First Workshop on Quark-Hadron Duality and the Transition to pQCD, Laboratori Nazionali di Frascati, 6-8 June 2005 MATCHING MESON RESONANCES TO OPE IN QCD A.A. Andrianov *#, V.A. Andrianov *, S.S. Afonin ** and D. Espriu ** # INFN, Sezione di Bologna * St. Petersburg State University ** Universitat de Barcelona Based on S.A., A.A., V.A., D.E., JHEP 0404, 039 (2004) Triggered by M.A.Shifman, hep-ph/ S. Beane, Phys. Rev. D64 (2001) M. Golterman and S. Peris, Phys. Rev. D67 (2003)

Linear mass spectrum with universal slope Large-N c QCD Introduction narrow resonances In two-point correlators of quark currents: Sum of narrow resonances Operator Product Expansion (OPE) Constraints on meson mass spectrum? Hadron string Nonlinear corrections to mass spectrum?

Two-point correlators in Euclidean space (q denotes u- or d-quarks): —residues— are related to some observables from weak decay constants—

In the vector and axial-vector cases the decay constants are normalized as follows: Relations with observables:

Operator Product Expansion Operator Product Expansion (chiral limit, large-N c ) gluon condensate four-quark condensate After summing over resonances and comparing with the OPE (at each power of ) one arrives at the so called asymptotic sum rules. from the pert. theory

In order to sum over resonances we use Euler-Maclaurin formula: where B 1 =1/6, B 2 =1/30,... (Bernoulli numbers)

Improving the linear ansatz … 1)2)  Phenomenologically it is plain that Regge trajectories are not linear for small “n”. However arbitrary ansätze for m 2 (n) and F 2 (n) result in appearance of terms which are absent in the standart OPE: 1) fractional or odd power of Q; 2) Q -2k ln(Λ 2 /Q 2 ).  We want to construct the parametrization that does not lead to the unwanted terms and reproduces the parton-model logarithm.  To reproduce the leading log 2) Condition 2) is satisfied only if where Δt(x) is a rapidly decreasing function to be determined.

If we do not consider the running coupling constant and anomalous dimensions, the direct expansion of the integral must be defined at any order [many proposals do not meet this criterium: Beane, Simonov,...] Apart from the constant solution (linear Regge ansatz) Δt(x) may drop as an exponential of some power of x (perhaps modulated by some powers of x) This is the simplest ansatz compatible with the OPE.

Let us consider the generalization of the Weinberg sum rules: Here the C (i) (i=0,1,...) represent the corresponding condensate. For the absolute convergence of the series at a given i one needs Consequently, for the convergence at any i one has to have m V = m A and it is natural to expect that δ(n) decrease exponentially too. Let us discuss corrections to the linear mass spectrum

Vector and axial-vector mesons Consider: String picture: universality (agrees with phenomenology – Pancheri, Anisovich) string tension Conditions: agreement with the analytical structure of the OPE & convergence of sum rules for П V (Q 2 ) – П A (Q 2 ) 1).linear trajectories degenerate spectrum 2). corrections to linear spectrum ( n is the principal quantum number)

Scalar and pseudoscalar mesons Following the same arguments (J=S,P): Important: sum rules over chiral partners (cutoff!) – there are two variants I. Linear σ-model: (π-in) II. Non-linear σ-model: π-meson is out of the trajectory, (π-out)

It is possible to use this analysis for some predictions of phenomenological interest. For instance:

An example of input masses (in MeV) for the mass spectra of our work and resulting constants. The corresponding experimental values (if any) are displayed in brackets.

Fitting parameters: VA channel

Mass spectrum (in MeV) and residues (in MeV) for the inputs from the previous table (for the first 4 states). π-in π-out

Remark 1: D-wave vector mesons D S V.V. Anisovich at al. D-wave vector states decouple!

Remark 2: dimension-two gluon condensate λ 2 In the OPE: VA-channels - no problem regular due to λrefers to pion only On the other hand:(from current algebra) If and π-meson belongs to the trajectory: Phenomenological bounds (B.L. Ioffe et al.): SP-channels:

Remark 3: perturbation theory Consider the vector correlator: Resonance saturation: where

Smoothness: Euler-Maclauren summation Result: Check for the first two states: Numerically (without the factor ):

One-loop: no anomalous dimensions, running α s (Q 2 ). In OPE: In order to reproduce this behaviour we should accept the following ansatz for the residues: The influence on the spectrum is negligible.