Multi-Mass Callan-Symanzik Equation. II

Abstract

The Callan-Symanzik equation is generalized so as to include several independent fermion masses. The resulting equation is applied to exploration of the possibility of dynamical generation of quark masses in the absence of Higgs fields. l ),2) describes the response of Green's func­ tions against infinitesimal changes of the bare fermion mass. In reality, however, there are many independent fermion masses, and it is desirable to generalize it to multi-mass cases. For that purpose we have to know the mechanism of how the fermion masses as well as other masses are generated. In the standard gauge theory it is a conventional wisdom to assume that they are generated by spontaneous symmetry-breaking through the Higgs fields, but there is an alternative possibility that they are generated dynamically without recourse to the Higgs fields. In the present article we shall look for the latter possibility by formulating the CS equation in the multi-mass case .. In such an approach we assume that the masses of other particles can be expressed in terms of quark masses,3) at least in principle. In § 2 we briefly review the derivation of the CS equation in the single mass case in such a way that it facilitates the generalization to multi-mass cases. In § 3 we immediately generalize the CS equation to the multi-mass case provided that scalar densities relevant to mass-insertion are multiplicatively renormalized. Quantum chromodynamics (QCD) belongs to this category. The multi-mass CS equation is inhomogeneous because of the presence of the mass-insertion term, so that we shall transform it into a homogeneous form by introducing Green's functions with exponentiated mass-insertion. This subject has already been discussed elsewhere,3) but we shall recapitulate it in § 4 for compl€teness. The physical mass and coupling constant are shifted as a result of mass-insertion, and we shall study in § 5 how these relevant quantities are determined from the CS equation in the single mass case. At this stage we shall ask if it is possible to obtain a finite physical mass starting from a massless theory. This problem is generalized to the multi-mass case and it is investigated how this approach is related to the idea of spontaneous chiral-symmetry-breaking in § 6. Finally in § 7 the CS equation is derived when scalar densities are mixed under renormalization.