11

Factorization Approach for Hadronic B Decays Hai-Yang Cheng

Factorization ( and history) General features of QCDF Phenomenology CPV, strong phases & FSIs

November 19, 2004, Mini-workshop on Flavor Physics

2

Two complementary approaches for nonleptonic weak decays of heavy mesons:

1. Model-independent diagrammatical approach

2. Effective Hamiltonian & factorization (QCDF, pQCD,…)

3

All two-body hadronic decays of heavy mesons can be expressed interms of six distinct quark diagrams [Chau, HYC(86)]

All quark graphs are topological and meant to have all stronginteractions included and hence they are not Feynmangraphs. And SU(3) flavor symmetry is assumed.

Diagrammatic Approach

(penguin) (or Pa)

(tree)

(color-suppressed) (exchange)

(annihilation)

Chiang,Gronau,Rosner,…

4

Effective Hamiltonian

Effective Hamiltonian for nonleptonic weak decays was first put forward by Gaillard, Lee (74), and developed further by Shifman, Vainshtein, Zakharov (75,77); Gilman, Wise (79). At scale , integrate out fermions & bosons heavier than

Heff=c()O() O(): 4-quark operator renormalized at scale

operators with dim > 6 are suppressed by (mh/MW)d-6

Why effective theory ?

When computing radiative corrections to 4-quark operators, the result will

depend on infrared cutoff and choice of gluon’s propagator, etc. The merit

of effective theory allows factorization: WCs c() do not depend on the

external states, while gauge & infrared dep. are lumped into hadronic m.e. Radiative correction to O1=(du)V-A(ub)V-A will induce O2=(db)V-A(uu)V-A

-- - -

25121 )1()( qqqq AV

5

Penguin Diagram

Penguin diagram [dubbed by John Ellis (77)] was first discussed by SVZ (75) motivated by solving I=1/2 puzzle in kaon decay

*

5

,,

,,,,,

))1()((82

isibi

aatcu

iii

sFeff

VVbcsduqwith

qqbsFGH

It is a local 4-quark operator since gluon propagator 1/k2 is cancelled by

(k k-gk2) arising from quark loop as required by gauge invariance

Responsible for direct CPV in K & B decays as dynamical phase can be

generated when k2>4m2 (time-like) Bander,Silverman,Soni (79)

Fierz transformation of (V-A)(V+A) -2(S-P)(S+P)

chiral enhancement of scalar penguin matrix elements

dominant contributions in many S=1 rare B decays

qsbqqqbsO AVAV )1()1(2)()( 556

6

QCD penguins

bcsduqsdqqqqqwith

qqbqOqqbqO

qqbqOqqbqO

OOOOqqbqP

AV

AVAVAVAV

AVAVAVAV

q

aa

,,,,' ,, ,)1()(

)''()( ,)''()(

)''()( ,)''()(

31

31'')1(

25121

65

43

'65435

EW penguins induce four more EW penguin operators

AVqAVAVqAV

AVqAVAVqAV

qqebqOqqebqO

qqebqOqqebqO

)''()(23 ,)''()(2

3

)''()(23 ,)''()(2

3

'10'9

'8'7

Effective Hamiltonian

0 ,

)(2

*)(

,

10

32211

tcudpspbp

cup iiit

ppp

Feff

VV

OcOcOcGH

Buras et al (92)

Gilman, Wise (79)

7

WC c()’s at NLO depend on the treatment of 5 in n dimensions:

i) NDR (naïve dim. regularization) {5, }=0

ii) HVBM (‘t Hooft, Veltman; Breitenlohner, Maison)

2- 4

0],ˆ[ ,0},~{ ,ˆ~55

mb LO NDR HV

c1 1.144 1.082 1.105

c2 -0.308 -0.185 -0.228

c3 0.014 0.014 0.013

c4 -0.030 -0.035 -0.029

c5 0.009 0.009 0.009

c6 -0.038 -0.041 -0.033

c7/ 0.045 -0.002 0.005

c8/ 0.048 0.054 0.060

c9/ -1.280 -1.292 -1.283

c10/ 0.328 0.263 0.266

Results of WCs ci(i=1,…,10) were

first obtained by Buras et al (92) For details about WCs, see

Buras et al. RMP, 68, 1125 (96) In s 0 limit, c1=1, ci=0 for i1

c3 c5 –c4/3 –c6/3

c9 is the biggest among EW

penguin WCs

8

Naïve Factorization

B M1

M2In mb limit, M2 produced in point-like interactions carries away energies O(mb) and will decouple from soft gluon effect

')1()'( |)(|0|)'(|||

5

1122221

qqqqwithBbqMqqMBOMM

M2 is disconnected from (BM1) system factorization

amplitude creation of M2 BM1 transition

decay constant form factor

Naïve factorization = vacuum insertion approximation

For a given effective Hamiltonian, how to evaluate the nonleptonic decay B M1M2 ?

