Calculation of the Low-EnergyS-Wave Pion-Pion Interaction from theKe4Decay Using Charge Commutation Relations

By making use of an appropriate charge commutation relation, we have computed the low-energy pion-pion scattering length directly from the rate of $K_{e4\mathrm{}}$-decay. The main differences from previous approaches are as follows: (a) Instead of using the current commutation relations (which have been extensively used with the soft-pion technique), we stick to the better-known charge commutation relations. This enables us, as in the Adler-Weisberger calculation, to avoid taking the usual soft-pion limit $k_{\mu }\to 0$; instead we let only $k^2(=-{m_{\pi }}^2)\to 0$, which is certainly a smaller extrapolation. (b) In contrast with previous calculations of $K_{e4\mathrm{}}$ decays, which utilized essentially the kaonic PCAC ($m_K\to 0$), we use the pionic PCAC, which involves a much smaller extrapolation ($m_{\pi }\to 0$). From the presently available $K_{e4\mathrm{}}$ decay rate, we have estimated the values of the $I=0\mathrm{}$ and $2S$-wave pion-pion scattering length as $a_0\simeq 0.18$ and $a_2\simeq -0.017\mathrm{}$. Although our numerical results for $a_0$, $a_2$, and the $K_{e4\mathrm{}}$-decay form factors turn out to be not very different from those obtained by using the current commutation relations with soft-pion techniques, we emphasize the important difference between the two approaches mentioned in (a) with respect to the extrapolation procedures.