ABSTRACT
If one neglects quark masses, then fields of the u, d, s quarks would not be distinguished and, as a consequence, the QCD Lagrangian would be invariant with respect to rotations in the quark flavors space. And at the same time, owing to the vector character of interaction between quarks and gluons, one may independently rotate left and right constituents of quark fields (qL and qR). Transformations of this kind are described by eight independent parameters ξaL for the left-handed particles and eight ones ξ
a R for the right-handed particles:
qL,R → ( 1 + i
∑ a
) qL,R, (5.1)
where λa operates in the flavor space of u, d, and s quarks. If ξ a R = ξ
a L, the transformations
(5.1) conserve the parity. Invariance with respect to these transformations takes place when the quark masses are nonzero but are equal to each other (originally just this possibility has been discussed). At current data on the masses of the u, d, and s quarks, there are no reasons to believe the approximation of equal quark masses to be better than that of zero quark masses. In the latter case the Lagrangian is also invariant with respect to the transformations with ξaR = −ξaL that do not conserve the parity (such transformations will be called chiral ones). From the mathematical point of view, the invariance with respect to the transformations (5.1) denotes the chiral SU(3)L × SU(3)R-symmetry of the strong interaction Lagrangian. Note, that the approximate SU(3)L×SU(3)R-symmetry had been discovered before QCD was formulated. As long as we do not observe any particles degeneration that may be associated with
the chiral symmetry, then this symmetry has to be violated. The existence of bosons set with masses being close to zero, three pi-mesons and the whole octet of 0−-mesons, is a physical manifestation of such a realization of the symmetry. In this chapter we will show that in definite kinematic limits matrix elements of these light pseudoscalar mesons may be calculated with the help of commutation relations and conservation laws for currents corresponding to the chiral symmetry. In accordance with the Noether theorem, exact symmetry results in conserved currents.
As this takes place, corresponding charges generate a group algebra associated with a given symmetry, that is, these charges are group generators. Thus, for example, in the case of the
