Ignasi RosellUniversidad CEU Cardenal HerreraIFIC, CSIC–Universitat de València

Revisiting the vector form

factor at NLO in 1/NC

QCD10, 29th June 2010

In collaboration with:A. Pich (IFIC)J.J. Sanz-Cillero (IFAE)

Work in progressRelated works:JHEP 07 (2008) 014 [arXiv:0803.1567]JHEP 01 (2007) 039 [hep-ph/0610290]JHEP 08 (2004) 042 [hep-ph/0407240]

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 L9 and (C88– C90)

7) Phenomenology

8) Summary

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell

1. Motivation

i) The amplitude:

ii) The framework:

Chiral Perturbation Theory (ChPT) up toO(p6) *

Resonance Chiral Theory (RChT)

i) Main aims: Physics in the resonance region and estimation of related LECs (L9)

in theresonanceregion up toO(NC0) **

at therhomesonpeak up to O(1/NC) ***in theresonanceregion up to O(1/NC) ****

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/NC, I. Rosell 3/15

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: LQCD ischiralinvariantin themasslesslimit.

•Organization in termsofincreasingpowersofmomentumormasses.

•GeV.

•Thenumberofcouplingsincreasesveryfast: 10 at NLO and 90 at NNLO.

• QCD at GeV.

• No natural expansionparameterandmanyresonanceswithclosemasses: a formal EFT approachisnotpossible.

•Mainfeatures:

•Chiralinvariant.•1/NC expansion.•Requirementof a good short-distancebehavior.

•Modeldependence: truncationofthetowerofresonances***.

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 4/15

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/NC, I. Rosell 5/15

• OneofthemajorproblemsofChiralPerturbationTheoryistheestimationofthelow-energyconstants (LECs).

• ThemostimportantcontributionstotheLECs come fromthephysicsoflow-lyingresonances.

• ResonanceChiralTheoryis a correctframeworktoincorporatetheresonancestateswithinaneffectivelagrangianformalism, ruled by the1/NC expansion.

• At leading-order(LO) in 1/NC resonancesaturationworksproperly.

• Large-NC 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/NC, I. Rosell 6/15

ChPT QCDRChT

predictions of LECs

reduction of the unknown couplings

Resonance saturation

Very low energies Resonance region High energies

20072008

2004 and this work2011 ?

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 7/15

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(p2) chiralstructures

Diagrammaticalcalculation

Removeof NLO operators by usingEquationsofMotion(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(p2) chiralstructures

Dispersive/diagrammaticalcalculation

Removeofsubtractionconstantsby absorptionintothetree-level NLO contributions

Well-behaved at highenergies

NO free NLO couplings

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 8/15

5. The Vector Form Factor within Resonance Chiral Theory

ChPT at NLO in 1/NC

i) Thelarge-NC limit in RChT (treelevel)Short-distancebehaviour

No couplingsand 1 mass !!!

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 9/15

ii) NLO corrections in RChT (one-looplevel) Single ResonanceApproximation

Operatorswith up toO(p2) chiralstructures

No needof *

Cutswith up tooneresonancefield

Dispersive/diagrammaticalcalculation

Absorptionofsubtractionconstantsinto

9 couplingsand 3 massesShort-distancebehaviour

* Portolés, IR & Ruiz-Femenia ’07* IR, Ruiz-Femenía& Sanz-Cillero ‘09

3 couplings (F,GV,FA) and3 masses (MV,MA,MS)

Consideringconstraintsfromother observables it can be reducedto 1 coupling (F) and 3 masses (MV,MA,MS).

** Ecker et al. ’89*** Guo et al. ‘07**** Pich, IR & Sanz-Cillero ‘08

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 10/15

6. The chiral couplings L9 and (C88–C90)

i) The LO estimation

ii) The NLO estimation

RChT at LO in 1/NC

ChPT at NLO in 1/NC

ChPT at LO in 1/NC

RChT at NLO in 1/NC

* Ecker et al. ’89** Cirigliano et al. ‘06

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 11/15

7. Phenomenology

Preliminary

i) Input ii) 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/NC, I. Rosell 12/15

8. Summary

2. Where?

NLO corrections3. Why?

1. What?

4. How?

The Vector Form Factor

Dispersive/diagrammatical calculation

RChTa) QCD at intermediate energies

b) An effective procedure to incorporate the mesonic states

c) Ingredients: 1/NC expansion and short-distance information

a) Improvement of thePhysics in the resonance regionb) 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/NC, I. Rosell 13/15

Thedeterminationof L9 at NLO step by step

i) Well-behavedspectralfunctionchannel by channel

ii) Well-behaved full Vector Form Factor

11couplingsand3masses 3couplingsand3masses

iii) MatchingbetweenChPTandRChT

iv) Literature

•Gasser &Leutwyler ’85

Preliminary

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 14/15

Nextsteps

Detaileduncertaintyestimate

Estimationof(C88– C90) at NLO

Analysisof experimental data (Physics in theresonanceregion)

Futurework

ScalarForm Factor

Pionscattering

Revisitingthe Vector Form Factor at NLO in 1/NC, I. Rosell 15/15