Once-Subtracted Dispersion Relations, Current Algebra, andKl3Form Factors

$K_{l3\mathrm{}}$ decay is treated using once-subtracted dispersion relations, current algebra, and partial conservation of axial-vector current. The dispersion integrals are evaluated by saturating them with the vector $K^{*}$ and and scalar $\kappa$ intermediate states. The subtraction point is chosen so that the subtraction constants may correspond to the soft-pion result. It is shown that if $f_{+}\mathrm{}\left(t\right)\mathrm{}$ obeys once-subtracted dispersion relations and $f_{-}\mathrm{}\left(t\right)\mathrm{}$ an unsubtracted one (scheme I), and if $\frac{f_K}{f_{\pi }}\simeq 1.16-1.28$, then $f_{+}\mathrm{}\left(0\right)\mathrm{}$ cannot be close to 1 unless a $\kappa$ meson exists. In this case it is also shown that both ${\lambda }_{+}$ and ${\lambda }_{-}$ are small, while $\xi =\frac{f_{-}\mathrm{}\left(0\right)\mathrm{}}{f_{+}\mathrm{}\left(0\right)\mathrm{}}$ is small and negative. We also consider the possibility of having once-subtracted dispersion relations for both $f_{+}\mathrm{}\left(t\right)\mathrm{}$ and $f_{-}\mathrm{}\left(t\right)\mathrm{}$ (scheme II). It is found that the results of schemes I and II are the same if either (a) there exists a $\kappa$ meson with mass around 1 BeV and $f_{+}\mathrm{}\left(0\right)\mathrm{}\simeq 1$, or (b) no $\kappa$ meson exists, but $f_{+}\mathrm{}\left(0\right)\mathrm{}\simeq \frac{f_K}{f_{\pi }}$. If, on the other hand, no $\kappa$ meson exists, and if $f_{+}\mathrm{}\left(0\right)\mathrm{}\simeq 1$, while $\frac{f_K}{f_{\pi }}\simeq 1.28$, then one is able to get ${\lambda }_{-}$ an order of magnitude bigger than ${\lambda }_{+}$ in scheme II. Thus there is a possibility for large ${\lambda }_{-}$ only in scheme II. Furthermore, in scheme I, using partial conservation of vector current for the strangeness-changing vector current, we obtain $\left(\frac{f_K}{f_{\pi }}\right){f_{+}}^{-1\mathrm{}}\mathrm{}\left(0\right)\mathrm{}\simeq \frac{{m_{\kappa }}^2}{({m_{\kappa }}^2-{m_K}^2)}$. For $\left(\frac{f_K}{f_{\pi }}\right){f_{+}}^{-1\mathrm{}}\mathrm{}\left(0\right)\mathrm{}\simeq 1.28$, we predict $m_{\kappa }\simeq 1.06$ BeV.