1. 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)

2. 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

3. 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

4. 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

5. 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, (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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

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

15. 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

16. 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

17. 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  106 (Belle 2005)
5.080.78  106 (BaBar 2005)
Large ACP = +30 11 (Belle)
ACP = +34 13 (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

18. 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

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

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

21. 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