  The fermion determinant in an instanton background for a quark field of
arbitrary mass is studied using the Schwinger proper-time representation with
WKB scattering phase shifts for the relevant partial-wave differential
operators. Previously, results have been obtained only for the extreme small
and large quark mass limits, not for intermediate interpolating mass values. We
show that consistent renormalization and large-mass asymptotics requires up to
third-order in the WKB approximation. This procedure leads to an almost
analytic answer, requiring only modest numerical approximation, and yields
excellent agreement with the well-known extreme small and large mass limits. We
estimate that it differs from the exact answer by no more than 6% for generic
mass values. In the philosophy of the derivative expansion the same amplitude
is then studied using a Heisenberg-Euler-type effective action, and the leading
order approximation gives a surprisingly accurate answer for all masses.
