$CP$ violation induced by the double resonance for pure annihilation decay process in Perturbative QCD

0.22 MB·

by  Gang Lü

CP violation induced by the double resonance for pure annihilation decay process in
Perturbative QCD
Gang Lu¨1 , Ye Lu2 , Sheng-Tao Li1 and Yu-Ting Wang1
∗ †
1College of Science, Henan University of Technology, Zhengzhou 450001, China
2Department of Physics, Guangxi Normal University, Guilin 541004, China
InPerturbativeQCD(PQCD)approachwestudythedirectCP violationinthepureannihilation
decay process of B¯s0 →π+π−π+π− induced by theρ and ω double resonance effect. Generally, the
7
1 CP violation is small in the pure annihilation type decay process. However, we find that the CP
0 violationcanbeenhancedbydoubleρ−ωinterferencewhentheinvariantmassesoftheπ+π− pairs
2
are in the vicinity of the ω resonance. For the decay process of B¯s0 → π+π−π+π−, the maximum
n
a CP violation can reach 28.64%.
J

8

] PACSnumbers: 11.30.Er,12.39.-x,13.20.He,12.15.Hh
h
p
-
p I. INTRODUCTION
e
h
[
CP violationis animportantareain searchingnew physics signalsbeyondthe standardmodel(SM). It is generally

1
believedthatthe B mesonsystemprovidesrichinformationaboutCP violation. The theoreticalworkhasbeendone
v
6 in this direction in the past few years. CP violation arises from the weak phase in the Cabibbo-Kobayasgi-Maskawa
6
9 (CKM) matrix [1, 2] in SM. Meanwhile, it is remarkable that CP violation can still be produced by the interference
1
effects between the tree and penguin amplitudes. Since the kinematic suppression, the strong phase associated with
0
1. long distance rescattering is generally neglected during the past decades. Recently, the LHCb Collaboration found
0
the large CP violation in the three-body decay channels of B π π+π and B K π+π [3–5]. Hence, the
7 ± → ± − ± → ± −
1 nonleptonic B meson decay from the three-body and four-body decay channels has been become an important area
:
v
in searching for CP violation.
i
X
A mixing between the u and d flavor leads to the breaking of isospin symmetry for the ρ ω system. The chiral
r −
a dynamics has been shown restore the isospin symmetry [6]. The ρ ω mixing matrix element Π˜ (s) gives rise to
ρω
−
isospin violation, where s is the Mandelstam variable. The magnitude has been extracted by the pion form factor
through the cross section of e+e π+π . We can separate the Π˜ (s) into two contribution of the direct coupling
− − ρω
→
of ω 2π and the mixing of ω ρ 2π. The emergence of Π˜ (s) arises from the inclusion of a nonresonant
ρω
→ → →
contribution to ω 2π. The appearance of the ρ and ω resonance is associated with complex strong phase from
→
relativelybroadρresonanceregion. Especially,thereisperhapslargerstrongphasefromdoubleρandω interference.
The CP violation origins from the weak phase difference and the strong phase difference. Hence, the decay process
of B¯0 π+π π+π is a great candidate for studying the origin of the CP violation.
s → − −
∗ Email: [email protected]
† Email: [email protected]
2
Meanwhile, it is known that the CP violation is extremely tiny from the pure annihilation decay process in ex-
periment. There is relatively large error in dealing with the decay amplitudes from the QCD factorization approach
[7]. The perturbative QCD (PQCD) factorization approach [8–11] is based on k factorization. The amplitude can
T
be divided into the convolution of the Wilson coefficients, the light cone wave function, and hard kernels by the low
energy effective Hamiltonian. The endpoint singularity can be eliminated by introducing the transverse momentum.
However, The transverse momentum integration leads to the double logarithm term which is resummed into the
Sudakovformfactor. The nonperturbativedynamicsareincludedinthe mesonwavefunctionwhichcanbe extracted
from experiment. The hard one can be calculated by perturbation theory.
The remainder of this paper is organized as follows. In Sec. II we present the form of the effective Hamiltonian.
