Symmetry Breaking of the Chiral $U(3) \otimes U(3)$ and the Quark Model

Through the recent analyses it has become apparent that strong interactions obey an approximate symmetry, chiral SU(3) Q9SU(3), which is realized by pseudoscalar Goldstone bosons in the symmetry limit. Nonvanishing masses of the actual pseudoscalar mesons are attributed to breaking of the symmetry. The main feature of the symmetry breaking can be well understood by assuming that the symmetry breaking part of the Hamiltonian density belongs to a single (3, 3*) EB (3*, 3) representation of SU(3) @SU(3) .) There still remain, however, some problems concerning r; and X mesons. When we consider the above-mentioned facts from the viewpoint of the quark model, Nambu's argumene) provides a very satisfactory explanation: In the symmetry limit, the quark has a vanishing bare mass, and its physical mass arises as a self-energy accompanied with the appearence of massless pseudoscalar mesons which are the bound-states of the quark-antiquark system. A simple explanation of the symmetry breaking term will be made by introducing a small quark mass. In this approach, however, it may be meaningful to make an investigation of transformation properties, including the chiral phase transformation q-? exp (icpJ..or5/2) q, i.e. extending the basic transformation group to U(3)@U(3). We will call the transformation property under chiral phase transformation, "chirality". For example, if a Hamiltonian has interactions of, at most, four-fermion type, SU(3) Q9SU(3) invariance means invariance under U(3) ® U(3). In this case, if we introduce the symmetry breaking term u 0 cu8 which has the same chirality as the quark mass term, we have