Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Charmless B to PP,PV, VV decays Based on the Six-quark
Effective Hamiltonian with Strong Phase
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
J.Phys.G38:015006,2011; Int.J.Mod.Phys.A25:69-111,2010;
KITPC/ITP-CAS
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Outline
1
Effective Hamiltonian of Six-Quark Operators
2
QCD Factorization based on effective six-quark operators
3
Treatment of singularities
4
Nonperturbative Corrections
5
Input parameters and numerical calculation
6
Conclusions and Discussions
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Four-quark operator effective Hamiltonian
"
#
10
X
GF X s
(q)
(q)
Heff = √
λq C1 (µ)O1 (µ) + C2 (µ)O2 (µ) +
Ci (µ)Oi (µ) + h.c. ,
2 q=u,c
i=3
∗ are products of the CKM matrix elements, C (µ) the Wilson
where λsq = Vqb Vqs
i
coefficient functions, and Oi (µ) the four-quark operators
(q)
O1 = (q̄i bi )V −A
(s̄j qj )V −A ,
X
O3 = (s̄i bi )V −A
(q̄j0 qj0 )V −A ,
(q)
O2 =X
(s̄i bi )V −A (q̄j qj )V −A ,
O4 =
(q̄i0 bi )V −A (s̄j qj0 )V −A ,
q0
0
O5 = (s̄i bi )V −A
q
X
(q̄j0 qj0 )V +A ,
O6 = −2
q0
X
(q̄i0 bi )S−P (s̄j qj0 )S+P ,
q0
X
3
(s̄i bi )V −A
eq 0 (q̄j0 qj0 )V +A ,
2
q0
X
3
O9 = (s̄i bi )V −A
eq 0 (q̄j0 qj0 )V −A ,
2
0
O7 =
q
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
O8 = −3
X
eq 0 (q̄i0 bi )S−P (s̄j qj0 )S+P ,
q0
O10 =
3X
eq 0 (q̄i0 bi )V −A (s̄j qj0 )V −A .
2 0
q
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Motivation of six-quark
1
Meson:
Quark-antiquark bound state;
2
B decays to two light mesons:
Three quark-antiquark pairs¶
3
Leading order:
One W boson and one gluon exchange;
4
The four quarks via W-boson exchange can be regarded as a local four quark
interaction at the energy scale much below the W-boson mass, while two QCD
vertexes due to gluon exchange are at the independent space-time points, the
resulting effective six quark operators are hence in general nonlocal;
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Six-quark operators
Figure:
Four different six-quark diagrams with a single W-boson exchange and a single gluon exchange
For example, the operator corresponding to the second diagram can be written asµ
(6)
Oq2
d4 k d4 p −i((x1 −x2 )p+(x2 −x3 )k) ¯0
1
e
(q (x3 )γν T a q 0 (x3 )) 2
(2π)4 (2π)4
k + i
p/ + mq1
(q̄2 (x2 ) 2
γ ν T a Γ1 q1 (x1 )) ∗ (q̄4 (x1 )Γ2 q3 (x1 )),
p − mq21 + i
ZZ
=
4παs
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
1
The full operators of six-quark shall sum over the four
diagramsµ
O (6) =
4
X
(6)
Oqj .
j=1
2
(6)
Actually, the initial six quark operators Oqj (j = 1, 2, 3, 4) can
be obtained from the following initial four quark operator via a
single gluon exchange:
O ≡ (q̄2 Γ1 q1 ) ∗ (q̄4 Γ2 q3 )
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
The possible six-quark diagrams at one loop level
(a)One-Loop
contributions
only to the effective weak
vertex(type I).
(b)One-Loop
contributions
only to the gluon vertexes
(type II).
(c)One-Loop contributions for
both weak and strong vertexes
(type III).
