1

QCD Factorization with Final-State Interactions

Chun-Khiang Chua

Academia Sinica, Taipei

3rd ICFP, Cung-Li, Taiwan

2

Factorization in B decays We basically have three

scales in a non-leptonic B decay:

mW >> mB >> QCD

Integrating out d.o.f. above mB:

H=ci() Qi()

Naïve factorization:

A BM1 0 M2

ai(cj) FFBM1 fM2

B M1

M2

')1()'(

|)(|0|)'(|||

5

1122221

qqqqwith

BbqMqqMBOMM

In mb limit, M2 produced in point-like interactions carries away energies O(mb) and will decouple from soft gluon effect Bjorken

3

Naïve factorization in B Decays For color allowed processes the naïve factorization a

pprox. works well.

However, Corrections (non factorization contributions) are incalculable.

Neglected. Dependence of scale in amp. from ai() cannot be cancell

ed.

BR(Theory)≈3 10-3

BR(Expt.)=(2.76±0.25)10-3

0||||)(~ 01 jBjDaAmp

4

sinsin )()(

)()(

fBfB

fBfBACP

B f

One needs at least two different B f paths with distinct weak & strong phases

strong phase weak phase

ei(+)

BaBar Belle Average

B0→K-+ -0.130.03

-0.100.25

-0.110.02

B0→+- -0.470.16

-0.530.30

-0.470.14

B0→+- 0.090.16

0.560.13

0.370.10

first confirmed DCPV (5.7) in B decays (2004)

_

__

We do have 2 different paths

Direct CP violations

strong phase ?

5

penguin corrections

Ali, Greub (98)

Chen,Cheng,Tseng,Yang (99)

Generalize factorization

For problem with scheme and scale dependence, consider vertex and penguin corrections to four-quark matrix elements

Strong phase from the BSS cut: k2~m2B/4 m2

B/2 gives large uncertainty

Corrections (non-fac. Contributions) are still incalculable. Parameterized.

6

QCD Factorization Beneke, Buchalla, Neubert, Sachrajda (99)

)()(1||0||

...)()()(),,(

)()(||

1122

21

21

2

1

2

b

QCDs

MMBII

MIBM

M

mOOBjMjM

yxyxdxdyTd

xxdxTFfBOMM

TI:

TII:hard spectator interactions

M(x): light-cone distribution amplitude (LCDA) and x the momentum fraction of quark in meson M

At O(s0) and mb, TI=1, TII=0, naïve factorization is recovered

At O(s), TI involves vertex and penguin corrections,

TII arises from hard spectator interactions (New)

7

Comparison between QCDF & generalized fac.QCDF is a natural extension of generalized factorization with the following improvements:

Corrections to naïve factorization are calculable [1+O(s)]

Hard spectator interaction, which is of the same 1/mb order as vertex &

penguin corrections, is included (new) crucial for a2 & a10

Include distribution of meson momentum fraction

1. a new strong phase from vertex corrections

2. fixed gluon virtual momentum in penguin diagram (imp.for CP)

Except a6 and a8 all effective wilson coefficients are gauge and scheme independent.

a6 and a8 come with /mB=m2/(mu+md) mB. Power correction.

QCDF is model independent in the large mB limit and

reduces to naïve fac. in the O(s0) limit.

8

Power corrections1/mb power corrections: twist-3 DAs, annihilation, FSIs,…

We encounter penguin matrix elements from O5,6 such as

formally 1/mb suppressed from twist-3 DA,

numerically important ( enhancement) :

(2GeV)m2/(mu+md) 2.6 GeV , 2 mb

For example, in the penguin-dominated mode B K

A(BK) a4+(2/mb) a6 where 2/mb 1 & a6/a4 1.7

Phenomenologically, power corrections should be taken into account

need to include twist-3 DAs p & systematically

02

05 ||0||

)(||0|| BVA

mmm

mBbuud

dub

OK for vertex & penguin corrections: (mb) a6,8: scale independent.

