Renormalizable sectors in resonance chiral theory: S→ππ decay amplitude

We develop a resonance chiral theory without any a priori limitation on the number of derivatives in the hadronic operators. Through an exhaustive analysis of the resonance Lagrangian within the chiral limit and by means of field redefinitions, we find that the number of independent operator contributing to the S→ππ decay amplitude is finite: there is only one single-trace operator (the cd term) and three multi-trace terms. The deep implication of this fact is that the ultraviolet divergences that appear in this amplitude at the loop level can only appear through these chiral invariant structures. Hence, a renormalization of these couplings renders the amplitude finite.