The $K\pi$ form factors from Analyticity and Unitarity

Analyticity and unitarity techniques are employed to obtain bounds on the shape parameters of the scalar and vector form factors of semileptonic $K_{l3}$ decays. For this purpose we use vector and scalar correlators evaluated in pQCD, a low energy theorem for scalar form factor, lattice results for the ratio of kaon and pion decay constants, chiral perturbation theory calculations for the scalar form factor at the Callan-Treiman point and experimental information on the phase and modulus of $K\pi$ form factors up to an energy $\text{tin}=1{\mathrm{G}\mathrm{e}\mathrm{V}}^2$. We further derive regions on the real axis and in the complex-energy plane where the form factors cannot have zeros.