Non-factorization of four-quark condensates at low energies within Chiral Perturbation Theory

TL;DR

This paper investigates the validity of the factorization hypothesis for four-quark condensates within Chiral Perturbation Theory, revealing non-factorization at higher orders due to divergent terms, except in the chiral limit.

Contribution

It demonstrates that four-quark condensates do not factorize at next-to-next-to-leading order in Chiral Perturbation Theory, except in the chiral limit, and discusses implications for large N_c limit.

Findings

Factorization holds at leading and next-to-leading order.

A non-factorization term causes divergence at next-to-next-to-leading order.

Factorization is restored in the large N_c limit if taken before renormalization.

Abstract

Four-quark correlators and the factorization hypothesis are analyzed in the meson sector within Chiral Perturbation Theory. Factorization of the four-quark condensate holds to leading and next to leading order. However at next to next to leading order, a term with a nontrivial space-time dependence in the four-quark correlator yields a divergent four-quark condensate, whereas the two-quark condensate and the scalar susceptibility are finite. Such a non-factorization term vanishes only in the chiral limit. We also comment on how factorization still holds in the large $N_c$ limit, provided such a limit is taken before renormalization.