Nonlocal Models of Goldstone Bosons in Asymptotically Free Gauge Theories
by John Terning
We develop a simple model Lagrangian which couples the pseudo-Goldstone bosons of QCD (pions, kaons, and the eta) to up, down, and strange quark fields. The model is a nonlocal generalization of the nonlinear sigma model, which incorporates a momentum dependent dynamical mass Sigma(p). Sigma(p) is an order parameter for spontaneous chiral symmetry breaking, and the Goldstone bosons are included as fluctuations of this order parameter in a manner consistent with global chiral symmetry. No kinetic energy terms or self-interactions for the Goldstone bosons are introduced, instead these arise dynamically. Explicit chiral symmetry breaking effects are also introduced by including current quark masses and external electroweak gauge fields. By integrating out the quark fields we obtain a chiral Lagrangian, including the Wess-Zumino term. The dynamical quark mass acts as a natural regulator, thereby making the calculations finite. All low energy parameters are expressed in terms of integrals of Sigma(p). The results are relatively insensitive to the precise form of Sigma(p), as long as Sigma(p) has the asymptotic behaviour required by QCD. We use physical values for masses and decay constants (m_pi^0, m_K^+, m_K^-, m_eta, f_pi, and f_K) to determine the five parameters of our model: the dynamical quark mass, the quark condensate, and the three current quark masses. We obtain the 14 parameters of the effective chiral Lagrangian and find that they are in good agreement with the experimental values.