Proof of the absence of multiplicative renormalizability of the Gross-Neveu model in the dimensional regularizationd=2+2ɛ

Abstract

We prove that the simplest four-fermion Gross-Neveu model with the dimensional regularization d=2+2ɛ is not multiplicatively renormalizable. This is due to the counter-term, generated by the three-loop vertex diagrams, that is proportional to the evanescent operator $$V_3 = (\bar \psi \gamma _{ikl}^3 \psi )^2 /2$$ . Here $$\gamma _{i_{1 \cdots i_n } }^{(n)} $$ is the totally antisymmetric product of n γ-matrices, which is not zero in a noninteger dimension as well. Therefore, calculations of the (2+ɛ)-expansion of the critical indices η and ν in the framework of the simple Gross-Neveu model are correct only up to e4 for η and to ɛ3 for ν. In higher orders, one must take into consideration the generation of other (not only V3) evanescent operators.