Some progress in PQCD approach

1. Some progressin PQCD approach Cai-Dian Lü (IHEP, Beijing) • Formalism of Perturbative QCD (PQCD) • Direct CP asymmetry • Polarization in BVV decays • Summary kT factorization ICFP3

2. Picture of PQCD Approach 4-quark operator Six quark interaction inside the dotted line ICFP3

3. PQCDapproach • A ~ ∫d4k1 d4k2 d4k3 Tr [ C(t)B(k1) (k2) (k3)H(k1,k2,k3,t) ] exp{-S(t)} • (k3) are the light-cone wave functions for mesons: non-perturbative, but universal • C(t)is Wilson coefficient of 4-quark operator • exp{-S(t)}is Sudakov factor，to relate the short- and long-distance interaction • H(k1,k2,k3,t)is perturbative calculation of six quark interaction channel dependent channel dependent ICFP3

4. Perturbative Calculation of H(t) in PQCD Approach Form factor—factorizable Non-factorizable ICFP3

5. Perturbative Calculation of H(t) in PQCD Approach Non-factorizable annihilation diagram Factorizable annihilation diagram D(*) D(*) ICFP3

6. Feynman Diagram Calculation Wave function k2=mB(y,0,k2T), k1=mB(0,x,k1T) k2·k1= k2+k1– - k2T·k1T ≈ mB2xy ICFP3

7. Endpoint Singularity The gluon propagator • x,y are integral variables from 01, singular at endpoint • In fact, transverse momentum at endpoint is not negligible then no singularity ICFP3

8. Endpoint Singularity • There is also singularity at non-factorizable diagrams • But they can cancel each other between the two diagrams，that is why QCD factorizationcan calculate these two without introducing kT ICFP3

9. Endpoint Singularity D meson with asymmetric wave function emitted, they are not canceled between the two diagrams that is why QCDF can not do this kind of decays It is also true for annihilation type diagram u D D u ICFP3

10. Sudakov factor The soft and collinear divergence produce double logarithm ln2Pb， Summing over these logs result a Sudakov factor. It suppresses the endpoint region ICFP3

11. Branching Ratios • Most of the branching ratios agree well with experiments for most of the methods • Since there are always some parameters can be fitted : • Form factors for factorization and QCD factorization • Wave functions for PQCD, but CP …. ICFP3

12. Direct CP Violation • Require two kinds of decay amplitudes with: • Different weak phases (SM) • Different strong phases– need hadronic calculation , usually non-perturbative ICFP3

13. B→  ,  K Have Two Kinds of Diagrams with different weak phase  (K) O1,O2 W b u Tree ∝ VubVud*(s) B d(s) (K) W b t Penguin∝VtbVtd* (s) B O3,O4,O5,O6 ICFP3

14. Direct CP Violation ICFP3

15. Strong phase is important for direct CP • But usually comes from non-perturbative dynamics, for example  K K D  K • For B decay, perturbative dynamic may be more important ICFP3

16. Main strong phase in FA When the Wilson coefficients calculated to next-to-leading order, the vertex corrections can give strong phase ICFP3

17. Strong phase in QCD factorization The strong phase of Both QCDfactorization and generalized factorizationcome from perturbative QCDcharm quark loop diagram It is small, since it is at αs order Therefore the CPasymmetry is small ICFP3

18. CP Violation in B  (K)(real prediction before exp.) (2001) ICFP3

19. B K puzzle • Their data differ by 3.6 A puzzle? • K+- and K+0 differ by subleading amplitudes Pew and C. Their CP are expected to be similar. ICFP3

20. Error Origin • The wave functions • The decay constants • CKM matrix elements • High order corrections See Kurimoto’s talk CP is sensitive to ICFP3

21. Next-to-leading order contribution • Vertex corrections, • quark loops, • magnetic penguins Li, Mishima, Sanda hep-ph/0508041 ICFP3

22. Branching ratio in NLO(10-6) Li, Mishima, Sanda hep-ph/0508041 ICFP3

23. NLO direct CP asymmetry ICFP3

24. How about mixing induced CP? • Dominant by the B-B bar mixing • Most of the approaches give similar results • Even with final state interactions: • B + –, K00, K, ’K … ICFP3

25. “Annihilation” Very important for strong phases Can not be universal for all decays, since not only one type ----sensitive to many parameters ICFP3

26. “Annihilation” W annihilation W exchange     Time-like penguin Space-like penguin ICFP3

27. ? Naïve Factorization fail Momentum transfer: ICFP3

28. For (V-A)(V-A), left-handed current spin (this configuration is not allowed) B fermion flow momentum p2 p1 Like Be e pseudo-scalar B requires spins in opposite directions, namely, helicity conservation Annihilation suppressed~1/mB ~10% ICFP3

29. PQCD Approach (K) Two diagrams cancel each other for (V-A)(V-A) current ICFP3

30. W exchange process Results: Reported by Ukai in BCP4 (2001) before Exps: ICFP3

31. Annihilation in Hadronic Picture Br(BD) ~10 –3Br(BDSK) ~10–5, 1-2 % Both Vcb ICFP3

32. BK+K– decay • Vtb*Vtd , small br, 10–8 Time-like penguin Also (V-A)(V-A) contribution K–   s s d d K+ u ICFP3

33. No suppression for O6 • Space-like penguin • Become (s-p)(s+p) operator after Fiertz transformation • No suppression, contribution “big” (20%) + (K+)  u d   – d ICFP3

34. Counting Rules for BVV Polarization See Yang’s talk • The fractions follow the counting rules, RL~O(1), R~R~O(mV2/mB2) from naïve factorization and kinematics. • The measured longitudinal fractions RL for B are close to 1. • RL~ 0.5 in  K*dramatically differs from the counting rules. • Are the K* polarizations understandable? ICFP3

35. Polarization for B()() 97 97 88 RL(exp) hep-ph/0508032 ICFP3

36. Penguin annihilation Naïve counting rules for pure-penguin modes are modified by annihilation from (S–P)(S+P) operator Annihilation contributes to all helicity amplitudes equally => Sizable deviation from RL~1 ICFP3

37. Large transverse component in BK* decays Annihilation can enhance transverse contribution: RL = 59% (exp:50%) and also right ratio of R=, R and right strong phase=,    H-n Li, Phys. Lett. B622, 68, 2005 s d  K* d ICFP3

38. Polarization of BK*() ICFP3 hep-ph/0508080

39. Time-like penguin in B  decays (10–8) • Transverse polarization is around 35% Eur. Phys. J. C41, 311-317, 2005    s d  s ICFP3

40. Polarization of BK*K* Tree dominant hep-ph/0504187 ICFP3

41. Summary • The direct CP asymmetry measured by B factories provides a test for various method of non-leptonic B decays • PQCD can give the right sign for CP asymmetry the strong phase from PQCD should be the dominant one. • The polarization in BVV decays can also be explained by PQCD Important role of Annihilation type diagram ICFP3

42. Thank you! ICFP3

43. QCD factorization approach • Based on naïve factorization，expand the matrix element in 1/mb and αs • <ππ|Q|B> = < π|j1|B> <π | j2 |0> [1+∑rn αsn+O(ΛQCD/mb)] • Keep only leading term in ΛQCD/mb expansion and the second order in αs expansion ICFP3

44. Polarization of BVV decays ICFP3

45. Contributions of different αsin H(t) calculation Fraction αs/ ICFP3

46. Naïve Factorization Approach Decay matrix element can be separated into two parts: • Short distance Wilson coefficients and • Hadronic parameters: form factor and decay constant + u B0 – d ICFP3