Structure of K3π Decay

Using the partially conserved axial-vector current and the commutation relations of the vector and axial-vector currents with themselves and with the Hamiltonian responsible for weak nonleptionic decays, we demonstrate how to expand weak-decay amplitudes in the momenta of pions involved in the decays. The resulting expansion gives the on-mass-shell decay amplitude up to first order in pion momenta. We apply these techniques to the three-pion decays of kaons and show that they quantitatively reproduce the experimental situation of an amplitude linearly dependent on the "odd"-pion kinetic energy. All three pions are treated on an equal footing, and on-the-mass-shell quantities are dealt with everywhere.