1. Resonance saturation at next-to-leading order
Euroflavour08, September 2008
Ignasi Rosell Universidad CEU Cardenal Herrera IFIC, CSIC-Universitat de València
In collaboration with:
J.J. Sanz-Cillero (IFAE)
P.D. Ruiz-Femenía (Aachen)
Phys.Rev.D 75 (2007)

2. OUTLINE Motivation A few words about Chiral Perturbation Theory
Resonance Chiral Theory
Resonance saturation at LO
Resonance saturation at NLO
Summary

3. Resonance saturation at NLO: an appealing task
1. Motivation
A lot of effort to determine chiral low-energy constants (LECs): difficult nonperturbative problem.
Phenomenology and theoretical arguments suggest that the most important contributions to the LECs come from the physics of low-lying resonances.
Resonance Chiral Theory (RChT) as a correct framework to incorporate the resonance states within an effective lagrangian formalism.
Main ingredients of RChT:
Use of large-NC ideas.
Matching procedure between the resonance region and the perturbative regime of QCD.
At leading-order (LO) in 1/NC resonance saturation works properly.
We would like to estimate LECs at next-to-leading order (NLO), since these subleading contributions can be sizable.
Resonance saturation at NLO: an appealing task
1. Motivation

4. 2. A few words about Chiral Perturbation Theory*
Asymptotic Freedom: pQCD
Confinement: non p-QCD
running of αs
( < 0)
a SOLUTION: Effective Field Theories
PROBLEM!!!
ChPT: EFT of QCD at very low energies
Massless limit → chiral invariant
Global symmetries → spectrum
CCWZ formalism → build effective lagrangians with SSB
representations
multiplet much lighter
* Weinberg ’79
* Gasser and Leutwyler ’84 ’85
* Bijnens, Colangelo and Ecker ’99 ‘00
2. A few words about Chiral Perturbation Theory

5. 2. A few words about Chiral Perturbation Theory
The ChPT Lagrangian
By using CCWZ*, the pGB (pion multiplet) can be parameterized with
, one can use only ChPT until scales with
Organization in terms of increasing powers of momentum,
The precision in present phenomenological applications makes necessary to include corrections of NLO: required LECs.
2. A few words about Chiral Perturbation Theory
* Callan et al. ’69
* Coleman et al.’69

6. 3. Resonance Chiral Theory*
Many resonances and no mass gap
1st Problem
QCD at
No FORMAL EFT approach
2nd Problem
No natural expansion parameter
MODEL DEPENDENT:
cut in the tower of
resonances
Tools
Phenomenological lagrangians**
Large-NC QCD***
Short-distance properties of QCD
WHY?
Technical reasons
Supported by phenomenology
Heavier contributions suppressed by their masses
* Ecker et al. ’89
* Cirigliano et al. ’06
** Weinberg ‘79
3. Resonance Chiral Theory
*** ‘t Hooft ’74
*** Witten ’79

7. Matching RChT ChPT QCD Very low energies Resonance region
High energies
RChT
ChPT
QCD
predictions of LECs
reduction of the unknown couplings
3. Resonance Chiral Theory

8. 4. Resonance saturation at LO
i) Main idea: LECs in terms of resonance parameters
ii) LECs
low-energy expansion of
resonance contributions
RChT
coupling
ChPT
coupling
iii) What’s the meaning of resonance saturation?
iv) What happens at LO?*
Short-distance constraints
is the key to get the
resonance saturation +
4. Resonance saturation at LO
* Ecker et al. ’89

9. 5. Resonance saturation at NLO
First result: one-loop renormalization of RChT with only scalar
and pseudoscalar resonances*
operators with
for
After short-distance
constraints
are implemented
Weinberg sum rules
SS-PP sum rules
Well-behaved form factors
Which constraints?
Why? Next slide
5. Resonance saturation at NLO
* Rosell et al. ’05

10. ii) 1st step: only scalar and pseudoscalars *
SS – PP correlators
1) Well-behaved form factors
2) Neither divergent nor finite
part in resonance contributions:
3) SS-PP sum rules and well-
behaved form factors at NLO:
Scalar form factors
Towards an understanging of resonance saturation at NLO
5. Resonance saturation at NLO
* Rosell et al. ’05

11. 5. Resonance saturation at NLO
iii) Which is the problem with vector and axial-vector resonances?
Spin-1 field propagator
“Bad-behaved”
piece at
two k’s
It breaks down our previous argumentation. For instance in the case of one resonance in the
absorptive cut one would have:
There are two possibilities for the resonance saturation at NLO:
“Extreme” version: , so
“Soft” version: , so but fixed by constraints and
without physical relevance.
5. Resonance saturation at NLO

12. Preliminary iv) 2nd step: one spin-1 field resonance in the cut
Only considered the “bad-behaved” piece of the propagator, since the other one follows previous argumentation.
It’s not needed to consider M2 in the numerator, since these contributions are well-behaved.
It’s not needed to consider k2 in the numerator, since these contributions are well-behaved.
They give either M2 multiplying the two-propagator integral (case 2) or they have the structure of the resonance propagators, so we have a one-propagator integral without scale, what vanishes.
At least two k’s in the numerator.
Then, one finally gets in the numerator
The result of the integration reads
Preliminary
“Extreme” version of resonance saturation
5. Resonance saturation at NLO

13. v) 3rd step: two spin-1 field resonances in the cut and only one “bad-behaved”
Only considered the “bad-behaved” piece of one propagator, “well-behaved” pieces follow previous argumentation (scalar and pseudoscalar case).
It’s not needed to consider M2 in the numerator, since these contributions are well-behaved.
At least two k’s in the numerator.
Then, one finally gets in the numerator
There’s a symmetry: (at least in the VV correlator)
The result of the integration reads
Preliminary
“Extreme” version of resonance saturation
5. Resonance saturation at NLO

14. vi) 4th step: two spin-1 field resonances in the cut and both “bad-behaved”
Only considered the “bad-behaved” piece of both propagators.
It’s not needed to consider M2 in the numerator, since these contributions follow argumentation of previous slide.
A global factor is found in the numerator within the RChT lagrangian (VV case).
Then, one finally gets in the numerator
There’s a symmetry:
The result of the integration reads…
Preliminary
vii) 5th step: future work → scattering
Which constraints?*
* Guo et al. ‘07

15. 6. Summary Where? RChT 2. Why? NLO corrections 3. What?
QCD at
An effective procedure to incorporate the mesonic states
Ingredients: 1/NC expansion and short-distance information
Where?
RChT
2. Why?
Improvement of the physics in the resonance region
Theoretical prediction of the LECs at NLO
NLO corrections
RChT
coupling
ChPT
coupling
3. What?
Resonance saturation
“Extreme” version:
“Soft” version: , but it’s fixed by the constraints and without physical relevance.
Weinberg sum rules
SS-PP sum rules
Well-behaved form factors
4. How?
Short-distance constraints
Preliminary result: towards an “extreme” version of saturation
6. Summary