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. E-mail address: [email protected]