1. Max BaakCKM Workshop 2006, Nagoya 1 1.Introduction 2.SU(3) breaking effects  Soft rescattering effects  W-exchange contributions  Non-factorizable contributions 3.Conclusion SU(3) breaking effects in the SU(3) determination of the amplitude ratio r D(*)h Max Baak, NIKHEF CKM Workshop 2006, Nagoya Thanks to Dan Pirjol!

2. Max BaakCKM Workshop 2006, Nagoya 2 sin(2  +  ) : determination of r D(*)h CP violation in B 0  D (*)  /  proportional to ratio r D(*)h of CKM-suppressed over CKM-favored amplitude. r D(*)h  |V * ub V cd / V cb V * ud |  0.02 Measured CP-asymmetries: 2 observables, 3 unknowns: Simultaneous determination of sin(2  +  ) and r D(*)h from time-evolution impossible with current statistics  need r D(*)h as external inputs ! Imposssible to measure r D(*)h from BR(B +  D +  0 ) (small: 5x10 -7 ) Estimate r D(*)h from B 0  D s (*)+  - /  - using SU(3) symmetry [1] Using: SU(3) [1] I. Dunietz, Phys. Lett. B 427, 179 (1998)

3. Max BaakCKM Workshop 2006, Nagoya 3 Caveat: possible SU(3) breaking Amplitude relation assumes factorization Not (yet) been proven to work for wrong-charm b  u transitions i.e. No theoretical handle on size of non-factorizable contributions involved Three potential sources of SU(3) breaking between D (*) h and D s (*) h : 1.Unknown SU(3) breaking uncertainty from non-factorizable contributions 2.Final state interactions: different rescattering diagrams 3.Missing W-exchange diagrams in calculation Accounted for by introducing theoretical uncertainty on amplitude ratio r D(*)h Size of uncertainty not well understood Typically guestimated to be 30% of size of amplitude ratio.  sin(2  +  ) measurements often excluded in averages of direct  measurements But: SU(3) breaking uncertainties have negligible impact on determination of gamma!

4. Max BaakCKM Workshop 2006, Nagoya 4 Theoretically ‘proven’ in SCET, and using limit N c  . Works experimentally at few % level for b  c color-allowed decays. No theoretical proof for factorization in wrong-charm b  u regime. Few wrong-charm b  u measurements exist yet to indicate whether factorization applies. Factorization B. Aubert et al., Phys.Rev.D74:012001 (2006) b  c transitions A.J. Buras, J-.M. Gerard, and R. Ruckl, Nucl. Phys. B268, 16 (1986) H.D. Politzer and M.B. Wise, Phys. Lett. B257, 399 (1991); M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. Lett. 87, 201806 (2001).

5. Max BaakCKM Workshop 2006, Nagoya 5 Naive factorization: b  u predictions  In analogy of relation of CKM-favored BRs of B  D (*)  /  to B  D (*) l...  No heavy quark symmetry used to obtain relations  Does not include rescattering effects.  B   D s * (B   D s * ) good test channel for factorization!

6. Max BaakCKM Workshop 2006, Nagoya 6  To good approximation, form factor models linear over q 2  [0,15] GeV 2  Average of partial branching fractions over q 2  [0,8] GeV 2 gives estimate at q 2 =m 2 D(*)(s)  Available B  l measurements already yield good prediction for BR(B   D s * ). More later... No need for both theoretical form factor predictions and large error on V ub. Results from B  l q 2 [GeV 2 ] BaBar CLEO Belle 10 4  B d  /dq 2 [GeV -2 ] T. Becher and R.J. Hill, Phys. Lett. B633, 61 (2006) Becirevic-Kaidalov FF model

7. Max BaakCKM Workshop 2006, Nagoya 7 Final state interactions (FSI)  Rescattering effects need not be suppressed at m B scale. [1]  Large rescattering suggested by large BFs of color-suppressed decays B 0  D (*)0  0 /  0 /   Example of SU(3) breaking rescattering diagram in r D(*)h :  No such “annihilation rescattering” for final state D s  /   Rescattering amplitudes hard to calculate. Interactions occur after hadron formation: i.e. rescattering amplitudes after CKM- favored decays B 0  D -  + same as after CKM-suppressed decays B 0  D -  +  Obtain rescattering amplitudes from CKM-favored decays! [1] J.F. Donoghue et al., Phys. Rev. Lett. 77, 2178 (1996)

8. Max BaakCKM Workshop 2006, Nagoya 8 Quasi-elastic rescattering Framework developed by C.-K. Chua, W.-S. Hou, and K.-C. Yang [1] Aim: seperate long-distance rescattering effects from ‘shorter’- distance interactions Rescattering occurs at m B >> m q (q=u,d,s): amplitudes should respect SU(3) to good degree. Extend rescattering between isospin-states to quasi-elastic rescattering between SU(3) states. 3 possible types of rescattering: (c-quark left unchanged) 1. quark exchange: r e 2. quark annihilation: r a 3. gluon exchange: r 0 [1] Chua, Hou, Yang, Phys. Rev. D65: 096007, 2002.

