Ignasi Rosell Universidad CEU Cardenal Herrera 2007 Determining chiral couplings at NLO: and JHEP 0408 (2004) 042 [hep-ph/ ] JHEP 0701 (2007) 039 [hep-ph/ ] In collaboration with: A. Pich (IFIC and Valencia U.) J.J. Sanz-Cillero (Peking U.)

OUTLINE 1) A few words about Chiral Perturbation Theory 2) Resonance Chiral Theory 3) Quantum loops in RChT 4) Form Factors and short distance constraints 5) Chiral couplings at NLO: and 6) Summary

1. A few words about Chiral Perturbation Theory Chiral Perturbation Theory (ChPT)* running of α s ( < 0) Asymptotic Freedom: pQCD Confinement: non p-QCD PROBLEM!!!a SOLUTION: Effective Field Theories ChPT: EFT of QCD at very low energies 1. Massless limit → chiral invariant 2. Global symmetries → spectrum 3. CCWZ formalism → build effective lagrangians with SSB representations multiplet much lighter * Weinberg ’79 * Gasser and Leutwyler ’84 ’85 * Bijnens, Colangelo and Ecker ’99 ‘00

1. 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 : required LEC’s * Callan et al. ’69 * Coleman et al.’69

2. Resonance Chiral Theory Resonance Chiral Theory (RChT)* QCD at 1st Problem 2nd Problem Many resonances and no mass gap No FORMAL EFT approach No natural expansion parameter RChT i)Phenomenological lagrangians ii)Large-N C QCD iii) Short-distance properties of QCD Improving Phenomenological lagrangians à la Weinberg** * Ecker et al. ’89 * Cirigliano et al. ’06 ** Weinberg ‘79

2. Resonance Chiral Theory Large-N C QCD* Assuming confinement, in the limit the Green Functions of QCD are described by the tree diagrams of an effective lagrangian with local vertices and meson fields. Higher corrections in are yielded by loops described within the same lagrangian theory. At the mesons and glue states are free, stable and non- interacting. Zweig’s rule is exact in the large-N C limit, mesons should be classified as nonets. In the limit of large number of colours, symmetry spontaneously breaks down to **. The number of meson states is infinite. * ‘t Hooft ’74 * Witten ’79 The expansion is observed to give a good quantitative approximation to the hadronic world. ** Coleman and Witten ‘80

2. Resonance Chiral Theory Constructing RChT 1) Phenomenological lagrangians à la Weinberg: –Considering the most general lagrangian, consistent with assumed symmetry principles. 2) Ruled by the 1/N C Expansion: –Degrees of freedom: pGB and resonances. –Mesons classified as nonets. –Only operators with one trace. 3) Matching – low energies: ChPT → chiral symmetry – high energies: OPE* and form factors** MODEL DEPENDENT: cut in the tower of resonances WHY? i)Supported by phenomenology ii)Heavier contributions suppressed by their masses iii)Technical reasons i)NO interactions with large number of derivatives ii)relation between couplings * Ecker et al. ’89 * Moussallam ’95 ’97 * De Rafael et al. ’98 ’02 * Knecht and Nyffeler ‘01 * Bijnens et al. ’03 * Cirigliano et al. ’04 ’05 ’06 * Ruiz-Femenía, Pich and Portolés ’03 ** Brodsky and Lepage ’79 ’80 ‘81

2. Resonance Chiral Theory ChPTQCDRChT predictions of the chiral LEC’s reduction of the unknown couplings within large-N C it is understood the saturation of the ChPT couplings LO prediction of LEC’s* * Ecker et al. ‘89

3. Quantum loops in Resonance Chiral Theory WHY QUANTUM LOOPS IN THE RChT?* Dyson-Schwinger resummation of subleading contributions to describe the amplitudes near the resonance peak Distinguish New Physics effects from Standard Model Resonance contribution to chiral LEC’s at NLO: control of μ dependence matching with ChPT Many phenomenological problems in the hadronic contributions First theoretical prediction of the ChPT L i at NLO * Rosell, Sanz-Cillero and Pich ‘04 Physics in the resonance region Improvement of the implementation of non-perturbative QCD

4. Form factors and short-distance constraints Form factors and short-distance constraints* Short-distance constraints from QCD OPE Form factors Resonances as asymptotic states? Motivated by the calculation of the vector form factor of the pion at NLO Correlators at NLO in the expansion Brodsky-Lepage i)Physical large- energy behaviour ii)Prediction of chiral LEC’s at NLO * Rosell, Sanz-Cillero and Pich ‘06

4. Form factors and short-distance constraints i) Two-point correlation functions of two QCD currents in the chiral limit ii) Associated spectral functions sum of positive contributions corresponding to the different intermediate states In the limit: a) tends to a constant* b) grows as * Brodsky-Lepage rules of the form factors vanishes as * behaviour of one-particle exchange VANISHING FORM FACTORS AT LARGE ENERGIES AS A REASONABLE ASSUMPTION * Floratos, Narison and De Rafael ’79 * Pascual and De Rafael ’82 * Shifman, Vainshtein and Zakharov ’79 * Jamin and Munz ‘95

5. Chiral couplings at NLO Towards a determination of the chiral LEC’s at NLO in : and i) The large-N C limit in RChT ChPT at NLO in Short-distance behaviour Low-energy expansion: LEC’s at LO

5. Chiral couplings at NLO ii) NLO corrections Short-distance behaviour Low-energy expansion: LEC’s at NLO phenomenology*: phenomenology: dispersive calculation Integration of the single field * Amorós, Bijnens and Talavera ‘01

5. Chiral couplings at NLO iii) Saturation at NLO ChPT LEC’s RChT couplings low-energy expansion of the resonance contributions SATURATION* i)determined value of ii)any value of ** a)At LO: asymptotic behaviour → →importance of high-energy constraints b)At NLO: one-loop diagrams generated by RChT lagrangian * Ecker et al. ’89 ** Catà and Peris ‘02 tree-level one-loop terms + + NO term a) do not run b) at NLO

6. Summary Summary Framework. QCD at. RChT as a procedure to incorporate the massive mesonic states within an effective lagrangian formalism. Phenomenological lagrangian à la Weinberg ruled by the expansion and improved by using the short- distance properties of QCD Motivation. Why quantum loops in the RChT? Improvement of the implementation of non-perturbative QCD. i)Distinguish New Physics effects from Standard Model. ii)Prediction of chiral LEC’s at NLO. Determination of chiral couplings at NLO. i) Vanishing form factors with resonances as asymptotic states at large energies as a reasonable assumption. ii)Determination of the chiral LEC’s at NLO in the expansion: