1. Theoretical tools for non-leptonic B decays
Ru Min Wang
Yonsei University

2. Outline Effective Hamiltonian
Theoretical tools for hadronic matrix elements
Naïve Factorization (NF)
Generalized Factorization (GF)
QCD Factorization (QCD)
Perturbative QCD (PQCD)
Soft-Collinear Effective Theory (SCET)
Summary

3. Effective Hamiltonian
Three fundamental scales:
weak interaction scale b-quark mass scale QCD scale
Two tools of quantum field theory:
Operator Product Expansion
To separate the full problem into two distinct parts
Long-distance physics at scale lower than µ is contained in operator matrix elements.
Short-distance physics at scale higher than µ is described by the Wilson coefficients.
Renormalization Group Improved Perturbation Theory
To transfer the coefficients from scale mW down to the appropriate low energy scale.

4. Fermi constant CKM matrix Wilson Coefficient Four-Fermion Operator
Low-energy effective weak Hamiltonian at the scale µ:
Separation of the hard and harder O(αs) contributions in effective field theory.
Full theory Effective theory
μindep Low-energy < μ high-energy > μ
4-fermion operator (Qi(μ)) Wilson coefficient (Ci (μ))
Sum asln(mW/)
to all orders
The factorization scale  is arbitrary,
and its dependence cancels between Ci() and Qi().
Fermi constant CKM matrix Wilson Coefficient Four-Fermion Operator

5. CKM matrix 1
10-1
10-2
10-3

6. Wilson Coefficients
Wilson coefficients can be calculated by Perturbation Theory
and Renormalization Group Improved Perturbation Theory.

7. α,β are the SU(3) color indices, and taαβ are the Gell-Mann matrices.
Current-current operators:
Magnetic penguins operators:
Electroweak penguins operators:
QCD penguins operators:
α,β are the SU(3) color indices, and taαβ are the Gell-Mann matrices.
The cross indicates magnetic penguins originate from the mass-term
on the external line in the usual QCD or QED penguin diagrams

8. Decay amplitude: Naïve Factorization (NF)
Is determined
Is calculated efficiently in the Renormalization
Group Improved Perturbation Theory
Hadronic matrix element
Naïve Factorization (NF)
Generalized Factorization (GF)
QCD Factorization (QCDF)
Perturbative QCD (PQCD)
Soft-Collinear Effective Theory (SCET)
Light-Cone Sum Rules (LCSR) ……
Different
approaches
have been
developed

9. Naïve Factorization Decay Amplitude:
M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29, 637 (1985),
M. Bauer, B. Stech and M.Wirbel, Z. Phys. C 34, 103 (1987).
Decay Amplitude:

10. Three classes of decaysb c d u
Two color traces,
Tr(I)Tr(I)=Nc2
Color-allowed b c d u
One color trace,
Tr(I)=Nc
Color-suppressed
Color flows (Q1(C))
Class I: Color-allowed A(B0→π+D–) ∝ C1 (µ)+ C2(µ)/Nc = a1
Class II: Color-suppressed A(B0→π0D0) ∝ C2(µ) + C1(µ)/Nc = a2
Class III: Color-allowed + Color-suppressed
A (B+→π+D0)∝[C2(µ) + C1(µ)] (1+/Nc ) = a1 + a2 -
NC is the number of colors with NC=3 in QCD
a1,a2 are universal parameters --- is not successful

11. The failure of NF implies the importance of non-factorizable corrections to color-suppressed modes
NF Non-Factorizable
a1(mb)≈ >>
a2(mb)≈ ~
Color-allowed
Color-suppressed

12. Summary of NF
NF is very simple and provides qualitative estimation of branching ratios of various two body nonleptonic B decays
Breaks the scale independence of decay amplitudes
Decay constant and form factor are not scale-dependent
Wilson coefficients Ci(µ) are scale-dependent
Physical decay amplitudes are dependent on renormalization scale and renormalization scheme
Non-factorizable amplitudes are not always negligible
Expected to work well for color-allowed modes
Fails for color-suppressed modes as expected
Evaluation of strong phases is ambiguous c q
unknown
Strong phase arises from charm penguin loop

13. Generalized Factorization
A. J. Buras and L. Silvestrini, Nucl. Phys. B 548, 293 (1999),
H. Y. Cheng, H. n. Li and K. C. Yang, Phys. Rev. D 60, (1999).
Scale independence: Before applying factorization, extract the scale  dependence from the matrix element. NF
 dependences cancel
Neff is treated as a phenomenological parameter which models the non-factorizable contributions to the hadronic matrix elements, which have not been includes into Cieff.
How to calculate g()?

