Understanding the Fierz Ambiguity of Partially Bosonized Theories

A useful tool in non perturbative studies of fermionic theories is partial bosonization. However, partial bosonization is often connected to an ambiguity due to Fierz rearrangement in the original theory. We discuss two different approximations for the calculation of the effective action $\mathrm{\Gamma }$ with respect to a spurious dependence on the choice of Fierz transformation: Mean field theory and the truncated flow of an exact renormalization group equation for the effective average action.