1. 1 Final state interactions in hadronic B decays Hai-Yang Cheng Academia Sinica FSIs BRs & CPV in B decays Polarization anomaly in B  K* QCD & Hadronic Physics, Beijing, June 16-20, 2005

2. 2 Importance of FSI in charm decays has long been recognized some nearby resonances exist at energies  m D charm is not very heavy General folklore for B decays: FSI plays a minor role due to large energy release in B decays and the absence of nearby resonances There are growing hints at some possible soft final-state rescattering effects in B physics

3. 3 B f One needs at least two different B  f paths with distinct weak & strong phases  strong phase  weak phase e i(  +  ) BaBar Belle Average B 0 →K -  + -0.13  0.03-0.10  0.25-0.11  0.02 B0→+-B0→+- -0.47  0.16-0.53  0.30-0.47  0.14 B0→+-B0→+- 0.09  0.16 0.56  0.13 0.37  0.10 first confirmed DCPV (5.7  ) in B decays (2004) 1. Direct CP violation _ _ _ Recall in kaon decays

4. 4 pQCD (Keum, Li, Sanda): A sizable strong phase from penguin- induced annihilation by introducing parton’s transverse momentum QCD factorization (Beneke, Buchalla, Neubert, Sachrajda): Because of endpoint divergences,  QCD /m b power corrections due to annihilation and twist-3 spectator interactions can only be modelled QCDF (S4 scenario): large annihilation with phase chosen so that a correct sign of A(K -  + ) is produced (  A =1,  A = -55  for PP,  A =-20  for PV and  A =-70  for VP) Comparison with theory: pQCD & QCDF input

5. 5 SD perturbative strong phases: penguin (BSS) vertex corrections (BBNS) weak strong Nonperturbative LD strong phases induced from power corrections especially from final-state rescattering annihilation (pQCD) Need sizable strong phases to explain the observed direct CPV If intermediate states are CKM more favored than final states, e.g. B  DD s  K   large phases  large corrections to rate

6. 6 2. Some color-suppressed or factorization-forbidden or penguin- dominated modes cannot be accommodated in the naïve factorization approach Some decay modes do not receive factorizable contributions e.g. B  K  c0 with sizable BR though  c0 |c   (1-  5 )c|0  =0. Color-suppressed modes e.g. B 0  D 0 h 0 (h 0 =  0, ,  0, ,  ’),  0  0,  0  0 have the measured rates larger than theoretical expectations. Penguin-dominated modes such as B  K * , K , K , K *  predicted by QCDF are consistently lower than experiment by a factor of 2  3  importance of power corrections (inverse powers of m b ) e.g. FSI, annihilation, EW penguin, New Physics, …

7. 7 Our goal is to study FSI effect on BRs and CPV (direct & indirect) in B decays (Polarization anomaly in B  K * will be briefly mentioned) LD rescattering can be incorporated in any SD approach but it requires modelling of 1/m b power corrections We would provide a specific model for FSI to compute strong phases so that we can predict (rather than accommodate) the sign and magnitude of direct CP asymmetries

8. 8  Regge approach [Donoghue,Golowich,Petrov,Soares] FSI phase is dominated by inelastic scattering and doesn’t vanish even in m b  limit  QCDF [Beneke,Buchalla,Neubert,Sachrajda] strong phase is O(  s,  /m b ): systematic cancellation of FSIs in m b   Charming penguin [Ciuchini et al.] [Colangelo et al.] [Isola et al.] long distance in nature, sources of strong phases, supported by SCET  One-particle-exchange model for LD rescattering has been applied to charm and B decays [Du et al.][Lu,Zou,..]  Quasi elastic scattering model [Chua,Hou,Yang] Consider MM  MM (M: octet meson) rescattering in B  PP decays  Diagrammatic approach [Chiang, Gronau, Rosner et al.] … Approaches for FSIs in charmless B decays

9. 9 All two-body hadronic decays of heavy mesons can be expressed in terms of six distinct quark diagrams [Chau, HYC(86)] All quark graphs are topological and meant to have all strong interactions included and hence they are not Feynman graphs. And SU(3) flavor symmetry is assumed. Diagrammatic Approach (penguin) (vertical W loop) (tree) (color-suppressed) (exchange) (annihilation)

10. 10 Global fit to B , K  data (BRs & DCPV) based on topological diagrammatic approach yields [Chiang et al.] consistent with that determined from B  D  decays

