N_f Dependence of the Quark Condensate from a Chiral Sum Rule

Information on how fast the $<\bar q q>$ condensate in QCD-like theories varies as a function of $N_f$, the number of different flavour massless quarks, can be inferred from real QCD using chiral perturbation theory. At order one loop, one must estimate the value of a single low-energy coupling constant, $L_6$, which we express in terms of a chiral sum rule. The spectral function is shown to obey a Weinberg-type sum rule which ensures dominance of the energy region $s<1 {\rm GeV^2}$. The spectral function is related to the $\pi\pi$ scattering T-matrix. Various models are confronted with the Weinberg sum rule constraint. The value of $L_6$ that we obtain is compatible with a strong $N_f$ dependence for $N_f=2-3$ possibly suggestive of the proximity of a chiral phase transition.