A~technique for calculating the $\gamma$-matrix structures of the diagrams of a~total four-fermion interaction with infinite number of vertices $d=2+\epsilon$ dimensional regularization

A. N. Vasil'ev ; S. È. Derkachev ; N. A. Kivel'

Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 2, pp. 179-191

Résumé

It is known [1] that in the dimensional regularization $d=2+\epsilon$ any four-fermion interaction generates an infinite number of the counterterms $(\bar \psi \gamma _{\alpha _1\dots \alpha _n}^{(n)}\psi )^2$, where $\gamma _{\alpha _1\dots \alpha _n}^{(n)}\equiv \operatorname {As}[\gamma _{\alpha _1}\dots \gamma _{\alpha _n}]$ is the antisymmetrized product of $\gamma$-matrices. A total multiplicatively renormalizable model includes all such vertices and, therefore, calculation of $\gamma$-matrix multipliers of the corresponding diagrams is a non-trivial task. An effective technique for performing such calculations is proposed. It includes: the realization of the $\gamma$-matrices by the operator free fermion field, utilization of generation functions and functionals and different versions of Wick theorem, reduction of the $d$-dimensional problem to $d=1$. The general method is illustrated by the calculations of $\gamma$-factors of one- and two-loop diagrams with an arbitrary set of vertices $\gamma ^{(n)}\otimes \gamma ^{(n)}$.