11. 11 quark exchange quark annihilation meson annihilation possible FSIs  W exchange Color suppressed C  At hadron level, FSIs manifest as resonant s-channel & OPE t-channel graphs B0D00B0D00 relevant for e.g. B 0 

12. 12 FSI as rescattering of intermediate two-body states [HYC, Chua, Soni]  FSIs via resonances are assumed to be suppressed in B decays due to the lack of resonances at energies close to B mass.  FSI is assumed to be dominated by rescattering of two-body intermediate states with one particle exchange in t-channel. Its absorptive part is computed via optical theorem: Strong coupling is fixed on shell. For intermediate heavy mesons, apply HQET+ChPT Form factor or cutoff must be introduced as exchanged particle is off-shell and final states are necessarily hard Alternative: Regge trajectory [Nardulli,Pham][Falk et al.] [Du et al.] …

13. 13 Dispersive part is obtained from the absorptive amplitude via dispersion relation  = m exc + r  QCD (r: of order unity)  or r is determined by a fit to the measured rates  r is process dependent  n=1 (monopole behavior), consistent with QCD sum rules Once cutoff is fixed  direct CPV can be predicted subject to large uncertainties and will be ignored in the present work Form factor is introduced to render perturbative calculation meaningful  LD amp. vanishes in HQ limit

14. 14 Inputs Form factors: covariant light-front approach: relativistic QM for s-wave to s- wave and p-wave transitions (HYC,Chua,Hwang 2004) CLFBall & ZwickyBeneke & Neubert F 0 B  (0) 0.25  0.030.258  0.0310.28  0.05 A 0 B  (0) 0.28  0.030.303  0.0280.37  0.06 SD approach: QCD factorization (default scenario) with  A =  H =0 in double counting problem is circumvented

15. 15 Theoretical uncertainties (SD) 1.variation of CKM parameters  =(63  15)  2.quark masses: m s (2 GeV)= 90  20 MeV 3.renormalization scale: from  =2m b to m b /2 4.heavy-to-light form factors: e.g. F B  (0)=0.25  0.03 5.meson distribution amplitudes

16. 16 Theoretical uncertainties (LD) 1. Model assumption multi-body contributions form-factor cutoff: i). n=1 ii).  = m exc + r  QCD (15% error assigned for  QCD ) r D =2.1, 1.6, 0.73, 0.67, respectively, for D , , K  modes varies for penguin-dominated PV modes dispersive contribution 2. Input parameters strong couplings of heavy mesons and their SU(3) breaking g(D * D  )=17.9  0.3  1.9 (CLEO) heavy-to-heavy form factors

17. 17 SD SD+LD Expt K0K0 5.6 +1.9 -1.8 8.6 +1.2+2.9 -1.2-1.8 8.3 +1.2 -1.0 K0K0 2.0 +3.5 -1.3 5.6 +2.9+3.7 -1.2-2.1 5.6  0.9 0K00K0 2.8 +3.2 -1.6 5.2 +3.2+2.6 -1.5-1.2 5.1  1.6  ’K 0 42.1 +45.6 -19.4 69.4 +51.3+50.4 -21.4-19.2 68.6  4.2 K0K0 1.8 +1.2 -0.9 1.8 +1.2+0.1 -0.8-0.0 <2.0 0K00K0 5.8 +5.5 -3.1 9.6 +5.5+8.4 -2.9-3.0 11.5  1.0 f0K0f0K0 8.1 +3.1 -2.6 8.1 +3.1+? -2.7-? 11.3  3.6 Br (10 -6 ) first error: SD, second error: LD LD uncertainties are comparable to SD ones & SD errors are affected only slightly by FSIs. No reliable estimate of LD rescattering effects for f 0 K S

18. 18 All rescattering diagrams contribute to penguin topology, dominated by charm intermediate states fit to rates  r D = r D*  0.67  predict direct CPV B  

19. 19 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt BB 16.6 22.9 +4.9 -3.1 24.1  1.3 0.01 0.026 +0.00 -0.002 -0.02  0.03 B0B0 13.7 19.7 +4.6 -2.9 18.2  0.8 0.03-0.15 +0.03 -0.01 -0.11  0.02 B0B0 9.3 12.1 +2.4 -1.5 12.1  0.8 0.17-0.09 +0.06 -0.04 0.04  0.04 B0B0 6.0 9.0 +2.3 -1.5 11.5  1.0 -0.040.022 +0.008 -0.012 -0.09  0.14 For simplicity only LD uncertainties are shown here FSI yields correct sign and magnitude for A(  + K - ) ! K  anomaly: A(  0 K - )  A(  + K - ), while experimentally they differ by 3.4   See Fleischer’s talk] _ _ _ _

