Theories of exclusive B meson decays

Hsiang-nan Li

Academia Sinica (Taiwan)

Presented at Beijing

Aug. 13-17, 2005

Titles of Lectures

• I: Naïve factorization and beyond

• II: QCDF and PQCD

• III: SCET

• IV: Selected topics in B Physics

• Will not cover SU(3), QCD sum rules, determination of CKM, specific modes, new physics,…

Lecture I

Naïve Factorization and beyond

Outlines

• Introduction

• Weak Hamiltonian

• Naïve factorization

• Diagnose FA

• A plausible proposal

• Summary

Introduction• Missions of B factories: Constrain standard-model parameters Explore heavy quark dynamics Search for new physics Must handle QCD eventually for precision measu

rement• Several theories have been developed recently,

which go beyond the naïve factorization.• Semileptonic B decays (B meson transition form

factors) are inputs to the above theories.• Predictions are then made for nonleptonic B dec

ays.• Will discuss their ideas, differences, applications

sin (21)=0.685± 0.032@LP 2005

Determine 1 using the golden mode B! J/ K. Penguin pollution ~5%. When reaching this precision, need a QCD theory.

Isospin relation

A(D+-)/ a1

p 2A(D00)/ a2A(D0-)

a1, a2: the BSW parameters|a2|=0.35-0.6, much larger than expectation.Arg(a2/a1)» 60o is generated by decay dynamics.Their understanding requires a theory.

60o

Mixing-induced CP

4 S0 due tonew physics?Need a theory for tree Pollution.

Penguin-dominated

Tree-dominated

Complexity

Weak Hamiltonian

Full theoryindep.

Effective theoryLow-energy< mW

IR finite differenceHigh-energy» mW

4-fermion operator O()

Wilson coefficient C()

Weak Hamiltonian Heff =

Sum ln(mW/) to all orders

The factorization scale is arbitrary, and its dependence cancels between C() and O()

Penguins

g

Wb s

q q

(1-5) ! V-A

(1+5)! V+A

At O(s) or O(), there are also penguin diagrams

QCD penguin: gEW penguin: g replaced by , Z

2 Color flows: 2TaijTa

kl = -ijkl/Nc + ilkj

b sb

q

q

s

q

q

Example: Heff for b! s

I, j label different color flows

Naïve factorization

h D|Heff|Bi/ C() hD|O()|Bi

perturbative nonperturbative

Must deal with the hadronic matrix element.

The factorization assumption (FA) was the first try.

The decay amplitude for B! D

f

BDF

Decay constant and form factor are physical. No dependence.To make physical prediction, mustassume C to be constant, and It is better to be universal.

Color flows

b c

du

O1(C)

two color traces, Tr(I)Tr(I)=Nc

2

Color-allowed

b

c

d

u

One color trace, Tr(I)=Nc1Color-suppressed

RHS is down by 1/N_c compared to LHS

Decay amplitudes

Df

BF

f

BDF

a1, a2: universal parameters

Class 1:Color-allowed

Class 2:Color-suppressed

B D

a1 and a2 seem to be universal!Success due to “color transparency”

Lorentz contractionSmall color dipole

Decoupling in space-timeFrom the BD system

FA is expected to work well for color-allowed modes witha light meson emitted from the weak vertex.

Success of FA

Large correction in color-suppressed modes due to heavy D, large color dipole

B D

a2(D)a2(J/ K) is not a surprise

Failure of FA

The failure of FA implies the importance of nonfactorizable correction to color-suppressed modes, for which a2(mb)» 0.1< a2(J/ K), a2(D)In terms of Feynman diagrams, nonfactorizablecorrection is not universal.

a1(mb)» 1.1

(J/)

(K)

Generalized naïve factorization

6~2 effCC NN

Exp shows that Wilson coefficients are not really universal

Due to nonfactorizable correction?

222111 , aaaa Fine tune the mode-dependent parameters to fit data

Equivalently, effective number of colors in

CNCCa )2(1)1(2)2(1

Not very helpful in understanding decay dynamicsHow to calculate nonfactorizable correction?

Nonfactorizable corrections

Diagnose FA

• FA should make some sense (color transparency).

• The assumption of constant a1, a2 is not successful.

• FA fails for color-suppressed modes as expected (small a2» nonfactorizable correction).

• Stop data fitting. How to go beyond FA?

Scale dependence

• Problem of FA

• Before applying factorization, extract the dependence from the matrix element

• The question is how to calculate g()

C()h O()i ¼ C()h O i|FA

Decay constant and form factor are physical. No dependence.

dependences cancel dependence in C() remains

|FAindependent

IR cutoff and gauge dependences• Look at the derivation of Weak Hamiltonian again

• Considering off-shell external quarks, the constant a is gauge dependent, which is also hidden into the matrix element.

• When extracting g(), one also extracts the dependences on cutoff and on gauge. The scale dependence is just replaced by the cutoff and gauge dependences. Dead end?

Evaluated betweenexternal quark states

ln(MW2/-p2)=ln(MW

2/2)+ln(2/-p2)

Absorbed into C(\mu)

-p2 is the off-shell IR cutoffhidden into matrix element

Strong phase and CP asymmetry

CP asymmetries in charmless decays can be measured at B factories

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 FA, strong phase comes from the BSS mechanism

It gives a small phase.Only source?Important source?

Bander-Silverman-Soni mechanism

Im/

Moreover, what is the gluon invariant mass q2? Can not compute thye strong phase unambiguoysly.

q

A plausible proposal• Recalculate O(s) corrections with on-shell

quarks (Cheng, Li, Yang, May, 99)

They are gauge invariant.But ln(2/-p2)! 1/IR

How to deal with this IR pole?

IR divergence• IR divergence is physical!

• It’s a long-distance phenomenon, related to confinement, the hadronic bound state.

• All physical hadronic high-energy processes involve both soft and hard dynamics.

q

q

g

t=1t=0

weak decayoccurs

Soft dynamics

Factorization theorem• The idea is to absorb IR divergence into m

eson distribution amplitudes • Factorization theorem

• f: factorization scale. Its role is like . mW

>>mb, mb>f>• H is the IR finite hard kernel.• The matrix element• A scale-independent, gauge-invariant, IR fi

nite theory is possible!

Scale dependence cancel

All allowed decay topologies

Factorization vs. factorization

• Factorization in “naïve factorization” means breaking a decay amplitude into decay constant and form factor.

• Factorization in “factorization theorem” means separation of soft and hard dynamics in decay modes.

• After 2000, factorization approach to exclusive B decays changed from 1st sense to 2nd.

Summary

• FA is a simple model for nonleptonic B decays based on color transparency.

• Its application is limited to branching ratios of color-allowed modes.

• It can not describe color-suppressed modes, neglects nonfactorizable contributions, and has incomplete sources of strong phases.

• Theoretically, it is not even a correct tool due to scale or gauge dependence.

• A proposal for constructing a theory with the necessary merits has been made.