The weak decay of pions into leptons and into leptons plus photon is examined in a simple model at the one loop level. A basic assumption made is that there is more than one pair of quarks to which the pion couples and the masses and coupling constants are such that all amplitudes are finite. The amplitudes are defined by means of dimensional regularization and gauge invariance is rigorously maintained throughout. The experimental lifetimes for charged and neutral pion decay give one constraint among the quark masses and the parameter γ, the ratio of structure dependent axial to vector form factor, gives a second. Quark masses can be found to satisfy either condition, but no solution exists for both constraints simultaneously.
M1 transitions in a quark loop model. [Cascade process]