1. * Collaborators: A. Pich, J. Portolés (Valencia, España), P. Roig (UNAM, México) Daniel Gómez Dumm * IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias Exactas Universidad de La Plata, Argentina Hadronic  decays in resonance chiral theory

2. Motivation A major task in particle physics: study of QCD interactions in the low / intermediate energy regime Hadronic  decays: ideal laboratory for the study of QCD currents Intermediate energies: m  = 1.777684(17) GeV Many channels: dynamics of different resonant hadron states involved Large amount of experimental data (BABAR, BELLE, CLEO...) Clean information on vector and axial-vector hadronic currents

3. : Factorizable interaction  τ Hadrons W Amplitude ( + EW loops )  decays into hadrons V – A hadronic current Nonperturbative QCD – What to do? Structure of the hadronic current: intermediate resonance states E.g.  

4. Asymptotic behaviour QCD Perturbative regime Chiral symmetry Chiral Perturbation Theory (ChPT) Resonance Chiral Theory (RChT)

5. Standard approach : ( e.g. KS model [J.H. Kühn, A. Santamaria, Z Phys. C 58 (1993) 445] ) “Broad” resonances  R (s) dependence needed (Ansatz based on general dynamical arguments) Everything introduced ad-hoc … Is this consistent with QCD ? Is it possible to address the problem from first principles ? General form of the hadronic tensor H μ Form factors matched with lowest order ChPT at threshold energies Peaks in the spectral functions (resonances) modelled with Breit-Wigner functions allowed Lorentz structureform factor

6. Resonance Chiral Theory: towards a model-independent approach  Extension of ChPT to intermediate energies  Most general Lagrangian compatible with QCD chiral symmetry  Resonances included as dynamical states  Asymptotic behaviour ruled by high energy QCD  Isospin symmetry  Expansion in 1/N C – resonances included only at tree level  Only lightest resonances taken into account  Fits to experimental data    2h   strongly dominated by well-described  and K * resonances Many hadronic  decay channels to analyze Our goal: study of    3h   processes ( 3h = , KK , , K ,  K , … )

7. Simplest processes:         ,       0    0   Intermediate resonances  and a 1 ( + excited states) poorly known –  (PDG) p 1 – p 2 symmetry Hadronic tensor given by only axial current allowed Goal: use our dynamical approach to determine

8. ChPT – O ( p 2 ) RChT + 1/N C Effective Lagrangian in RChT ... plus masses and widths of resonances (coming through resonance propagators) in terms of six parameters: couplings F V, G V, F A, ,,  linear combinations of the i ’s

9. QCD constraints Asymptotic behaviour of the axial two-point function Asymptotic behaviour of two-pion vector form factor Asymptotic behaviour of the  VAP  Green function (only first resonances) Free couplings remaining: F V, F A Perturbative QCD: behaviour of two-point Green functions of vector and axial-vector currents in the large Q 2 limit One gets: [ V. Cirigliano et al., Phys. Lett. B 596 (2004) 96 ] – plus resonance masses and widths

10. a 1 meson width : similar procedure (more involved – two loop diagrams) Off-shell resonance widths: resummations in RChT (one loop level) where One gets [ D.G.D., A. Pich, J. Portolés, Phys. Rev. D 62 (2000) 054014 ] E.g.  meson width : absorptive parts of pion loops [ D.G.D., A. Pich, J. Portolés, Phys. Rev. D 69 (2004) 073002 ] – Previous analyses with ad-hoc a 1 width

11. Fit to experimental data: BR and spectral function d  /dQ 2 for       From pion vector form factor : F V = 0.180 GeV F A = 0.149 GeV M a 1 = 1.12 GeV Fit results : PDG: M a 1 = 1.23  0.04 GeV,  a 1 = 250 to 600 MeV a 1 on-shell width (prediction):  a 1 = 480 MeV [ ALEPH: R. Barate et al., Eur. Phys. C 4 (1998) 409 ] [ D.G.D., P. Roig, A. Pich, J. Portolés, Phys. Lett. B 685 (2010) 158 ] Inclusion of the  (1450) resonance to improve the fit at large Q 2 (mixing coefficient  =  0.25)

12. WZW contribution + tree level diagrams with one / two vector resonances ( , , , K * )               ’         channels dominated by the vector contribution to the V – A hadronic amplitude. Form factor Other strangeness-conserving procresses:   ’     ,       Effective Lagrangian for the vector current: new couplings g i, c i, d i Total of 16 new coupling parameters – but only 10 combinations in

13. Explicit form of the interaction terms:

14. Same procedure as before allows to reduce degrees of freedom : Only two free couplings, c 3 and d 2  F V, G V from       0    0   analysis  Ten combinations of c i, d i, g i in the vector form factor  Asymptotic behaviour leads to seven constraints  Additional constraint on VPPP couplings from   3  decay width  Moreover:           differential decay width related to e + e        cross section : where [ D.G.D., P. Roig, Phys. Rev. D 86 (2012) 076009 ]

15. Constraints in the c 3 – d 2 plane from BR(           ) and fit to the spectral function Fits to experimental data: Predictions: e + e        low energy cross section and BR(     ’       )

16.        channels:      K + K        K  K 0    ,      K  0 K 0     Spectral functions given by W A : same parameters as in    Both vector and axial-vector contributions to the V – A hadronic amplitude  Threshold at Q 2 = ( 2 m K + m  ) 2 = 1.28 GeV 2 W S : negligibly small W V : tree level diagrams with one / two vector resonances ( , , , K * ) [ D.G.D., P. Roig, A. Pich, J. Portolés, Phys. Rev. D 81 (2010) 034031 ]

17. Asymptotic behaviour: constraints consistent with  3  ,   decays Vector form factor related with I = 1 component of e + e   K K  cross section :  Axial-vector form factor fully determined  Vector form factor in terms of couplings c 4 and g 4     K K   branching ratios e + e   K K  cross sections Present experimental information : (not trivial to disentangle the I = 1 component)

18. Results for c 4 =  0.05, g 4 =  0.48 Predictions: vector vs. axial-vector contributions to the spectral function Comparison with KS model (    K + K     ) Comparative results

19. Summary Proper low-energy behavior of form factors General Lagrangian compatible with QCD symmetries – avoids ad-hoc assumptions Asymptotic behavior of form factors ruled by QCD Consistent treatment of off-shell resonance widths We consider a theoretical description of hadronic  decays within a chiral theory that includes light vector and axial-vector resonances as dynamical states Theoretical advantages :  Agreement with experimental data for    3  ,      Predictions for V and A spectral functions of    K K     Analysis of new data expected – Further channels to be studied theoretically (  S = 1 )  Inclusion of heavier resonances would imply to lose predictivity (higher states encoded in effective couplings). Limit in the range of applicability of the approach  Difficulty in the estimation of systematic errors

20. Low energy expansion in   amplitude: problems with KS approach KS model not consistent with low energy QCD chiral expansion at O (p 4 ) 