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
When ignoring the type III diagrams, we arrive at an approximate six quark operator
effective Hamiltonian as follows:
(6)
Heff
=
4
GF X X s(d)
(q)(6)
(q)(6)
√
{
λq [C1 (µ)O1qj (µ) + C2 (µ)O2qj (µ)]
2 j=1 q=u,c
+
10
X
s(d)
λt
(6)
qj (µ)}
Ci (µ)Oi
+ h.c. + . . . ,
i=3
with:
(6)
qj (µ)(j
Oi
= 1, 2, 3, 4)
are six quark operators which may effectively be obtained from the corresponding four
quark operators Oi (µ)(i = 1 − 10) at the scale µ via the effective gluon exchanging
interactions between one of the external quark lines of four quark operators and a
spectator quark line at the same scale µ.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
From six-quark operators to hadronic matrix elements
For example, we examine the hadronic matrix elements of B → π 0 π 0 decay for a
(6)
typical six-quark operator OLL µ
(6)
OLL
ZZ
=
1
d4 k d4 p −i((x1 −x2 )p+(x2 −x3 )k) 1
e
(2π)4 (2π)4
k 2 p 2 − md2
[d̄k (x2 )(p/ + md )γ ν Tkia γ µ (1 − γ 5 )bi (x1 )]
a
[d̄j (x1 )γµ (1 − γ 5 )dj (x1 )][d̄m (x3 )γν Tmn
dn (x3 )],
(6)
which is actually a part of the six quark operator O4q2 in the effective Hamiltonian. Its
hadronic matrix elements are the most abundant for B → π 0 π 0 decay.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Reduction of hadronic matrix elements
Its hadronic matrix element for B → π 0 π 0 decay leads to the following most general
terms in the QCD factorization approach:
(6)
O
MLL
(Bππ) ≡< π 0 π 0 | OLL | B¯0 >
ZZ
d4 k d4 p −i((x1 −x2 )p+(x2 −x3 )k) 1
1
=
e
(2π)4 (2π)4
k 2 p 2 − md2
< π 0 π 0 | [d̄k (x2 )(p/ + md )γ ν Tkia γ µ (1 − γ 5 )bi (x1 )]
a
[d̄j (x1 )γµ (1 − γ 5 )dj (x1 )][d̄m (x3 )γν Tmn
dn (x3 )] | B¯0 >
O(1)
≡ MLL
O(2)
+ MLL
O(3)
+ MLL
O(4)
+ MLL ,
O(i)
MLL (i = 1, 2, 3, 4) correspond to the four different diagrams of reductionµ
Figure: Different ways of reducing hadronic matrix element by QCDF
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Explicit structure of MLL
O(1)
For example, the expression of MLL
O(1)
MLL
ZZ
=
ν
is as follows:
d 4 k d 4 p −i((x1 −x2 )p+(x2 −x3 )k)
1
a a
e
T T
(2π)4 (2π)4
k 2 (p 2 − md2 ) ki mn
µ
5
5
σγ
δρ
βα
[(p
/ + md )γ γ (1 − γ )]ρσ [γµ (1 − γ )]αβ [γν ]γδ MBim (x1 , x3 )Mπnk (x3 , x2 )Mπjj (x1 , x1 ),
with:
βα
=
βα
=
MBnm (xi , xj )
Mπnm (xi , xj )
α
β
0
< 0 | d̄m (xj )bn (xi ) | B̄ (PB ) >
Z 1
iFB δmn
−i(u P + xj +(PB −u P + ) xi )
βα
B
B
du e
MB (u, PB ),
=−
4 Nc 0
Z 1
iFπ δmn
0
α
β
−i(x P xj +(1−x)P xi )
βα
< π (P) | d̄m (xj )dm (xi ) | 0 >=
dx e
Mπ (x, P),
4 Nc 0
with FM (M = B, π) the decay constants. Here MBβα (u, PB ) and Mπβα (x, P) are the
spin structures for the bottom meson and light meson π and characterized by the
corresponding distribution amplitudes.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
After performing the integration over space-time and momentum, the amplitude is
simplified to be:
(6)
O
MLL
(Bππ) =< π 0 π 0 | OLL | B¯0 >=
(1)
(u PB+
+
1
Z
Z
1
1
Z
du dx dy
0
0
0
(2)
MLL
MLL
1
[
+
]
− (1 − x)P1 )2 (P1 − u PB+ )2 − md2
((1 − x)P1 + y P2 − u PB+ )2 − md2
(3)
(4)
MLL
MLL
1
[
+
] ,
2
2
2
(xP1 + (1 − y )P2 ) (x P1 + P2 ) − md
(x P1 + (1 − y )P2 − u PB+ )2 − md2
with:
(1)
MLL
=
......
CF
∗ FB Fπ2 Tr[MB (u, PB )γν Mπ (x, P1 )γ ν (P
/ 1− u P/B++ md )γµ (1 − γ 5 )]
NC
CF
3
Tr[Mπ (y , P2 )γ µ (1 − γ 5 )] = i
FB Fπ2 φB (u)mB
µπ φπ (y )φpπ (x),
4NC
(i)
where MLL (i = 1, 2, 3, 4) are obtained by performing the trace of matrices and
determined by the distribution amplitudes.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Four kinds of diagrams
Figure: Four types of effective six quark diagrams lead to sixteen diagrams for hadronic two body
decays of heavy meson via QCD factorization
The first diagram is known as the factorizable one, the second is the non-factorizable
one and color suppressed. The third is the factorizable annihilation diagram and
color suppressed, and the fourth is an annihilation diagram and its matrix element
vanishes.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Independent terms
The effective four quark vertexes concern three types of current-current interactions:
(V − A) × (V − A) or (LL), (V − A) × (V + A) or (LR), (S − P) × (S + P) or (SP).
