On the fulfillment of Ward identities in the functional renormalization group approach

I consider the fulfillment of conservation laws and Ward identities in the one- and two-loop functional renormalization group approach. It is shown that in a one-particle irreducible scheme of this approach Ward identities are fulfilled only with the accuracy of the neglected two-loop terms O(V^3) at one-loop order, and with the accuracy O(V^4) at two-loop order (V is the effective interaction vertex at scale \Lambda). The one-particle self-consistent version of the two-loop RG equations which leads to smaller errors in Ward identities due to the absence of the terms with non-overlapping loops, is proposed. In particular, these modified equations exactly satisfy Ward identities in the ladder approximation.