20. 20 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt B 0  +   8.3 8.7 +0.4 -0.2 10.1  2.0 -0.01 -0.43  0.11 -0.47 +0.13 -0.14 B 0    + 18.0 18.4 +0.3 -0.2 13.9  2.1 -0.02 -0.25  0.06-0.15  0.09 B 0  0  0 0.44 1.1 +0.4 -0.3 1.8  0.6 -0.005 0.53  0.01 -- B     0 12.3 13.3 +0.7 -0.5 12.0  2.0 -0.04 0.37  0.10 0.16  0.13 B      6.9 7.6 +0.6 -0.4 9.1  1.3 0.06 -0.58  0.15-0.19  0.11 Sign and magnitude for A(  +  - ) are nicely predicted ! DCPVs are sensitive to FSIs, but BRs are not (r D =1.6) For  0  0, 1.4  0.7 BaBar Br(10 -6 )= 5.1  1.8 Belle 1.6 +2.2 -1.6 CLEO Discrepancy between BaBar and Belle should be clarified. ﹣ _ _ B   _

21. 21 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt B    K *0 4.4 9.9 +3.6 -2.7 9.76 +1.16 -1.22 0.01 0.026  0.003 -0.14 +0.09 -0.11 B0+K*B0+K* 3.8 9.9 +3.7 -2.8 12.7 +1.8 -1.7 0.15-0.44   -0.25  0.17 BK*BK* 2.8 5.6 +1.8 -1.4 ? 0.17 -0.39  0.01 0.04  0.29 B 0    *0 1.3 4.4 +1.8 -1.4 1.7  0.8 -0.08 0.066 +0.005 -0.001 -0.01 +0.27 -0.26 B   B   * _ _ BaBar, hep-ex/0504009  Br(B -  0 K * - )=(6.9  2.4)10 -6 For  0 K *0, Br(10 -6 )= 3.0  1.0 BaBar 0.4 +1.9 -1.7 Belle _ _ K.F. Chen (CKM2005): BaBar 6.9  2.4

22. 22 Comparison with other approaches All known existing models fit the data of BRs and DCPV and then make predictions for mixing (indirect) CPV. e.g. 1. charming penguin (Ciuchini et al. and many others) Consider charmless B decay B  K  with B  DD s  K   charming penguin is CKM doubly enhanced & gives dominant LD corrections  S(  0 K 0 )=0.77  0.04 CKM2005 a fit result

23. 23 2. Fit QCDF to data  fix unknown power correction parameters  A,  H,  A,  H Aleksan et al. (hep-ph/0301165) S4 scenario of Beneke & Neubert (hep-ph/0308092) Leitner, Guo, Thomas (hep-ph/0411392) Cottingham et al. (hep-ph/0501040)

24. 24 Mixing-induced CP violation [HYC,Chua,Soni] It is expected in SM that -  f S f  sin2  0.726  0.037 with deviation at most O(0.1) in B 0  K S,  K S,  0 K S,  ’K S,  0 K S, f 0 K S, K + K - K S, K S K S K S [London,Soni; Grossman, Gronau, Ligeti, Nir, Rosner, Quinn,…

25. 25 G. Kane (and others): The 2.7-3.7  anomaly seen in b→s penguin modes is the strongest hint of New Physics that has been searched in past many many years… It is extremely important to examine how much of the deviation is allowed in the SM and estimate the theoretical uncertainties as best as we can. A current hot topic

26. 26 In general, S f  sin2  eff  sin(2  +  W ). For b  sqq modes, Since a u is larger than a c, it is possible that S will be subject to significant “tree pollution”. However, a u here is color-suppressed. Penguin contributions to  K S and  0 K S are suppressed due to cancellation between two penguin terms (a 4 & a 6 )  relative importance of tree contribution  large deviation of S from sin2  Time-dependent CP asymmetries:

27. 27 SD Expt SD Expt KSKS 0.747 +0.002 -0.039 0.35  0.20 1.4 +0.3 -0.5 4  17 KSKS 0.850 +0.052 -0.055 0.55  0.31 -7.3 +3.5 --2.6 48  25 0KS0KS 0.635 +0.028 -0.067 -- 9.0 +2.2 -4.6 --  ‘K S 0.737 +0.002 -0.038 0.43  0.11 1.8  0.4 4  8 KSKS 0.793 +0.017 -0.044 -- -6.1 +5.1 -2.0 -- 0KS0KS 0.787 +0.018 -0.044 0.34  0.28 -3.4 +2.1 -1.1 8  14 f0KSf0KS 0.749 +0.002 -0.039 0.39  0.26 7.7  0.1 14  22 S(  K S )>sin2 , S(  0 K S )<sin2  FSI can bring in additional weak phase via K * ,K  intermediate states (even when tree is absent at SD) -n f S f A f (%) (see also Beneke)