Therefore, there are totally 4 × 4 × 3 kinds of hadronic matrix elements involved in the
QCD factorization approach, while only half of them are independent with the
following relations:
a1
MLL
a1
MLR
a1
MSP
b1
MLL
b1
MLR
b1
MSP
c1
MLL
c1
MLR
c1
MSP
d1
MLL
d1
MLR
d1
MSP
=
=
=
=
=
=
=
=
=
=
=
=
F ;
TLLa
F ;
TLRa
F ;
TSPa
F ;
TLLb
F ;
TLRb
F ;
TLLb
0;
0;
0;
0;
0;
0;
a2
MLL
a2
MLR
a2
MSP
b2
MLL
b2
MLR
b2
MSP
c2
MLL
c2
MLR
c2
MSP
d2
MLL
d2
MLR
d2
MSP
=
=
=
=
=
=
=
=
=
=
=
=
F /N ;
TLLa
C
F /N ;
TSPa
C
F /N ;
TLRa
C
N /N ;
TLLb
C
N
TSPb /NC ;
N /N ;
TLRb
C
N /N ;
TLLa
C
N /N ;
TSPa
C
N /N ;
TLRa
C
F /N ;
TLLb
C
F /N ;
TSPb
C
F /N ;
TLRb
C
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
a3
MLL
a3
MLR
a3
MSP
b3
MLL
b3
MLR
b3
MSP
c3
MLL
c3
MLR
c3
MSP
d3
MLL
d3
MLL
d3
MLL
=
=
=
=
=
=
=
=
=
=
=
=
AN
LLa /NC ;
AN
SPa /NC ;
AN
LRa /NC ;
AFLLb /NC ;
AFSPb /NC ;
AFLRb /NC ;
AFLLa /NC ;
AFSPa /NC ;
AFLRa /NC ;
AN
LLb /NC ;
AN
SPb /NC ;
AN
LRb /NC ;
a4
MLL
a4
MLR
a4
MSP
b4
MLL
b4
MLR
b4
MSP
c4
MLL
c4
MLR
c4
MSP
d4
MLL
d4
MLL
d4
MLL
=
=
=
=
=
=
=
=
=
=
=
=
0;
0;
0;
0;
0;
0;
AFLLa ;
AFLRa ;
AFSPa ;
AFLLb ;
AFLRb ;
AFSPb .
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Treatment of Singularities
There are two kinds of singularities in the evaluation of hadronic matrix elements:
Infrared divergence of gluon exchanging interaction
On mass-shell divergence of internal quark propagator
Figure: Definition of momentum in B → M1 M2 decay. The light-cone coordinate is adopted with
(n+ , n− , ~
k⊥ ).
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Treatment of Singularities
1
Quark propagator
Applying the Cutkosky rule to deal with a physical-region singularity of all
propagators, the following formula holds:
1
1
=P 2
−iπδ[p 2 − mq2 ],
2
2
2
p − mq + i
p − mq
which is known as the principal integration method, and the integration with the
notation of capital letter P is the so-called principal integration.
2
Gluon propagator
We prefer to add the same dynamics mass for both gluon and light quark to
investigate the infrared cut-off dependence of perturbative theory prediction:
1 p/ + mq
k 2 (p 2 − mq2 )
→
1
p/ + µq
(q is a light quark).
(k 2 − µ2g + i) (p 2 − µ2q + i)
Here the energy scale µg , µq plays the role of infrared cut-off but preserving
gauge symmetry and translational symmetry of original theory.Use of this effective
gluon propagator is supported by the lattice and the field theoretical studies,
which have shown that the gluon propagator is not divergent as fast as k12 .
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Applying this prescription to the amplitude illustrated in previous section, we have
(6)
O
MLL
(Bππ) =< π 0 π 0 | OLL | B¯0 >=
1
Z
1
Z
Z
1
du dx dy
0
1
[
(u PB+ − (1 − x)P1 )2 −µ2g + i (P1 −
0
(1)
MLL
u PB+ )2 −
0
mq2 +i
(2)
+
MLL
((1 − x)P1 + y P2 − u PB+ )2 − mq2 +i
]
(3)
+
MLL
1
[
(xP1 + (1 − y )P2 )2 −µ2g + i (x P1 + P2 )2 − mq2 +i
(4)
+
MLL
(x P1 + (1 − y )P2 − u PB+ )2 − mq2 +i
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
] .
