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

2. Contents: Introduction. Introduction. Method of calculations. Method of calculations. Applications to B  ππ and B  ρρ decays. Applications to B  ππ and B  ρρ decays. Conclusions. Conclusions.

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

4. How to calculate FSI? For single channel case from unitarity follows Migdal-Watson theorem: the phase of the matrix element for the phase of the matrix element for the decay X  ab is equal to the phase the decay X  ab is equal to the phase of the elastic scattering amplitude δ ab. of the elastic scattering amplitude δ ab. Generalization to isospin. Generalization to isospin. δ I (K  ππ)= δ I ( ππ) δ I (K  ππ)= δ I ( ππ)

5. 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 M Q. Arguments (J.D.Bjorken). Arguments (J.D.Bjorken). b) FSI do not decrease with M Q. (at least for two-body final states) (at least for two-body final states)

6. Classification of FSI in 1/N-expansion. In 1/N-expansion the following diagrams with FSI are possible: 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 b)~1/N and decreases as 1/MQ (reggeon). A.Kaidalov(1989) A.Kaidalov(1989) Similar conclusions: Similar conclusions: J.P.Donoghue et al.(1996)

7. Experimental results on FSI phases in D, B-decays Data on D  ππ branching ratios lead to: δ 2 – δ 0 = (86º±4º) δ 2 – δ 0 = (86º±4º) In B-decays: From B  Dπ decays FSI difference between From B  Dπ decays FSI difference between I=1/2 and I=3/2 amplitudes δ Dπ = 30º±7º I=1/2 and I=3/2 amplitudes δ Dπ = 30º±7º From analysis of B  ππ decays: From analysis of B  ππ decays: δ 2 – δ 0 = (53º±7º) (talk by M.Vysotsky) δ 2 – δ 0 = (53º±7º) (talk by M.Vysotsky) Large FSI phases! Large FSI phases! However small phases in B  ρρ However small phases in B  ρρ (ππ,ρρ-puzzle). (ππ,ρρ-puzzle).

8. Unitarity based approaches. Formal solution of unitarity: M X  ab = Σ m M 0 (X  m)S½ m,ab M X  ab = Σ m M 0 (X  m)S½ m,ab Difficult to apply for realistic multi- channel S-matrix. Some recent work along these lines: A. Deandrea et al., A. Deandrea et al., L. Wolfensten, F. Wu L. Wolfensten, F. Wu

9. 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/M² i(k) ). strongly suppressed (as powers of 1/M² i(k) ). The states with M i  1 GeV are taken into account.

10. Method of calculations (cont). Transforming ∫dk  ∫d²k t dM i ²dM k ² We obtain M I (B  ab)=∑ M I º(B  ik)( δ ia δ kb + i T I (ik  ab)) T I (ik  ab) is J=0 projection of the T I (ik  ab) is J=0 projection of the corressponding scattering amplitude. Note that for real T I (ik  ab) this formula gives Note that for real T I (ik  ab) this formula gives the same result as unitarity condition. However the same result as unitarity condition. However at high energies amplitudes have substantial at high energies amplitudes have substantial imaginary parts. imaginary parts.

11. Method of calculations (cont). 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:

12. Reggeization of t-channel exchanges. For exchange by an elementary ρ-meson in 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 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- α ρ (0))ln(s))~1/s½~1/M Q as exp(-(1- α ρ (0))ln(s))~1/s½~1/M Q for α ρ (0)=0.5 for α ρ (0)=0.5

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

14. Applications to B  ππ and B  ρρ decays. For B  ππ and B  ρρ decays the ππ, ρρ and πA 1 intermediate states were used. For B  ππ and B  ρρ decays the ππ, ρρ and πA 1 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  ρρ π- exchange gives the main contribution to the longitudinally polarized rho. In the amplitudes of ππ  πA 1 ρ - exchange contributes.

15. Applications to B  ππ and B  ρρ decays. The pion exchange in contribution of ρρ The pion exchange in contribution of ρρ intermediate state (neglected by other authors ~1/M² Q ) plays an important role in the resolution of ππ-ρρ puzzle. The pomeron and f-exchanges does not The pomeron and f-exchanges does not contribute to the phase difference of contribute to the phase difference of amplitudes with I=0 and I=2 and it decreases amplitudes with I=0 and I=2 and it decreases as ~1/M Q for M Q  ∞. as ~1/M Q for M Q  ∞. Vertices of reggeons with pions were taken from analysis of πN, NN-scattering and Regge from analysis of πN, NN-scattering and Regge factorization. factorization.

16. Results. Branching B  ρ + ρ- ≈ 6 times 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  ρρ decays. Final result is: B  ππ: δ 0 = 24º ; δ 0 - δ 2 =40º (±15º) B  ππ: δ 0 = 24º ; δ 0 - δ 2 =40º (±15º) δ 2 =-16º δ 2 =-16º B  ρρ: δ 0 = 7º ; δ 0 - δ 2 =13º (±5º) B  ρρ: δ 0 = 7º ; δ 0 - δ 2 =13º (±5º) δ 2 =-6º δ 2 =-6º

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