9. Max BaakCKM Workshop 2006, Nagoya 9 Example rescattering matrix  Strong interactions rescattering matrix:  Decompose:   ’ falls outside SU(3) octet  independent rescattering amplitudes for D (*)  ’  Similar rescattering matrices for B  D * P and B  DV modes.  T, C, E: ‘bare’ tree, color-suppressed, W-exchange amplitudes. Calculate T amplitude using factorization. Fit for C (a 2 ) and E amplitudes.  Fit model requires many input parameters (masses, form factors, decay constants, CKM-constants). See backup slides for numbers used. Rescattering matrix for B  DP Bare decay amplitudes X

10. Max BaakCKM Workshop 2006, Nagoya 10 Fit parameters of S  Change of parameters to facilitate calculation of S 1/2 :  To obtain S 1/2, divide all phases by two: r i  r i ’ Fit Parameters # D P / D * P rescattering angles3 D V rescattering angle1 a 2 Wilson param. (col.-sup.)1 W-exchange amplitudes3  -  ’ mixing angle (fixed) - Free parameters8 BR measurements26 DP, D * P, DV No annihilation rescattering for B  DV 1. 2. [1] Chua, Hou, Yang, Phys. Rev. D65: 096007, 2002. zero phases: no rescattering

11. Max BaakCKM Workshop 2006, Nagoya 11 Fit results 1  Good agreement between all rescattering predictions and measured branching fractions. Fac.:  2 = 351.3 Fac.+Resc.:  2 /d.o.f. = 18.2/18  Rescattering contributions dominate color-suppressed BRs.  Fit consistent with single a 2 for all color-suppressed amplitudes C.  Value of a 2 (0.22  0.02) consistent with naive factorization. (a 1  1.02)  Good predictive power! PRD65, 096007 (2002) already made correct predictions for B  D s (*) K, D (*)0 K (*) BR (x10 -4 )

12. Max BaakCKM Workshop 2006, Nagoya 12 Fit results 2 Cross checks: Rescattering angles found compatible with Chua, Hou, Yang [1] Fit for a 1 : 1.06  0.03 (fixed: 1.02) Fit for eta-eta’ mixing angle : -20  28  (fixed: -16) [1] Chua, Hou, Yang, Phys. Rev. D65: 096007, 2002.

13. Max BaakCKM Workshop 2006, Nagoya 13 BR predictions for b  u decays  Assuming naive factorization and rescattering results...  Ignoring possible non- factorizable contributions.  Large rescattering contributions to B +  D (*)0 K (*)+  large rescattering phases.  Good agreement within errors.  Form-factor uncertainties dominant for most BR predictions.  B 0  D s *  : ‘golden’ decay channel to determine size of non-factorizable contributions!  Low BR(B 0  D s  ) explained BaBar Belle BR (x10 -5 )

14. Max BaakCKM Workshop 2006, Nagoya 14 SU(3) correction from rescattering  More b  u decays (V cd )  Hardly any rescattering contributions from color-suppressed decays to color- allowed decays.  i.e. To good approximation, color-allowed BRs only affected by diagonal rescattering matrix elements  Naive amplitude ratio :  SU(3) rescattering correction factor R i to amplitude ratio r D(*)h :  Values of R i : 1. No annihilation rescattering for B 0  h - D s (*)+ 2. Kinematic factor

15. Max BaakCKM Workshop 2006, Nagoya 15 W-exchange amplitudes  SU(3) breaking error on r[D (*) h] from missing exchange diagram:  Ignores rescattering contribution to D s K  overestimation of E  W-exchange amplitudes from rescattering fit consistent with naive factorization estimates! No exchange diagram for final state D s  /  b  u transition  Large uncertainty on |E/T| estimate for b  u transition: 1.Factorization uncertainty for b  u 2.Value of Callan-Treiman prediction: |E/T|<5.0%  Add 200% error on predicted ratio: b  c transition

16. Max BaakCKM Workshop 2006, Nagoya 16  Non-factorizable b  u contributions  Relative size of non-factorizable amplitude:  Limit should improve with updates of: BR(B  l ), BR(B 0  - D s *+ ), f Ds.  Two definitions to describe SU(3) breaking from non-factorizable corrections: typical SU(3) perturbation parameter SU(3) breaking parameter in amplitude ratio r D(*)h (contains 1 unit of 2m s /   ) 1. 2.

