Proof of the absence of multiplicative renormalizability of the Gross–Neveu model in dimensional regularization $d=2+2\varepsilon$

We prove that the simplest four-fermion Gross–Neveu model with dimensional regularization $d=2+2\varepsilon$ is not multiplicatively renormalizable due to the counterterm generated by the three-loop vertex diagrams that is proportional to the evanescent operator [1] $V_3=(\bar\psi\gamma_{ikl}^{(3)}\psi )^2/2$, where $\gamma_{i_1\dots i_n}^{(n)}$ is the fully antisymmetric product of $n$ $\gamma$-matrices and is not zero in arbitrary dimensions. Therefore, calculations of the $(2+\varepsilon)$-expansion of the critical indices $\eta$ and $\nu$ in the framework of the simple Gross–Neveu model are correct only to $\varepsilon^4$ for $\eta$ and to $\varepsilon^3$ for $\nu$. In higher orders, one must take into consideration the generation of other (not only $V_3$) evanescent operators.