Instanton Determinant with Arbitrary Quark Mass: WKB Phase-shift Method   and Derivative Expansion
  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. 
Gerald V. Dunne, Jin Hur, Choonkyu Lee and Hyunsoo Min