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

Comments: 22 pages, 4 figures. A few improvements and corrections made