Lecture II
Factorization Approaches
QCDF and PQCD
Outlines
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Introduction
Factorization theorem
QCD-improved factorization (QCDF)
Perturbative QCD (PQCD)
Power counting
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  S , 1 mb
respect the factorization limit…
explain observed data
predict not yet observed modes
• 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.
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+…
• 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 (processindependent).
• 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.
Twist expansion
• Twist=dim-spin, usually higher twist corresponds
to stronger power suppression.
• Fock-state expansion of a light meson bound
state
2-parton
3-parton
start with twsit-2 start with twist-3
• Concentrate only on two-parton. 3-parton
contribution is negligible.
Pion distribution amplitudes
• Pion DAs up to two-parton twist-4
twist-4
twist-3
Chiral scale
d/dx
Integration by parts
• Model DAs
From derivative of x(1-x)
• Gegenbauer polynomials
• Asymptotic behavior
• Also from equation (neglect 3-parton)
QCDF
• The plausible proposal was realized by
BBNS
(P2)
(P1)
AB       TI  F B    TII  B  
• Form factor F, DAs  absorb IR
divergences. T are the hard kernels.
Hard kernels I
• TI comes from vertex corrections
x
q
1
Magnetic penguin O8g
• The first 4 diagrams are IR finite, extract
the  dependence of the matrix element.
• q=P1+xP2 is well-defined, q2=xmB2
• IR divergent, absorbed into F
g
Wilson coefficients
• Define the standard combinations,
• Upper (lower) sign for odd (even) i
• Adding vertex corrections
Scale independence
Dotted: no VC; solid: Re part with VC; dashed: Im part with VC
Scale independence
The  dependence of most ai
is moderated. That of a6, a8 is not.
It will be moderated by combining
m0().
Hard kernels II
• TII 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!
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.
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.
B!  ,  K branching ratios
For Treedominated
modes,
close to FA
For penguindominated
modes, larger
than FA by a
factor 2 due
to O8g.
The central values are enhanced by b! sg*g* (Y.D. Yang’ talk).
B! ,  K direct CP asy.
b-b
b+b
In FA, direct CP asy.» 0
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.
PQCD
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.
Same end-point singularity appears in the factorizable
emission diagram. Why are emission and annihilation
treated In different ways?
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.
Want to calculate subleading correction?.....
An end-point singularity means breakdown
of simple collinear factorization
Use more conservative kT factorization
Include parton kT to smear the singularity

1
0
1
dx
x  kT2 mB2
The same singularity in the form factor is also smeared
Then the form factor also becomes factorizable
The 1st amplitude in QCDF is further factorized:
b
b
F B
(a )
F B
(a )
(b)
But kT » , not helpful?
(b)
PQCD factorization picture
Sudakov factors S
Describe the parton
Distribution in kT
Always collinear gluons
g
Large kT
Small b
g
kT accumulates after infinitely many gluon exchanges
Similar to the DGLAP evolution up to kT~Q
• behavior of Sudakov factor
Suppression at large b
becomes stronger at
larger x
• 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.
PQCD predictions (NLO)
Sources of strong phase
See Y. Li’s talk.
Different sources lead to different direct CP asy.
Why is there the difference?
Power counting in QCDF
Annihilation is power suppressed
Due to helicity conservation
Power counting in PQCD
Vertex correction is NLO
• (V-A) and (V+A) currents for annihilation
• For (V-A)(V-A), left-handed current
spin (this configuration is not allowed)
fermion flow
B
p2
p1
momentum
• Pseudo-scalar B requires spins in opposite
directions, namely, helicity conservation
1=s1¢ p1=(-s2 ) ¢ (-p2 ) =2 .
• For (V-A)(V+A)=(S-P)(S+P), scalar current
B
Survive helicity conservation,
But S is twist-3 DA, down by m0/mB
Scales and penguin enhancement
Fast
partons
In QCDF
this gluon is off-shell by O(mB2 )
F B
In PQCD
this gluon is off-shell by
b
O (  mB )
Slow parton
Fast parton
For penguin-dominated
modes,
Br from PQCD are larger
than Br from QCDF. See
B!  K.
For penguin-dominated
VP, VV modes,
PQCD
QCDF
~ 1.5  2
2
More detail when discussing SCET, where different
scales are treated carefully.