Pion mass and decay constant

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Independent of assumptions about the form of the quark-antiquark scattering kernel we derive the explicit relation between the pion Bethe-Salpeter amplitude, Gammaₚi, and the quark propagator in the chiral limit; Gammaₚi necessarily involves a non-negligible gamma₅ gamma.P term (P is the pion four-momentum). We also obtain exact expressions for the pion decay constant, fₚi, and mass, both of which depend on Gammaₚi; and demonstrate the equivalence between fₚi and the pion Bethe-Salpeter normalisation constant in the chiral limit. We stress the importance of preserving the axial-vector Ward-Takahashi identity in any study of the pion itself, and in any study whose goal is a unified understanding of the properties of the pion and other hadronic bound states.