14. Gauge invariant Scale invariant
Infrared cutoff and gauge dependences
The extraction of g() involves the one-loop corrections
to the four quark vertex, which are infrared divergent.
Cieff are μ and renormalization scheme independent,
but are both gauge and infrared regulator dependent.
Nceff
The scale dependence is just replaced by the cutoff and gauge dependences.
One proposal by Cheng, Li, Yang in Phys.Rev.D60:094005,1999.
Gauge dependent
assume external quarks are on shell
Infrared cutoff
absorb Infrared divergence into meson
wave functions
Gauge invariant Scale invariant
Infrared finite

15. Summary of GF NCeff=2 may explain many nonleptonic B decays
Decays of Class I and III are consistent with data
GF can not work well in Decays of Class II
GF can not offer convincing means to analyze the
physics of non-factorizable contributions to non-
leptonic decays
Strong phases still come from charm penguin loop
CPV can not be estimated correctly

16. QCD Factorization Beneke, Buchalla, Neuber and Sachrajda
BBNS approach: PRL 83,1914(1999); NPB 591,313(2000).
Naïve Factorization
QCDF = NF + as and ΛQCD/mb non-factorizable corrections
Form Factor term
Spectator scattering
Up to power
Corrections
ofΛQCD/mb

17. TI,II are the hard scattering kernels
includes:
: tree diagram
:non-factorizable
gluon exchange
: hard spectator scattering
They are suppressed by ΛQCD/mb
Other corrections
which aren’t included in

18. At twist-3 for spectator amplitudes Also in annihilation amplitudes
Summary of QCDF
At the zeroth order of , QCDF reduces to naive factorization.
At the higher order, the corrections can be computed systematically.
The renormalization scheme and scale dependence of is restored. In the heavy quark limit, the “non-factorizable” contributions is calculable perturbatively. It does not need to introduce NCeff.
Non-factorizable contribution to NF and strong phase from the penguin loop diagrams can be computed.
QCDF is a breakthrough!
End-point divergences
At twist-3 for spectator amplitudes
Also in annihilation amplitudes
Parameterization (Phase parameters are arbitrary)

19. Has Naïve Factorization been so successful
that what we need to do is only small corrections ?
One 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.

20. Perturbative QCD Φ: Meson distribution amplitude
H-n. Li and H. L. Yu, Phys. Rev. Lett. 74, 4388; Phys. Lett. B 353, 301.
Y. Y. Keum, H-n. Li, and A. I. Sanda, Phys. Lett. B 504, 6; Phys. Rev. D 63,
C. D. Lu, K. Ukai, and M. Z. Yang, Phys. Rev. D 63,
Φ: Meson distribution amplitude
H: Hard scattering kernel
e-S: Sudakov factor
Describe the meson Distribution in kT
kT accumulates after infinitely many
gluon exchanges

21. x,y are integral variables from 0 to 1
There is no end-point singularity because of the Sudakov factor.
The gluon propagator:
x,y are integral variables from 0 to 1
singularity
at endpoint
The arbitrary cutoffs in QCDF are not necessary.
M. Beneke (Flavour Physics and CP violation, Taipeh, May 5-9, 2008):

22. The Feynman diagrams for calculating H
Factorizable Non-factorizable
Factorizable annihilation Non-factorizable annihilation

23. Power and Order Counting Rule
Foctorizable : Annihilation : Non-Factorizable
in PQCD
PQCD F Results
mb→∞
m0 is the chiral symmetry breaking scale (~1.8GeV)
in QCDF
QCDF F Results
as→0

24. Summary of PQCD
Incorporating Sudakov suppression makes form factors perturbatively calculable.
Hard spectator scattering and annihilation topology are calculable, no end-point divergences.
The prediction of large direct CP asymmetries are strongly model dependent.
The assumption of Sudakov suppression in hadronic B decays is questionable: PQCD calculation are very sensitive to details of kT dependence of the wave function

25. Soft Collinear Effective Theory
C.W. Bauer, S. Fleming, M. Luke, Phys. Rev. D 63, (2001).
C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63,
New approach to the analysis of infrared divergences in QCD.
SCET has a transparent power counting in λ = ΛQCD/Q
(or λ = (ΛQCD/Q)1/2) so that power corrections can be investigated in a systematic way.
Also introduces transverse momentum kT.
Many proofs of factorization are already complete:

26. Summary
The calculation of hadronic matrix elements is an great challenge for non-perturbative.
NF was employed for a long time (since 80s).
Different approaches have been developed: GF, QCDF, PQCD, SCET......

27. Thanks!