28. 28 FSI effect is tiny due to small source (K* ,K  ) amplitudes (Br~10 -6 ) compared to D s *D (Br~10 -2,-3 ). It tends to alleviate the deviation from sin2  For  0 K S,  S=S-sin2  <0 at SD but it becomes positive after including FSI.  S f is positive and less than 0.1 in SM, while experimentally  S f is always negative SD SD+LD Expt SD SD+LD Expt KSKS 0.747 0.759 +0.009 -0.041 0.35  0.20 1.4 -2.6 +1.9 -1.3 4  17 KSKS 0.850 0.736 +0.033 -0.38 0.55  0.31 -7.3 -13.2 +4.4 -3.8 48  25 0KS0KS 0.635 0.761 +0.102 -0.127 -- 9.0 46.6 +12.9 -15.8 --  ‘K S 0.737 0.734 +0.004 -0.037 0.43  0.11 1.8 2.1 +0.5 -0.3 4  8 KSKS 0.793 0.802 +0.025 --0.046 -- -6.1 -3.7 +4.6 -3.0 -- 0KS0KS 0.787 0.770 +0.016 --0.046 0.34  0.28 -3.4 3.7 +2.7 -2.0 - 8  14 f0KSf0KS 0.749 0.749 +0.002 -0.039 0.39  0.26 0.8 0.8  0.1 -14  22 -n f S f A f (%)

29. 29 Effective sin2  in K + K - K S & K S K S K S For K + K - K S, S= -(2f + -1)sin2  eff (f + : CP-even fraction) For K S K S K S, S= -sin2  eff theory expt theory expt K+K-KSK+K-KS 0.830 +0.063 -0.086 0.60 +0.22 -0.20 0.74 +1.79 -1.18 -9  10 KSKSKSKSKSKS 0.749 +0.003 -0.039 0.26  0.34 0.75 +0.09 -0.13 41  21  K + K - K S is subject to large tree pollution from color-allowed tree diagrams  K S K S K S is very clean for testing SM sin2  eff A f (%)

30. 30 Short-distance induced transverse polarization in B  V 1 V 2 (V: light vector meson) is expected to be suppressed Get large transverse polarization from B  D s * D * and then convey it to  K * via FSI Polarization anomaly in B   K * f L (D s * D * )  0.51 contributes to f  only f ||  0.41, f   0.08 [HYC, Chua, Soni] Confirmed for B  with f L  0.97 but for B  K *  f L  0.50, f ||  0.25, f   0.25

31. 31  very small perpendicular polarization, f   2%, in sharp contrast to f   15% obtained by Colangelo, De FArzio, Pham ＋  0 ! Large cancellation occurs in B  {D s * D,D s D*}  K* processes. This can be understood as CP & SU(3) symmetry While f T  0.50 is achieved, why is f  not so small ? Cancellation in B  {VP,PV}  K* can be circumvented in B  {SA,AS}  K*. For S,A=D**,D s **  f   0.22 It is very easy to explain why f L  0.50 by FSI, but it takes some efforts to understand why f   f ||

32. 32 Conclusions  DCPV in charmless B decays is significantly affected by LD rescattering. Correct sign and right magnitude of DCPV in K -  + and  +  - are obtained after including FSI.  For penguin-dominated MK S modes, FSI tends to alleviate the deviation from sin2 .  Large transverse polarization f T  0.50 can be obtained from final-state rescattering of B  D s *D*  K *

33. 33 The subleading amplitudes in QCDF develop end-point singularities in twist-3 nonspectator and in annihilation An end-point singularity means breakdown of simple collinear factorization Use more conservative k T factorization Include parton k T to smear the singularity Perturbative QCD approach [Keum, Li, Sanda; Lu, Yang, Ukai] Collinear vs. k T factorization

34. 34 k T factorization Sudakov factors S describe the parton distribution in k T K T accumulates after infinitely many gluon exchanges Similar to the DGLAP evolution up to k T ~Q Parton-level diagrams Bound-state distribution amplitude

