  We derive fermionic Green's functions in the background of the Euclidean
solitons describing false vacuum decay in a prototypal Higgs-Yukawa theory. In
combination with appropriate counterterms for the masses, couplings and
wave-function normalization, these can be used to calculate radiative
corrections to the soliton solutions and transition rates that fully account
for the inhomogeneous background provided by the nucleated bubble. We apply
this approach to the archetypal example of transitions between the
quasi-degenerate vacua of a massive scalar field with a quartic
self-interaction. The effect of fermion loops is compared with those from
additional scalar fields, and the loop effects accounting for the spacetime
inhomogeneity of the tunneling configuration are compared with those where
gradients are neglected. We find that scalar loops lead to an enhancement of
the decay rate, whereas fermion loops lead to a suppression. These effects get
relatively amplified by a perturbatively small factor when gradients are
accounted for. In addition, we observe that the radiative corrections to the
solitonic field profiles are smoother when the gradients are included. The
method presented here for computing fermionic radiative corrections should be
applicable beyond the archetypal example of vacuum decay. In particular, we
work out methods that are suitable for calculations in the thin-wall limit, as
well as others that take account of the full spherical symmetry of the
solution. For the latter case, we construct the Green's functions based on spin
hyperspherical harmonics, which are eigenfunctions of the appropriate angular
momentum operators that commute with the Dirac operator in the solitonic
background.