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
For example, the explicit expression for the hadronic matrix element of the factorizable
emission contributions for the (V − A) × (V − A) effective four-quark vertexes:
FM1 M2
TLLa
(M)
=
Z 1Z 1Z 1
1 CF
2
du dx dy mB
φM (u)
i
FM FM1 FM2
4 NC
0
0
0
mB (2mb − mB x)φM1 (x) + µM1 (2mB x − mb )
F
[φpM (x) − φT
M1 (x)] φM2 (y )hTa (u, x),
1
The functions hF (u, x)TA arise from propagators of gluon and quark and has the
following explicit form:
F
hTa
(u, x) =
1
,
2 − µ2 + i)(xm2 − m2 + i)
(−u(1 − x)mB
g
B
b
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Nonperturbative Corrections I: Emission Diagrams
The vertex corrections were proposed to improve the scale dependence of Wilson
coefficient functions of factorizable emission amplitudes in QCDF. Those coefficients
C
are always combined as C2n−1 + CN2n and C2n + 2n−1
, which, after taken into account
NC
C
the vertex corrections, are modified to:
C2n
C2n
αs (µ)
C2n (µ)
(µ) → C2n−1 (µ) +
(µ) +
CF
V2n−1 (M2 ) ,
NC
NC
4π
Nc
C2n−1
C2n−1
αs (µ)
C2n−1 (µ)
C2n (µ) +
(µ) → C2n (µ) +
(µ) +
CF
V2n (M2 ) ,
NC
NC
4π
Nc
C2n−1 (µ) +
with n = 1, ..., 5, M2 being the meson emitted from the weak vertex.In the naive
dimensional regulation (NDR) scheme, Vi (M) are given by:

R1
mb
for i = 1 − 4, 9, 10 ,

 12 ln( µ ) − 18 + R0 dx φa (x) g (x) ,
Vi (M) =
−12 ln( mµb ) + 6 − 01 dx φa (1 − x) g (1 − x) , for i = 5, 7 ,

R

−6 + 01 dx φb (x) h(x) ,
for i = 6, 8 ,
where φa (x) and φb (x) denote the leading-twist and twist-3 distribution amplitudes
for a pseudoscalar meson or a longitudinally polarized vector meson, respectively.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
To further improve our predictions, we shall examine an interesting case that vertexes
receive additional large non-perturbative contributions, namely the Wilson coefficients
C
ai = Ci + Ni±1 are modified to be the following effective ones:
C
Ci±1
αs (µ)
Ci±1 (µ)
e1 (M2 )) ,
(µ) +
CF
(Vi (M2 ) + V
NC
4π
Nc
αs (µ)
Ci±1 (µ)
Ci±1
e2 (M2 )) ,
(µ) +
CF
(Vi (M2 ) + V
= Ci (µ) +
NC
4π
Nc
ai → aieff = Ci (µ) +
(i = 1 − 4, 9, 10)
ai → aieff
(i = 5 − 8)
e1 (M2 ) and V
e2 (M2 ) depend on whether the meson M2 is a
The corrections V
pseudoscalar or a vector. It could be causedπ from the higher order non-perturbative
e1 (P) = 26e − 3 i , V
e2 (P) = −26, V
e1 (V ) = 15e π8 i , and
non-local effects. Adopting V
πi
e2 (V ) = −15e 8 , both the branching ratios and CP asymmetries of most
V
B → PP, PV , VV decay modes are improved.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Nonperturbative Corrections II:Annihilation Diagrams
Most of the annihilation contributions are from factorizable annihilation diagrams with
the (S − P) × (S + P) effective four-quark vertex:
Z
(µP1 + µP2 )y (1 − y )
1 P2
,
AP
dxdy
SP (M) ∼
2 − µ2 + i)((1 − y )m2 − m2 + i)
(x(1 − y )mB
g
q
B
Z
(µP1 − 3(2x − 1)mV2 )y (1 − y )
1 V2
AP
dxdy
,
SP (M) ∼
2 − µ2 + i)((1 − y )m2 − m2 + i)
(x(1 − y )mB
g
q
B
Z
(−3(1 − 2x)mV1 − µP2 )y (1 − y )
1 P2
AV
,
dxdy
SP (M) ∼
2 − µ2 + i)((1 − y )m2 − m2 + i)
(x(1 − y )mB
g
q
B
Z
3(1 − 2x)(−mV1 + 3(2x − 1)mV2 )y (1 − y )
1 V2
dxdy
AV
.
SP (M) ∼
2 − µ2 + i)((1 − y )m2 − m2 + i)
(x(1 − y )mB
g
q
B
1
Since the contributions of these amplitudes are dominated by the area x ∼ 0 or y ∼ 1,
P P
P V
V P
V V
ASP1 2 (M) and ASP1 2 (M) have the same sign, while ASP1 2 (M) and ASP1 2 (M) have a
P P2
(M)
different sign from ASP1
2
P P2
(M)
As a result, we use the same strong phase for ASP1
another one for
V P
ASP1 2 (M)
and
V V
ASP1 2 (M)(θ1a
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
P V2
(M)(θ1a
and ASP1
∼ 5o ), and
o
∼ 60 ).
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Input Parameters
1
Light-cone distribution amplitudes
2
CKM Matrix Elements,Decay constants etc.
