Abstract
We calculate the divergent part of the one-loop effective action in curved spacetime for a particular class of second-order vector field operators with a degenerate principal part. The principal symbol of these operators has the structure of a longitudinal projector. In this case, standard heat-kernel techniques are not directly applicable. We present a method which reduces the problem to a nondegenerate scalar operator for which standard heat-kernel techniques are available. Interestingly, this method leads to the identification of an effective metric structure in the longitudinal sector. The one-loop divergences are compactly expressed in terms of invariants constructed from this metric.
Highlights
Perturbative calculations in quantum field theory, especially in curved spacetime, are efficiently performed by a combination of the background field method and the heatkernel technique [1,2,3,4,5,6,7,8,9,10,11,12,13]
The principal part of the associated fluctuation operator is degenerate; the gauge symmetry implies that the total operator is degenerate
By a formal manipulation of the vector trace (11), the calculation could be reduced to the evaluation of the functional trace for a minimal second-order scalar operator (22)
Summary
Perturbative calculations in quantum field theory, especially in curved spacetime, are efficiently performed by a combination of the background field method and the heatkernel technique [1,2,3,4,5,6,7,8,9,10,11,12,13]. For the minimal second-order operator, a closed algorithm for the calculation of the one-loop divergences, proposed by DeWitt, is available [1]. In many cases the gauge freedom is sufficient to choose a particular simple, minimal gauge In these cases, the generalized Schwinger-DeWitt technique becomes applicable again. There are, many interesting models—gauge as well as nongauge theories—which lead to fluctuation operators for which the degeneracy of the principal part cannot be removed This happens, for example, in softly broken gauge theories where no gauge fixing is available and in higher-derivative theories, where only some of the degrees of freedom (d.o.f.) are propagating with higher derivatives. A systematic classification of vector field operators according to their degeneracy structure has been developed in Ref. V and give a brief outline of future generalizations of the obtained results
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