Pion-Pion Interactions inτandτ′Decays

The final-state interactions in $\tau$ and ${\tau }^{\prime }$ decays of $K^{+}$ mesons are studied by means of a Mandelstam representation. It is assumed that only the $S$-wave pion-pion scattering amplitude is large enough to have an appreciable imaginary part. Coupled, linear, singular, integral equations are found for the amplitudes describing $K^{+}+\pi \to \pi +\pi$ in the physical and unphysical regions. From the solutions to these equations the $\tau$ and ${\tau }^{\prime }$ decay matrix elements may be constructed. In an approximation the equations are solved in terms of pion-pion phase shifts. Comparison with experiment is made using Chew and Mandelstam's solution of the corresponding nonlinear equations for $\pi -\pi$ scattering. Consistency with experiment is found for values of the coupling constant implying repulsive $T=0\mathrm{}$ and $T=2\mathrm{}$ phase shifts. Independently of the approximation the equations show that the implications in $\tau$ and ${\tau }^{\prime }$ decay of the rule, $\Delta T=\frac{1}{2}, \frac{3}{2}$, are identical with those of the $\Delta T=\frac{1}{2}$ rule. Hence the energy spectra in ${\tau }^{\prime }$ decay cannot be a critical test of the $\Delta T=\frac{1}{2}$ rule. Also, independently of the approximation, it is shown that the decay matrix element cannot be expanded in a series of integral powers of the pion kinetic energies.