1. 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/0009131 S. Beane, Phys. Rev. D64 (2001) 116010 M. Golterman and S. Peris, Phys. Rev. D67 (2003) 096001

2. 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?

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

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

5. 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

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

7. 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.

8. 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.

9. 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

10. 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)

11. 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)

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

13. 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.

14. Fitting parameters: VA channel

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

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

17. 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:

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

19. Smoothness: Euler-Maclauren summation Result: Check for the first two states: Numerically (without the factor 10 -2 ):

20. 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.