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We use one-loop $\SU(2)_L\times \SU(2)_R$ chiral perturbation theory($\SU(2)$ ChPT) to study the behaviour of the form-factors for semileptonic$K\to\pi$ decays with the pion mass at $q^2=0$ and at$q^2_{\textrm{max}}=(m_K-m_\pi)^2$, where $q$ is the momentum transfer. At$q^2=0$, the final-state pion has an energy of approximately $m_K/2$ (for$m_K\gg m_\pi$) and so is not soft, nevertheless it is possible to compute thechiral logarithms, i.e. the corrections of $O(m_\pi^2\log(m_\pi^2))$. Weenvisage that our results at $q^2=0$ will be useful in extrapolating latticeQCD results to physical masses. A consequence of the Callan-Treiman relation isthat in the $\SU(2)$ chiral limit ($m_u=m_d=0$), the scalar form factor $f^0$at $\qsqmax$ is equal to $f^{(K)}/f$, the ratio of the kaon and pion leptonicdecay constants in the chiral limit. Lattice results for the scalar form factorat $\qsqmax$ are obtained with excellent precision, but at the masses at whichthe simulations are performed the results are about 25% below $f^{(K)}/f$ andare increasing only very slowly. We investigate the chiral behaviour of$f^0(\qsqmax)$ and find large corrections which provide a semi-quantitativeexplanation of the difference between the lattice results and $f^{(K)}/f$. Westress the generality of the relation $f^0_{P\to\pi}(\qsqmax)=f^{(P)}/f$ in the$\SU(2)$ chiral limit, where $P=K,D$ or $B$ and briefly comment on thepotential value of using this theorem in obtaining physical results fromlattice simulations.

SU(2) chiral perturbation theory for Kl3 decay amplitudes