Formula omitted] chiral perturbation theory for [formula omitted] decay amplitudes

Abstract

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 q 2 = 0 and at q max 2 = ( m K − m π ) 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 ≫ m π ) and so is not soft, nevertheless it is possible to compute the chiral logarithms, i.e. the corrections of O ( m π 2 log ( m π 2 ) ) . We envisage that our results at q 2 = 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 ( m u = m d = 0 ), the scalar form factor f 0 at q max 2 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 q max 2 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 f 0 ( q max 2 ) 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 f P → π 0 ( q max 2 ) = 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.