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Some progress in PQCD approach
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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
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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