SU(2) chiral perturbation theory for Kℓ3 decay amplitudes

We use one-loop SU(2)L×SU(2)R chiral perturbation theory (SU(2) ChPT) to study the behaviour of the form-factors for semileptonic K→π decays with the pion mass at q2=0 and at qmax2=(mK−mπ)2, where q is the momentum transfer. At q2=0, the final-state pion has an energy of approximately mK/2 (for mK≫mπ) and so is not soft, nevertheless it is possible to compute the chiral logarithms, i.e. the corrections of O(mπ2log(mπ2)). We envisage that our results at q2=0 will be useful in extrapolating lattice QCD results to physical masses. A consequence of the Callan-Treiman relation is that in the SU(2) chiral limit (mu=md=0), the scalar form factor f0 at qmax2 is equal to f(K)/f, the ratio of the kaon and pion leptonic decay constants in the chiral limit. Lattice results for the scalar form factor at qmax2 are obtained with excellent precision, but at the masses at which the simulations are performed the results are about 25% below f(K)/f and are increasing only very slowly. We investigate the chiral behaviour of f0(qmax2) and find large corrections which provide a semi-quantitative explanation of the difference between the lattice results and f(K)/f. We stress the generality of the relation fP→π0(qmax2)=f(P)/f in the SU(2) chiral limit, where P=K, D or B and briefly comment on the potential value of using this theorem in obtaining physical results from lattice simulations.