1. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Recovering QCD at large N C : Two-point Green-functions J. J. Sanz Cillero, IN2P3 - Orsay

2. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Outline: DUALITY QCD at large N C (QCD ∞ )  Resonance Theory (R  T ∞ ) Analysis of QCD correlators: a)- Physical QCD content b)- Relation R  T Sum-rules c)- Truncated R  T + “pQCD Continuum” Conclusions

3. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Duality in QCD

4. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay RT∞RT∞RT∞RT∞ …… R  T (n) QCD ∞ Infinite resonance Regularization CUT-OFFRegularization

5. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay LOCAL DUALITY R  T ∞  QCD ∞ pQCDpQCD Local Duality NOT POSSIBLE at LO in 1/N C ! …

6. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay QCD Green-functions NOT BEING order-parameters

7. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay - UV divergent - Scale dependent - Logarithmic behaviour - Physical & finite  independent - Logs within  s pQCDShort distance behaviour in pQCD for (V+A): Left-left correlator in QCD ∞ [ Shiftman et al. ’76 ] & AA correlator & VV correlator

8. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay UV finite when M ∞ 2  ∞ with q 2 = -Q 2 < 0 UVDIVERGENT when M ∞ 2  ∞ Resonance Effective Theory R  T at Large N C Resonance description:Resonance description:

9. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay From series to integral  and the interspacing with the function Integral representation of discrete series:Integral representation of discrete series: [ Peris et al. ´98 ] [ Beane ’ 01 ] [ Golterman & Peris ‘ 01 ] …

10. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Relation with sum-rules: R  T Infinite series from R  T Short distance QCD : Information about higher multiplets Short distance QCD : Information about higher multipletspQCD QCD dispersion relations

11. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay High mass resonances: Origin of the “Continuum” Convergence of H(t) around Truncated R  T “pQCD Continuum” s Mc2Mc2Mc2Mc2 “CONTINUUM”

12. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay High and low energy Hierarchies Analytical Non-analytical Analytical pQCD:  PT: DOMINANT DOMINANT

13. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Phenomenology: V+A Adler-function (without pion pole) NO “CONTINUUM” Truncated R  T + “CONTINUUM” “CONTINUUM”

14. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Conclusions

15. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay QCD R  T (local & global) -Duality: QCD R  T (local & global) -QCD at large N C : Infinite resonance summation Hierarchy Truncated R  T + pQCD continuum Logarithmic & UV behaviour Phenomenologicalsuccess Short distances: Short distances: Continuum dominance Long distances: Long distances: Lightest resonances

16. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay OUTLOOK: Form-factors (Correlators at NLO in 1/N C ) Form-factors (Correlators at NLO in 1/N C ) Other non-order-parameter Green-funtions Other non-order-parameter Green-funtions Scattering amplitudes Scattering amplitudes ? ? ?

17. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay

18. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay In addition: In addition: LOCAL DUALITY AVERAGED (GLOBAL) DUALITY DUALITY In general:In general:  for the scalar correlator Example  for the scalar correlator

19. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay High and low energy Hierarchies

20. Frascati-EU4CM, February 8 th 2005J.J. Sanz Cillero, IN2P3-Orsay Phenomenology: Vector Adler-function (without pion pole) NO CONTINUUM Truncated R  T + CONTINUUM CONTINUUM