Pion mass dependence of the $K_{l3}$ semileptonic scalar form factor within finite volume

Understand

We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer $t_{m} = (m_{K} - m_{\pi})^2$, using chiral perturbation theory.

• At first we obtain the hadronic matrix element to be calculated in finite volume.
• We then evaluate the finite size effects for two volumes with $L = 1.83 fm$ and $L= 2.73 fm$ and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in \cite{Boyle2007a}.
• It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections.