2. C.D. LuICFP31 Some progress in PQCD approach Cai-Dian Lü (IHEP, Beijing) Formalism of Perturbative QCD (PQCD) Direct CP asymmetry Polarization in B  VV decays Summary k T factorization

3. C.D. LuICFP32 Picture of PQCD Approach Six quark interaction inside the dotted line 4-quark operator

4. C.D. LuICFP33 PQCD approach A ~ ∫d 4 k 1 d 4 k 2 d 4 k 3 Tr [ C(t)  B (k 1 )   (k 2 )   (k 3 ) H(k 1,k 2,k 3,t) ] exp{-S(t)}   (k 3 ) 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(k 1,k 2,k 3,t) is perturbative calculation of six quark interaction channel dependent

5. C.D. LuICFP34 Perturbative Calculation of H(t) in PQCD Approach Form factor — factoriz able Non- factori zable

6. C.D. LuICFP35 Perturbative Calculation of H(t) in PQCD Approach Non- factorizable annihilation diagram Factorizable annihilation diagram D (*)

7. C.D. LuICFP36 Feynman Diagram Calculation Wave function k 2 =m B (y,0,k 2 T ), k 1 =m B (0,x,k 1 T ) k 2 ·k 1 = k 2 + k 1 – - k 2 T ·k 1 T ≈ m B 2 xy

8. C.D. LuICFP37 Endpoint Singularity x,y are integral variables from 0  1, singular at endpoint In fact, transverse momentum at endpoint is not negligible then no singularity The gluon propagator

9. C.D. LuICFP38 Endpoint Singularity There is also singularity at non-factorizable diagrams But they can cancel each other between the two diagrams ， that is why QCD factorization can calculate these two without introducing k T

10. C.D. LuICFP39 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 DD u u Endpoint Singularity

11. C.D. LuICFP310 Sudakov factor The soft and collinear divergence produce double logarithm ln 2 Pb ， Summing over these logs result a Sudakov factor. It suppresses the endpoint region

12. C.D. LuICFP311 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 ….

13. C.D. LuICFP312 Direct CP Violation Require two kinds of decay amplitudes with: Different weak phases (SM) Different strong phases – need hadronic calculation, usually non-perturbative

14. C.D. LuICFP313 B→  ,  K Have Two Kinds of Diagrams with different weak phase W b u Tree ∝ V ub V ud * (s) B  d(s)  (K ) W b t Penguin ∝ V tb V td * (s) B  O 3,O 4,O 5,O 6 O 1,O 2  (K )

15. C.D. LuICFP314 Direct CP Violation

16. C.D. LuICFP315 Strong phase is important for direct CP But usually comes from non- perturbative dynamics, for example D K K  K  For B decay, perturbative dynamic may be more important

17. C.D. LuICFP316 Main strong phase in FA When the Wilson coefficients calculated to next-to-leading order, the vertex corrections can give strong phase

18. C.D. LuICFP317 Strong phase in QCD factorization It is small, since it is at α s order Therefore the CP asymmetry is small The strong phase of Both QCD factorization and generalized factorization come from perturbative QCD charm quark loop diagram

19. C.D. LuICFP318 CP Violation in B    (K) (real prediction before exp.) CP(%)FABBNSPQCDExp  + K – +9±3+5±9–17±5–11.5±1.8  + K 0 1.7 ± 0.11 ±1– 1.0 ±0.5– 2 ±4  0 K + +8 ± 27 ±9– 13 ±4 +4 ± 4  +  – –5±3–6±12+30±10+37±10 (2001)

20. C.D. LuICFP319 B  K  puzzle Their data differ by 3.6  A puzzle? K +  - and K +  0 differ by subleading amplitudes P ew and C. Their CP are expected to be similar.

