Exact Bounds forKl3Decay Parameters

Abstract

We derive exact bounds for the $K_{l3\mathrm{}}$ decay form factors $f_{\pm }\mathrm{}\left(t\right)\mathrm{}$. Particularly, we find the bound $({m_K}^2-{m_{\pi }}^2)\left|f_{+}\mathrm{}\right(0\left)\right|\le 16{\left[\frac{1}{3}\pi \Delta \mathrm{}\right(0\left)\right]}^{\frac{1}{2}}{(m_K+m_{\pi })}^{\frac{1}{2}}{({m_K}^{\frac{1}{2}}+{m_{\pi }}^{\frac{1}{2}})}^{-1\mathrm{}}$, where $\Delta \mathrm{}\left(0\right)\mathrm{}$ is the propagator of the divergence of the strangeness-changing current at zero momentum. If we further assume the Hamiltonian of Gell-Mann, Oakes, and Renner in order to estimate $\Delta \mathrm{}\left(0\right)\mathrm{}$, we obtain $\left|f_{+}\mathrm{}\right(0\left)\right|\lesssim 1.0$. Similarly, an inequality testing the standard $K_{l3\mathrm{}}$ soft-pion theorem is found to be well satisfied. In addition, a new inequality involving derivatives of $f_{+}\mathrm{}\left(t\right)\mathrm{}$ is derived. Taking ${\lambda }_{+}\sim 0.02$, this inequality leads to $\left|f_{-}\mathrm{}\right(0\left)\right|\lesssim 0.33$.