Chiral Symmetry Breaking for Fundamental Fermions

Abstract

Massive fermions have long been a problem in gauge theories. Unification of electromagnetic and weak forces was once hindered by the fact that the introduction of mass terms broke the gauge invariance of the theory. This problem was solved by the introduction of the Higgs field. Spontaneous breakdown of the SU(2) × U(1) symmetry then takes place. The gauge bosons gain mass and the masses for the fermions are generated through their Yukawa interaction with this Higgs field. However, there has been a widespread dissatisfaction with this mechanism since the masses are not predictable. Rather, they must be fixed by experiment. Studying the non-perturbative behaviour of gauge theories provides an alternative. If the interactions are strong enough, they are capable of generating masses for the particles dynamically even if they start with zero bare mass. Moreover, experiment tells us that the top quark is very heavy and so the Yukawa coupling g t for top-Higgs interaction is O(1). Then one naturally expects that non-perturbative effects become important. Indeed, it has been suggested [1] that the top quark may acquire mass non-perturbatively through four-fermion interactions, and the Higgs can then be viewed as the condensate of the top and the antitop. However, in an attempt to include the effects of gauge boson exchange term, one loses gauge invariance of the physical quantities. Of course, physical quantities must be gauge independent. This motivates the study of how to achieve this in non-perturbative calculations. Quenched QED provides a toy model in which to study this problem, as we discuss.