Abstract
We study renormalizability aspects of the spectral action for the Yang–Mills system on a flat 4-dimensional background manifold, focusing on its asymptotic expansion. Interpreting the latter as a higher-derivative gauge theory, a power-counting argument shows that it is superrenormalizable. We determine the counterterms at one-loop using zeta function regularization in a background field gauge and establish their gauge invariance. Consequently, the corresponding field theory can be renormalized by a simple shift of the spectral function appearing in the spectral action.This manuscript provides more details than the shorter companion paper, where we have used a (formal) quantum action principle to arrive at gauge invariance of the counterterms. Here, we give in addition an explicit expression for the gauge propagator and compare to recent results in the literature.
Highlights
Noncommutative geometry [9] has been shown to be capable of describing Yang–Mills theories on the classical level, which further extends to the full Standard Model of highenergy physics [8]
Results from [12,13,14,4,15] on BRST-cohomology for Yang–Mills type theories ascertain that the only BRST-closed functional of order 4 or less in the fields is represented by δ Z Fμν F μν for some constant δ Z. We will confirm this through an explicit calculation using zeta function regularization in background field gauge in the remaining part of this paper
By naive power counting we found that this higher-derivative field theory is superrenormalizable
Summary
Noncommutative geometry [9] has been shown to be capable of describing Yang–Mills theories on the classical level, which further extends to the full Standard Model of highenergy physics [8]. D. van Suijlekom (BRST-invariance of the one-loop effective action) Since this is the only counterterm needed, and is proportional to the Yang–Mills action, this establishes renormalizability of the corresponding gauge field theory. We use zeta function regularization in a background field gauge – exploiting the explicit forms for the heat invariants for higher-order Laplacians derived by [17] – to determine in Sect. The authors in [21] consider the spectral action S[A] as defined in Eq (1) (without expanding in ) This defines a different – non-local – field theory, with correspondingly different large momentum behaviour
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