Rigorous Bound onKl3Decay Amplitudes

We derive a rigorous bound on the $K_{l3\mathrm{}}$ decay form factors $f_{\pm }\mathrm{}\left(t\right)\mathrm{}$. With $\xi \equiv \frac{f_{-}\mathrm{}\left(0\right)\mathrm{}}{f_{+}\mathrm{}\left(0\right)\mathrm{}}$ and ${\lambda }_{+}\equiv \frac{{m_{\pi }}^2{f_{+}}^{\prime }\mathrm{}\left(0\right)\mathrm{}}{f_{+}\mathrm{}\left(0\right)\mathrm{}}$, our bound is $|\xi +\frac{({m_K}^2-{m_{\pi }}^2){\lambda }_{+}}{{m_{\pi }}^2}|<~\frac{8{\Delta }^{\frac{1}{2}}\mathrm{}\left(0\right)\mathrm{}{I}^{\frac{1}{2}}}{\sqrt{3}{m_K}^2\left|f_{+}\mathrm{}\right(0\left)\right|}$, where $I=0.584\mathrm{}$ and $\Delta \mathrm{}\left(0\right)\mathrm{}$ is the propagator of the divergence of the strangeness-changing current at zero momentum. If one further assumes for the Hamiltonian density $H\left(x\right)=H_0\mathrm{}\left(x\right)+{\epsilon }_0{S}_0\mathrm{}\left(x\right)+{\epsilon }_8{S}_8\mathrm{}\left(x\right)\mathrm{}$, with $H_0\mathrm{}\left(x\right)\mathrm{} \mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ invariant and $S_{0,8}$ transforming like members of the $(\overline{3},3)+(3,\overline{3})$ representation, then one can prove that $\Delta \mathrm{}\left(0\right)=\frac{3{\epsilon }_8{〈S_8〉}_0}{4}$, so $\Delta \mathrm{}\left(0\right)\mathrm{}$ is proportional to the product of $\mathrm{SU}\left(3\right)\mathrm{}$ violations in the Hamiltonian times those in the vacuum, and is hence expected to be very small. We estimate ${\Delta }^{\frac{1}{2}}\sim m_{\pi }{f}_{\pi }$, and our bound implies...