Chiral Lagrangian from gauge invariant, nonlocal, dynamical quark model

Parameters of Gasser Leutwyler chiral Lagrangian are proved saturated by dynamical quark self energy Σ(k2) in a gauge invariant, nonlocal, dynamical quark model. Much of low-energy QCD can be encoded into a series parameters appearing in a chiral Lagrangian, expanded to some finite order of low energy expansion. Attempts have been made to understand these parameters: it is shown that low lying vector mesons will saturate the parameters [1]. To go beyond phenomenological level, the anomaly contribution was taken as the main source of the parameters [2], we call this type investigation the anomaly approach,it leads result 8L1 = 4L2 = −2L3 = 24L7 = −8L8 = L9 = −2L10 = Nc 48π2 (1) which are close to experiment result except L7 and L8 which have wrong signs. The deficiency of this calculation lies in its independence of interaction: if we switch off the strong interaction and discuss a system of free quark field with external sources, the anomaly calculation can still be performed without any change. Then it seems that (1) is not due to strong interaction among quarks and gluons, but rather an artificial result. Another type research, we call it dynamical approach, mainly consider the dynamical effect [3], in which the main source of the parameters is from dynamical quark self energy Σ(k). This approach has advantage of maintaining chiral symmetry and momentum dependence of dynamical quark