Abstract
Using the method of covariant symbols we compute the divergent part of the effective action of the Proca field with non-minimal mass term. Specifically a quantum abelian vector field with a non-derivative coupling to an external tensor field in curved spacetime in four dimensions is considered. Relatively explicit expressions are obtained which are manifestly local but non polynomial in the external fields. Our result is shown to reproduce existing ones in all particular cases considered. Internal consistency with Weyl invariance is also verified.
Highlights
The difference between our calculation and that in [13] is that we use throughout the original metric gμν, with the exception of a term for which a different metric is clearly superior, and in any case just one metric is present in each single term of the final result
Instead of the heat kernel, we use the method of covariant symbols, which seems quite appropriate for this kind of problems
There we summarize the method of covariant symbols, which is already applied in that section for some of the terms
Summary
The goal is to obtain the divergent part of the effective action, div, of an abelian vector field Aμ(x) in curved spacetime. The kinetic term is gauge invariant, implying that fluctuations with large wavenumbers are not suppressed for the longitudinal polarization. To cope with this problem we follow [10] and apply Stückelberg’s method. Let us note the Weyl symmetry present in the action, namely S is invariant under the local rescaling gμν (x) → gμν (x) = 2(x) gμν (x), Mμν (x) → (M )μν (x) = −4(x) Mμν (x). This symmetry can be secured in the final result by using Weyl-invariant combinations, for instance This choice corresponds to the prescription det gμν = det gμν, where gμν stands for the inverse matrix of gμν. An exception is taken in the case of G since there the advantages of using gμν are overwhelming
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