Abstract
We revisit the semi-classical calculation of the size distribution of instantons at finite temperature in non-abelian gauge theories in four dimensions. The relevant functional determinants were first calculated in the seminal work of Gross, Pisarski and Yaffe and the results were used for a wide variety of applications including axions most recently. In this work we show that the uncertainty on the numerical evaluations and semi-analytical expressions are two orders of magnitude larger than claimed. As a result various quantities computed from the size distribution need to be reevaluated, for instance the resulting relative error on the topological susceptibility at arbitrarily high temperatures is about 5% for QCD and about 10% for SU(3) Yang-Mills theory. With higher rank gauge groups this discrepancy is even higher. We also provide a simple semi-analytical formula for the size distribution with absolute error 2 · 10−4. In addition we also correct the over-all constant of the instanton size distribution in the overline{mathrm{MS}} scheme which was widely used incorrectly in the literature if non-trivial fermion content is present.
Highlights
It is useful to compare the non-perturbative lattice results both in pure Yang-Mills and in full QCD with the semi-classical results
In addition we correct the over-all constant of the instanton size distribution in the MS scheme which was widely used incorrectly in the literature if non-trivial fermion content is present
In this work we report the correct zero temperature expression in the MS-scheme which was frequently used incorrectly once light fermions were included and secondly we report that the factor responsible for the temperature dependence which was numerically evaluated in [19] has an uncertainty that is two orders of magnitude larger than claimed
Summary
We will consider SU(N ) gauge theory with Nf flavors of light fermions in the fundamental representation. Note that the constant C in principle depends on mi given by the contribution of the non-zero mode fermion determinant [23,24,25,26] but the massless limit can be taken for the asymptotically high temperature limit. In the present paper we show that the actual numerical accuracy of (2.9) is two orders of magnitude larger than claimed in [19] at around λ ∼ O(1) leading to about a 5% mismatch for the topological susceptibility in QCD or 10% mismatch for SU(3) pure Yang-Mills. Before calculating A(λ) accurately in section 4 we make a comment on the constant C appearing in (2.5) It turns out its flavor number dependence in the MS scheme was widely used incorrectly in the literature, especially recently in axion mass estimates as well as comparisons with lattice results
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