Towards a determination of the chiral couplings at NLO in 1/N_C: L₈^r(μ) and C₃₈^r(μ)

Abstract

We present a dispersive method which allows to investigate the low-energy couplings of chiral perturbation theory at the next-to-leading order (NLO) in the 1/N C expansion, keeping full control of their renormalization scale dependence. Using the resonance chiral theory Lagrangian, we perform a NLO calculation of the scalar and pseudoscalar twopoint functions, within the single-resonance approximation. Imposing the correct QCD short-distance constraints, one determines their difference Π(t) ≡ Π S (t) − Π P (t) in terms of the pion decay constant and resonance masses. Its low momentum expansion fixes then the low-energy chiral couplings L 8 and C 38 . At µ 0 = 0.77 GeV, we obtain L r 8 (µ 0 ) SU (3) = (0.6 ± 0.4) · 10 −3 and C r 38 (µ 0 ) SU (3) = (2 ± 6) · 10 −6 .

The vector form factor at the next-to-leading order in 1/N C : chiral couplings L9(μ) and C88(μ) − C90(μ)

“…Nonetheless, on the contrary to what happens with other matrix elements (e.g. the S − P correlator [17]), the spin-1 two-point functions do not gain contributions from the U(3)-singlet chiral pseudo-Goldstone; the η 1 does neither enter at tree-level nor in the one-loop correlators. Therefore, the corresponding LECs are identical in both theories at leading and next-to-leading order in 1/N C :…”

“…[17,18], our results can be also derived in an utterly equivalent way through a Feynman diagram computation and the standard renormalization procedure. This derivation is slightly more complex and its detailed explaination is relegated to appendix E.…”

“…for a convenient renormalization scheme of this combination of vector couplings [17,18,19] (see appendix E for further details).…”

“…Using dispersion relations we can avoid the technicalities associated with the renormalization procedure [15,17,18]. This allows one to understand the underlying physics in a much more transparent way.…”

“…Following these ideas we determined, up to NLO in 1/N C , the couplings L 8 (µ) and C 38 (µ) in ref. [17] and L 10 (µ) and C 87 (µ) in ref. [18].…”