We formally derive the chiral Lagrangian for low lying pseudoscalar mesons from the first principles of QCD considering the contributions from the normal part of the theory without taking an approximation. The derivation is based on the standard generating functional of QCD in the path integral formalism. The gluon-field integration is formally carried out by expressing the result in terms of the physical Green’s functions of the gluon. To integrate over the quark field, we introduce a bilocal auxiliary field $\Phi \mathrm{}(x,y)\mathrm{}$ representing the mesons. We then develop a consistent way of extracting the local pseudoscalar degree of freedom $U\left(x\right)\mathrm{}$ in $\Phi \mathrm{}(x,y)\mathrm{}$ and integrating out the rest degrees of freedom such that the complete pseudoscalar degree of freedom resides in $U\left(x\right).$ With certain techniques, we work out the explicit $U\left(x\right)\mathrm{}$ dependence of the effective action up to the $p^4$ terms in the momentum expansion, which leads to the desired chiral Lagrangian in which all the coefficients contributed from the normal part of the theory are expressed in terms of certain quark Green’s functions in QCD. Together with the exsisting QCD formulas for the anomaly contributions, the present results lead to the complete effective chiral Lagrangian for pseudoscalar mesons. The final result can be regarded as the fundamental QCD definition of the coefficients in the chiral Lagrangian. The relation between the present QCD definition of the $p^2$-order coefficient $F_0^2$ and the well-known appoximate result given by Pagels and Stokar is discussed.

• Received 1 March 1999

DOI:https://doi.org/10.1103/PhysRevD.61.054011