C.D. Lu ICFP3 1

Some progress in PQCD approach

Cai-Dian Lü (IHEP, Beijing)

Formalism of Perturbative QCD (PQCD) Direct CP asymmetry Polarization in BVV decays Summary

kT factorization

C.D. Lu ICFP3 2

Picture of PQCD Approach

Six quark interaction inside the dotted line

4-quark operator

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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

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Perturbative Calculation of H(t) in PQCD Approach

Form factor—factorizable

Non-factorizable

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Perturbative Calculation of H(t) in PQCD Approach

Non-factorizable annihilation diagram

Factorizable annihilation diagram

D(*) D(*)

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Feynman Diagram Calculation

21

5221

24

14 )1(

)( pk

itr

kk

ikdkd B

Wave function

221

22

21

221 22)( Bxym

i

kkkk

i

kk

i

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

T)

k2·k1= k2+k1

– - k2T·k1

T ≈ mB2xy

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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 kkxym

i

kk

i

2221 2)( Bxym

i

kk

i

then no singularity

The gluon propagator

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Endpoint Singularity

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

i

kk

i

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D meson with asymmetric wave function emitted,

they are not canceled between the two diagrams

that is why QCDF can not do this kind of decays

It is also true for annihilation type diagram

D Du uc c

Endpoint Singularity 22

21 2)( Bxym

i

kk

i

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Sudakov factor

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

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Branching Ratios Most 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 ….

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Direct CP Violation Require two kinds of decay

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

hadronic calculation , usually non-perturbative

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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)

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Direct CP Violation

)1( )()()()( 112211 iiii reTePeTeB

)1( )()()()( 112211 iiii reTePeTeB

12

12 TPr /

)]cos(21[)()( 22* rrTBBB

)]cos(21[)()( 22* rrTBBB

)()(

)()(

BB

BBACP

coscos21

sinsin22 rr

rACP

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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

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Main strong phase in FA

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

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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

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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)

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B K puzzle Their data differ by 3.6 A puzzle?

K+- and K+0 differ by subleading

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

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Error Origin

The wave functions The decay constants CKM matrix elements High order corrections

CP is sensitive toSee Kurimoto’s talk

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Next-to-leading order contribution

Vertex corrections, quark loops, magnetic penguins

Li, Mishima, Sanda hep-ph/0508041

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

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NLO direct CP asymmetry

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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 + –, K00, K, ’K …

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“ Annihilation”

Very important for strong phases

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

----sensitive to many parameters

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“ Annihilation”

W annihilation W exchange

Time-like penguin

Space-like penguin

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Naïve Factorization fail

Bf

22BMQ

?Bf

Momentum transfer:

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

momentum

Bfermion flow

spin (this configuration is not allowed)

p1p2

Annihilation suppressed~1/mB ~ 10%

Like Be e

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

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PQCD Approach

Two diagrams cancel each otherfor (V-A)(V-A) current

(K)

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

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Annihilation in Hadronic Picture

DSD

K

0B 0*K

Br(BD) ~10 –3 Br(BDSK) ~10–5, 1-2 %Both Vcb

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Vtb*Vtd , small br, 10–8

bu

ds

u K+

BK+ K– decay

K–b

ds

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

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No suppression for O6

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

transformation No suppression, contribution “big” (20%)

b)(sd

du

d

+ (K+)

–

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Counting Rules for BVV Polarization

The fractions follow the counting rules, RL~O(1), R~R~O(mV

2/mB2) from naïve

factorization and kinematics. 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?

See Yang’s talk

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Polarization for B()()

hep-ph/0508032

97

97

88

RL(exp)

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Penguin annihilation

Naïve counting rules for pure-penguin modes are modified by annihilation from (S–P)(S+P) operator

Annihilation contributes to all helicity amplitudes equally => Sizable deviation from RL~1

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

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

bs

ds

d

Large transverse component in BK* decays

K*

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

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Polarization of BK*()

Decay modes

RL(exp)

RL R= R

66% 76-82% 13% 11%

96% 78-87% 11% 11%

78-89% 12% 10%

72-78% 19% 9%

0*KB

0 *KB

*KB0

*KB

hep-ph/0508080

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Transverse polarization is around 35%

bs

ds

s

Time-like penguin in B decays (10–8 )

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

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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 hep-ph/0504187

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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

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Thank you!

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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 αsn+O(ΛQCD/mb)]

Keep only leading term in ΛQCD/mb expansion and the second order in αs expansion

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Polarization of BVV decays

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Contributions of different αs in H(t) calculation

Fraction

αs/

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Naïve Factorization Approach

+

u

B0 –u

d

d

b

Decay matrix element can be separated into two parts:

Short distance Wilson coefficients and

Hadronic parameters: form factor and decay constant