Fermion Green Functions in Non-abelian Gauge Theories in Four Dimensions

Previous results on fermion chirality-ipping four-point functions are extended to non-abelian gauge theories. The problem is purely non-perturbative, and the analytical formalism used is based on the Schwinger-Dyson hierarchy. This is truncated, and the resulting equations are solved numerically by relaxation techniques. Taking the large-N limit in SU(N) theories simpliies the problem substantially, allowing the formulation of conjectures on the behavior of n-point and chirality-ipping fermion Green functions for general n. In this work we study the dynamical generation of fermion chirality ipping four-point functions within the framework of non-abelian gauge theories. These are generated in a purely non-perturbative manner, so in our analysis we have to use an adequate formalism like the one based on the Schwinger-Dyson (SD) hierarchy. The resulting equations are analytically intractable and very challenging even numerically. However, a certain limit in the theory allows a drastic simpliication. Our interest is focused on two objectives. One is to estimate the critical value of the gauge coupling, below which the four-point functions are equal to zero. We would like to study this in a case where four-fermion condensates develop on scales higher than the scale of two-fermion-condensate formation. This would make it consistent to treat the four-fermion condensate problem independently of the mass generation problem. Our other goal is to study the momentum dependence of the fermion four-point function. If such an object could generate a fermion or a gauge-boson mass, then two or 4 lines of the four-point function would be closed oo into a loop or into another four-point function. It would therefore be interesting to study the details of the momentum dependence, such as the relative size of the four-point function when diierent pairs of momenta are large. In the present study we use a one gauge-boson exchange approximation, where one gauge boson can attach to any pair of the four legs. The present work constitutes a clear progress with respect to our previous study 2], since, apart from considering a general non-abelian group, it includes a treatment of non-linearities, it does not neglect terms proportional to external momenta while at the same time exploring the full available momentum space, and in a certain limit allows us to draw conclusions on the behavior of general n-point functions. 1 Talk given at the 17th annual MRST meeting in Rochester (May 1995), based on work in collaboration with B. Holdom 1].