Nonvanishing pion masses for vanishing bare quark masses

Abstract

It is generally thought that pion masses vanish when bare light quark masses $m_q$ vanish. The pions are Nambu-Goldstone bosons of chiral $SU_{A}(2)$ symmetry. We discuss a possibility that even when $m_q=0$, the chiral symmetry is explicitly broken in strong coupled QCD, while it is not in weakly coupled QCD. The point is that QCD monopoles are dynamical degrees of freedom in the strong coupled QCD. We show that the monopole quark interactions break the chiral $U_5(1)$ symmetry as well as the chiral $SU_{A}(2)$ symmetry. They are weak relative to the other chiral symmetric interactions of quarks and gluons. These weak interactions produce small pion masses as well as quark masses $\sim 20$MeV in the phase with the monopole condensation. The presence of the interactions modifies the Gell-Mann Oakes Renner relation. Using the relation we predict the value $(-\langle\bar{q}q\rangle)^{1/3}\simeq 160$MeV. We also show that the monopole quark interactions do not prevent the monopole condensation in dual superconducting model.