hadronic b decays in perturbative qcd approach

C.D. Lu Sino-German 1

Hadronic B Decays in Perturbative QCD Approach

Formalism of perturbative QCD (PQCD) based on kT factorization Direct CP asymmetry Polarization in BVV decays Summary

Thank colleagues: Keum, Li, Sanda, Ukai, Yang, …

Cai-Dian Lü (IHEP, Beijing)


Naïve Factorization Approach

+

uB0 –

u

d

d

b

Decay matrix element can be separated into two parts:

Short distance Wilson coefficients and

Hadronic parameters: form factor and decay constantIdea borrowed from

semi-leptonic decay

(BSW)


QCD factorization approach

Based on naïve factorization ， expand the matrix element in 1/mb and αs

<ππ|Q|B> = < π|j1|B> < π | j2 |0> [1+∑rn αs

n+O(ΛQCD/mb)] Keep only leading term in ΛQCD/mb

expansion and sub-leading order in αs expansion


QCDF OCD-improved factorization =

naïve factorization + QCD correctionIIT

Factorizable emission

Leading

Vertex Non-spectator Exchange &correction Annihilation

Sub-leading

IT


BIIB

I TFTfBA

Two concerns:

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 sub-leading correction ?

QCDF amplitude:

Both answers are “No”


Picture of PQCD Approach

Six quark interaction inside the dotted line

4-quark operator

b

B


PQCD approach A ~ ∫d4k1 d4k2 d4k3 Tr [ C(t) B(k1) (k2) (k3)

H(k1,k2,k3,t) ] exp{-S(t)} (k3) are the light-cone wave functions for

mesons: non-perturbative, but universal C(t) is Wilson coefficient of 4-quark operator exp{-S(t)} is Sudakov factor ， to relate the short-

and long-distance interaction H(k1,k2,k3,t) is perturbative calculation of six quark

interaction

channel dependent

channel dependent


Perturbative Calculation of H(t) in PQCD Approach

Form factor—factorizable

Non-factorizable


Perturbative Calculation of H(t) in PQCD Approach

Non-factorizable annihilation diagram

Factorizable annihilation diagram

D(*) D(*)


Do not need form factor inputs

All diagrams using the same wave functions

(same order in s expansion) All channels use same wave functions Number of parameters reduced


Feynman Diagram Calculation

21

5221

24

14 )1(

)( pkitr

kkikdkd B

Wave function

221

22

21

221 22)( Bxym

ikkkk

ikk

i

k2=mB(y,0,k2T), k1=mB(0,x,k1

T)

k2·k1= k2+k1

– - k2T·k1

T ≈ mB2xy


Endpoint Singularity

x,y are integral variables from 01, singular at endpoint

In fact, transverse momentum at endpoint is not negligible

221

2221 )(2)( TT

B kkxymi

kki

2221 2)( Bxym

ikk

i

then no singularity

The gluon propagator


After including the quark transverse momentum

there is no endpoint singularity large double logarithm are produced after radiative corrections, they should be resummed to generate the Sudakov form factor


Sudakov factor

The soft and collinear divergence produce double logarithm ln2Pb ，Summing over these logs result a Sudakov factor. It suppresses the endpoint region


There is also singularity at non-factorizable diagrams

But they can cancel each other between the two diagrams ， that is why QCD factorization can calculate these two without introducing kT

2221 2)( Bxym

ikk

i


D(K) meson with asymmetric wave function emitted, they are not canceled between the two diagramsthat is why QCDF can not do this kind of decays It is also true for annihilation type diagrams

D Du uc c

Endpoint Singularity 22

21 2)( Bxymi

kki


Endpoint singularity in collinear factorization

The sub-leading calculation shows an end-point singularity

Need to introduce arbitrary cutoffs


Power Counting--QCDF

Form factor diagrams are leading All others are s suppressed Annihilation-type are even power suppressed (small)


Power Counting--PQCD All diagrams are at the same order of s

Some non-factorizable diagram contributions are suppressed due to cancellations

power suppressed


QCDF vs PQCD Form factor input No need form factor Wave function input Wave function input Parameterize

Annihilation /exchange diagram

Annihilation /exchange diagram calculable BD0 pi0 not

calculable Most modes

calculable


Two operators contribute to decay:

B0 0 B0 0

color enhanced color suppressed C1 ~ – 0.2 ~ 1/3 C2 ~ 1/3

b d db

c

d d

0D

0D

)()(1 dbcuO )()(2 cbduO

000 DB

cu

u


47.02 a

arg (a2/a1) ~ – 41°

For B0 D00, non-factorizable diagrams do not cancel

PQCD Exp.

