Hadronic B Decays in Perturbative QCD Approach

1. Hadronic B Decays in Perturbative QCD Approach Cai-Dian Lü (IHEP, Beijing) • Formalism of perturbative QCD (PQCD) based on kT factorization • Direct CP asymmetry • Polarization in BVV decays • Summary Thank colleagues: Keum, Li, Sanda, Ukai, Yang, … Sino-German

2. Naïve Factorization Approach (BSW) Decay matrix element can be separated into two parts: • Short distance Wilson coefficients and • Hadronic parameters: form factor and decay constant + u B0 – d Idea borrowed from semi-leptonic decay Sino-German

3. 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 • sub-leading order in αs expansion Sino-German

4. QCDF OCD-improved factorization = naïve factorization + QCD correction Factorizable emission Leading Vertex Non-spectator Exchange & correction Annihilation Sub-leading Sino-German

5. QCDF amplitude: Two concerns: The emission diagram is certainly leading…. But why must it be written in the BSW form ? Has naïve factorization been so successful that what we need to do is only small sub-leading correction ? Both answers are “No” Sino-German

6. Picture of PQCD Approach 4-quark operator b Six quark interaction inside the dotted line Sino-German

7. 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 Sino-German

8. Perturbative Calculation of H(t) in PQCD Approach Form factor—factorizable Non-factorizable Sino-German

9. Perturbative Calculation of H(t) in PQCD Approach Non-factorizable annihilation diagram Factorizable annihilation diagram D(*) D(*) Sino-German

10. Do not need form factor inputs • All diagrams using the same wave functions • (same order in sexpansion) • All channels use same wave functions • Number of parameters reduced Sino-German

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

12. 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 Sino-German

13. After including the quark transverse momentum there is no endpoint singularity large double logarithm are produced after radiative corrections, they should be resummed to generate the Sudakov form factor Sino-German

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

15. 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 Sino-German

16. Endpoint Singularity D(K) 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 diagrams u D D u Sino-German

17. The sub-leading calculation shows an end-point singularity Endpoint singularity in collinear factorization Need to introduce arbitrary cutoffs Sino-German

18. Power Counting--QCDF • Form factor diagrams are leading • All others are s suppressed • Annihilation-type are evenpower suppressed (small) Sino-German

19. Power Counting--PQCD • All diagrams are at the same order ofs • Some non-factorizable diagram contributions are suppressed due to cancellations power suppressed Sino-German

20. Form factor input No need form factor Wave function input Wave function input QCDFvs PQCD Parameterize Annihilation /exchange diagram Annihilation/exchangediagram calculable BD0 pi0 not calculable Most modes calculable Sino-German

21. Two operators contribute todecay: u B00 B00 color enhancedcolor suppressed C1 ~ – 0.2 ~ 1/3 C2 ~ 1/3 u d d Sino-German

22. For B0 D00, non-factorizable diagrams do not cancel arg (a2/a1) ~ – 41° Sino-German

23. Branching Ratios • Some 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 …. Sino-German

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

25. 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 Sino-German

26. 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 Sino-German

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

28. 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 Sino-German

29. Inclusive Decay ~ Cut quark diagram ~ Sum over final-state hadrons Off-shell hadrons On-shell Sino-German

30. Annihilation-Type diagram Very important for strong phases Can not be universal for all decays, since not only one type ----sensitive to many parameters Sino-German

31. Annihilation-Type diagram W annihilation W exchange     Time-like penguin Space-like penguin Sino-German

32. ? Naïve Factorization fail Momentum transfer: Sino-German

33. 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 suppression ~ 1/mB ~10% Sino-German

34. PQCD Approach (K) Two diagrams cancel each other for (V-A)(V-A) current —dynamical suppression Sino-German

35. W Exchange Process Vcb* Vud ~ 2 Sino-German

36. W exchange process Results: Reported by Ukai in BCP4 (2001) before Exps: Sino-German

37. BK+K– decay Comparing B(B pi pi): 10–6, 1% • Vtb*Vtd , small br, 10–8 Time-like penguin Also (V-A)(V-A) contribution K–   s s d d K+ u Sino-German

38. Chiral Enhancement R,L=15 • Two penguin operators: • O4~(V-A)(V-A) • O6~(V-A)(V+A) b s Fiertz trans. t (S+P)(S-P) 2(mK2/ms) x 1/mB O(1) q Sino-German

39. No suppression for O6 • Space-like penguin • Become (s-p)(s+p) operator after Fiertz transformation Chirally enhanced • No suppression, contribution “big” (20-30%) + (K+)  u d   – d Sino-German

40. CP Violation in B  (K)(real prediction before exp.) (2001) Sino-German

41. Annihilation in QCDF • Power (1/mB) suppressed • and s suppressed • Should not be large • But has to be large from exp. Sino-German

42. Operator O6 is very important • Important for I = 1/2 rule in history • B ,  K -- direct CP •   K* •   K •   K* •   K* branching ratio too small in QCDF polarization problem Sino-German

43. 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, ’K , KKK … Sino-German

44. For Example: (From Yossi Nir) Sino-German

45. Polarization of BVV decays Sino-German

46. Helicity flip suppression of the transverse polarization amplitude H = MN  MT Naïve counting rule Sino-German

47. Counting Rules for BVV Polarization • 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? Sino-German

48. Theoretical attempts to solve these puzzles Nonperturbative corrections: a) the charming penguin b) the final state interactions …… Currents that breaks the naïve cutting rule: a) new physics b) the magnetic penguin c) the annihilation diagrams …… Sino-German

49. There are still problems for some of the explanations The perpendicular polarization is given by: Naïve Babar and Belle Avg. Final state interaction can not explain RN = RT and some others are difficult to explain the relative phase Sino-German

50. The annihilation diagram Fierz Transformation The (S+P)(S-P) current can break the counting rule, The annihilation diagram contributes equally to the three polarization amplitudes Sino-German