9

mb/2 mb 2mb

a1 1.073+ i0.048 -0.086 0.986+ i0.048

1.054+ i0.026 -0.061 0.993+ i0.026

1.037+ i0.015 -0.045 0.992+ i0.015

a2 -0.039- i0.113 0.231 0.192-i0.113

0.005-i0.084 0.192 0.197-i0.084

0.045-i0.066 0.167 0.212-i0.066

a4u -0.031+i0.023

0.004 -0.027+i0.023

-0.029+i0.017 0.003 -0.026+i0.017

-0.027+i0.014 0.002 -0.025+i0.014

a5 -0.011+i0.005 0.016 0.004+i0.005

-0.007+i0.003 0.010 0.003+i0.003

-0.004+i0.001 0.008 0.004+i0.001

a6u -0.052+i0.017

-0.052+i0.017 -0.052+i0.018 -0.052+i0.018

-0.052+i0.019 -0.052+i0.019

a10/ 0.062+i0.168 -0.221 -0.161+i0.004

0.018+i0.121 -0.182 -0.164+i0.121

-0.028+i0.093 -0.157 -0.185+i0.093

black: vertex & penguin, blue: hard spectator green: total

ai for B K at different scales

10

)(

12)(

)1)(1(

)()(

)0( 11

2

1

2

02

yx

x

my

yx

xdxdy

d

Fm

ffH p

MB

MM

BBMB

MB

Endpoint divergence in hard spectator and annihilation interactions

The twist-3 term is divergent as p(y) doesn’t vanish at y=1: Logarithmic divergence arises when the spectator quark in M1 becomes soft

Not a surprise ! Just as in HQET, power corrections are a priori nonperturbative in nature. Hence, their estimates are model dependent & can be studied only in a phenomenological way

BBNS model the endpoint divergence by

with h being a typical hadron scale 500 MeV.

For annihilation contributions endpoint divergence starts at twist-2 term.

Both endpoint divergences occur as 1/mB power corrections (model dependent).

FSI could be important. Several hints…

10 ,1ln1

1

0

H

i

H

h

BH

Hem

y

dyX

11

1037

47

211

(%) Expt

0

1314

0

0

B

B

KB

723

7.1

517

pQCD

1.02.0

6.5

0.6

4.5

QCDF

2.131.00.31.28.123.08.21.2

5.111.03.12.07.111.06.11.0

7.85.02.21.15.96.05.21.1

10.3

9.12

4.1

QCDF(S4)

pQCD (Keum, Li, Sanda): A sizable strong phase from penguin-induced annihilation by introducing parton’s transverse momentum

QCD factorization (Beneke, Buchalla, Neubert, Sachrajda):Because of endpoint divergences, QCD/mb power corrections due to annihilation and twist-3 spectator interactions can only be modelled

QCDF (S4 scenario): large annihilation with phase chosen so that a correct sign of A(K-+) is produced (A=1, A= -55 for PP, A=-20 for PV and A=-70 for VP)

1. Large strong phases in charmless modes are needed

1 with )1(ln HA,,

0

,,

HAiHA

BHA e

m

y

dyX

input

12

Some decay modes do not receive factorizable contributions

e.g. B Kc0 with sizable BR though c0|c(1-5)c|0=0. Color-suppressed modes: B0 D0 h0 (0,,0,,’), 00, 00 have the measured rates larger than theoretical expectations.

Penguin-dominated modes such as BK*, K, K, K* predicted by QCDF are consistently lower than experiment by a factor of 2 3

importance of power corrections (inverse powers of mb) e.g. FSI, annihilation, EW penguin, New Physics, …

2. Rate enhancements in color-suppressed, fac.-forbidden or penguin-dominated modes

13

FSI as rescattering of intermediate two-body states

[Cheng, CKC, Soni]

FSIs via resonances are assumed to be suppressed in B decays due to the lack of resonances at energies close to B mass.

FSI is assumed to be dominated by rescattering of two-body intermediate states with one particle exchange in t-channel. Its absorptive part is computed via optical theorem:

i

ifTiBMfBMm )()( 2

• Strong coupling is fixed on shell. For intermediate heavy mesons,

apply HQET+ChPT

• Form factor or cutoff must be introduced as exchanged particle is

off-shell and final states are necessarily hard

Alternative: Elastic Rescattering [CKC, Hou Yang] Regge trajectory [Nardulli,Pham][Falk et al.] [Du et al.] …

14

Dispersive part is obtained from the absorptive amplitude via dispersion relation

''

)'( )(

0

22 ds

ms

sMmPmMe

s BB

= mexc + rQCD (r: of order unity)

or r is determined by a fit to the measured rates

r is process dependent

n=1 (monopole behavior), consistent with QCD sum rules

Once cutoff is fixed direct CPV can be predicted

subject to large uncertainties and will be ignored in the present work

Form factor is introduced to render perturbative calculation meaningful

n

QCD

n

t

m

t

mtF

2

22

)(

LD amp. vanishes in HQ limit

15

BR

SD

(10-6)

BR

with FSI

(10-6)

BR

Expt

(10-6)

DCPV

SD

DCPV

with FSI

DCPV

Expt

B 16.6 22.9+4.9-3.1 24.11.3 0.01 0.026+0.00

-0.002 -0.020.03

B0 13.7 19.7+4.6-2.9 18.20.8 0.03 -0.15+0.03

-0.01 -0.110.02

B0 9.3 12.1+2.4-1.5 12.10.8 0.17 -0.09+0.06

-0.04 0.040.04

B0 6.0 9.0+2.3-1.5

11.51.0 -0.04 0.022+0.008-0.012 -0.090.14

For simplicity only LD uncertainties are shown here

FSI yields correct sign and magnitude for A(+K-) !

