Broken Chiral Symmetry. I. Continuous Transitions between Subgroups

We investigate the general properties of the Gell-Mann model for chiral $U\left(3\right)\mathrm{}\otimes U\left(3\right)\mathrm{}$ symmetry breaking. From a study of the two-point functions, we find that the symmetry-breaking parameters cannot assume arbitrary values, but must be confined in specified domains. The boundaries of these domains are related to several interesting subgroup symmetries. We present arguments to show that one must have essential singularities at those values of the symmetry-breaking parameter which correspond to subgroup symmetries realized via the emergence of zero-mass bosons. In a suitable singularity-free range of physical interest, we next discuss the possibility of continuous transitions between different symmetry subgroups, and show how, with the use of a variational principle, one can obtain some mass formulas and relations between other physically relevant quantities in a nonperturbative manner. In particular, the relation obtained by Gell-Mann, Oakes, and Renner for the symmetry-breaking parameter is obtained naturally in this manner. Also, it is shown that this formalism requires the existence of scalar mesons.