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
Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 2, pp. 179-191
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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 $\gamma ^{(n)}\otimes \gamma ^{(n)}$.
@article{TMF_1995_103_2_a0,
author = {A. N. Vasil'ev and S. \`E. Derkachev and N. A. Kivel'},
title = {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},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {179--191},
year = {1995},
volume = {103},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1995_103_2_a0/}
}
TY - JOUR AU - A. N. Vasil'ev AU - S. È. Derkachev AU - N. A. Kivel' TI - 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 JO - Teoretičeskaâ i matematičeskaâ fizika PY - 1995 SP - 179 EP - 191 VL - 103 IS - 2 UR - http://geodesic.mathdoc.fr/item/TMF_1995_103_2_a0/ LA - ru ID - TMF_1995_103_2_a0 ER -
%0 Journal Article %A A. N. Vasil'ev %A S. È. Derkachev %A N. A. Kivel' %T 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 %J Teoretičeskaâ i matematičeskaâ fizika %D 1995 %P 179-191 %V 103 %N 2 %U http://geodesic.mathdoc.fr/item/TMF_1995_103_2_a0/ %G ru %F TMF_1995_103_2_a0
A. N. Vasil'ev; S. È. Derkachev; N. A. Kivel'. 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. Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 2, pp. 179-191. http://geodesic.mathdoc.fr/item/TMF_1995_103_2_a0/
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