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The fermion determinant in an instanton background for a quark field of arbitrary mass is determined exactly using an efficient numerical method to evaluate the determinant of a partial-wave radial differential operator. The bare sum over partial waves is divergent but can be renormalized in the minimal subtraction scheme using the result of WKB analysis of the large partial-wave contribution. Previously, only a few leading terms in the extreme small and large mass limits were known for the corresponding effective action. Our approach works for any quark-mass and interpolates smoothly between the analytically known small and large mass expansions.

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The precise quark mass dependence of the one-loop effective action in an instanton background has recently been computed [1]. The result interpolates smoothly between the previously known extreme small and large mass limits. The computational method makes use of the fact that the single instanton background has radial symmetry, so that the computation can be reduced to a sum over partial waves of logarithms of radial determinants, each of which can be computed numerically in an efficient manner. The bare sum over partial waves is divergent and must be regulated and renormalized. In this paper we provide more details of this computation, including both the renormalization procedure and the numerical approach. We conclude with comparisons of our precise numerical results with a simple interpolating function that connects the small and large mass limits, and with the leading order of the derivative expansion.

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