In Sec. III we give the calculating formalism and calculation details of CP violation from ρ ω mixing in the
−
B¯0 ρ0(ω)ρ0(ω) π+π π+π decay. In Sec. IV we show input parameters. We present the numerical results in
s → → − −
Sec. V. Summary and discussion are included in Sec. VI. The related function defined in the text are given in the
Appendix.
II. THE EFFECTIVE HAMILTONIAN
With the operator product expansion, the effective weak Hamiltonian can be written as [12]
10
G
= F V V C (µ)Qu(µ)+C (µ)Qu(µ) V V C (µ)Q (µ) +H.c., (1)
Heff √2( ub u∗q 1 1 2 2 − tb t∗q i i )
h i hXi=3 i
where q =(d,s), G represents Fermi constant, C (i=1,...,10) are the Wilson coefficients, V (q and q represent
F i q1q2 1 2
quarks) is the CKM matrix element, and O is the four quark operator. The operators O have the following forms:
i i
Ou = d¯ γ (1 γ )u u¯ γµ(1 γ )b ,
1 α µ − 5 β β − 5 α
Ou = d¯γ (1 γ )uu¯γµ(1 γ )b,
2 µ − 5 − 5
O = d¯γ (1 γ )b q¯γµ(1 γ )q ,
3 µ 5 ′ 5 ′
− −
q′
X
O4 = d¯αγµ(1−γ5)bβ q¯β′γµ(1−γ5)qα′ ,
q′
X
O5 = d¯γµ(1 γ5)b q¯′γµ(1+γ5)q′,
−
q′
X
O = d¯ γ (1 γ )b q¯ γµ(1+γ )q ,
6 α µ − 5 β β′ 5 α′
q′
X
3
O7 = d¯γµ(1 γ5)b eq′q¯′γµ(1+γ5)q′,
2 −
q′
X
3
O8 = 2d¯αγµ(1−γ5)bβ eq′q¯β′γµ(1+γ5)qα′ ,
q′
X
3
O9 = d¯γµ(1 γ5)b eq′q¯′γµ(1 γ5)q′,
2 − −
q′
X
3
O10 = 2d¯αγµ(1−γ5)bβ eq′q¯β′γµ(1−γ5)qα′ , (2)
q′
X
3
where α and β are color indices, and q = u,d,s,c or b quarks. In Eq.(2) Ou and Ou are tree operators, O –O are
′ 1 2 3 6
QCDpenguin operatorsandO –O arethe operatorsassociatedwith electroweakpenguin diagrams. C (m )canbe
7 10 i b
written [11],
C = 0.2703, C =1.1188,
1 2
−
C = 0.0126, C = 0.0270,
3 4
−
C = 0.0085, C = 0.0326,
5 6
−
C = 0.0011, C =0.0004,
7 8
C = 0.0090, C =0.0022. (3)
9 10
−
So, we can obtain numerical values of a . The combinations a of Wilson coefficients are defined as usual [9]:
i i
a = C +C /3, a =C +C /3,
1 2 1 2 1 2
a = C +C /3, a =C +C /3,
3 3 4 4 4 3
a = C +C /3, a =C +C /3,
5 5 6 6 6 5
a = C +C /3, a =C +C /3,
7 7 8 8 8 7
a = C +C /3, a =C +C /3. (4)
9 9 10 10 10 9
III. CP VIOLATION IN B¯s0 →ρ0(ω)ρ0(ω)→π+π−π+π−
A. Formalism
The amplitudes Aσ of the process B¯ (p) V (p ,ǫ )+V (p ,ǫ ) can be written [13]
s 1 1 1 2 2 2
→
b ic
Aσ =ǫ (σ)ǫ (σ)(agµν + pµpν + ǫµναβp p ) (5)
∗1µ ∗2ν m m m m 1α 2β
1 2 1 2
where σ is the helicity of the vector meson. ǫ (p ) and ǫ (p ) are the polarization vectors (momenta) of V and
1 1 2 2 1
V , respectively. m and m refer to the masses of the vector mesons V and V . The invariant amplitudes a, b, c
2 1 2 1 2
are associated with the amplitude A ( i refer to the three kind of polarizations, longitudinal (L), normal (N) and
i
transverse (T)). Then we have
Aσ =M2 A +M2 A ǫ (σ =T) ǫ (σ =T)+iA ǫαβγρǫ (σ)ǫ (σ)p p (6)
Bs L Bs N ∗1µ · ∗2µ T ∗1α ∗2α 1γ 2ρ
The longitudinal H , transverse H of helicity amplitudes can be expressed H = M2 A , H = M2 A
0 ± 0 Bs L ± Bs N ∓
m m √r2 1A . The decay width is written
1 2 T
−
P P
Γ= c A(σ)+A(σ) = c H 2+ H 2+ H 2. (7)
8πM2 8πM2 | 0| | +| | −|
Bs Bs
The interaction of the photon and the hadronic matter can be described by the vector meson dominance model
