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Understanding Duality in QCD∞ Resonance Theory with RcT∞

This study delves into the analysis of QCD correlators, exploring their physical content and relation to RcT sum rules. Examining truncated RcT with pQCD continuum, it discusses local duality challenges and resonance theory implications in QCD at large NC. The convergence of high mass resonances and continuum behaviors are highlighted, emphasizing the dominance of pQCD in analytical frameworks and the phenomenological relevance of specific correlators.

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Understanding Duality in QCD∞ Resonance Theory with RcT∞

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  1. Recovering QCD at large NC:Two-point Green-functions J. J. Sanz Cillero, IN2P3 - Orsay

  2. DUALITY QCD at large NC (QCD∞) Resonance Theory (RcT∞) Outline: • Analysis of QCD correlators: • a)- Physical QCD content • b)- Relation RcT Sum-rules • c)- Truncated RcT + “pQCD Continuum” • Conclusions

  3. Duality in QCD

  4. QCD∞ Infinite resonance Regularization CUT-OFF Regularization …… RcT∞ RcT(n)

  5. pQCD pQCD LOCAL DUALITY RcT∞QCD∞ … Local Duality NOT POSSIBLE at LO in 1/NC !

  6. QCD Green-functions NOT BEING order-parameters

  7. UV divergent • Scale dependent • Logarithmic behaviour • Physical & finite • m independent • Logs within as Left-left correlator inQCD∞ & AA correlator & VV correlator • Short distance behaviour inpQCDfor (V+A): [ Shiftman et al. ’76 ]

  8. UV DIVERGENT when M∞2∞ UV finite when M∞2∞ • Resonance description: Resonance Effective Theory with q2= -Q2 < 0 RcT at Large NC

  9. [ Peris et al. ´98 ] [ Beane ’ 01 ] [ Golterman & Peris ‘ 01 ] … • Integral representation of discrete series: From series to integral with the function and the interspacing

  10. Short distance QCD : Information about higher multiplets pQCD Relation with sum-rules: QCDdispersion relations Infinite series fromRcT

  11. High mass resonances: Origin of the “Continuum” Convergence of H(t) around Truncated RcT “CONTINUUM” “pQCD Continuum” s Mc2

  12. High and low energy Hierarchies Analytical pQCD: DOMINANT Non-analytical cPT: Analytical DOMINANT

  13. Phenomenology: V+A Adler-function NO “CONTINUUM” (without pion pole) Truncated RcT + “CONTINUUM” (without pion pole)

  14. Conclusions

  15. Duality: QCDRcT (local & global) QCDat large NC : Infinite resonance summation Hierarchy Truncated RcT + pQCD continuum Logarithmic & UV behaviour Short distances: Continuum dominance Long distances: Lightest resonances Phenomenological success

  16. OUTLOOK: • Form-factors (Correlators at NLO in 1/NC) • Other non-order-parameter Green-funtions • Scattering amplitudes ? ? ?

  17. In general: Example  for the scalar correlator • In addition: LOCAL DUALITY AVERAGED (GLOBAL) DUALITY

  18. High and low energy Hierarchies

  19. Phenomenology: Vector Adler-function NO CONTINUUM (without pion pole) Truncated RcT + CONTINUUM (without pion pole)

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