17. Max BaakCKM Workshop 2006, Nagoya 17 Non-factorizable SU(3) breaking  SU(3) breaking in amplitude ratio r from non-factorizable contributions:  Additional SU(3) breaking proportional to non-factorizable contributions times perturbation parameter  Assuming up to 3 times typical SU(3) breaking scale for B 0  - D (s) *+ : < 0.17 @ 68.3% CL Absorbs one unit of 2m s /   Non-factorizable amplitude B 0  h - D (*)+

18. Max BaakCKM Workshop 2006, Nagoya 18 Amplitude ratios  Amplitude ratios after rescattering correction:  New since PDG ‘06: large uncertaintainty from V cs  We add 9% Gaussian errors for SU(3) from non- factorizable contributions and 5% flat errors for SU(3) breaking from W-exchange diagrams. No SU(3) uncertainties included

19. Max BaakCKM Workshop 2006, Nagoya 19 Impact on   Extraction dominated by experimental uncertainties  Improved SU(3) breaking errors have negligible impact on determination of  !  With SU(3) breaking:   [-0.28,1.91] rad @ 90% CL  No SU(3) breaking:   [-0.27,1.90] rad @ 90% CL HFAG ICHEP 2006: a,c CP asymmetries CKMfitter frequentist recipe 3 SU(3) breaking error scenarios f: flat error g: Gaussian error  [rad]

20. Max BaakCKM Workshop 2006, Nagoya 20 Conclusions  Improved SU(3) breaking uncertainties have negligible impact on determination of  !  Partial branching fractions of B  l provide good handle on non-factorizable contributions to B 0   - D s *+  Soft rescattering amplitudes can be determined precisely from CKM-favored decays B  D (*)  and B  D  using a quasi-elastic rescattering framework. Small SU(3) breaking corrections to r D(*)h.  Simple use of BR(B 0  D s -  + ) gives overestimate of E/T. Naive conversion ignores dominant contribution from rescattering.  Estimates of E from rescattering fit consistent with naive factorization estimates.  |E/T| < 5%  SU(3) breaking uncertainty to r D(*)h from non-factorizable contributions suppressed with SU(3) perturbation scale  uncertainty < 9% @ 68% CL Include sin(2  +  ) measurements in world average of  !

21. Max BaakCKM Workshop 2006, Nagoya 21 CP violation in B 0  D (*)  /   CP violation through B 0 -B 0 mixing and interference of amplitudes: Large BF’s, at level of 1% No penguin pollution  theoretically clean  CP violation proportional to ratio r D(*)h of amplitudes  Small: r D(*)h  |V * ub V cd / V cb V * ud |  0.020  Relative weak phase  from b  u transition  Relative strong phase  Strong phase difference   CKM Unitarity Triangle Favored amplitude Suppressed amplitude through b  u transition u,c,t

22. Max BaakCKM Workshop 2006, Nagoya 22 World Average CP-asymmetries  3.4  CP violation in B 0  D *  !  The only solution with  is statistics... HFAG ICHEP 2006 Pity to ignore these measurements!

23. Max BaakCKM Workshop 2006, Nagoya 23 Rescattering formalism 1.Time-reversal operator gives rescattering relation: 2.Strong interactions rescattering matrix S 3.Rescattering relation can formally be solved as: A i b represent real, ‘bare’ decay amplitudes – in absence of rescattering effects.  For A i b we test the predictions from naive factorization

24. Max BaakCKM Workshop 2006, Nagoya 24 Linearity FF models B   l B   l

25. Max BaakCKM Workshop 2006, Nagoya 25 SU(3) breaking in B  D (s) (*) D (*)

26. Max BaakCKM Workshop 2006, Nagoya 26 Decay Constants 1  Alle numbers taken from PDG06 where possible...  From factor values from: - Melikhov & Stech - Braun & Zwicky - Ball

27. Max BaakCKM Workshop 2006, Nagoya 27 Decay Constants 2 Callan-Treiman relation