4
(VMD) [14]. The photon can couple to the hadronic field through a ρ meson. The mixing matrix element Π (s) is
ρω
extracted from the data of the cross section for e+e π+π [15, 16]. The nonresonant contribution of ω π+π
− − −
→ e→
has been effectively absorbed into Π which leads to the explicit s dependence of Π [17]. We can make the
ρω ρω
expansion Π (s) = Π (m2)+(s m )Π (m2). However, one can neglect the s dependence of Π in practice.
ρω ρω ω −e ω ′ρω ω e ρω
The ρ ω mixing parameters were determined in the fit of Gardner and O’Connell [18]:
− e e e e
ReΠ (m2) = 3500 300MeV2,
ρω ω − ±
ImΠ (m2) = 300 300MeV2,
eρω ω − ±
Π (m2) = 0.03 0.04. (8)
e′ρω ω ±
e
The formalism of the CP violation is presented for the B¯0 meson decay process in the following. The amplitude A
s
(A¯) for the decay process B¯0 π+π π+π (B0 π+π π+π ) can be written as:
s → − − s → − −
A=<π+π−π+π−|HT|B¯s0 >+<π+π−π+π−|HP|B¯s0 >, (9)
A¯=<π+π π+π HT B0 >+<π+π π+π HP B0 >, (10)
− −| | s − −| | s
where HT and HP refer to the tree and penguin operators in the Hamiltonian, respectively. We define the relative
magnitudes and phases between the tree and penguin operator contributions as follows:
A= π+π−π+π−|HT|B¯s0 [1+rei(δ+φ)], (11)
A¯=(cid:10)π+π−π+π−|HT|Bs0(cid:11)[1+rei(δ−φ)], (12)
(cid:10) (cid:11)
whereδandφarestrongandweakphases,respectively. Theweakphasedifferenceφcanbeexpressedasacombination
of the CKM matrix elements: φ = arg[(V V )/(V V )]. The parameter r is the absolute value of the ratio of tree
tb t∗s ub u∗s
and penguin amplitudes:
π+π π+π HP B¯0
r − −| | s . (13)
≡(cid:12)(cid:12)(cid:10)π+π−π+π−|HT|B¯s0(cid:11)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:10) (cid:11)(cid:12)
The parameter of CP violating asymmetry, A (cid:12), can be written as (cid:12)
cp
A2 A¯2 2(T2r sinδ +T2r sinδ +T2r sinδ )sinφ
ACP = |A|2+−|A¯|2 = − 0 0 0 T2(+1++r2++2r co−sδ−cosφ)− , (14)
| | | | i=0+− i i i i
P
where
A2 = A(σ)+A(σ) = H 2+ H 2+ H 2 (15)
0 +
| | | | | | | −|
σ
X
and T (i = 0,+, ) represent the tree-level helicity amplitudes. We can see explicitly from Eq. (14) that both weak
i
−
5
and strong phase differences are responsible for CP violation. ρ ω mixing introduces the strong phase difference
−
andwellknowninthe three bodydecayprocessesofthe bottomhadron[19–25]. Due to ρ ω interferencefromthe u
−
and d quark mixing, we canwrite the following formalismin an approximatefrom the first orderof isospin violation:
2g2 g2
π+π π+π HT B¯0 = ρ Π t + ρt , (16)
− −| | s s2s ρω ρω s2 ρρ
ρ ω ρ
(cid:10)π+π π+π HP B¯0(cid:11)= 2gρ2 Πe p + gρ2p , (17)
− −| | s s2s ρω ρω s2 ρρ
ρ ω ρ
(cid:10) (cid:11)
e
where t (p ) and t (p ) are the tree (penguin) amplitudes for B¯ ρ0ρ0 and B¯ ρ0ω, respectively, g is
ρρ ρρ ρω ρω s s ρ
→ →
the coupling for ρ0 π+π , Π is the effective ρ ω mixing amplitude which also effectively includes the direct
− ρω
→ −
coupling ω π+π . s , m and Γ (V=ρ or ω) is the inverse propagator,mass and decay rate of the vector meson
− V V V
→ e
V, respectively.
