Final-State Interactions in B  ( )S,P K and B  ( )S,P K Decays: f0 and  Resonances B. El-Bennich, A. Furman, R. Kamiński, L. Leśniak, B. Loiseau (LPNHE-Paris and INP-Kraków Collaboration)

Motivation Experimental results at SLAC and KEK (charmless three-body decays more frequent than two-body ones) Direct CP-violation Establish the role of hadronic long-distance effects (charming penguins, meson-meson interactions) Final-state interactions in unitarized description (replacement of a sum of Breit-Wigner terms by analytical meson-meson amplitude) Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Menu du Jour Summary of B ()S K results by A. Furman et al. Phys. Lett. B622, 207 (2005) Extension to  and interactions in P-wave Current improvements Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

I. Three-Body Decay Reactions in S-Wave Examples of quasi two-body reactions: B±  f0(980)K± with subsequent f0(980) (+–)S or f0(980)  (K+K–)S , where (+–)S and (K+K–)S mean +– and pairs in S-wave I=0 state. The effective mass range is: 2m < m < 1.2 GeV Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Decay Amplitudes for B  ()S K and B  (KK)S K reactions Weak decay amplitudes and in QCD factorization approximation built from NLL Wilson coefficients following Beneke et al. Nucl. Phys. B606, 245 (2001) and de Groot et al., Phys. Rev. D 68, 113005 (2003) Hard scattering with spectator quark and annihilation topologies not included (later… ) Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Quark-Line Topologies for B f0(980)K  Example:  For B0 decays no tree diagram (a), only penguin diagrams similar to ones in (b) or (c) Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Formation of Meson Pairs The or transition into or described by 4 scalar form factors by Meißner and Oller, Nucl. Phys. A679, 671 (2001): f = 92.4 MeV Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Complete Amplitude for B  ()SK Normalization constant  fitted to branching ratio B±  f0(980)K± but can also be estimated from: Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Charming Penguins C(m) is too small by a factor of 4 : cancellation due to and r ≈ 1 If Long distance contribution, considered by N. de Groot et al. to improve their fit to hadronic charmless strange and non-strange two-body decays B-decay data: CHARMING PENGUINS: ENHANCED CHARM QUARK LOOPS M. Ciuchini, E. Franco, G. Martinelli, M. Pierini and L. Silvestrini, Phys. Lett. B515, 33 (2001) P1GIM, P1 are complex parameters determined by de Groot et al. and also in by M. Ciuchini et al. to fit some charmless two-body B-decay data. m = m or mK , FB  (0): B  transition form factors They could correspond to Parametrized as: Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Final-State Interactions There are two coupled channels in the S-wave (I = 0) state:  and KK   T() T( ) T( ) T(  ) The 4 scalar form factors are incorporated in the unitarity-constraint formulae: Model of R. Kamiński, L. Leśniak and B. Loiseau, 1997 and 1999 Gi (m): free Green’s function. Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Production functions initial formation of meson pair prior to scattering, derived by Meißner and Oller in one-loop chiral perturbation theory: Using on shell contribution for Below threshold: Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Energy dependence of I=0 S-wave: a)  phase shifts and b)  inelasticity Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Results for B  f0(980)K Average branching fractions in units of 10-6 Model I: C(m) of de Groot et al. Model II: C(m) of Ciuchini et al. Average branching fractions in units of 10-6 Model errors from uncertainty on C(m); A(t) = S sin (∆mt) + A cos (∆mt) Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Comparison with BaBar (2005) Model I Comparison with Belle (2005) Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Comparison with BaBar (2005) Model I  If C(m)=0, prediction drops by a factor of 18: near cancellation of the two-penguin diagrams + absence of tree diagram Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Comparison with the spectrum of Belle (2004) for Model I Note non-zero contribution at f0(600) in S-wave. Presence of  (770) (= /2) Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

II. Three-Body Decay Reactions in P-Wave Clear presence of (770) in effective mass m distribution In previous work P-wave interactions not included Experimental branching ratio for B (770)0 K : 3.890.470.29  106 (Belle 2005) 5.080.780.39  106 (BaBar 2005) Large ACP = +30 113.0+11 (Belle) ACP = +34 136.0+15 (BaBar) QCDF predictions for Br [B 0 K] are lower: 1.54  106, Leitner et al., J. Phys. G31, 199 (2005) 2.6  106, Beneke and Neubert, Nucl. Phys. B675, 333 (2003) +0.32 0.29 +0.22 0.66 4 20 Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Quark-Line Topologies for B 0K, 0()P  One additional tree diagram where U and W are functions of CKM elements and weak decay effective coefficients ai() in QCD factorization, C(m) possible charming penguin contributions. with The amplitude is Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

For the  vertex function a Breit-Wigner function is used where m = 775.8 MeV,  = 150.3 MeV, g = 3m2  /(2p2) = 6 Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

Adding the S- wave contributions: Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006

III. Current Improvements Introduction of annihilation diagrams Taking into account hard-scattering with spectator quark Study of importance of charming penguins Bruno EL-Bennich, Three Body Charmless B-Decays Workshop, LPNHE Paris, February 2006