3
Running scale, cut-off scale
4
Strong Phases
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Light-cone distribution amplitudes
1
B meson wave function
For the B-meson wave function, we take the following standard form in our
numerical calculations:
"
#
1 xmB 2
φB (x) = NB x 2 (1 − x)2 exp −
,
2
ωB
where the shape parameter of B meson is ωB = 0.25GeV , and NB is a
normalization constant.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
1
Input parameters
Numerical results
Light mesons
The light-cone distribution amplitudes (LCDAs) for pseudoscalar and vector
mesons are as below:
twist-2
φP (x, µ)
=
∞
X
P
3/2
6x(1 − x) 1 +
an (µ)Cn (2x − 1) ,
=
∞
X
V
3/2
6x(1 − x) 1 +
an (µ)Cn (2x − 1) ,
=
∞
X
T ,V
3/2
6x(1 − x) 1 +
an (µ)Cn (2x − 1) ,
n=1
φV (x, µ)
n=1
T
φV (x, µ)
n=1
twist-3
φp (x, µ)
φν (x, µ)
=
1,
=
φσ (x, µ) = 6x(1 − x),
3 2x − 1 +
∞
X
T ,V
an
(µ)Pn+1 (2x − 1)
n=1
φ+ (x)
=
2
3(1 − x) ,
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
2
φ− (x) = 3x ,
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
CKM Matrix Elements, Hadronic Input Parameters and Decay constants
As for the CKM matrix elements, we use the Wolfenstein parametrization with the
four parameters chosen as:
+0.00083
+0.035
A = 0.798+0.023
−0.017 , λ = 0.2252−0.00082 , ρ̄ = 0.141−0.021 , η̄ = 0.340 ± 0.016
And the hadronic input parameters and the decay constants taken from the QCD sum
rules and Lattice theory are as below:
τB ±
1.638ps
mc
1.5GeV
mω
1.7GeV
fB
0.210GeV
fKT∗
0.185GeV
τ Bd
1.525ps
ms
0.122GeV
mφ
1.8GeV
fπ
0.130GeV
fφT
0.186GeV
mB
5.28GeV
mπ±
0.140GeV
mK ∗±
300MeV
fK
0.16GeV
fρT
0.165GeV
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
mb
4.4GeV
mπ0
0.135GeV
mK ∗0
0.78GeV
fρ
0.216GeV
mt
173.3GeV
mK
0.494GeV
µπ
1.02GeV
fω
0.187GeV
mu
4.2MeV
mρ0
0.775GeV
µK
0.892GeV
fK ∗
0.220GeV
md
7.6MeV
mρ±
0.775GeV
fφ
0.215GeV
fωT
0.151GeV
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
1
Input parameters
Numerical results
Running scale
In our numerical calculations, the running scale is taken to be
p
µ = 1.5 ± 0.1GeV ∼ 2ΛQCD mb .
The scale of αs (µ) in the six-quark operator effective Hamiltonian is also taken at
µ = 1.5GeV . Mass of b quark running to µ = 1.5 ± 0.1 GeV is
mb (µ) ' 5.54GeV .
2
Cut-off scale
The infrared cut-offs for gluon and light quarks are the basic scale to determine
annihilation diagram contributions and are set to be µq = µg = 0.37GeV .
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Strong phases
In general, a Feynman diagram will yield an imaginary part for the decay
amplitudes when the virtual particles in the diagram become on mass-shell,which
generates the strong phase of the process and the resulting diagram can be
considered as a genuine physical process.
The calculation of strong phase from nonperturbative QCD effects is a hard task,
there exist no efficient approaches to evaluate reliably the strong phases caused
from nonperturbative QCD effects, so we set the strong phase as an input
parameter in our framework.
We adopt different strong phases of annihilation diagrams θa in
B → PP, PV , VV decay modes respectively to get the reasonable results.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Numerical Calculation: Form factors
The method developed based on the six quark effective Hamiltonian allows us to
calculate the relevant transition form factors via a simple factorization approach. They
are calculated by the following formalisms
F0B→M1 =
4πα(µ)CF FM1 M2
TLL
(B)(M1 , M2 = P),
2F
Nc mB
M2
V B→M1 =
2 (m + m
mB
4πα(µ)CF FM1 M2
B
M1 )
TLL,⊥ (B)
(M1 , M2 = V ),
2F
2 −m
Nc mB
m
(m
M2
M2
M1 mM2 )
B
A0B→M1 =
4πα(µ)CF FM1 M2
TLL
(B)(M1 = V , M2 = P),
2F
Nc mB
M2
A1B→M1 =
2
mB
4πα(µ)CF FM1 M2
TLL,// (B)
(M1 , M2 = V )
2
Nc mB FM2
mM2 (mB + mM1 )
with:
TLL,⊥ =
1
(TLL,+ − TLL,− ),
2
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
CF =
Nc2 − 1
,
2Nc
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Table: The B → P, V form factors at q 2 = 0 in QCD Sum Rules, Light Cone and our work.
