1. Ignasi Rosell Universidad CEU Cardenal Herrera IFIC, CSIC–Universitat de València Revisiting the vector form factor at NLO in 1/N C QCD10, 29th June 2010 In collaboration with: A. Pich (IFIC) J.J. Sanz-Cillero (IFAE) Work in progress Related works: JHEP 07 (2008) 014 [arXiv:0803.1567] JHEP 01 (2007) 039 [hep-ph/0610290] JHEP 08 (2004) 042 [hep-ph/0407240]

2. 2/15 OUTLINE 1)Motivation 2) The framework:ChPT and RChT 3) Towards a determination of thechiral LECs 4)Why revisiting the Vector Form Factor? 5)The Vector Form Factor within RChT 6) The chiral couplings L 9 and (C 88 – C 90 ) 7)Phenomenology 8)Summary Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell

3. 1. Motivation i)The amplitude: ii)The framework: Chiral Perturbation Theory (ChPT) up toO(p 6 ) * Resonance Chiral Theory (RChT) i)Main aims: Physics in the resonance region and estimation of related LECs (L 9 ) in theresonanceregion up toO(N C 0 ) ** at therhomesonpeak up to O(1/N C ) *** in theresonanceregion up to O(1/N C ) **** Correct framework to incorporate the resonance states within an effective lagrangian formalism. Needto be improved * Gasser &Leutwyler ’84 * Bijnens et al. ’98 ’02 ** Ecker et al. ‘89 *** Guerrero & Pich ’97 **** IR, Sanz-Cillero & Pich ‘04 Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 3/15

4. 2. The framework: ChPT and RChT ChiralPerturbationTheory * ResonanceChiralTheory ** * Weinberg ’79 * Gasser &Leutwyler ‘84 ‘85 * Bijnens et al. ‘99 ‘00 ** Ecker et al. ’89 ** Cirigliano et al. ’06 *** Knecht& de Rafael ‘97 EffectiveFieldTheory (EFT) ofQCD at very-lowenergies. Key-point: L QCD ischiralinvariantin themasslesslimit. Organization in termsofincreasingpowersofmomentumorm asses. GeV. Thenumberofcouplingsincreasesveryfast: 10 at NLO and 90 at NNLO. QCD at GeV. No natural expansionparameterandmanyresonanceswithclos emasses: a formal EFT approachisnotpossible. Mainfeatures: Chiralinvariant. 1/N C expansion. Requirementof a good short- distancebehavior. Modeldependence: truncationofthetowerofresonances***. Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 4/15

5. 2. The framework: ChPT and RChT 3 couplings !!! 9 couplingsand 3 masses !!! ChiralPerturbationTheory * ResonanceChiralTheory ** * Weinberg ’79 * Gasser &Leutwyler ‘84 ‘85 * Bijnens et al. ‘99 ‘00 ** Ecker et al. ’89 ** Cirigliano et al. ‘06 Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 5/15

6. OneofthemajorproblemsofChiralPerturbationTheoryistheestimationofthelow- energyconstants (LECs). ThemostimportantcontributionstotheLECs come fromthephysicsoflow- lyingresonances. ResonanceChiralTheoryis a correctframeworktoincorporatetheresonancestateswithinaneffectivelagrangi anformalism, ruled by the1/N C expansion. At leading-order(LO) in 1/N C resonancesaturationworksproperly. Large-N C estimates are unableto control therenormalization- scaledependenceoftheLECs, which may produce sizablevariations. 3. Towards a determination of the chiral LECs Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 6/15

7. ChPT QCD RChT predictions of LECs reduction of the unknown couplings Resonance saturation Very low energies Resonance regionHigh energies 20072008 2004 and this work 2011 ? Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 7/15