s =s m2 +im Γ , (18)
V − V V V
with √s being the invariant masses of the π+π pairs. There are double ρ ω interference in the decay process of
−
−
B¯0 ρ0(ω)ρ0(ω) π+π π+π . Hence, a factor of 2 appears in Eqs. (16), (17) compared with the case of single
s → → − −
ρ ω interference [19–27]. From Eqs. (9)(11)(16)(17) one has
−
2Π p +s p
reiδeiφ = ρω ρω ω ρρ, (19)
2Π t +s t
ρω ρω ω ρρ
e
Defining e
p t p
ρω r′ei(δq+φ), ρω αeiδα, ρρ βeiδβ, (20)
t ≡ t ≡ p ≡
ρρ ρρ ρω
where δ , δ and δ are strong phases, one finds the following expression from Eqs. (19)(20):
α β q
reiδ =r′eiδq2Πρω+βeiδβsω. (21)
2Πρωαeiδα +sω
e
In order to obtain the CP violating asymmetry in Eq. (1e4), sinφ and cosφ are needed, where φ is determined by the
CKM matrix elements. In the Wolfenstein parametrization [28], one has
η
sinφ = ,
− ρ2+η2
ρ
cosφ = p . (22)
− ρ2+η2
p
B. Calculation details
We candecompose the decayamplitude for the decay processB¯0 ρ0(ω)ρ0(ω) in terms of tree-levelandpenguin-
s →
level contributions depending on the CKM matrix elements of V V and V V . Due to the equations (14)(19)(20),
ub u∗s tb t∗s
6
we calculate the amplitudes t , t , p and p in perturbative QCD approach. The F and M function associated
ρρ ρω ρρ ρω
with the decay amplitudes can be found in the appendix from the perturbative QCD approach.
There are four types of Feynman diagrams contributing to B¯ M M (M ,M =ρ or ω) annihilation decay mode
s 2 3 2 3
→
at leading order. The pure annihilation type process can be classifiedinto factorizable diagramsand non-factorizable
diagrams [29, 30]. Through calculating these diagrams, we can get the amplitudes A(i), where i = L,N,T standing
forthe longitudinalandtwotransversepolarizations. Becausethese diagramsarethe sameas thoseofB K φand
∗
→
B K ρ decays [29, 30], the formulas of B¯ ρρ or B¯ ρω are similar to those of B K φ and B K ρ. We
∗ s s ∗ ∗
→ → → → →
just need to replace some corresponding wave functions, Wilson coefficients and corresponding parameters.
WiththeHamiltonian(1),dependingonCKMmatrixelementsofV V andV V ,thedecayamplitudesA(i)(i=
ub u∗s tb t∗s
L,N,T) for B¯0 ρ0ρ0 in PQCD can be written as
s →
√2A(i)(B¯s0 →ρ0ρ0)=VubVu∗stiρρ−VtbVt∗spiρρ, (23)
The tree level amplitude t can written as
ρρ
G
ti = F f FLL,i[a ]+MLL,i[C ] , (24)
ρρ √2 Bs ann 2 ann 2
n o
where f refers to the decay constant of B¯ meson.
