theories of exclusive b meson decays
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Theories of exclusive B meson decays
Hsiang-nan Li
Academia Sinica
Presented at Mini-workshop
Nov. 19, 2004
Outlines
• 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)
Df
BF
f
BDF
B
DBD FfaFfaDBA 21)(
,1a : universal Wilson coefficients
Color-allowed Color-suppressed
2a
B D
Success due to “color transparency”
To be quantitative, nonfactorizable correction?
Lorentz contractionSmall color dipole
Decoupling in space-timeFrom the BD system
Large correction in color-suppressed modes due to heavy D, large color dipole
Generalized naïve factorization
6~2 effCC NN
Exp shows that the Wilson coefficients are not really universal
Due to nonfactorizable correction?
222111 , aaaaFine tune the mode-dependent parameters to data
Equivalently, effective number of colors in CNCCa )2(1)1(2)2(1
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
b
W
u
g
Wb d
q q
Tree Penguin
2sinsin CPA
2)( 0 ii ePeTBA
Interference of T and P
Data
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
bm1,SFactorization limit…
Explain observed data
Predict not yet observed modes
QCDF vs. PQCD
• OCD-improved factorization=naïve factorization + QCD correction
IIT
BF)(a
BF
)(b
b
)(c
b
)(d
Factorizable emission
Leading
Vertex Non-spectator Exchange &correction Annihilation
Sub-leading
IT
BIIB
I TFTBA
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 ?
QCDF amplitude:
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()(,)(
210 xxx
x
xdx
Same singularity appears in the form factorThis is the reason the form factor is not factorizable (calculable), and treated as a soft object (BSW form)
in twist-3 nonspectator and in annihilationNeed to introduce arbitrary cutoffs
Curiosity:Why are the form factor and the annihilation ,though none is calculable ?
)( 0SO )( SO
AH iA
BiH
BC e
me
mx
1ln,1ln
An end-point singularity means breakdown of simple collinear factorizationUse more conservative kT factorizationInclude parton kT to smear the singularity
)(
)(22
10
BT mkxx
xdx
The same singularity in the form factor is also smeared
Want to calculate subleading correction?.....
Then the form factor also becomes factorizable
b
)(a
BF)(a
BF
)(b
b
)(b
Perturbative QCD approach
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 SDescribe the parton Distribution in kT
kT accumulates after infinitely many gluon exchangesSimilar 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 kT-depe
ndent wave function• Then hard kernels H are gauge-invariant• Convolution of H with WF models (predicti
on) 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 gauge-invariant kT-dependent WF from proof
• …….
Scales and penguin enhancement
b
BF
)( BmO
Fastpartons
In QCDFthis gluon is off-shell by
In PQCDthis gluon is off-shell by
)( 2BmO
Slow parton Fast parton
PQCD QCDF 25.1~ 2
For penguin-dominated modes,
Strong phase and CP asymmetry
Annihilation is similar to BSS mechanism
Sudakov gluons
Loop linecan go on-shell
kT
kT: loop momentum with the weight (Sudakov) factor
Strong phase
Pinch-induced strong phase=FSI?lXB u
b
u
Inclusive decay
Cut quark diagram ~ Sum over final-state hadrons
np,,,
~
On-shell
Off-shell hadrons
Our concerns in 2000
• Is kT factorization an appropriate theory?• Is a pinched singularity the correct way to produ
ce 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 fiel
ds, soft fields,…• Expansion of Lagrangian in 1/E in terms of
effective operators• Wilson coefficients: hard kernels• Convenient for factorization proof. Effectiv
e operators define nonlocal matrix elements (wave functions)
QCDBm
BmEffective (soft) operator for energy <
At lower energy, detailed structure of form factor can be seen
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.
0.016-0.064BBNS 04
Acc is large
In fact, charming penguin is factorizable(no IR divergence) and smallLi, Mishima 04BBNS 04
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 parameter
s.
pi+pi0: T+C
pi+pi-: T+Ppi0pi0: 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
14.02.023.0 )5143(07.005.0
ioe
Exact solution
PQCD
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. T exp(i phi3)
P
T exp(-i phi3)
K+pi-
(T+C) exp(i phi3)Pew
(T+C) exp(-i phi3)
K+pi0
Large strong phase between P and T is confirmed
Buras’s picture
T exp(i phi3)
P
T exp(-i phi3)
K+pi0
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
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