K anomaly: A(0K-) A(+ K-), while experimentally they differ

by 3.4SD effects?Fleischer et al, Nagashima Hou Soddu, H n Li et al.]

Final state interaction is important.

_

_

_

_

16

BR

SD

(10-6)

BR

with FSI

(10-6)

BR

Expt

(10-6)

DCPV

SD

DCPV

with FSI

DCPV

Expt

B0+ 8.3 8.7+0.4-0.2 10.12.0 -0.01 -0.430.11 -0.47+0.13

-0.14

B0+ 18.0 18.4+0.3-0.2 13.92.1 -0.02 -0.250.06 -0.150.09

B000 0.44 1.1+0.4-0.3 1.80.6 -0.005 0.530.01 -0.49+0.70

-0.83

B0 12.3 13.3+0.7-0.5 12.02.0 -0.04 0.370.10 0.010.11

B 6.9 7.6+0.6-0.4

9.11.3 0.06 -0.580.15 -0.07+0.12-0.13

Sign and magnitude for A(+-) are nicely predicted !

DCPVs are sensitive to FSIs, but BRs are not (rD=1.6)

For 00, 1.40.7 BaBar

Br(10-6)= 3.11.1 Belle

1.6+2.2-1.6 CLEO

Discrepancy between BaBar and Belle should be clarified.

﹣

__

B B B

_

17

Mixing induced CP violation

)0(0 tB

0B

,...

,/

S

SCP

K

KJf

Oscillation, eim t

(Vtb*Vtd)2

=|(Vtb*Vtd)2| e-i 2

)(||

)(SWi

CP

eA

fBA

)(||

)(SW

CP

if

CP

eA

fBA

)(2sin

,sincos

))(())((

))(())((00

00

Wff

ff

CPCP

CPCPf

CPCP

CPCP

CP

S

mtSmtC

ftBftB

ftBftBa

Bigi, Sanda 81

Quantum Interference

18

sin2eff CKM phase is dominated.

Look for small effects.

Measuring the deviation of sin2eff in charmonium and penguin modes (w0) is important in the search of NP [new physics (phase)]

Deviation NP

How robust is the argument?

Originally, FSI was totally ignored.

19

In general, Sf sin2eff sin(2+W). For bsqq modes,

cuib

ccscb

uusub

aAaeRA

aVVaVVfBA24

**0

)(

Since au is larger than ac, it is possible that S will be subject to significant “tree pollution”. However, au here is color-suppressed.

Penguin contributions to KS and 0KS are suppressed due to cancellation between two penguin terms (a4 & a6)

relative importance of tree contribution

may have large deviation of S from sin2

mtAmtSftBftB

ftBftBff

cossin))(())((

))(())((

Time-dependent CP asymmetries:

20

FSI effects on sin2eff (Cheng, CKC, Soni 05) FSI can bring in additional

weak phase

-- B→K*, K contain tree

Vub Vus*=|Vub Vus|e-i

21

FSI effects in rates

FSI enhance rates though rescattering of charmful intermediate states [rates are used to fixed cutoffs (=m + r QCD, r~1)].

6.33.116.86.8

0.15.115.111.6

3.32.632.635.44'

9.19.1

6.11.51.52.3

6.07.47.412

3.83.864

)10(

0.03.30.08.2

3.38.20

0

3.110.61.41.3

9.53.3

00

3.353.533.133.22

5.486.20

0

1.03.10.09.0

3.19.0

0

5.13.46.01.2

8.39.1

00

8.23.35.12.1

8.34.1

0

2.10.1

6.86.05.39.0

9.16.1

0

6

fK

K

K

K

K

.K

.K

ExptFSIFSInoBr

22

FSI effects on direct CP violation

Large CP violation in the K mode.

2167.07.0

1487.323

771.271'

9.37.5

9.4887

23445.1386

1493.280

. (%)

0.01.00.01.0

1.01.00

17.09.17.16.1

1.13.2

0

1.02.04.05.0

4.03.0

5.28.16.10.5

0.25.5

8.58.155.127.13

5.40.2

0

4.25.35.17.5

4.20.4

2.29.01.50.1

5.02.0

S

S

S

S

S

S

S

Kf

.K

.K

K

.K

.K

.K

ExptFSIFSInoA

23

FSI effect on S

Theoretically and experimentally cleanest modes: ’Ks Ks

Tree pollutions are diluted for non pure penguin modes. KS, 0KS

24.075.0024.00240

26.031.0040.00600

09.050.0003.00130'

073.00660

035.00770

30.063.0023.01190

19.047.0035.00200

.