Mode
B → K∗
B→ρ
B→ω
B→π
B→K
F(0)
V
A0
A1
V
A0
A1
V
A0
A1
F0
F0
QCDSR
0.411
0.374
0.292
0.323
0.303
0.242
0.293
0.281
0.219
0.258
0.331
LC
0.339
0.283
0.248
0.298
0.260
0.227
0.275
0.240
0.209
0.247
0.297
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
LC(HQEFT)
0.331
0.280
0.274
0.289
0.248
0.239
0.268
0.231
0.221
0.285
0.345
PQCD
0.406
0.455
0.30
0.318
0.366
0.25
0.305
0.347
0.30
0.292
0.321
This work
0.277
0.328
0.220
0.233
0.280
0.193
0.206
0.251
0.170
0.269
0.349
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Amplitudes of charmless B Decays
Take B → ππ decay as example,the decay amplitudes can be expressed as follows:
A(B 0 → π + π − )
=
A(B + → π + π 0 )
=
A(B 0 → π 0 π 0 )
=
∗
Vtd Vtb
[PTππ (B) +
2 C ππ
P
(B) + PEππ (B) + 2PAππ (B)
3 EW
1 ππ
1 Aππ
∗
(B) − AEEW
(B)] − Vud Vub
[T ππ (B) + E ππ (B)],
+ PEW
3
3
1
∗
ππ
C ππ
∗
√ {Vtd Vtb
[PEW
(B) + PEW
(B)] − Vud Vub
[T ππ (B) + C ππ (B)]},
2
1
1 C ππ
∗
ππ
√ {−Vtd Vtb
[PTππ (B) − PEW
(B) − PEW
(B) + PEππ (B)
3
2
1 Aππ
1 E ππ
+2PAππ (B) + PEW
(B) − PEW
(B)]
3
3
∗
ππ
ππ
+Vud Vub [−C (B) + E (B)]},
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Totally,there’re 11 amplitudes, which are defined as follows:
T
M1 M2
(M)
=
C
M1 M2
(M)
=
M M2
(M)
=
PEW1
......
GF 1
FM M
4παs (µ) √ [C1 (µ) +
C2 (µ)]TLL 1 2 (M)
NC
2
1
NM M
+
C2 (µ)TLL 1 2 (M) ,
NC
GF 1
FM M
4παs (µ) √ [C2 (µ) +
C1 (µ)]TLL 1 2 (M)
NC
2
1
NM M
+
C1 (µ)TLL 1 2 (M) ,
NC
GF 3 1
FM M
4παs (µ) √
[C9 (µ) +
C10 (µ)]TLL 1 2 (M)
NC
22
1
1
NM M
FM M
+
C10 (µ)TLL 1 2 (M) + [C7 (µ) +
C8 (µ)]TLR 1 2 (M)
NC
NC
1
NM M
+
C8 (µ)TSP 1 2 (M)},
NC
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Numerical results for B to PP decays
Table: The branching ratios (in units of 10−6 ) and direct CP asymmetries in B → πK decays.
The central values are obtained at µq = µg =0.37GeV.(Penguin dominate)
Mode
B + → π+ K 0
B + → π0 K +
B 0 → π− K +
B 0 → π0 K 0
ACP (π + K 0 )
ACP (π 0 K + )
ACP (π − K + )
ACP (π 0 K 0 )
S 0
π KS
Data[HFAG]
23.1 ± 1.0
12.9 ± 0.6
19.4 ± 0.6
9.8 ± 0.6
0.009 ± 0.025
0.050 ± 0.025
−0.098 ± 0.012
−0.01 ± 0.10
0.58 ± 0.17
NLO+Vertex
22.5
12.8
19.2
8.3
−0.006
-0.053
−0.118
−0.052
0.699
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
NLOeff
21.4
12.5
19.5
8.4
-0.006
0.012
-0.139
-0.139
0.760
This work
NLOeff (−10◦ )
19.0
11.2
17.4
7.4
-0.006
0.003
-0.158
-0.143
0.768
NLOeff (5◦ )
22.6
13.1
20.5
8.9
-0.007
0.018
-0.131
-0.138
0.756
NLOeff (20◦ )
25.9
14.9
23.3
10.2
-0.007
0.034
-0.105
-0.137
0.745
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Table: B → ππ, KK decay modes (Tree dominate)
Mode
Data[HFAG]
NLO+Vertex
NLOeff
This work
NLOeff (−40◦ )
NLOeff (5◦ )
NLOeff (50◦ )
B 0 → π− π+
B + → π+ π0
B 0 → π0 π0
B + → K + K̄ 0
B 0 → K 0 K̄ 0
B0 → K +K −
5.16 ± 0.22
5.59 ± 0.40
1.55 ± 0.19
1.36 ± 0.28
0.96 ± 0.20
0.15 ± 0.10
7.1
4.1
0.3
1.7
1.5
0.09
6.5
5.5
1.0
1.6
1.4
0.09
6.00
5.51
1.11
1.0
0.7
0.09
6.6
5.5
1.0
1.7
1.5
0.09
7.6
5.5
1.0
2.2
2.2
0.09
ACP (π − π + )
ACP (π + π 0 )
ACP (π 0 π 0 )
Sππ
ACP (K + K̄ 0 )
ACP (K 0 K̄ 0 )
ACP (K + K − )
0.38 ± 0.06
0.06 ± 0.05
0.43 ± 0.25
−0.61 ± 0.08
0.12 ± 0.17
−0.58 ± 0.7
-
0.206
−0.000
0.382
−0.504
0.101
0.000
−0.184
0.266
-0.001
0.453
-0.506
0.098
0.000
-0.184
0.239
-0.001
0.272
-0.353
0.041
0.000
-0.184
0.260
−0.001
0.485
−0.524
0.101
0.000
−0.184
0.141
-0.001
0.789
-0.638
0.106
0.000
-0.184
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Numerical results for B to PV decays
Table: The branching ratios (in units of 10−6 ) and direct CP asymmetries in penguin dominated
B → PV decays
.
