Massive field contributions to the QCD vacuum tunneling amplitude

For the one-loop contribution to the QCD vacuum tunneling amplitude by quarks of generic mass value, we make use of a calculational scheme exploiting a large mass expansion together with a small mass expansion. The large mass expansion for the effective action is given by a series involving higher-order Seeley-DeWitt coefficients, and we carry this expansion up to order $1/(m\rho)^8$, where $m$ denotes mass of the quark and $\rho$ the instanton size parameter. For the small mass expansion, we use the known exact expression for the particle propagation functions in an instanton background and evaluate explicitly the effective action to order $(m\rho)^2$. A smooth interpolation of the results from both expansions suggests that the quark contribution to the instanton tunneling amplitude have a relatively simple $m\rho$-dependent behavior.