000.0007.0000.0018.0

007.0018.00

016.0014.0017.0024.0

027.0026.0

0

004.0010.0004.0021.0

009.0021.0

003.0034.0006.0028.0

027.0026.0

095.0069.0119.0104.0

027.0104.0

0

027.0029.0018.0036.0

064.0046.0

010.0014.0015.0024.0

007.0019.0

.Kf

.K

.K

.K

.K

.K

.K

ExptSFSIFSInoS

S

S

S

S

S

S

S

sin2 =0.6850.032Input CKM sin2 =0.724

24

FSI effects in mixing induced CP violation of penguin modes are small The reason for the smallness of the deviations:

The dominated FSI contributions are of charming penguin like. Do not bring in any additional weak phase.

The source amplitudes (K*,K) are small (Br~10-6) compare with Ds*D (Br~10-2,-3)

The source with the additional weak phase are even smaller (tree small, penguin dominate)

If we somehow enhance K*,Kcontributions ⇒ large direct CP violation (AKs). Not supported by data

25

Conclusion QCDF improve naïve and generalized factorizations. It is model independent in the large mB limit.

FSI should play some (sub-leading) role in B decays. (finite mB) Rates are enhanced: PP modes K, ’K…;

PV modes 00 K, K, 0K… Large direct CP violation in K-K

The deviation of sin2eff from sin2 = 0.6850.032 are at most O(0.1) in penguin-dominated B0 KS, KS, 0KS, ’KS, 0KS, f0KS (w/wo FSI)

sin2eff on penguin modes are still good places to look for new phase.

We should also try to look for them in other places.

26

Back up slides

27

...]0|)0()(|0|)0()(|[4

10|)0()(|

'2

1'''''4

55

5555

uxduxduxd

qqqqqqqqqqqq

6/)( )(0|)0()(|)(

)( 0|)0()(|)(

)( 0|)0()(|)(

5

5

1

0

5

ueduxpxpifuxdp

ueduifuxdp

uedupifuxdp

xiup

pxiup

xiup

twist-2 & twist-3 LCDAs:

Twist-3 DAs p & are suppressed by /mb with =m2/(mu+md)

)(1)1(6)(

)(1)(

)12(1)1(6)(

2/3

2/1

2/3

uCDuuu

uCBu

uCBuuu

nn

nnp

nn

with 01 du (u)=1, 0

1 du p,(u)=1

Cn: Gegenbauer poly.

28

In mb limit, only leading-twist DAs contribute

BjMjMBjMjMBjjMM

njjaT

VVBTMMGBHMM

nnnn

dpspbppcup

pF

||0||or ||0||||

8,6 with

,||2/||

211211222121

10

121

*)(21

,21

The parameters ai are given by

ixx

xxg

ixgxdxm

ixgxdxm

V

iPPHN

VC

N

c

N

cca

Mb

Mb

i

iiic

isF

c

i

c

iii

3ln1

213)(

7,5 )1()(6ln12

4,9,10-1 )()(18ln12

4,6,8,10for 0 )4

(4

2

2

211

strong phase from vertex corrections

ai are renor. scale & scheme indep except for a6 & a8

29

Hard spectator interactions (non-factorizable) :

)1)(1(

)()()(

)0(21

1

2

02 yx

yxdxdy

d

Fm

ffH MM

BBMB

MB

not 1/mb2 power suppressed:

i). B() is of order mb/ at =/mb d/ B()=mB/B

ii). fM , fB 3/2/mb1/2, FBM (/mb)3/2

H O(mb0) [ While in pQCD, H O(/mb) ]

Penguin contributions Pi have similar expressions as before except that G(m) is replaced by

Gluon’s virtual momentum in penguin graph is thus fixed, k2 xmb2

)(])1()/ln[()1(4)(1

0

1

0

2 xxuummuduudxmG Mb

responsible for enhancement of color-suppressed graphs (see a2 below)

30

Annihilation topology

Weak annihilation contributions are power suppressed

yyxxwith

yxyxyyxdxdy

N

Cfff

GA MMs

c

FMMB

Fann

1 ,1

...1

)1(

1)()(

2

1

022 2121

ann/tree fBf/(mB2 F0

B /mB

Endpoint divergence exists even at twist-2 level. In general, ann. amplitude contains XA and XA

2 with XA 10 dy/y

Endpoint divergence always occurs in power corrections

While QCDF results in HQ limit (i.e. leading twist) are model independent,

model dependence is unavoidable in power corrections