B0 D–+ 2.8±0.4 3.0±0.4B+ D0+ 5.5±0.4 5.3±0.5B0 D00 0.26±0.05 0.29±0.05


Branching Ratios Some of the branching ratios agree

well with experiments for most of the methods

Since there are always some parameters can be fitted :

Form factors for factorization and QCD factorization

Wave functions for PQCD, but CP ….


Direct CP Violation Require two kinds of decay

amplitudes with: Different weak phases (SM) Different strong phases – need

hadronic calculation , usually non-perturbative


B→ , K Have Two Kinds of Diagrams with different weak phase

W b u Tree ∝ VubVud*(s)

B

d(s) (K) W b t Penguin∝VtbVtd* (s)

B

O3,O4,O5,O6

O1,O2 (K)


Strong phase is important for direct CP

But usually comes from non-perturbative dynamics, for example

DK

K

K

For B decay, perturbative dynamic may be more important


Main strong phase in FAWhen the Wilson coefficients calculated to next-to-leading order, the vertex corrections can give strong phase


Strong phase in QCD factorization

It is small, since it is at αs order

Therefore the CP asymmetry is small

The strong phase of Both QCD factorization and generalized factorization come from perturbative QCD charm quark loop diagram


lXB u

bu

Cut quark diagram ~ Sum over final-state hadrons

np,,,

~

On-shell

Off-shell hadrons

Inclusive Decay


Annihilation-Type diagram

Very important for strong phases

Can not be universal for all decays, since not only one type

----sensitive to many parameters


Annihilation-Type diagram

W annihilation W exchange

Time-like penguin

Space-like penguin


Naïve Factorization failBf

22BMQ

?Bf

Momentum transfer:


pseudo-scalar B requires spins in opposite directions, namely, helicity conservation

momentum

Bfermion flow

spin (this configuration is not allowed)

p1p2

Annihilation suppression ~ 1/mB ~ 10%

Like Be e

For (V-A)(V-A), left-handed current


PQCD Approach

Two diagrams cancel each other for (V-A)(V-A) current — dynamical suppression

(K)


W Exchange Process

Vcb* Vud ~ 2


W exchange process

5*0

58.06.0

0

10)6.07.2()(

10)6.4()(

KDBBr

KDBBr

S

S

BaBarKDBBr

BelleKDBBr

S

S

,10)0.10.12.3()(

,10)3.16.4()(50

52.16.0

0

ResultResults:s:

Reported by Ukai in BCP4 Reported by Ukai in BCP4 (2001)(2001) before Exps before Exps::


Vtb*Vtd , small br, 10–8

b u

d s

u K+

BK+ K– decay

K–b

d s

Time-like penguinAlso (V-A)(V-A) contribution

Comparing B(B pi pi): 10–6, 1%


Chiral Enhancement Two penguin operators: O4~(V-A)(V-A) O6~(V-A)(V+A)

b s

q q

t

RqqLbdOq6

LqqLbdOq4

R,L=15

Fiertz trans.

O(1)

(S+P)(S-P) 2(mK2/ms) x 1/mB


No suppression for O6

Space-like penguin Become (s-p)(s+p) operator after Fiertz

transformation Chirally enhanced No suppression, contribution “big” (20-30%)

b)(sd

d u

d

+ (K+)

–


CP Violation in B (K)(real prediction before exp.)

