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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