Fulfillment of Ward identities in the functional renormalization group approach

We 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_Λ³) at one-loop order, and with the accuracy O(V_Λ⁴) at two-loop order (V_Λ is the effective interaction vertex at scale Λ). The one-particle self-consistent version of the two-loop renormalization group equations which leads to smaller errors in Ward identities due to the absence of the terms with nonoverlapping loops, is proposed. In particular, these modified equations exactly satisfy Ward identities in the ladder approximation.