A mechanism of the power-counting breaking in chiral perturbation theory

Abstract

Chiral perturbation theory is a power tool for studying the low-energy strong interactions. In the conventional chiral expansion, according to the power counting rule, the higher-order corrections for physical observables are expected to decrease order by order as the chiral dimension increases. However, most previous study of $πK$ scattering lengths by different references showed that the $\mathcal{O}(p^6)$ correction to the scattering lengths is anomalously large and even exceeds the $\mathcal{O}(p^4)$ one. It poses a serious challenge to the convergence of chiral expansion. In order to explore the underlying origin of this phenomenon, the contributions of two-loop diagrams, one-loop diagrams and tree diagrams for the $πK$ scattering amplitude are detailed analyzed. The reducible two-loop integrals, sunset integrals and vertex integrals from the $\mathcal{O}(p^6)$ order all violate the power counting rule. The sunset and vertex integrals can be largely canceled numerically, but not enough. Furthermore, $πK$ scattering is compared with $ππ$ scattering, which seems not violate the power counting rule. The analysis indicates that $ππ$ scattering contains a higher symmetry, although each $\mathcal{O}(p^6)$-order part also violates the power counting rule, but the violation is much smaller than the one from $πK$ scattering. Large kaon mass also makes a contribution to the power-counting breaking, but it is not a major origin. Hence, it needs to pay more attention to the many-loop low-symmetry diagrams in future work. These diagrams may give large contribution and cannot be ignored.