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arXiv:hep-ph/0205154

Understanding the Fierz Ambiguity of Partially Bosonized Theories

Joerg Jaeckel

Published May 15, 2002• 7 citations
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Abstract

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 $Γ$ 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.

DOI
10.48550/arxiv.hep-ph/0205154

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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 ΓΓΓ 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.
We compare different versions of a bosonic description for systems of interacting fermions, with particular emphasis on the free-energy functional. The bosonic effective action makes the issue of symmetries particularly transparent. A systematic expansion for the bosonic effective action starts with a solution to the lowest-order Schwinger-Dyson or gap equation. We propose a two-particle irreducible formulation of an exact functional renormalization group equation for computations beyond leading order. On this basis we suggest a renormalized gap equation. This approach is compared with functional renormalization in a partially bosonized setting.
The application of the exact renormalization group to a many-fermion system with a short-range attractive force is studied. We assume a simple ansatz for the effective action with effective bosons, describing pairing effects and derive a set of approximate flow equations for the effective coupling including boson and fermionic fluctuations. The phase transition to a phase with broken symmetry is found at a critical value of the running scale. The mean-field results are recovered if boson-loop effects are omitted. The calculations with two different forms of the regulator were shown to lead to similar results.
We compare different methods used for nonperturbative calculations in strongly interacting fermionic systems. Mean field theory often shows a basic ambiguity related to the possibility to perform Fierz transformations. The results may then depend strongly on an unphysical parameter which reflects the choice of the mean field, thus limiting the reliability. This ambiguity is absent for Schwinger-Dyson equations or fermionic renormalization group equations. Also renormalization group equations in a partially bosonized setting can overcome the Fierz ambiguity if the truncation is chosen appropriately. This is reassuring since the partially bosonized renormalization group approach constitutes a very promising basis for the explicit treatment of condensates and spontaneous symmetry breaking even for situations where the bosonic correlation length is large.