9

Consider B--0 and H=c1O1+c2O2=c1(du)(ub)+c2(db)(uu)

B-

-

0b

u

u

ud

B-

0

bu

u

du

-

BOuubdN

c

BbuudcBOc

c

|~))((1|

|)(|0|)(|||

10

1

011

01

BOudbuN

c

BbduucBOc

c

|~))((1|

|)(|0|)(|||

20

2

022

02

))((21~ ),)((

21~

21 udbuOuubdO aaaa

BbduuNcc

BbuudNccBH

c

c

|)(|0|)(|

|)(|0|)(|||

012

021

0

Neglect nonfactorizable

contributions from O1,2

- - - -

~

from O1

from O2

color allowed

color suppressed

10

ff BOaBOaBH |||| || 20

210

10

decay suppressed-colorfor /diagram treeallowed-colorfor /

122

211

c

c

NccaNcca

Two serious problems with naïve factorization: Empirically, it fails to describe color-suppressed modes

for c1(mc)=1.26 and c2(mc)=-0.51, while Rexpt=0.55

Theoretically, scheme and scale dependence of ci() doesn’t get compensation from Of as V and A are renor. scale & scheme independent unphysical amplitude from naïve factorization

42

1

20

000

104.321

)()(

aa

KDKDR

11

How to overcome aforementioned difficulties ?

Bauer, Stech, Wirbel (87) proposed to treat ai’s as effective parameters and extract them from experiment. (Of course, they should be renor. scale & scheme indep.)

If ai’s are universal (i.e. channel indep) generalized factorization

Test of factorization means a test of universality of a1,2

Problems:

Penguin ai’s are difficult to determine

Cannot predict CPV

How to predict ai from a given effective Hamiltonian ?

12

For problem with color-suppressed modes, consider nonfactorizable contributions

)2/(~ ),/1(

)2/(~ ),/1(

2122122

1211211

fc

fc

OONcca

OONcca

To accommodate DK data -0.35

In late 70’s & early 80’s, it was found empirically by several groups that discrepancy is greatly improved if Fierz-transformed 1/Nc terms are dropped so that a1 c1, a2 c2. Note that c2+c1/Nc=-0.09 vs. c2=-0.51

[Fukugita et al (77); Tadic & Trampetic (82); Bauer & Stech (85)]

This is understandable as 1/Nc+ 0 !

Buras, Gerard, Ruckl large-Nc (or 1/Nc) approach (86)

for charm decays has been estimated by Shifman & Blok (87) using

QCD sum rulesNowadays, it is known that one needs sizable nonfactorizable effects & FSIs to describe hadronic D decays

13

If large-Nc approach is applied to B decays

a1eff=c1(mb) 1.10, a2

eff=c2(mb) -0.25

destructive interference in B-D0- just like D+K0+

A(B-D0-)= a1O1+a2O2, while A(B0D-+) = a1O1

supported by sum-rule calculations (Blok, Shifman; Khodjamirian, Ruckl; Halperin)

BigBig surprisesurprise from CLEO (93): constructive interference as B-D0- > B0D-+

Generalized factorization (I) [HYC (94), Kamal (96)]

with 1/Nceff=1/Nc+ determined from experiment

For BD decays, Nceff 2 rather than , is positive !

/ ,/ 122211effc

effc NccaNcca

_

14

For problem with scheme and scale dependence, consider vertex and

penguin corrections to four-quark matrix elements

penguin corrections

treejeffj

treejij

ee

ss

iii

treejij

ee

ss

i

Oc

OMMcOc

OMMO

)(4

)(4

1)()()(

)(4

)(4

1)(

Apply factorization to Otree rather than to O()

15

)()()ln(4

)(

ijij

TV

bTV

si

effi Pcrmcc

2

22

6543

97211

)1(ln)1(4)( ,31

31

))((49

4)(4

)7ln6(4

)37ln2(

41)(

xxkmxdxxmGOOOOP

OOmGPmGOmOmO ue

usbsbs

Compute corrections to 4-quark matrix elements in the same 5 scheme as ci() : NDR or ‘t Hooft-Veltman

Then, in generalAli, Greub (98)

Chen,HYC,Tseng,Yang (99)

