International Conference on QCD and Hadronic Physics,

1. International Conference on QCD and Hadronic Physics, Beijing, China , 16-20 June2005 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. Introduction narrow resonances Large-Nc QCD In two-point correlators of quark currents: Operator Product Expansion (OPE) Sum of narrow resonances Constraints on meson mass spectrum? Linear mass spectrum with universal slope Hadron string Nonlinear corrections to mass spectrum?

3. Two-point correlators in Euclidean space (q denotes u- or d-quarks): are related to some observables from — residues — — 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 (chiral limit, LO of PT, large-Nc ) four-quarkcondensate gluoncondensate from the pert. theory After summing over resonances and comparing with the OPE (at each power of ) one arrives at the so called asymptotic sum rules.

6. In order to sum over resonances we use Euler-Maclaurin formula: where B1=1/6, B2=1/30, ... (Bernoulli numbers)

7. Improving the linear ansatz … • To reproduce the leading log Condition 2) is satisfied only if where Δt(x) is a rapidly decreasing function to be determined. • Phenomenologically it is plain that Regge trajectories are not linear for small “n”. However arbitrary ansätze for m2(n) and F2(n) result in appearance of terms which are absent in the standart OPE: 1) fractional or odd power of Q; 2) Q-2kln(Λ2/Q2). • We want to construct the parametrization that does not lead to the unwanted terms and reproduces the parton-model logarithm.

8. If we do not consider the running coupling constant and anomalous dimensions, the direct expansion of the integral 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. must be defined at any order [many proposals do not meet this criterium: Beane, Simonov, ...]

9. Let us discuss corrections to the linear mass spectrum 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 mV = mA and it is natural to expect that δ(n) decrease exponentially too.

10. Vector and axial-vector mesons Consider: (n is the principal quantum number) corrections to linear spectrum String picture: universality (agrees with phenomenology – Pancheri, Anisovich) string tension Conditions: agreement with the analytical structure of the OPE & convergence of sum rules for ПV(Q2) – ПA(Q2) degenerate spectrum 1). linear trajectories 2).

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

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

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

16. due to λ regular SP-channels: On the other hand: (from current algebra) refers to pion only If and π-meson belongs to the trajectory: Phenomenological bounds (B.L. Ioffe et al.): Remark 2: dimension-two gluon condensate λ2 In the OPE: VA-channels - no problem

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

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

19. One-loop: no anomalous dimensions, running αs(Q2). In OPE: In order to reproduce this behaviour we should accept the following ansatz for the residues: The influence on the spectrum is negligible.