Lecture II

Factorization Approaches

QCDF and PQCD

Outlines

• Introduction

• Factorization theorem

• QCD-improved factorization (QCDF)

• Perturbative QCD (PQCD)

• Power counting

Inroduction

• Nonleptonic decays involve much abundant QCD dynamics of heavy quarks.

• Naïve factorization was employed for a long time (since 80s).

• Need a systemic, sensible, and predictive theory:

expansion in respect the factorization limit… explain observed data predict not yet observed modes

,S bm1

• The complexity of nonleptonic decays drags theoretical progress till year 2000, when one could really go beyond naïve factorization.

• Different approaches have been developed: QCDF, PQCD, SCET, LCSR...

• The measurement of nonleptonic decays could discriminates different approaches.

Factorization theorem• High-energy (Q! 1) QCD processes involve both

perturbative and nonperturbative dynamics.• The two dramatically different dynamics (charact

erized by Q and by a hadronic scale , respectively) factorize.

• The factorization holds up to all order in s, but to certain power in 1/Q.

• Compute full diagrams of *! , and determine DA at quark level (the IR regulartor) / 1/IR+…

• Difference between the full diagram and the effective diagram () gives the IR finite hard kernel H (Wilson coefficient).

• Fit the factorization formula F= H to the *! data. Extract the physical pion DA

• Subtract the previous IR regulator from the full diagrams for *!, and determine the hard kernel H’.

• H’ should be IR finite. If not, factorization theorem breaks down.

• That is, should be universal (process-independent).

• Predict *!using the extracted and the factorization formula F’= H’.

• This is how factorization theorem has a predictive power.

• The precision can be improved by computing H and H’ to higher orders in s, and by including contributions from higher twists.

Twist expansion

• Twist=dim-spin, usually higher twist corresponds to stronger power suppression.

• Fock-state expansion of a light meson bound state

• Concentrate only on two-parton. 3-parton contribution is negligible.

2-parton 3-partonstart with twsit-2 start with twist-3

Pion distribution amplitudes

• Pion DAs up to two-parton twist-4

twist-4

twist-3

Chiral scaleIntegration by parts

d/dx

• Model DAs

• Gegenbauer polynomials

• Asymptotic behavior

• Also from equation (neglect 3-parton)

From derivative of x(1-x)

QCDF• The plausible proposal was realized by

BBNS

• Form factor F, DAs absorb IR divergences. T are the hard kernels.

BIIB

I TFTBA

(P1)

(P2)

Hard kernels I• TI comes from vertex corrections

• The first 4 diagrams are IR finite, extract the dependence of the matrix element.

• q=P1+xP2 is well-defined, q2=xmB2

• IR divergent, absorbed into F

Magnetic penguin O8g

q1

x

g

Wilson coefficients• Define the standard combinations,

• Upper (lower) sign for odd (even) i

• Adding vertex corrections

Scale independence

Dotted: no VC; solid: Re part with VC; dashed: Im part with VC

Scale independence

The dependence of most ai

is moderated. That of a6, a8 is not.It will be moderated by combining m0().

Hard kernels II• TII comes from spectator diagrams

• Nonfactorizable contribution to FA and strong phase from the BSS mechanism can be computed.

• QCDF=FA + subleading corrections, respects the factorization limit.

• QCDF is a breakthrough!

End-point singularity• Beyond leading power (twist), end-point

singularity appears at twist-3 for spectator amplitudes.

• Also in annihilation amplitudes

• parameterizationPhase parameters are arbitrary.

Predictive power• For QCDF to have a predictive power, it is better

that subleading (singular) corrections, especially annihilation, are small.

• Predictions for direct CP asymmetries from QCDF are then small, close to those from FA.

• Large theoretical uncertainty from the free parameters.

B! , K branching ratios

For Tree-dominatedmodes, close to FA

For penguin-dominated modes, larger than FA by afactor 2 due to O8g.

The central values are enhanced by b! sg*g* (Y.D. Yang’ talk).

B! , K direct CP asy.

In FA, direct CP asy.» 0

b-bb+b

Direct CP asy. datab-bb+b

Opposite to QCDF predictions!!To explain data, subleading corrections must be large,Which, however, can not be reliably computed in QCDF.

The emission (1st) diagram in QCDF is certainly leading…But why must it be written in the BSW form (F )?The factorization limit is still respcted.

Has naïve factorization been so successfulthat what we need to do is only small correction ?

CLY’s proposal could be realized in an alternative way,the perturbative QCD approach.The leading term is further factorized, and naïve factorization prediction could be modified greatly.

PQCD

Same end-point singularity appears in the factorizable emission diagram. Why are emission and annihilation treated In different ways?

An end-point singularity means breakdown of simple collinear factorizationUse more conservative kT factorizationInclude parton kT to smear the singularity 22

10

1

BT mkxdx

The same singularity in the form factor is also smeared

Want to calculate subleading correction?.....

Then the form factor also becomes factorizableThe 1st amplitude in QCDF is further factorized:

b

)(a

BF)(a

BF

)(b

b

)(b

But kT » , not helpful?

PQCD factorization picture

Sudakov factors SDescribe the parton Distribution in kT

kT accumulates after infinitely many gluon exchangesSimilar to the DGLAP evolution up to kT~Q

Large kT

Small b

Always collinear gluons

gg

• behavior of Sudakov factor

• Physical picture for Sudakov suppression: large b means large color dipole. Large dipole tends to radiate during hard scattering. No radiation in exclusive processes, which then prefer small b configuration.

Suppression at large bbecomes stronger at larger x

PQCD predictions (NLO)

Sources of strong phase

Different sources lead to different direct CP asy.Why is there the difference?

See Y. Li’s talk.

Power counting in QCDF

Annihilation is power suppressedDue to helicity conservation

Power counting in PQCD

Vertex correction is NLO

• (V-A) and (V+A) currents for annihilation

• For (V-A)(V-A), left-handed current

• Pseudo-scalar B requires spins in opposite directions, namely, helicity conservation 1

=s1¢ p1=(-s2 ) ¢ (-p2 ) =2 .

• For (V-A)(V+A)=(S-P)(S+P), scalar current

momentum

Bfermion flow

spin (this configuration is not allowed)

B

p1p2

Survive helicity conservation,But S is twist-3 DA, down by m0/mB

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

Br from PQCD are larger than Br from QCDF. See B! K.

For penguin-dominatedmodes,

PQCD QCDF 25.1~ 2

For penguin-dominatedVP, VV modes,

More detail when discussing SCET, where different scales are treated carefully.