On equivalence of renormalizations for standard and dimensional regularizations of $2D$ four-fermion interactions

We discuss the problem of equivalence between the dimensional regularization in $d=2+\varepsilon$ and some other regularization in $d=2$ for the $U_N$-symmetric four fermion interactions. To make the multiplicative renormalization in $d=2+\varepsilon $ one need to introduce the infinite system of couplings $g\equiv \{g_n,n=0,1,\dots\}$ but in $d=2$ there are three couplings only. We prove that after MS renormalization one can make UV-finite renormalization of couplings and field in such a way that after taking the limit of $\varepsilon=0$ all Green's functions will depend on three couplings only $g'_n(g)$ with $n=0,1,2$. This gives the equivalence between the renormalizations in $d=2+\varepsilon$ and $d=2$. We calculate the two-loop $\beta$-functions and three-loop anomalous dimension of field in MS-scheme and give the rigorous proof of the so called “projection technique” which allows one to rewrite the higher renormalized operators through lower renormalized operators after the taking the limit of $\varepsilon =0$.