Three loops calculation of the field anomalous dimension in the full four-fermion $U_N$-symmetrical model

We present a three-loop calculation for the contribution of the anomalous dimension $\gamma\psi$ in the RG function for the full $U_N$-symmetric model using dimensional regularization with $d=2+\varepsilon$. This model contains an infinite number of independent four-fermion interactions and couplings $g_n$. In the MS scheme, the three-loop contribution in $\gamma\psi$ depends on all of the charges. In the symmetric subtraction scheme for $\varepsilon=0$, only the dependence on the three lower charges $g'_{0,1,2}$ remains. The results of a three-loop calculation of $\gamma\psi$ for the Thirring model and for the Gross–Neveu model are discussed; they seem to contradict the equivalence of all four-fermion interactions for $d=2$, $N=1$. It is shown that this equivalence appears only for the symmetric subtraction scheme. The precise meaning of $(2+\varepsilon)$-expansions of the critical dimensions is discussed for when the multiplication renormalizability is absent in the Gross–Neveu model.