Matching Meson Resonances to Ope in QCD

Abstract

We investigate the possible corrections to the linear Regge trajectories for the lightquark meson sector by matching two-point correlators of quark currents to the Operator Product Expansion. We find that the allowed modifications to the linear behavior must decrease rapidly with the principal quantum number. After fitting the lightest states in each channel and certain low-energy constants the whole spectrum for meson masses and residues is obtained in a satisfactory agreement with phenomenology. The perturbative corrections to our results are discussed.Keywords: QCD; sum rules; large-Nc.The observed masses squared of mesons with given quantum numbers form linear trajectories 1,2 depending on the number of radial excitation n. This is a strong indication that QCD admits an effective string description, as this type of spectrum is characteristic e.g. of the bosonic string. In the bosonic string model the slope of all trajectories must be equal since this quantity is proportional to the string tension depending on gluedynamics only. However, there exist sizeable deviations from the string picture. In the present analysis we examine possible corrections to the linear trajectories in the vector (V), axial-vector (A), scalar (S), and pseudoscalar (P) channels 3 . Our method is based on the consideration of the two-point correlators of V,A,S,P quark currents in the large-N c limit of QCD 4 . On the one hand, by virtue of confinement they are saturated by an infinite set of narrow meson resonances, that is, they can be represented by the sum of related meson poles in Euclidean space:expressing the quark-hadron duality 5 . Here J ≡ S, P, V, A; Γ = i, γ 5 , γ µ , γ µ γ 5 . Further we denote F S,P ≡ G S,P m S,P . On the other hand, their high-energy asymptotics 1

Scalar-isoscalar states in the large-NcRegge approach

“…Only that way the leading logarithmic behavior (22) can be reproduced. Similar studies have been carried out in the past for other correlators in the Regge approach [75][76][77][78][79][80][81][82][83][84][85]. Here we focus on the Π ΘΘ correlator defined in Eq.…”

“…[104]; it was also briefly discussed in Ref. [81]. A more quantitative analysis was carried out by us in Refs.…”

Weinberg like sum rules revisited

“…The only exception is paper [74], where the problem was addressed for the case of infinite number of resonances, but the discussions of that paper remain relevant for our case. The doubling of V-states forces to replace the sum in Eq.…”

“…Identifying now we arrive at the standard pattern of resonance saturation, so the usual formulae remain formally valid as if the D-wave states were decoupled. A qualitative argument in favour of D-wave decoupling presented in [74] relied, in essence, on the fact that the vector interpolating current (19) couples to the e + e -annihilation, which is a point-like process, hence, the extended objects like the D-wave states should decouple, i.e. their residues vanish rapidly.…”

“…Recently the resonance sum rules were employed to demonstrate that the chiral symmetry is realized in the Wigner-Weil mode in the upper part of meson spectrum [73][74][75], the first attempt of this kind seems to go back to [76], where the baryon sector was analyzed. These attempts boiled down to justification of parity doubling among the highly excited states, the procedure turned out to be model-dependent, thus not replying unambiguously whether the chiral symmetry is restored or not.…”