SU(2) chiral perturbation theory for Kl3 decay amplitudes

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We use one-loop (2)ₗ× (2)ᵣ chiral perturbation theory ((2) ChPT) to study the behaviour of the form-factors for semileptonic K→π decays with the pion mass at q²=0 and at q²max=(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(mπ²(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 (2) chiral limit (mᵤ=md=0), the scalar form factor f⁰ at 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 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() 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 f⁰ₚ→π()=f⁽ᵖ⁾/f in the (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.