Usage of REDUCE for computations of group-theoretical weight of Feynman diagrams in non-Abelian gauge theories

Abstract

Intr oduction Among modern physical theories, non-Abel i-an gauge fiel d theories are most important, The development of these theories and the explanation of experimental data necessitated the higher-or der pertur bation cal culations of Feynman diagrams. As opposed to the us-al quantum electr odynamics, in non-Abelian gauge theories it is necessary to calculate the factor associated with the gauge gr oup. In what foll ows this factor will be r effered to as a group-theoretic weight. Knowledge of this factor is essential for solving the problem of summation of some classes of Feynman diagrams. In some cases the group-theoretic weight allows one to estimate the asymptotic behaviour of Feynman diagrams in the l/N expansion The pr esent paper describes the pr og-ram COLOR which realizes the Cvitanovic al gor ithm /2/ of computation of the gr oup-theoretic weight for the gauge groups SU(n) and SO(n). The pr ogram is written in a symbolic mode of the REDUCE 1 an-sage Ill. 2. Cvitanovic algor ithm. The problem of the group-theor etic weight for Feynman diagrams arises when treating non-Abelian gauge theories. Q CD describing n quarks and N gl uons is an exampl e of such theories. This theor y Lagr angian is Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specfic permission. q ar e the quark fiel ds transformed as a fundamental repr esentation of a Lie group, Aip are gluon fiel ds transformed as its adjoint representation, Ti are generator s of a Lie group.