CP(%) FA BBNS PQCD Exp

+K – +9±3 +5±9 –17±5 –11.5±1.8

+K 0 1.7± 0.1 1 ±1 –1.0±0.5 –2 ±4

0K + +8 ± 2 7 ±9 –13 ±4 +4 ± 4

+ – –5±3 –6±12 +30±10 +37±10

(2001)


Annihilation in QCDF Power (1/mB) suppressed and s suppressed

Should not be large But has to be large from exp.


Operator O6 is very important

Important for I = 1/2 rule in history B , K -- direct CP K* K K* K*

branching ratio too small in QCDF

polarization problem


How about mixing induced CP?

Dominant by the B-B bar mixing Most of the approaches give similar

results Even with final state interactions: B + –, K, ’K , KKK …


For Example:

(From Yossi Nir)


Polarization of BVV decays


Helicity flip suppression of the transverse polarization amplitude

Naïve counting ruleH = MN MT


Counting Rules for BVV Polarization

The measured longitudinal fractions RL for B are close to 1.

RL~ 0.5 in K* dramatically differs from the counting rules.

Are the K* polarizations understandable?


Theoretical attempts to solve these puzzles

Currents that breaks the naïve cutting rule: a) new physics b) the magnetic penguin c) the annihilation diagrams ……

Nonperturbative corrections: a) the charming penguin b) the final state interactions ……


There are still problems for some of the explanations

The perpendicular polarization is given by:

Final state interaction can not explain RN = RT and some others are difficult to explain the relative phase

Naïve Babar and Belle Avg.


Fierz Transformation

The annihilation diagram

The (S+P)(S-P) current can break the counting rule, The annihilation diagram contributes equally to the three polarization amplitudes


No suppression for O6

Space-like penguin Become (s-p)(s+p) operator after Fiertz

transformation Chirally enhanced No suppression, contribution “big” (20-30%)

bs

d s

d

K*


Annihilation can enhance transverse contribution: RL = 59% (exp:50%)

and also right ratio of R=, R and right strong phase =, b

s

d s

d

Large transverse component in BK* decays

K*

H-n Li, Phys. Lett. B622, 68, 2005


Alex Kagan’s study in QCDF


Polarization for B()()

Phys.Rev.D73:014024,2006


Polarization of BK*()Decay modes

RL(exp) RL R= R

66% 76% 13% 11%96% 78% 11% 11%

78% 12% 10%72% 19% 9%

0*KB0 *KB *KB0

*KBPhys.Rev.D73:014011,2006


Transverse polarization is around 35%

bs

d s

s

Time-like penguin in B decays (10–8 )

Eur. Phys. J. C41, 311-317, 2005


Vtb*Vtd , small br, 10–7

b u

d s

u K*

BK* K* decay

K*b

d s

Time-like penguinAlso (V-A)(V-A) contribution


Polarization of BK*K*

Decay modes RL R= R

67% 18% 15%75% 13% 12%99% 0.5% 0.5%

000 ** KKB

0** KKB

** KKB 0

Tree dominant Phys.Rev.D72:054015,2005


andis the first measured channel in Bs decays, which is useful to determine the Bs wave function

Experiment:


Summary The direct CP asymmetry measured by B

factories provides a test for various method of non-leptonic B decays

PQCD can give the right sign for CP asymmetry the strong phase from PQCD should be the dominant one.

The polarization in BVV decays can also be explained by PQCD

Important role of Annihilation type diagram


Thank you!Vielen

Dank ！谢谢！


Contributions of different αs in H(t) calculation

Fraction

αs/


Error Origin in PQCD The wave functions The decay constants CKM matrix elements High order corrections

CP is sensitive to


Branching ratio in NLO(10-6)Li, Mishima, Sanda hep-ph/0508041


NLO direct CP asymmetry


B K puzzle Their data differ by 3.6 A puzzle?

K+- and K+0 differ by sub-leading

amplitudes Pew and C. Their CP are expected to be similar.

hadronic b decays in perturbative qcd approach

Documents

subleading order

quark transverse momentum

quark operatorexp

small subleading correction

leading term

bsw form

nonfactorizable diagramsbut

form factor inputsall