21. C.D. LuICFP320 Error Origin The wave functions The decay constants CKM matrix elements High order corrections CP is sensitive to See Kurimoto’s talk

22. C.D. LuICFP321 Next-to-leading order contribution Vertex corrections, quark loops, magnetic penguins Li, Mishima, Sanda hep-ph/0508041

23. C.D. LuICFP322 Branching ratio in NLO(10 -6 ) Li, Mishima, Sanda hep-ph/0508041

24. C.D. LuICFP323 NLO direct CP asymmetry

25. C.D. LuICFP324 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   +  –, K 0  0,  K,  ’K …

26. C.D. LuICFP325 “ Annihilation ” Very important for strong phases Can not be universal for all decays, since not only one type ----sensitive to many parameters

27. C.D. LuICFP326 “ Annihilation ”     W annihilation W exchange Time-like penguin Space-like penguin

28. C.D. LuICFP327 Naïve Factorization fail ? Momentum transfer:

29. C.D. LuICFP328 pseudo-scalar B requires spins in opposite directions, namely, helicity conservation momentum B fermion flow spin (this configuration is not allowed) p1p1 p2p2 Annihilation suppressed~1/m B ~ 10% Like B  e e For (V-A)(V-A), left-handed current

30. C.D. LuICFP329 PQCD Approach Two diagrams cancel each other for (V-A)(V-A) current (K)

31. C.D. LuICFP330 W exchange process Results: Reported by Ukai in BCP4 (2001) before Exps:

32. C.D. LuICFP331 Annihilation in Hadronic Picture Br(B  D  ) ~10 –3 Br(B  D S K) ~10 –5, 1-2 % Both V cb

33. C.D. LuICFP332 V tb * V td, small br, 10 – 8 d s u K+K+ B  K + K – decay K–K– d s  Time-like penguin Also (V-A)(V-A) contribution

34. C.D. LuICFP333 No suppression for O 6 Space-like penguin Become (s-p)(s+p) operator after Fiertz transformation No suppression, contribution “big” (20%) d u d  + (K + ) ––  

35. C.D. LuICFP334 Counting Rules for B  VV Polarization The fractions follow the counting rules, R L ~O(1), R  ~R  ~O(m V 2 /m B 2 ) from na ï ve factorization and kinematics. The measured longitudinal fractions R L for B   are close to 1. R L ~ 0.5 in  K * dramatically differs from the counting rules. Are the  K* polarizations understandable? See Yang’s talk

36. C.D. LuICFP335 Polarization for B   (  )  (  ) hep-ph/0508032 97 88 R L (exp)

37. C.D. LuICFP336 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 R L ~1

38. C.D. LuICFP337 Annihilation can enhance transverse contribution: R L = 59% (exp:50%) and also right ratio of R =, R  and right strong phase  =,   d s d    Large transverse component in B   K * decays K*K* H-n Li, Phys. Lett. B622, 68, 2005

39. C.D. LuICFP338 Polarization of B  K *  (  ) Decay modesR L (exp)RLRL R=R= RR 66%76-82%13%11% 96%78-87%11% 78-89%12%10% 72-78%19%9% hep-ph/0508080

40. C.D. LuICFP339 Transverse polarization is around 35% d s s   Time-like penguin in B    decays ( 10 – 8 )  Eur. Phys. J. C41, 311-317, 2005

41. C.D. LuICFP340 Polarization of B  K * K * Decay modesRLRL R=R= RR 67%18%15% 75%13%12% 99%0.5% Tree dominant hep-ph/0504187

42. C.D. LuICFP341 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

43. C.D. LuICFP342 Thank you!

44. C.D. LuICFP343 QCD factorization approach Based on naïve factorization ， expand the matrix element in 1/m b and α s = [1+∑r n α s n +O(Λ QCD /m b )] Keep only leading term in Λ QCD /m b expansion and the second order in α s expansion

45. C.D. LuICFP344 Polarization of B  VV decays

46. C.D. LuICFP345 Contributions of different α s in H(t) calculation Fraction αs/αs/

47. C.D. LuICFP346 Naïve Factorization Approach  + u B 0  – d Decay matrix element can be separated into two parts: Short distance Wilson coefficients and Hadronic parameters: form factor and decay constant