V: anomalous dim., rV: scheme-dep constant, Pi: penguin

Z,

Gauge & infrared problems with effective WCs [Buras, Silvestrini (99)]

are resolved using on-shell external quarks [HYC,Li,Yang (99)]

16

8,6 ,)36ln36( 6ln12

;7,5 6ln12 4,9,10;-1 18ln12

4,6,8,10for 0 4

1

11

imccm

imimV

iPPVCNc

Ncca

b

i

ib

bbi

iiisF

c

i

c

iii

Scale independence of ai or cieff

0ln

4

12ln

)3/( 4ln

)( 34

43

ddac

NC

dccdc

ddc

c

Fsi

Tij

si

Scheme independence can be proved analytically for a1,2 and

checked numerically for other ai’s

CF=(Nc2-1)/(2Nc)

A major progress before 1999!

It is more convenient to define ai=ci+ci1/Nc for odd (even) i

(Vertex & penguin corrections have not been considered in pQCD approach)

17

Generalized Factorization (II)

effc

effi

effi

effi Ncca /1Generalized factorization (II):

Some of nonfactorizable effects are already included in cieff

Difficulties:

Gluon’s momentum k2 is unknown, often taken to be mB2/2. It is OK

for BRs, but not for CPV as strong phase is not well determined

a6 & a8 are associated with matrix elements in the form mP2/[mb()mq()],

which is not scale independent !

a2,3,5,7,10 (especially a2, a10) are sensitive to Nceff. For example,

Nceff 2 3 5

a2(=mb) 0.219 0.024 -0.131 -0.365

Expt’l data of charmless B decay a2 0.20 Nceff 2

18

QCD Factorization

Beneke, Buchalla, Neubert, Sachrajda (BBNS) PRL, 83, 1914 (99)

)()(1||0||

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

)()(||

1122

21

21

2

1

2

b

QCDs

MMBII

MIBM

M

mOOBjMjM

yxyxdxdyTd

xxdxTFfBOMM

TI:

TII: hard spectator interactions

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

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

19

...]0|)0()(|0|)0()(|[410|)0()(|

'21'''''4

55

5555

uxduxduxd

qqqqqqqqqqqq

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

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

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

5

5

1

05

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.

20

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

ixxxxg

ixgxdxm

ixgxdxm

V

iPPHN

VC

Nc

Nc

ca

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

21

Hard spectator interactions (non-factorizable) :

)1)(1()()(

)()0(

21

1

2

02 yx

yxdxdyd

Fmff

H MMBBM

B

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

22

Power corrections

1/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 very important due to chiral enhancement: m

2/(mu+md) 2.6 GeV at =2 GeV

Consider penguin-dominated mode B K

A(BK) a4+2a6/mb where 2/mb 1 & a6/a4 1.7 Phenomenologically, chirally enhanced power corrections should be taken into account

need to include twist-3 DAs p & systematically

02

05 ||0||

)(||0|| BVA

mmmmBbuud

dub

OK for vertex & penguin corrections

23

)(1)(

)1)(1()(

)()0( 11

2

1

2

02 y

xxry

yxx

dxdydFm

ffH p

MMM

BBMB

MB

Not OK for hard spectator 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.

Relevant scale for hard spectator interactions

h=(h)1/2 (hard-collinear scale), s=s(h)

as the hard gluon is not hard enough

k2=(-pB+xp1)2 xmB2 QCDmb 1 GeV2

10 ,1ln1

1

0

Hi

Hh

BH

Hemy

dyX

24

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

25

Annihilation topology

Weak annihilation contributions are power suppressed

yyxxwith

yxyxyyxdxdy

NCfffGA 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

26

Classify into (i) (V-A)(V-A), (ii) (V-A)(V+A), (iii) (S-P)(S+P) (V-A)(V-A) annihilation is subject to helicity suppression,

in analog to the suppression of e relative to Helicity suppression is not applicable to (V-A)(V+A) & penguin-

induced (S-P)(S+P) annihilation dominant contributions Since k2 xymB

2 with x,y O(1), imaginary part can be induced

from the quark loop bubble when k2> mq2/4

Gerard & Hou (91)

27

Comparison between QCDF & generalized factorization

QCDF is a natural extension of generalized factorization with the following improvements: Hard spectator interaction, which is of the same 1/mb order as vertex &

penguin corrections, is included crucial for a2 & a10

Include distribution of momentum fraction

1. a new strong phase from vertex corrections

2. fixed gluon virtual momentum in penguin diagram

For a6 & a8, V=6 without log(mb/ dependence ! So unlike other ai’s,

a6 & a8 must be scale & scheme dependent

)(4

6ln

)( as

scheme and scale renor. oft independen is )()(

)(2

6

mCddm

mmma

Fs

qb

P

Contrary to pQCD claim, chiral enhancement is scale indep.

