1 factorization approach for hadronic b decays hai-yang cheng factorization ( and history) general...
DESCRIPTION
3 All two-body hadronic decays of heavy mesons can be expressed in terms of six distinct quark diagrams [Chau, HYC(86)] All quark graphs are topological and meant to have all strong interactions included and hence they are not Feynman graphs. And SU(3) flavor symmetry is assumed. Diagrammatic Approach (penguin) (or P a ) (tree) (color-suppressed) (exchange) (annihilation) Chiang,Gronau, Rosner,…TRANSCRIPT
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