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