Mode
B + → K ∗0 π +
B + → K ∗+ π 0
B 0 → K ∗− π +
B 0 → K ∗0 π 0
B + → φK +
B 0 → φK 0
ACP (K ∗0 π + )
ACP (K ∗+ π 0 )
ACP (K ∗− π + )
ACP (K ∗0 π 0 )
ACP (φK + )
ACP (φK 0 )
Data[HFAG]
9.9 ± 0.8
6.9 ± 2.3
8.6 ± 0.9
2.4 ± 0.7
8.30 ± 0.65
8.3 ± 1.1
−0.038 ± 0.042
0.04 ± 0.29
−0.23 ± 0.08
−0.15 ± 0.12
0.23 ± 0.15
−0.01 ± 0.06
NLO+Vertex
10.3
6.2
8.8
3.5
9.3
8.9
−0.017
−0.224
−0.357
−0.067
−0.022
0
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
NLOeff
9.0
5.5
8.3
3.6
6.9
6.6
-0.018
-0.123
-0.355
-0.125
-0.025
0
This work
NLOeff (−10◦ )
7.6
4.8
7.2
3.1
5.6
5.4
-0.020
-0.164
-0.415
-.114
-0.028
0
NLOeff (5◦ )
9.8
5.9
8.9
3.9
7.6
7.3
-0.017
-0.103
-0.327
-0.129
-0.023
0
NLOeff (20◦ )
11.8
7.1
10.6
4.6
9.6
9.2
-0.015
-0.045
-0.251
-0.143
-0.020
0
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Mode
B + → ρ+ K 0
B + → ρ0 K +
B 0 → ρ− K +
B 0 → ρ0 K 0
B + → ωK +
B 0 → ωK 0
ACP (ρ+ K 0 )
ACP (ρ0 K + )
ACP (ρ− K + )
ACP (ρ0 K 0 )
ACP (ωK + )
ACP (ωK 0 )
Input parameters
Numerical results
Data[HFAG]
8.0 ± 1.45
3.81 ± 0.48
8.6 ± 1.0
4.7 ± 0.7
6.7 ± 0.5
5.0 ± 0.6
−0.12 ± 0.17
0.37 ± 0.11
0.15 ± 0.06
0.06 ± 0.20
0.02 ± 0.05
0.32 ± 0.17
NLO+Vertex
5.2
3.0
5.4
2.8
2.4
1.9
0.016
0.635
0.605
0.056
0.453
−0.011
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
NLOeff
7.1
3.3
6.2
3.9
3.6
3.2
0.013
0.727
0.549
-0.136
0.404
0.117
This work
NLOeff (45◦ )
7.1
2.8
7.3
4.9
4.3
4.1
0.014
0.594
0.373
-0.044
0.167
0.048
NLOeff (60◦ )
6.8
2.6
7.3
5.0
5.3
4.8
0.014
0.463
0.290
-0.015
0.091
0.026
NLOeff (75◦ )
6.3
2.4
7.2
5.0
5.2
4.6
0.014
0.285
0.196
0.015
0.015
0.002
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Table: Tree dominant B → PV decay modes
.
Mode
Data[HFAG]
This work
12.0
5.2
19.6
6.2
0.2
0.6
0.6
0.255
−0.308
default
13.9
7.4
17.4
6.5
1.3
0.3
0.4
0.199
-0.344
(−45◦ ,0◦ )
14.2
7.0
16.5
6.5
1.5
0.3
0.2
0.196
−0.330
NLOeff
(45◦ ,0◦ )
13.5
7.8
19.1
6.5
1.2
0.3
0.8
0.133
-0.269
(0◦ ,−45◦ )
13.7
7.2
17.5
6.1
1.5
0.3
0.8
0.195
-0.309
(0◦ ,45◦ )
14.2
7.4
17.4
7.5
1.1
0.3
0.1
0.131
-0.285
0.120
−0.281
0.058
0.191
0.000
0.126
-0.282
0.187
0.257
0
0.108
−0.283
0.112
−0.342
0
0.066
-0.281
0.381
0.205
0
0.121
-0.217
0.258
0.112
0
0.127
-0.176
-0.008
0.837
0
NLO+Vertex
B + → ρ+ π 0
B + → ρ0 π +
B 0 → ρ+ π −
B 0 → ρ− π +
B 0 → ρ0 π 0
B + → K̄ ∗0 K +
B 0 → K ∗0 K̄ 0
ACP (ρ+ π 0 )
ACP (ρ0 π + )
ACP (ρ+ π − )
ACP (ρ− π + )
ACP (ρ0 π 0 )
ACP (K̄ ∗0 K + )
ACP (k ∗0 K̄ 0 )
10.9 ± 1.5
8.3 ± 1.3
15.7 ± 1.8
7.3 ± 1.2
2.0 ± 0.5
0.68 ± 0.19
< 1.9
0.02 ± 0.11
0.18+0.09
−0.17
0.11 ± 0.06
−0.18 ± 0.12
−0.30 ± 0.38
-
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Input parameters
Numerical results
Numerical results for B to VV decays
Table: Branching ratios for B → VV decay modes (in unit of 10−6 ) which includes the
contribution of effective Wilson coefficients and effect of different strong phase θ a = 60◦ ± 15◦ )
for annihilation diagram.
