Hadronic B Decays in
Perturbative QCD Approach
Cai-Dian Lü (IHEP, Beijing)




Formalism of perturbative QCD (PQCD)
based on kT factorization
Direct CP asymmetry
Polarization in BVV decays
Summary
Thank colleagues: Keum, Li, Sanda, Ukai, Yang, …
C.D. Lu
Sino-German
1
Naïve Factorization Approach
(BSW)

d
u
b
B0
u
–
d
Idea borrowed from
semi-leptonic decay
C.D. Lu
+


Decay matrix element
can be separated into
two parts:
Short distance Wilson
coefficients and
Hadronic parameters:
form factor and decay
constant
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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
sub-leading order in αs expansion
C.D. Lu
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QCDF
OCD-improved factorization =
naïve factorization + QCD correction
TI
TII
b
b
F
B
(a )
Factorizable
emission
Leading
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F B
(c)
(b)
Vertex
correction
Non-spectator
(d )
Exchange &
Annihilation
Sub-leading
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QCDF amplitude:
AB     f  TI  F B    TII  B  
Two concerns:
The emission diagram is certainly leading….
But why must it be written in the BSW form ?
Has naïve factorization been so successful
that what we need to do is only small sub-leading
correction ?
Both answers are “No”
C.D. Lu
Sino-German
5
Picture of PQCD Approach
4-quark
operator

b

B
Six quark interaction inside the dotted line
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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-quarkdependent
operator
channel
exp{-S(t)} is Sudakov factor，to relate the shortand long-distance interaction
H(k1,k2,k3,t) ischannel
perturbative
calculation of six
dependent
quark interaction
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Perturbative Calculation of H(t)
in PQCD Approach
Form
factor—
factoriz
able
Nonfactori
zable
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Perturbative Calculation of H(t)
in PQCD Approach
Nonfactorizable
annihilation
diagram
D(*)
D(*)
Factorizable
annihilation
diagram
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



All diagrams using the same wave
functions
(same order in s expansion)
All channels use same wave functions
Number of parameters reduced
Do not need form factor inputs
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Feynman Diagram
Calculation

i
i 


 d k1d k2 (k1  k2 )2 tr      B (1   5 ) k1  p 2 
4
4
Wave function
i
i
i
 2

2
2
2
(k1  k2 )
k1  k2  2k1  k2  2 xymB
k2=mB(y,0,k2T),
k1=mB(0,x,k1T)
k2·k1= k2+k1– - k2T·k1T ≈ mB2xy
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Endpoint
Singularity
The gluon propagator
i
i

2
2
(k1  k 2 )
 2 xymB


x,y are integral variables from 01, singular
at endpoint
In fact, transverse momentum at endpoint is
not negligible
then no
i
i

singularity
2
2
T
T 2
(k1  k 2 )
 2 xymB  (k1  k 2 )
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After including the quark transverse
momentum
there is no endpoint singularity
large double logarithm are produced after
radiative corrections,
they should be resummed to generate the
Sudakov form factor
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Sudakov factor
The soft and collinear
divergence
produce
2
double logarithm ln Pb，
Summing over these logs result a Sudakov
factor. It suppresses the endpoint region
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i
i

2
2
(k1  k 2 )
 2 xymB


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
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Endpoint
Singularity
i
i

2
2
(k1  k 2 )
 2 xymB
D(K) 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 diagrams
c
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D
u
c
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D
u
16
The sub-leading calculation shows an
end-point singularity
Endpoint singularity in
collinear factorization
Need to introduce arbitrary cutoffs
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Power Counting--QCDF



Form factor diagrams are leading
All others are s suppressed
Annihilation-type are even power suppressed
(small)
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Power Counting--PQCD


All diagrams are at the same order of s
Some non-factorizable diagram contributions
are suppressed due to cancellations
power suppressed
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QCDF vs PQCD

Form factor input
No need form factor

Wave function input
Wave function input
Parameterize
Annihilation
/exchange diagram
Annihilation
/exchange
diagram calculable
BD0 pi0 not
calculable
Most modes
calculable

C.D. Lu
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Two operators contribute to
0
0
0
B   D decay:
c
D
u
b
d
B0
D
0
c
d
b
0
B0
color enhanced
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u
d
d
0
O2  (ud )  (bc)
O1  (uc)  (bd )
C1 ~ – 0.2
0
color suppressed
~
1/3 C2 ~ 1/3
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For B0 D00, non-factorizable
diagrams do not cancel
a2  0.47
arg (a2/a1) ~ – 41°
PQCD
Exp.
B0 D–+ 2.8±0.4
3.0±0.4
B+ D0+ 5.5±0.4
5.3±0.5
B0 D00 0.26±0.05 0.29±0.05
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Branching Ratios




Some 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 ….
C.D. Lu
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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
C.D. Lu
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B→  ,  K Have Two Kinds of
Diagrams with different weak phase
W
 (K)
O1,O2
u
b

B
d(s)
Tree ∝ VubVud*(s)
 (K)
O3,O4,O5,O6
W
b
t
Penguin∝VtbVtd* (s)
B

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Strong phase is important
for direct CP

But usually comes from non-perturbative
dynamics, for example
K
D

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
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
The strong phase of Both QCD factorization
and generalized factorization come from
perturbative QCD charm quark loop diagram
It is small, since it is at αs order
Therefore the CP asymmetry is small
C.D. Lu
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Inclusive Decay
 ,  , p, n
B  X u l
u
~
b
Cut quark diagram ~ Sum over final-state hadrons
Off-shell
hadrons



On-shell
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Annihilation-Type diagram
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-Type diagram
W annihilation
W exchange

