Final state interactions in heavy mesons decays.

1. Final state interactions in heavy mesons decays. A.B.Kaidalov and M.I. Vysotsky ITEP, Moscow

2. Contents: • Introduction. • Method of calculations. • Applications to Bππ and Bρρdecays. • CPV in Bππ and Bρρdecays. • Conclusions.

3. Introduction. • Importance of theoretical understanding of phases due to final state interactions in hadronic B-decays. • a) For extraction of CKM parameters from B-decays. • b) A good laboratory to study large distance aspects of QCD.

4. Unitarity triangle

5. The diagrams for B decays

6. The ρρ – ππ puzzle. A large difference in +-/00 branching ratios for ππ and ρρ–decays.

7. Matrix elements for Bππ (ρρ) decays.

8. Structure of the matrix elements are the phases due to final state interactions for tree (I=0,2) and penguin diagrams correspondingly. bandgare CKM phases. P-contribution to these decays is rather small (P/T ~ 0.1), but it is important for CP-violation.

9. Determination of penguin contribution. The values of P can be determined, using SU(3)-symmetry, from , -decays, where penguin diagrams give dominant contributions. M.Gronau, J.Rosner

10. Isospin analysis Neglecting by P-term From data on Bππ decays we get With account of P-term

11. Large deviation from factorization in phases. For Bρρ decays FSI-phases are smaller: Below the model will be presented, which explains the pattern of FSI- phases in Bππ , Bρρ decays.

12. How to calculate FSI? For single channel case from unitarity follows Migdal-Watson theorem: the phase of the matrix element for the decay Xab is equal to the phase of the elastic scattering amplitude δab. Generalization to isospin. δI(Kππ)= δI(ππ) At MK δ0(ππ)=(35±3)º , δ2(ππ)=(-7±2)º

13. Application to heavy (D,B)-mesons. For heavy mesons there are many open channels and application of unitarity is not straightforward. Different ideas about FSI for heavy quark decays. a) Effects of FSI should decrease with the mass of heavy quark MQ. Arguments (J.D.Bjorken). b) FSI do not decrease with MQ. (at least for two-body final states)

14. In 1/N-expansion the following diagrams with FSI are possible: The diagram a)~ 1/N² and does not decrease with MQ (pomeron), while the diagram b)~1/N and decreases as 1/MQ (reggeon). A.Kaidalov(1989) Similar conclusions: J.P.Donoghue et al.(1996) Classification of FSI in 1/N-expansion.

15. Experimental results on FSI phases in D, B-decays Data on Dππ branching ratios lead to: Iδ2 – δ0 I= (86º±4º) In B-decays: From BDπ decays FSI difference between I=1/2 and I=3/2 amplitudes δDπ= 30º±7º From analysis of Bππ decays: Iδ2 – δ0I= (37º±10º) Large FSI phases! However small phases in Bρρ (ππ,ρρ-puzzle).

16. Method of calculations. We use Feynman diagrams approach, which is often applied to high-energy hadronic interactions. Amplitudes for the transitions abik with large masses of the states i,k should be strongly suppressed (as powers of 1/M²i(k) ). It is possible to prove that M²i(k) ~LMB. The states with Mi ~ 1 GeV are taken into account.

17. Method of calculations (cont). Transforming ∫dk  ∫d²ktdMi²dMk² We obtain MI(Bab)=∑ MIº(Bik)(δiaδkb + i TI(ikab)) TI(ikab) is J=0 projection of the corressponding scattering amplitude. Note that for real TI(ikab) this formula gives the same result as unitarity condition. However at high energies amplitudes have substantial imaginary parts.

18. There are many papers on this subject, which use two-body intermediate states for calculations of effects due to FSI. For example: H-Y. Cheng, C-K. Chua and A.Soni The diagrams of the following type are used: Triangle diagram Method of calculations (cont).