Mode
.
B + → ρ+ ρ0
B 0 → ρ+ ρ−
B 0 → ρ0 ρ0
B + → K ∗0 ρ+
B + → K ∗+ ρ0
B 0 → K ∗+ ρ−
B 0 → K ∗0 ρ0
B + → K̄ ∗0 K ∗+
B 0 → K̄ ∗0 K ∗0
B 0 → K ∗+ K ∗−
B + → φK ∗+
B 0 → φK ∗0
B + → ωK ∗+
B 0 → ωK ∗0
Data[HFAG]
24.0 ± 2.0
24.2 ± 3.1
0.73 ± 0.27
9.2 ± 1.5
< 6.1
< 12
3.4 ± 1.0
1.2 ± 0.5
1.28 ± 0.35
< 2
10.0 ± 1.1
9.8 ± 0.7
< 7.4
2.0 ± 0.5
NLO+Vertex
13.4
22.3
0.4
16.2
9.9
13.9
5.6
0.9
0.8
0.07
19.4
18.7
5.6
6.2
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
NLOeff
16.8
19.8
0.92
14.0
9.0
13.0
5.2
0.8
0.6
0.07
15.2
14.8
4.2
4.1
This work
NLOeff (45◦ )
16.8
21.7
0.67
9.6
6.4
9.1
3.6
0.6
0.5
0.07
10.9
10.5
3.3
2.8
NLOeff (60◦ )
16.8
22.3
0.61
8.3
5.6
7.9
3.1
0.5
0.5
0.007
9.5
9.2
3.0
2.5
NLOeff (75◦ )
16.8
22.7
0.57
7.2
5.0
6.9
2.7
0.4
0.5
0.07
8.4
8.1
2.8
2.2
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Conclusions
Based on the approximate six-quark operator effective Hamiltonian derived from
perturbative QCD, the QCD factorization approach has been naturally applied to
evaluate the hadronic matrix elements for charmless two body B-meson decays. It
is shown that, with annihilation contribution and extra strong phase, our
framework provides a simple way to evaluate the hadronic matrix elements of two
body decays.
For B → PP final states, our predictions for branching ratios and CP asymmetries
are generally consistent with the current experimental data within their respective
uncertainties, once the effective Wilson coefficients and annihilation amplitude
with small strong phase (θa = 5◦ ) are adopted. Especially for the branching ratio
of B → π 0 π 0 mode, our result, although being still smaller than the data, is
consistent with that in QCDF.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
As for B → PV decays, similar conclusions are found. The exceptions here are
the branching ratio of ρ+ π 0 and B → K ∗0 π 0 modes, which are bigger than the
data. An interesting point should be noted that our predictions are also
consistent with the ones in QCDF. Since the current data on CP asymmetries
have large uncertainties in these modes, more precise experimental data are
expected to further test our framework.
In B → VV decay modes, we have shed light on the polarization anomalies
observed in B → φK ∗ , ρK ∗ , ωK ∗ decays. It is noted that these anomalies could
be explained in our framework when considering annihilation contributions with a
big strong phase (θa = 60◦ ). Moreover, the annihilation contributions with a
strong phase have remarkable effects on the branching ratios and CP
asymmetries, especially on the observables of penguin dominated decay modes.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Discussions
It is noted that the method developed in this paper allows us to calculate the
relevant transition form factors. Our predictions(for B to light mesons form
factors) are consistent with the results of light-core QCD sum-rules and pQCD.
A complete six-quark operator effective Hamiltonian may involve more effective
operators from the type III diagrams and lead to a non-negligible contribution to
hadronic B meson decays when evaluating the hadronic matrix elements of six
quarkpoperator effective Hamiltonian around the energy scale
µ ∼ 2ΛQCD mb ∼ mc ∼ 1.5 GeV where the nonperturbative effects may play
the role.
Through this approach, we find that the factorization is a natural result and it is
more clear to see what approximation is made(such as the Type III diagrams),
which makes the physics of long-distance contribution much understandable.
For the calculation of strong phase from nonperturbative QCD effects is a hard
task, and there exist no efficient approaches to evaluate reliably the strong phases
caused from nonperturbative QCD effects, they are treated as free parameters in
this frame work, and the values differ for different decay modes in order to get
reasonable predictions.
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt
Effective Hamiltonian of Six-Quark Operators
QCD Factorization based on effective six-quark operators
Treatment of Singularities
Vertex Corrections
Input parameters and numerical calculation
Conclusion and discussion
Thanks for attention!
Ci Zhuang in collaboration with Su Fang, Y.L.Wu, Y.B.Yang
Charmless B to PP,PV, VV decays Based on the Six-quark Effective Hamilt