Pion Phase-Shift Information fromKl4Decays

The structure of the $K_{l4\mathrm{}}$ decay spectrum is displayed in its full generality as a function of a complete set of five kinematic variables; polarization effects are in part similarly described. The nontrivial dynamical aspects of $K_{l4\mathrm{}}$ decay reside in certain form factors, which depend on only three of the variables. It is shown that, with large cuts at the two remaining variables, the measurement of the decay spectrum alone suffices to overdetermine the form factors. On certain standard assumptions, these form factors carry information relating to the phase shifts for pion-pion scattering. It is an important practical matter to extract something of this information under conditions of limited statistics, where the spectra have to be treated in partially integrated form. We find that this can be accomplished, with surprising economy, for the energy-dependent phase-shift difference ${\delta }_s-{\delta }_p$, on the single additional assumption that the dipion system is produced chiefly in $s$- and $p$-wave states. The information emerges from the intensity spectrum for $K_{e4\mathrm{}}$ decay, and again from the polarization spectrum for $K_{\mu 4}$ decay, the spectra being treated as functions of only three variables. Moreover, the structure in two of these variables is simple and therefore relatively undemanding in a statistical sense.