 
Time-like penguin
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
Space-like penguin
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Naïve Factorization fail
fB
?
fB
Momentum transfer:
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Q M
2
2
B
32
For (V-A)(V-A), left-handed current
B
p2
spin (this configuration is not allowed)
fermion flow
p1 momentum
Like Be 
e
pseudo-scalar B requires spins in opposite
directions, namely, helicity conservation
Annihilation suppression ~ 1/mB ~ 10%
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PQCD Approach
(K)
Two diagrams cancel each other for
(V-A)(V-A) current — dynamical suppression
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W Exchange Process
Vcb* Vud ~ 2
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W exchange process
Results:
Br ( B 0  DS K  )  (4.600..86 ) 105
Br ( B 0  DS* K  )  (2.7  0.6) 105
Reported by Ukai in BCP4 (2001) before Exps:

S


S

Br ( B  D K )  (4.6
0
1.2
0.6
5
 1.3) 10 , Belle
5
Br ( B  D K )  (3.2  1.0  1.0) 10 , BaBar
0
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+
BK
–
K
decay
Comparing B(B pi pi): 10–6, 1%

Vtb*Vtd , small br, 10–8
b
d
C.D. Lu
 
u K–
Time-like penguin
Also (V-A)(V-A)
contribution
s
u K
+
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Chiral Enhancement R,L=15



Two penguin operators:
O4~(V-A)(V-A)
b
O6~(V-A)(V+A) Fiertz trans.
s
t
(S+P)(S-P) 2(mK2/ms) x 1/mB
O4  d   Lb   q    Lq
q
O(1)
O6  d   Lb   q    Rq
q
q
q
C.D. Lu
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No suppression for O6



Space-like penguin
Become (s-p)(s+p) operator after Fiertz
transformation Chirally enhanced
No suppression, contribution “big” (20-30%)
b
d
C.D. Lu


d (s )
+
+
 (K )
u
d 
–
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CP Violation in B   (K)
(real prediction before exp.)
CP(%)
FA
BBNS
PQCD
Exp
(2001)
 +K –
+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
 +
–
C.D. Lu
+9±3
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Annihilation in QCDF




Power (1/mB) suppressed
and s suppressed
Should not be large
But has to be large from exp.
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Operator O6 is very
important






Important for I = 1/2 rule in history
B  ,  K -- direct CP
  K*
branching ratio too small
in QCDF
K
  K*
  K*
polarization problem
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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 + –, K, ’K , KKK …
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For Example:
(From Yossi Nir)
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Polarization of BVV decays
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Helicity flip suppression of the
transverse polarization amplitude
Naïve counting rule
C.D. Lu
H = MN  MT
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Counting Rules for BVV
Polarization



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?
C.D. Lu
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Theoretical attempts to
solve these puzzles
Nonperturbative corrections:
a) the charming penguin
b) the final state interactions
……
Currents that breaks the naïve cutting rule:
a) new physics
b) the magnetic penguin
c) the annihilation diagrams
……
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There are still problems for
some of the explanations
The perpendicular polarization is given by:
Naïve Babar and Belle Avg.
Final state interaction can not explain
RN = RT and some others are difficult to
explain the relative phase
C.D. Lu
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The annihilation diagram
Fierz Transformation
The (S+P)(S-P) current can break the counting
rule,
The annihilation diagram contributes equally to
the three polarization amplitudes
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No suppression for O6



Space-like penguin
Become (s-p)(s+p) operator after Fiertz
transformation Chirally enhanced
No suppression, contribution “big” (20-30%)
b
d
C.D. Lu


s

s
d K*
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51
Large transverse component
in BK* decays
Annihilation can enhance transverse
contribution: RL = 59% (exp:50%)
and also right ratio of R=, R and right
strong phase =, 
b
d
C.D. Lu


s

s
H-n Li, Phys. Lett.
B622, 68, 2005
*
K
d
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Alex Kagan’s study in QCDF
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Polarization for B()()
Phys.Rev.D73:014024,2006
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Polarization of BK*()
Decay modes RL(exp)
B K 

*0

B K 
0
B K 


0
*
*
B   K * 
RL
R=
R
66%
76%
13%
11%
96%
78%
11%
11%
78%
12%
10%
72%
19%
9%
Phys.Rev.D73:014011,2006
C.D. Lu
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Time-like penguin in B 
decays (10–8 )

Transverse polarization is around 35%
b
d
 
s 
s
s
C.D. Lu
Eur. Phys. J. C41,
311-317, 2005

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56
*
BK

*
K
decay
Vtb*Vtd , small br, 10–7
b
d
 
Time-like penguin
Also (V-A)(V-A)
u K* contribution
s
*
u
C.D. Lu
K
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57
Polarization of BK*K*
Decay modes
RL
R=
R
0
*0
*0
67%
18%
15%

*
*0
75%
13%
12%
99%
0.5%
0.5%
B K K
B K K
*
B K K
0
C.D. Lu
*
Tree dominant
Phys.Rev.D72:054015,2005
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and
Experiment:
C.D. Lu
is the first measured channel in Bs decays,
which is useful to determine the Bs wave
function
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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 BVV decays can also be
explained by PQCD
Important role of Annihilation type diagram

C.D. Lu
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Thank you!
Vielen Dank！
谢谢！
C.D. Lu
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61
Contributions of different αs in
H(t) calculation
Fraction
αs/
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Error Origin in PQCD




The wave functions
The decay constants
CKM matrix elements
High order corrections
CP is sensitive to
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Branching ratio in
-6
NLO(10 )
Li, Mishima, Sanda hep-ph/0508041
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NLO direct CP asymmetry
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B K puzzle


Their data differ by 3.6 A puzzle?
K+- and K+0 differ by sub-leading
amplitudes Pew and C. Their CP are
expected to be similar.
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