Theories of exclusive B
meson decays
Hsiang-nan Li
Academia Sinica
Presented at Mini-workshop
Nov. 19, 2004
Outlines
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Naïve factorization and beyond
QCDF vs. PQCD
Parton kT?
Scales and penguin enhancement
Strong phase and CP asymmetry
SCET
Remarks
Naïve factorization and beyond
Naïve factorization (BSW)
f
fD
F BD
F B
Color-allowed
A( B  D )  a1 f F
a1 , a2
Color-suppressed
BD
 a2 f D F
: universal Wilson coefficients
B
Success due to “color transparency”

Lorentz contraction
Small color dipole
D
B
Decoupling in space-time
From the BD system
To be quantitative, nonfactorizable correction?
Large correction in color-suppressed modes due to heavy D,
large color dipole
Generalized naïve factorization
Exp shows that the Wilson coefficients are not really universal
Due to nonfactorizable correction?
a1  a1  1 , a2  a2   2
Fine tune the mode-dependent parameters

Equivalently, effective number of colors in
a1( 2)  C2(1)  C1( 2) N C
to data
N C  N Ceff  2 ~ 6
Not very helpful in understanding decay dynamics
Strong phase and CP asymmetry
When entering the era of B factories, CP asymmetries in
charmless decays can be measured
u
b
W
b
d
g
W
q
Tree
q
Penguin
Interference of T and P
A( B 0     )  T  Pei ei2
Data
ACP  sin  sin 2
Theory
Extraction
In naïve factorization, strong phase comes from the BSS mechanism
Only source?
Important source?
Nonfactorizable correction, strong phase,…
Need a systemic, sensible, and predictive theory
Expansion in
 S , 1 mb
Factorization limit…
Predict not yet observed modes
Explain observed data
QCDF vs. PQCD
• OCD-improved factorization=naïve
factorization + QCD correction
TII
TI
b
b
F
B
(a)
Factorizable
emission
Leading
F B
(b)
Vertex
correction
(c )
Non-spectator
Sub-leading
(d )
Exchange &
Annihilation
QCDF amplitude:
AB       TI  F B    TII  B  
Two questions:
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 correction ?
Both answers are “No”
There is another option for factorizing the leading term,
and naïve factorization prediction could be modified.
However, the subleading calculation shows an end-point singularity

1
0
 ( x)
dx 2  ,
x
 ( x)  x(1  x)
in twist-3 nonspectator and in annihilation
Need to introduce arbitrary cutoffs



mB
mB
i H
xC  ln
1   H e , ln
1   Aei A



Same singularity appears in the form factor
This is the reason the form factor is not factorizable (calculable),
and treated as a soft object (BSW form)
Curiosity:
0
Why are the form factor O( S ) and the annihilation O( S ) ,
though none is calculable ?
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
 ( x)
dx
x( x  kT2 mB2 )
The same singularity in the form factor is also smeared
Then the form factor also becomes factorizable
b
b
F B
(a)
F B
(a )
(b)
Perturbative QCD approach
(b)
Parton kT?
Beneke’s 6 comments (ICHEP, Osaka, 2000)
1.Parton kT must be small, no help
2.kT breaks gauge invariance
3.kT factorization needs a proof
4.Twist-3 contribution is not complete
5.DA models should come from sum rules
6…..Could not remember all of them
1.Parton kT must be small, no help?
Sudakov factors S
Describe the parton
Distribution in kT
kT accumulates after infinitely many gluon exchanges
Similar to the DGLAP evolution up to kT~Q
2.kT breaks gauge invariance?
• kT factorization still starts with on-shell
external particles
• Decay amplitudes are gauge invariant
• Parton kT is gained by exchanging gluons
• Try to construct a gauge-invariant kTdependent wave function
• Then hard kernels H are gauge-invariant
• Convolution of H with WF models
(prediction) is gauge-invariant
3.kT factorization needs a proof
• Have proved it for semileptonic decays
• Leading-power proof is easy: dynamics of
different scales decouples
• Proof for nonleptonic decays follows
• Learned how to construct a gaugeinvariant kT-dependent WF from proof
• …….
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,
PQCD
QCDF
~ 1.5  2
2
Strong phase and CP asymmetry
Annihilation is similar to BSS mechanism
Loop line
can go
on-shell
Strong phase
kT
Sudakov gluons
kT: loop momentum with the weight (Sudakov) factor
Pinch-induced strong phase=FSI?
Inclusive decay B  X u l
 ,  , p, n
u
~
b
Cut quark diagram ~ Sum over final-state hadrons
Off-shell
hadrons



On-shell
Our concerns in 2000
• Is kT factorization an appropriate theory?
• Is a pinched singularity the correct way to
produce the strong phase?
• Is the annihilation the only important source of
strong phases?
• Do we have the guts to present the prediction,
large CP asymmetries with definite signs?
Soft-collinear Effective Theory
• An effective theory at large energy E
• Effective degrees of freedom: collinear
fields, soft fields,…
• Expansion of Lagrangian in 1/E in terms of
effective operators
• Wilson coefficients: hard kernels
• Convenient for factorization proof.
Effective operators define nonlocal matrix
elements (wave functions)
mB  QCD
At lower energy, detailed
structure of form factor
can be seen
Effective (soft) operator for energy <
mB
nonpert
SCET
• SCET is more careful in scale separation.
• A form factor is split into two pieces:
soft and hard contributions.
• No annihilation contribution.
• Need Acc (nonperturbative charming
penguin) to introduce large strong phases.
• All the above parameters are from fitting.
T can be chosen to be real, and C is assumed to be real.
0.016-0.064
BBNS 04
In fact, charming
penguin is factorizable
(no IR divergence)
and small
Li, Mishima 04
BBNS 04
Acc is large
My personal comments
• A bit disappointed by that SCET was led to this direction.
• I can get the same “prediction” using T, C, P, assuming C
to be real---4 parameters with 4 inputs.
• The pi0pi0 amplitude is fixed by the isospin relation.
• A stringent test will be Kpi modes. Need more
parameters.
pi+pi-: T+P
pi+pi0: T+C
pi0pi0: C-P
Amplitude topologies
Remarks
• Compard to HQET, exclusive theories are
still not yet well established:
Matrix elements (wave function) not known
Subleading corrections not clear
Mechanism not explored completely
………
• It is definitely a much richer and
challenging field.
Experimental data
Exact
solution
PQCD
0.23 e
0.07
0.05
(1435) o i
0.2
0.14
pi0pi0 branching ratio gets smaller. P/T approaches theory.
New data:
~0.38
B->K pi amplitudes and data
K pi data imply large Pew ?
• The updated data imply a large C, instead
of a large Pew.
K+piLarge strong
phase between P
and T is confirmed
T exp(i phi3)
P
T exp(-i phi3)
(T+C) exp(i phi3)
Pew
K+pi0
(T+C) exp(-i phi3)
Buras’s picture
K+pi0
T exp(i phi3)
P
T exp(-i phi3)
Pew
This is a possible solution, but ruled out by the pi pi data
• Charming penguin: need many Acc for
each polarization and for each mode.
• Rescattering: hard to accommodate rho K*,
phi K* simultaneously.
• b->sg: negligible due to G parity.
• Annihilation: not sufficient for phi K*, but
able to explain rho K*.
•
rho+ K*0: P
rho0 K*+: P+T
• Interference between P and T enhances
the longitudinal polarization