19. Reggeization of t-channel exchanges. For exchange by an elementary ρ-meson in the t-channel the partial wave amplitudes do not decrease as energy increases. However it is well known from phenomenology of high-energy binary reactions that ρ-exchange should be reggeized. In this case its contribution to the FSI decreases as exp(-(1-αρ(0))ln(s))~1/s½~1/MQ for αρ(0)=0.5

20. Situation is even more drastically changed for D*-trajectory withαD*(0) ≈ -0.8 . We approximate high- energy scattering amplitudes by exchanges of Regge poles. Amplitudes at high energies in Regge model. Reggeization of t-channel exchanges.

21. Applications to Bππ and Bρρdecays. For Bππand Bρρdecays the ππ, ρρ and πA1 intermediate states were used. In the amplitudes of ππ ππ P, f and ρ - exchanges have been taken into account. In the amplitudes of ππ ρρπ-exchange gives the main contribution to the longitudinally polarized rho. In the amplitudes of ππ πA1 ρ -exchange contributes.

22. Applications to Bππ and Bρρdecays. The pion exchange in contribution ofρρ intermediate state (neglected by other authors ~1/M²Q) plays an important role in the resolution of ππ-ρρ puzzle. The pomeron and f-exchanges do not contribute to the phase difference of amplitudes with I=0 and I=2 and it decreases as ~1/MQ for MQ∞. Vertices of reggeons with pions were taken from analysis of πN, NN-scattering and Regge factorization.

23. Results. Branching Bρ+ρ- ≈5times larger than the one for Bπ+π- and ρρ –intermediate state is very important in Bππ decays, while ππ – intermediate state plays a minor role in Bρρdecays. Final result is: Bππ: δ0= 30º ; δ0- δ2=40º (±15º) δ2=-10º Bρρ: δ0= 11º ; δ0- δ2=15º (±5º) δ2=-4º

24. CP-violation in B-decays Time-dependent CP assymetry ACP(Dt)=Ssin(DmDt) – Ccos(DmDt) ↑ ↑ Mixing induced CPV Direct CPV

25. Direct CPV in Bππ CPV-parameter C is sensitive to a magnitude of penguin contribution and phases.

26. Direct CPV in Bππ The ratios of amplitudes was determined above: A0/A2 = 0.8±0.09, P/A2 = 0.092±0.02 Phases δ0 andδ2 were calculated. What about phases of penguin contribution? In PQCD the phase δP is ~ 10º and positive. The sign of the phase for contribution of intermediate state in Regge model depends on the intercept of D*-trajectory.

27. For linear D*-trajectory with ά=0.5 GeV- : αD*(0)= -0.8 and δP is negative. δP ~ - 10º In leading log PQCD calculation αD*(0)≥ 0 and δP is positive. Thus the sign of δP gives an important information on dynamics of high-energy interactions.

28. If δP is positive it is possible to obtain the lower bound for C+-: C+- > - 0.18 Belle and BABAR give different results for this quantity: C+-(Belle)=-0.55(0.09), C+-(BaBar)=-0.21(0.09) Using d s symmetry and C from BKπ decay one obtains

29. If negative δP is allowed, then it is possible to obtain C+- closer to Belle result. For Bπºπº decay we have

30. Very large lC00l is predicted. It is not sensitive to δP. Present experimental error is too big: C00= -0.36 ± 0.32 Belle and BABAR agree on the S+- S+-=-0.59 ± 0.09 Without P-contribution sin 2αT = S+- , αT= (108± 3)º With P: α= (88± 4(exp) ± 5(th))º

31. Relative smallness of P-contribution in Bρρ–decays allow us to determine αwith better theoretical accuracy. From experimental result (S+-)ρρ=-0.06 ± 0.18 we obtainα= (87± 5(exp) ± 1(th))º These values of α are in a good agreement with results of a global fit of unitarity triangle αUTfit= (94± 4)º

32. Conclusions. • FSI play an important role in two-body hadronic decays of heavy mesons. • Theoretical estimates with account of the lowest intermediate states give a satisfactory agreement with experiment and provide an explanation of a difference between the properties of ππand ρρ –final states in B decays.

33. Conclusions • C+- gives an important information on FSI and, in particular, on the phase of the penguin diagram. • It is important to resolve the problem with difference in experimental measurments of C+- in Bππdecays.