1. Lecture II Factorization Approaches QCDF and PQCD

2. Outlines Introduction Factorization theorem QCD-improved factorization (QCDF) Perturbative QCD (PQCD) Power counting

3. Inroduction Nonleptonic decays involve much abundant QCD dynamics of heavy quarks. Naïve factorization was employed for a long time (since 80s). Need a systemic, sensible, and predictive theory: expansion in respect the factorization limit… explain observed data predict not yet observed modes

4. The complexity of nonleptonic decays drags theoretical progress till year 2000, when one could really go beyond naïve factorization. Different approaches have been developed: QCDF, PQCD, SCET, LCSR... The measurement of nonleptonic decays could discriminates different approaches.

5. Factorization theorem High-energy (Q! 1) QCD processes involve both perturbative and nonperturbative dynamics. The two dramatically different dynamics (characterized by Q and by a hadronic scale , respectively) factorize. The factorization holds up to all order in  s, but to certain power in 1/Q. Compute full diagrams of  *  ! , and determine DA at quark level (the IR regulartor)  / 1/  IR +…

6. Difference between the full diagram and the effective diagram (  ) gives the IR finite hard kernel H (Wilson coefficient). Fit the factorization formula F=   ­ H to the  *  !  data. Extract the physical pion DA   Subtract the previous IR regulator from the full diagrams for  *  ! , and determine the hard kernel H’. H’ should be IR finite. If not, factorization theorem breaks down. That is,  should be universal (process- independent).

7. Predict  *  !  using the extracted   and the factorization formula F’=   ­ H’. This is how factorization theorem has a predictive power. The precision can be improved by computing H and H’ to higher orders in  s, and by including contributions from higher twists.

8. Twist expansion Twist=dim-spin, usually higher twist corresponds to stronger power suppression. Fock-state expansion of a light meson bound state Concentrate only on two-parton. 3-parton contribution is negligible. 2-parton 3-parton start with twsit-2 start with twist-3

9. Pion distribution amplitudes Pion DAs up to two-parton twist-4 twist-4 twist-3 Chiral scale Integration by parts d/dx

10. Model DAs Gegenbauer polynomials Asymptotic behavior Also from equation (neglect 3-parton) From derivative of x(1-x)

11. QCDF The plausible proposal was realized by BBNS Form factor F, DAs  absorb IR divergences. T are the hard kernels. (P 1 ) (P 2 )

12. Hard kernels I T I comes from vertex corrections The first 4 diagrams are IR finite, extract the  dependence of the matrix element. q=P 1 +xP 2 is well-defined, q 2 =xm B 2 IR divergent, absorbed into F Magnetic penguin O 8g q 1 x g

13. Wilson coefficients Define the standard combinations, Upper (lower) sign for odd (even) i Adding vertex corrections

14. Scale independence Dotted: no VC; solid: Re part with VC; dashed: Im part with VC

15. Scale independence The  dependence of most a i is moderated. That of a 6, a 8 is not. It will be moderated by combining m 0 (  ).

16. Hard kernels II T II comes from spectator diagrams Nonfactorizable contribution to FA and strong phase from the BSS mechanism can be computed. QCDF=FA + subleading corrections, respects the factorization limit. QCDF is a breakthrough!

17. End-point singularity Beyond leading power (twist), end-point singularity appears at twist-3 for spectator amplitudes. Also in annihilation amplitudes parameterization Phase parameters are arbitrary.

18. Predictive power For QCDF to have a predictive power, it is better that subleading (singular) corrections, especially annihilation, are small. Predictions for direct CP asymmetries from QCDF are then small, close to those from FA. Large theoretical uncertainty from the free parameters.

19. B!  ,  K branching ratios For Tree- dominated modes, close to FA For penguin- dominated modes, larger than FA by a factor 2 due to O 8g. The central values are enhanced by b! sg * g * (Y.D. Yang’ talk).

20. B! ,  K direct CP asy. In FA, direct CP asy.» 0 b-b b+b

21. Direct CP asy. data b-b b+b Opposite to QCDF predictions!! To explain data, subleading corrections must be large, Which, however, can not be reliably computed in QCDF.

22. The emission (1st) diagram in QCDF is certainly leading… But why must it be written in the BSW form (F­  )? The factorization limit is still respcted. Has naïve factorization been so successful that what we need to do is only small correction ? CLY’s proposal could be realized in an alternative way, the perturbative QCD approach. The leading term is further factorized, and naïve factorization prediction could be modified greatly. PQCD Same end-point singularity appears in the factorizable emission diagram. Why are emission and annihilation treated In different ways?

23. An end-point singularity means breakdown of simple collinear factorization Use more conservative kT factorization Include parton kT to smear the singularity The same singularity in the form factor is also smeared Want to calculate subleading correction?..... Then the form factor also becomes factorizable The 1st amplitude in QCDF is further factorized: But k T » , not helpful?

24. PQCD factorization picture Sudakov factors S Describe the parton Distribution in kT kT accumulates after infinitely many gluon exchanges Similar to the DGLAP evolution up to k T ~Q Large k T Small b Always collinear gluons g g

25. behavior of Sudakov factor Physical picture for Sudakov suppression: large b means large color dipole. Large dipole tends to radiate during hard scattering. No radiation in exclusive processes, which then prefer small b configuration. Suppression at large b becomes stronger at larger x

26. PQCD predictions (NLO)

27. Sources of strong phase Different sources lead to different direct CP asy. Why is there the difference? See Y. Li’s talk.

28. Power counting in QCDF Annihilation is power suppressed Due to helicity conservation

29. Power counting in PQCD Vertex correction is NLO

30. (V-A) and (V+A) currents for annihilation For (V-A)(V-A), left-handed current Pseudo-scalar B requires spins in opposite directions, namely, helicity conservation 1 =s 1 ¢ p 1 =(-s 2 ) ¢ (-p 2 ) = 2. For (V-A)(V+A)=(S-P)(S+P), scalar current momentum B fermion flow spin (this configuration is not allowed) B p1p1 p2p2 Survive helicity conservation, But  S is twist-3 DA, down by m 0 /m B

31. Scales and penguin enhancement Fast partons In QCDF this gluon is off-shell by In PQCD this gluon is off-shell by Slow partonFast parton Br from PQCD are larger than Br from QCDF. See B!  K. For penguin-dominated modes,

32. PQCD QCDF For penguin-dominated VP, VV modes, More detail when discussing SCET, where different scales are treated carefully.