  We derive the decay rate of an unstable phase of a quantum field theory in
the presence of an impurity in the thin-wall approximation. This derivation is
based on the how the impurity changes the (flat spacetime) geometry relative to
case of pure false vacuum. Two examples are given that show how to estimate
some of the additional parameters that enter into this heterogeneous decay
rate. This formalism is then applied to the Higgs vacuum of the Standard Model
(SM), where baryonic matter acts as an impurity in the electroweak Higgs
vacuum. We find that the probability for heterogeneous vacuum decay to occur is
suppressed with respect to the homogeneous case. That is to say, the
conclusions drawn from the homogeneous case are not modified by the inclusion
of baryonic matter in the calculation. On the other hand, we show that Beyond
the Standard Model physics with a characteristic scale comparable to the scale
that governs the homogeneous decay rate in the SM, can in principle lead to an
enhanced decay rate.