28

Form factors

B D form factor due to hard gluon exchange is suppressed by wave function mismatch dominated by soft process For B , k2 h

2 mb. Let FB=Fsoft+Fhard

• It was naively argued by BBNS that Fhard=s(h)(/mB)3/2 & Fsoft=(/mB)3/2 so that B to form factor is dominated by soft process

• In soft-collinear effective theory due to Bauer,Fleming,Pirjol,Stewart(01),

B light M form factor at large recoil obeys a factorization theorem

Writing FB(0)=+J, Bauer et al. determined & J by fitting to B data

and found J (/mb)3/2

• In pQCD based on kT factorization theorem, <<J

0

1

0

),,()()()()()( uEJudufdEECEF MMBMMBM

Beneke,Feldmann (01)

29

In short, for B M form factor

QCDF: Fsoft>> Fhard, SCET: Fsoft Fhard, pQCD: Fsoft<< Fhard

However, BBNS (hep-ph/0411171) argued that Fsoft>>Fhard even in SCET

We compute form factors & their q2 dependence using covariant light-front model [HYC, Chua, Hwang, PR, D69, 074025 (04)]

CLF BSW MS LCSR

FB(0) 0.25 0.33 0.29 0.31

FBK(0) 0.35 0.38 0.36 0.35

A0B(0) 0.28 0.28 0.29 0.37

A0BK*(0) 0.31 0.32 0.45 0.47

BSW=Bauer,Stech,Wirbel

MS=Melikhov,Stech

LCSR=light-cone sum rule

B+ +0 F0B(0) 0.25

B0 A0B(0) 0.29

Light meson in B M transition at large recoil (i.e. small q2) can be highly relativistic importance of relativistic effects

30

Phenomenology: B PP

0.280.38 0.3 0.31.5

1.03.8 5.1 0.65.5

2.58.5 7.6 0.44.6

40~ 40~ 4.577.6 '

1.86.3 6.3 0.15.11

3.211.0 9.7 8.01.12

3.316.0 13.9 8.02.18

6.020.3 17.8 3.11.24

PQCD QCDF Average

00

0

00

0

0

K

K

K

K

K For FB(0)=0.25, predicted BRs for K modes are (15-30)% smaller than expt.

A longstanding puzzle for the enormously large rate of K’. Same puzzle occurs for f0(980)K. Note that ’ & f0

(980) are SU(3) singlet

A LD rescattering (e.g. B DD+-) is needed to interfere destructively with +-. This will give rise to observed BR of 00 (annihilation doesn’t help)

BRs in units of 10-6

31

Phenomenology: B VP

3.011.1 4.5 0.79.4

3.2 3.5 0.75.1

2.5 4.6 12

5.4 7.4 1.69.9

2.2 2.6 0.95.2

3.0 5.8 48

1.33.1 0.7 1.03.0

1.89.8 3.3 1.812.6

1.63.6 3.3 31

2.212.4 3.6 1.29.8

0.2~ 0.6 2.19.1

6.8 3.11.9

12.9 0.20.12

26.3 5.20.24

PQCD QCDF Average

00

0

0

00*

*

0*

0*

00

0

0

K

K

K

K

K

K

K

K

K

K

For penguin-dominated modes, VP < PP due to destructive interference between a4 & a6 terms (K) or absence of a6 terms (K*)

Br(00)=1.40.7<2.9 by BaBar & 5.1 1.8 by Belle Final-state rescattering will enhance 00 from 0.6 to 1.30.3. The pQCD prediction 0.2 is too small QCDF predictions for penguin dominated modes K*, K are consistently too small

power corrections from penguin-induced annihilation and/or FSIs such as LD charming penguins

32

Phenomenology: B VV

average QCDF pQCD (a) (b)

QCDF results from HYC & Yang, PL, B511, 40(a): BSW, (b): LCSR

4.215.7 8.7 4.6 0.99.5

4.316.9 9.3 4.3 1.59.7

3.1 1.9 3.5

3.7 2.2 3.4

1.0 0.7 2.6

4.8 3.1 16.3

5.6 3.0 10.6

6.7 4.0 4.79.6

21.0 13.8 12.6

0.3 0.2 1.1

21.8 12.8 26.4

35.0 21.2 630

0*

*

6.21.2

*

8.16.1

0*

00*

9.56.8

*

8.35.3

0*

0*

0.47.3

00

6.14.6

0

K

K

K

K

K

K

K

K

Tree-dominated modes tend to have large BRs

BRs can differ by a factor of 2 in different form factor models

The predicted K* & K*

by QCDF are too small

33

Direct CP violation in B decays

Direct CPV (5.7) in B0 K+- was established by BaBar and Belle

sinsin)()()()(

fBfBfBfBACPDirect CPV:

