Dynamical Chiral Symmetry Breaking in Effective Models of QCD in the Bethe-Salpeter Approach

Dynamical breaking of chiral symmetry in effective models of QCD is studied. Introducing a cut-off function or a non-local interaction, the Noether current must be modified and thus the Ward-Takahashi identity and the PCAC relation are modified accordingly. We point out that the pion decay constant must be defined consistently with the Noether current so that the low-energy relations are satisfied. We define the proxy of the Noether current for general effective models, which is consistent with loop expansion of the Cornwall-Jackiw-Tomboulis effective action. A general formula for the pion decay constant in terms of the Bethe-Salpeter amplitude is derived. The effective Pagels-Stokar formula is proposed which is useful to estimate the decay constant without solving the Bethe-Salpeter equation.