35. 35 Scales and penguin enhancement In PQCD this gluon is off-shell by Slow partonFast parton PQCD QCDF For penguin-dominated modes, the branching ratios Wilson coefficients

36. 36 Recent progress on PQCD Nonfactorizable contributions are important for color-suppressed modes---explained B! D 0  0, (J/   c0,c1,  c ) K (*) branching ratios, helicity amplitudes (Keum, Kurimoto et al.; Chen, Li). Annihilation lowers longitudinal polarization in B! VV. Also predicted pure-annihilation modes, which cannot be done in FA (Lu et al.). Predicted CP asymmetry, isospin breaking of B! K *  (Matsumori et al.). NLO PQCD enhances C, and resolves B! K  puzzle (Li, Mishima, Sanda): LO NLO Data A cp (K +  - ) -0.13 -0.11 -0.109  0.019 A cp (K +  0 ) -0.09 +0.03 0.04  0.04 Annihilation generates large strong phase, which explains direct CP asymmetries.

37. 37 Baryonic B decays 3-body baryonic B deacys were found to have larger BRs than 2-body decays There are extensive studies of baryonic B decays in Taiwan both experimentally and theoretically B - →ppK - : first evidence of charmless baryonic B decay B→pp(K,K *,  ) →  p( ,K) →  K B→pp, , p  stringent limits) B→p  : first evidence of b→s  penguin in baryonic B decays Expt. Theory Chua, Geng, Hou, Hsiao, Tsai, Yang, HYC,… Publication after 2001: (hep-ph) 0008079, 0107110, 0108068, 0110263, 0112245, 0112294, 0204185, 0204186, 0208185, 0210275, 0211240, 0302110, 0303079, 0306092, 0307307, 0311035, 0503264 0201015, 0405283 Belle group at NTU first paper on radiative baryonic B decays C.Q. Geng, this afternoon

38. 38 Back-up slides

39. 39 Regge approach  In evaluating absorptive part via replace Feynman-diagram strong scattering amplitude T by Regge amplitude R(s,t):  (t): residual function, linear Regge trajectory  t)=  0 +  ’t, intercept  0 =0.45 for , K *,…, -1.8 for D, D *  suppression of FSI with increasing energy s  suppression of charming penguin relative to the light Regge exchanges  uncertainties: t-dep. of  (t), BR sensitive to unknown  ’ R(s,t) is valid at large s and t  0

40. 40 Long-distance contributions to B  D 

41. 41 without FSI with FSI expt C/T 0.200.30exp(-i50  ) E/T 0(by hand)0.14exp(-i84  ) (C-E)/(T+E) 0.200.43exp(-i69  ) (0.46  0.05)exp(-i61  ) Br(B 0 →D +  - ) 3.2  10 -3 3.1  10 -3 (2.76  0.25)  10 -3 Br(B 0 →D 0  0 ) 0.6  10 -4 (2.7 +2.3 1.3  10 -4 (2.9  0.2)  10 -4 Br(B - →D 0  - ) 4.9  10 -3 (5.0 +0.2 -0.1  10 -3 (4.98  0.29)  10 -3 Even if short-distance W-exchange vanishes (i.e. E SD =0), final-state rescattering does contribute to weak annihilation B 0  D s + K - proceeds via W-exchange

42. 42 Cutoff scale is fixed by B  K  via SU(3) symmetry  too large  +  - (  9  10 -6 ) and too small  0  0 (  0.4  10 -6 ) An additional rescattering contribution unique to  but not available to K  is needed to suppress  +  - and enhance  0  0 D+(+)D+(+) D-(-)D-(-) ++ -- B  B 0  DD(  )  has the same topology as vertical W- loop diagram V _

43. 43 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt B0+B0+ 7.6 4.6 +0.9 -0.7 4.6  0.4 -0.05 0.37  0.10 (input) 0.56  0.13 (Belle) 0.09  0.16 (BaBar) B000B000 0.3 1.5 +0.3 -0.2 1.5  0.3 0.56 -0.45 +0.08 -0.06 0.28  0.39 B0B0 5.1 5.3  0.0 5.5  0.6 5x10 -5 -0.003  0.001-0.02  0.07 Need to fit to rates and CPV of  +  - simultaneously Charming penguin alone doesn’t suffice to explain  0  0 rate (r D =0.67) Sign of A(  0  0 ) can be used to discriminate between different models W-exchange can receive LD contributions from FSI Define T eff =T+E+V, C eff =C-E-V  C eff /T eff =(0.90  0.02) exp[-i(88  2)  ]