First confirmed DCPV observed in B decays ! 2nd evidence at Belle !! Combined BaBar & Belle data 3.6 DCPV in B0 -+

34

Direct CP violation in QCDF

For DCPV in B +-, 5.2 effect claimed by Belle(03), not yet confirmed by BaBar

723 6.5 2437

7.1 0.6 48

517 4.5 211

PQCD QCDF Expt(%)

7.133.13

0

1.02.0

6.118.11

1415

0

1.99.9

0

B

B

KB

QCDF predictions for DCPV disagree with experiment !

35

“Simple” CP violation from perturbative strong phases:

penguin (BSS) vertex corrections (BBNS) annihilation (pQCD)

“Compound” CP violation from LD rescattering: [Atwood,Soni]

weak

strong

ACP sin sin : weak phase : strong phase

36

723 5.2 6.5 2437

7.1 9.12 0.6 48

517 1.4 4.5 211

PQCD QCDF(S4) QCDF Expt(%)

7.133.13

0

1.02.0

6.118.11

1415

0

1.99.9

0

B

B

KB

Beneke & Neubert: Penguin-dominated VP modes & DCPV can be accommodated by having a large penguin-induced annihilation topology with

A=1, A=-55 (PP), A=-20 (PV), A=-70 (VP)

Sign of A is chosen so that sign of A(K+-) agrees with data

Difficulties: The origin of strong phase is unknown & its sign is not predicted The predicted ACP(K+)=0.10 is in wrong sign: expt= -0.510.19

Annihilation doesn’t help explain tree-dominated modes 00 & 00

necessity of another power correction: FSI

37

FSI as rescattering of intermediate two-body states [HYC, Chua, Soni; hep-ph/0409317] Strong phases O(s,1/mb)

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 (for soft Goldstone boson)

• Cutoff must be introduced as exchanged particle is off-shell

and final states are hard

Alternative: Regge trajectory [Nardulli,Pham][Falk et al.] [Du et al.] …

38

n

tmtF

2

22

)(

Dispersive part is obtained from the absorptive amplitude via dispersion relation

''

)'( 1)( 22 ds

mssMmmMe

s BB

= mexc + rQCD (r: of order unity)

or r is determined form a 2 fit to the measured rates

r is process dependent n=1 (monopole behavior), consistent with QCD sum rules

Once cutoff is fixed 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

39

Penguin-dominated B K, K’, K*, K, K, K* receive significant

LD charm intermediate states (i.e. charming penguin) contributions.

Such FSIs contribute to penguin-induced annihilation topologies Tree-dominated B 00 is enhanced by LD charming penguins to

(1.30.3)10-6 to be compared with (1.91.2)10-6: (1.4 0.7)<2.9 10-6 from

BaBar & (5.11.8)10-6 from Belle Charming penguin contributions to B 00 are CKM suppressed. B0D00 and its strong phase relative to B0D-+ are well accounted for

by FSI non-negligible annihilation E/T = 0.14 exp(i96)

B0D-sK+ can proceed only via annihilation is

well predicted FSI can be neglected for tree-dominated color-allowed modes

Final-state rescattering effects on decay rates

40

Strong phases are governed by final-state rescattering. Signs of DCPV are in general flipped by FSIs.

212 624 915

1030 30 3928

723 64 2437

1.7 1143 48

517 14 211

PQCD FSIQCDF Expt(%)

0

14

000

38

0

1.02.0

1415

0

13

0

B

B

B

B

KB

Final-state rescattering effects on DCPV

41

QCDF by BBNS:NP, B591, 313 (00): B DNP, B606, 245 (01): B K, NP, B651, 225 (03): B P’NP, B675, 333 (03): B, Bs PP, VP & DCPV

QCDF by Du et al.:PR, D64, 014036 (01): B PP (a detailed derivation of ai)PR, D65, 074001 (02): B PPPR, D65, 094025 (02): B VPPR, D68, 054003 (03): Bs PP,VP

K.C. Yang, HYC:PR, D63, 074011 (01) : B J/KPR, D64, 074004 (01) : B KPL, B511, 40 (01) : B VV

References for QCDF