Lecture III

Factorization approaches

SCET

Outlines

• Introduction

• B! form factor

• Nonleptonic decays

• Charming penguin

• Summary

Introduction• An effective theory by integrating out high energ

y (E) modes.• Effective degrees of freedom: collinear fields, sof

t fields,…• Express an amplitude in 1/E in terms of the effec

tive operators.• The Wilson coefficients of these operators are th

e hard kernels. • The (nonlocal) matrix element of the operators a

re DAs (or form factors).• Convenient for deriving factorization theorem.

B! form factor• Kinematics

• Soft spectator in B, r» • If p2» mb, pg

2=(p2-r)2=-2r¢ p2» O(mb)

• Then the internal quark is off-shell by (mbv+k+pg)2-mb

2» O(mb2 )

• SCET is careful in matching at different scales.

Matching• Demonstrate the matching in SCET

• Full theory! SCETI: integrating out the lines off-shell by mb

2

C()J(0)()! C()()J(0)

gT(0)J(1)(0)

b

W

mb2

mb

Wilson coeff at SCETI2

1/mb suppressedcurrent

Power scaling mb-3/2

from HQET

• SCETI! SCETII: integrating out the lines off-shell by mb

J(0,)O()

2

Jet=Wilson coeff at SCETII

=hard kernel in PQCD

! T(0)J(0,)M()B()

Power scaling mb-1/2

Two terms have the same power scaling.

Comparison

• The fundamental inputs, B meson transition forjm factors, are treated differently in different approaches.

• fNF is not calculable, so FB is not in QCDF. No matching at mb. Just input it from sum rules.

• FNF is factorizable in kT factorization theorem, so FB is in PQCD. No matching at mb. Input B from sum rules, and compute FB.

• FB contains both fNF and fF, so it is a mixture in SCET. They are determined from the fit to the B! data.

Nonleptonic decays

• SCET can be applied to nonleptonic decays. The result for B! MM’ is

Charming penguin

• SCET gives another example that the leading amplitude in a nonleptonic decay does not need to be in the BSW form.

• At leading power, no alrge source of strong phases in SCET (no annihilation) .

• Long-distance charming penguin is then introduced, parameterized as Acc.

Decay amplitude• SCET factotrization formula for B! M1M2

Color-allowed

Color-suppressed

factorizable

Wilson coeff

Wilson coeff

Comment on charming penguin• Charming penguin is factorizable at leadin

g power (see BBNS). • Compute one-loop correction to the charm

loop, and see no IR divergence.• No need for additional nonperturbative par

ameter at leading power.• IR divergence could appear at next-to-lead

ing power.• Then annihilation should be also formulate

d into SCET.

Fit to data• Do not compute the jet function J(s(mb))

• Determine complex Acc, real B, real JB=

s dz JB(z) from the B!data,

Absorb JB+, from somewhere

Results

• Small FB

• Acc dominates penguin contribution

• Predict

Why is Pso large?

Amplitude parameterization• I can get the same “prediction” using T, C, P, assuming

C to be real, same as in SCET---4 parameters with 4 inputs.

• The 00 amplitude is fixed by the isospin relation.• A stringent test will be B! K modes. Need more parame

ters.

p2+0: T+C

+-: T+Pp200: P-C

Amplitude topologies

Summary (Beneke at CKM 2005)

SCET

QCDF/PQCD

QCDF/SCET

PQCD

Summary

• QCDF, PQCD, SCET go beyond FA.• They have different assumptions, whose

verification or falsification may not be easy.• They all have interesting phenomenological

applications.• Huge uncertainty from QCDF is annoying. Input

from time-like form factor for annihilation?• NLO correction in PQCD needs to be checked. • SCET should be applied to explore heavy quark

decay dynamics more.