8. 4. Why revisiting the Vector Form Factor? I.R., Sanz-Cillero & Pich ‘04 Thiswork Ourfirstapproachto NLO calculationsin ResonanceChiralTheory DeterminationofLECskeeping a full control oftherenormalizationscaledependenceμ Single ResonanceApproximation Twoflavoursin thechirallimit Operators up tooneresonancefield LO operatorswithup toO(p 2 ) chiralstructures Diagrammaticalcalculation Removeof NLO operators by using EquationsofMotion(fieldredefinitions) Bad-behaved at highenergies Free NLO couplings No ourfirstapproachto NLO calculationsin ResonanceChiralTheory DeterminationofLECskeeping a full control oftherenormalizationscaledependenceμ Single ResonanceApproximation Threeflavoursin thechirallimit Cutswith up tooneresonancefield LO operatorswithup toO(p 2 ) chiralstructures Dispersive/diagrammaticalcalculation Removeofsubtractionconstantsby absorptionintothetree-level NLO contributions Well-behaved at highenergies NO free NLO couplings Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 8/15

9. 5. The Vector Form Factor within Resonance Chiral Theory ChPT at NLO in 1/N C i) Thelarge-NC limit in RChT (treelevel) Short-distancebehaviour No couplingsand 1 mass !!! Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 9/15

10. ii) NLO corrections in RChT (one-looplevel) Single ResonanceApproximation Operatorswith up toO(p 2 ) chiralstructures No needof * Cutswith up tooneresonancefield Dispersive/diagrammaticalcalculation Absorptionofsubtractionconstantsinto 9 couplingsand 3 masses Short-distancebehaviour * Portolés, IR & Ruiz-Femenia ’07 * IR, Ruiz-Femenía& Sanz-Cillero ‘09 3 couplings (F,G V,F A ) and 3 masses (M V,M A,M S ) Consideringconstraintsfromot her observables it can be reducedto 1 coupling (F) and 3 masses ( M V,M A,M S ). ** Ecker et al. ’89 *** Guo et al. ‘07 **** Pich, IR & Sanz-Cillero ‘08 Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 10/15

11. 6. The chiral couplings L 9 and (C 88 –C 90 ) i) The LO estimation ii) The NLO estimation RChT at LO in 1/N C ChPT at NLO in 1/N C ChPT at LO in 1/N C RChT at NLO in 1/N C * Ecker et al. ’89 ** Cirigliano et al. ‘06 Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 11/15

12. 7. Phenomenology Preliminary i) Inputii) Ouput (high- andlow- energycontributions) iv) Literature * Gasser &Leutwyler ’85 ** Bijnens& Talavera ’02 *** Sanz-Cillero & Pich ’03 **** Gonzalez-Alonso et al. ’09 ***** Kaiser ‘05 iii) Ouput (final number) Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 12/15

13. 8. Summary 2. Where? NLO corrections 3. Why? 1. What? 4. How? The Vector Form Factor Dispersive/diagrammatical calculation RChT a)QCD at intermediate energies b)An effective procedure to incorporate the mesonic states c)Ingredients: 1/N C expansion and short-distance information a)Improvement of thePhysics in the resonance region b)Theoretical prediction of the LECs at NLO a)Again? TheestimationoftheLECs as a majorproblemofChPT ThemostimportantcontributionstotheLECs come fromthelightestresonances Therenormalization-scaledependenceissizable Cutswith up tooneresonancefield Well-behavedat highenergies NO free NLO couplings Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 13/15

14. Thedeterminationof L 9 at NLO step by step i) Well-behavedspectralfunctionchannel by channel ii) Well-behaved full Vector Form Factor 11couplingsand3masses3couplingsand3masses iii) MatchingbetweenChPTandRChT iv) Literature Gasser &Leutwyler ’85 Preliminary Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 14/15

15. Nextsteps Detaileduncertaintyestimate Estimationof(C 88 – C 90 ) at NLO Analysisof experimental data (Physics in theresonanceregion) Futurework ScalarForm Factor Pionscattering Revisitingthe Vector Form Factor at NLO in 1/N C, I. Rosell 15/15