$N_f$ dependence of the quark condensate from a chiral sum rule

Abstract.

How fast does the quark condensate in QCD-like theories vary as a function of $N_f$ is inferred from real QCD using chiral perturbation theory at order one loop. A sum rule is derived for the single relevant chiral coupling constant, $L_6$. A model independent lower bound is obtained. The spectral function satisfies a Weinberg-type superconvergence relation. It is discussed how this, together with chiral constraints, allows a solid evaluation of $L_6$, based on experimental $\pi\pi$–$K\bar K$ S-wave T-matrix input. The resulting value of $L_6$ is compatible with a strong $N_f$ dependence possibly suggestive of the proximity of a chiral phase transition.