From QCD to the low-energy effective action through composite fields: Goldstone’s theorem andfπ
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Abstract
We elaborate upon a formal derivation from QCD of a composite-field (bilocal) effective chiral Lagrangian which we presented in a previous work. We give here a detailed treatment of the transition to the local form of the chiral Lagrangian in the low-energy limit, when chiral symmetry is spontaneously broken. We find an explicit form of the Goldstone-boson mode wave function and use it to derive expressions for the pion decay constant in terms of the quark dynamical mass \ensuremath{\Sigma} by two different methods. Using several models for \ensuremath{\Sigma} appropriate to QCD and one appropriate to technicolor, we compare the values given by our two different ${f}_{\ensuremath{\pi}}$ expressions with each other and with the values for ${f}_{\ensuremath{\pi}}$ given by an expression due to Pagels and Stokar. We find good agreement among the various methods of calculating ${f}_{\ensuremath{\pi}}$, and the agreement with the experimental value is satisfactory for a condensate mass scale in the range 0.5--1.0 GeV.
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