Kl5Form Factors from Partially Conserved Axial-Vector Current and Current Algebra

Abstract

The form factors for the $K_{l5\mathrm{}}$ decay ($K\to \pi \pi \pi l{\nu }_l$) are derived through the use of the algebra of currents and the hypothesis of partially conserved axial-vector current. In obtaining the results, two different methods were used: the single-soft-pion method, in which the momentum of only one pion at a time is set equal to zero, and the multi-soft-pion method, in which all pions in the matrix element are taken off the mass shell simultaneously. The results obtained by the two methods are consistent one with the other; the existence of a pole in the form factors in the limit of two soft pions indicates, however, that the matrix element obtained in the limit of three soft pions is not a valid approximation to the matrix element in the physical region. The $K\pi$ and $\pi \pi$ scattering amplitudes and the transformation properties and matrix elements of the $\sigma$ field are also discussed, since they are intimately connected with the derivation of the $K_{e5\mathrm{}}$ form factors. The rates obtained for the four possible $K_{e5\mathrm{}}$ decay modes were found to be ∼${10}^{-3\mathrm{}}$-${10}^{-4\mathrm{}}$ ${\sec }^{-1\mathrm{}}$.