
Summary
The effective action provides rich insight into the low energy regime of an underlying quantum field theory. The Fock–Schwinger proper-time approach and its generalizations, which appear in both perturbative and nonperturbative calculations of effective Lagrangians, are especially useful because they are symmetry-preserving and can be applied to theories involving the totally antisymmetric tensor μ1···μn .18. Proper-time techniques have been used to compute the radiatively induced effect of adding Lorentz- and CPT-violation to QED,[24,25,26,27] as well as the effective action for the Yukawa model in curved space–time.[28] In the nonperturbative regime, heat-kernel methods were used to obtain the worldline path integral for fermions with general scalar, pseudoscalar, and vector couplings.[29,30]. We provide a simple derivation of the effective action for fermions in the oneloop and constant background field approximations. The odd portion of the effective action is proportional to the CP-odd Lorentz scalar F Fand the angle between the effective scalar and pseudoscalar Yukawa terms
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