Form-factors and current correlators: chiral couplings L_10(mu) and   C_87(mu) at NLO in 1/N(C)
  Using the resonance chiral theory Lagrangian, we perform a calculation of the vector and axial-vector two-point functions at the next-to-leading order (NLO) in the 1/N(C) expansion. We have analyzed these correlators within the single-resonance approximation and have also investigated the corrections induced by a second multiplet of vector and axial-vector resonance states. Imposing the correct QCD short-distance constraints, one determines the difference of the two correlators Pi(t) = Pi_VV(t)- Pi_AA(t) in terms of the pion decay constant and resonance masses. Its low momentum expansion fixes then the low-energy chiral couplings L_10 and C_87 at NLO, keeping full control of their renormalization scale dependence. At mu_0=0.77 GeV, we obtain L_10(mu_0) = (-4.4 \pm 0.9)10^{-3} and C_87^r(mu_0)=(3.1 \pm 1.1)10^{-5}. 
A. Pich, I. Rosell, J.J. Sanz-Cillero