Revisiting the vector form factor at next-to-leading order in 1/N(C)

Using the Resonance Chiral Theory lagrangian, we perform a calculation of the vector form factor of the pion at the next-to-leading order (NLO) in the $1/N_C$ expansion. Imposing the correct QCD short-distance constraints, one determines it in terms of $F$, $G_V$, $F_A$ and resonance masses. Its low momentum expansion fixes then the low-energy chiral couplings $L_{9}$ and $C_{88}-C_{90}$ at NLO, keeping full control of their renormalization scale dependence. At $μ_0=0.77$ GeV, we obtain $L_{9}^r(μ_0) = (7.6 \pm 0.6)\cdot 10^{-3}$ and $C_{88}^r(μ_0)-C_{90}^r(μ_0)=(-4.5 \pm 0.5)\cdot 10^{-5}$.