Bs s
The penguin level amplitude are expressed in the following
G 1 1
pi = F f FLL,i 2a + a +f FLR,i 2a + a
ρρ √2 Bs ann 3 2 9 Bs ann 5 2 7
(cid:26) (cid:20) (cid:21) (cid:20) (cid:21)
1 1
+MLL,i 2C + C +MSP,i 2C + C . (25)
ann 4 2 10 ann 6 2 8
(cid:20) (cid:21) (cid:20) (cid:21)(cid:27)
The decay amplitude for B¯0 ρ0ω can be written as
s →
2A(i)(B¯0 ρ0ω) = V V ti V V pi . (26)
s → ub u∗s ρω − tb t∗s ρω
We can give the tree level the contribution in the following
G
ti = F f FLL,i[a ]+MLL,i[C ] , (27)
ρω √2 Bs ann 2 ann 2
n o
and the penguin level contribution are given as following
G 3 3
pi = FV V f FLL,i a +f FLR,i a
ρω √2 tb t∗s Bs ann 2 9 Bs ann 2 7
n (cid:20) (cid:21) (cid:20) (cid:21)
3 3
+MLL,i C +MSP,i C + ρ0 ω . (28)
ann 2 10 ann 2 8 ↔
(cid:20) (cid:21) (cid:20) (cid:21)o (cid:2) (cid:3)
7
Based on the definition of (20), we can get
t
αeiδα = ρω, (29)
t
ρρ
p
βeiδβ = ρρ, (30)
p
ρω
p V V
r′eiδq = ρω tb t∗s , (31)
t × V V
ρρ (cid:12) ub u∗s(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
V V ρ2+η2
tb t∗s = . (32)
V V λ2(ρ2+η2)
(cid:12) ub u∗s(cid:12) p
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
IV. INPUT PARAMETERS
The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein
parameters A, ρ, λ and η [28]:
1 1λ2 λ Aλ3(ρ iη)
− 2 −
λ 1 1λ2 Aλ2 , (33)
 − − 2 
Aλ3(1 ρ iη) Aλ2 1
 − − − 
 
where (λ4) corrections are neglected. The latest values for the parameters in the CKM matrix are [31]:
O
λ=0.22537 0.00061, A=0.814+0.023,
± −0.024
ρ¯=0.117 0.21, η¯=0.353 +0.013. (34)
± ±
where
λ2 λ2
ρ¯=ρ(1 ), η¯=η(1 ). (35)
− 2 − 2
From Eqs. (34) ( 35) we have
0.121<ρ<0.158, 0.336<η <0.363. (36)
The other parameters and the corresponding references are listed in Table.1.
V. THE NUMERICAL RESULTS OF CP VIOLATION IN B¯s0 →ρ0(ω)ρ0(ω)→π+π−π+π−
In the numerical results, we find that the CP violation can be enhanced via double ρ ω mixing for the pure
−
annihilation type decay channel B¯0 ρ0(ω)ρ0(ω) π+π π+π when the invariant mass of π+π is in the vicinity
s → → − − −
of the ω resonance within perturbative QCD scheme. The CP violation depends on the weak phase difference from
8
TABLE I: Input parameters used in this paper.
Parameters Input data References
Fermi constant (in GeV−2) GF =1.16638 10−5 [32]
×
mB0 =5.36677, τB0 =1.512 10−12s
s s ×
mρ0(770) =0.77526, Γρ0(770) =0.1491,
Masses and decay widths m =0.78265, Γ =8.49 10 3, [32]
ω(782) ω(782) −
×
(in GeV) m =0.13957, m =80.385,
π W
m =0.0023, m =0.0048,
u d
m =0.095, m =1.275,
s c
m =173.21, m =4.18,
t b
Decay constants f =209 2, fT =165 9, [32, 33]
ρ ± ρ ±
(in MeV) f =195.1 3, fT =145 10,
ω ± ω ±
CKMmatrixelementsandthe strongphasedifference whichisdifficult to control. The CKMmatrixelements,which
relate to ρ, η, λ and A, are given in Eq.(34). The uncertainties due to the CKM matrix elements come from ρ, η, λ
and A. In our numerical calculations, we let ρ, η, λ and A vary among the limiting values. The numerical results
are shown from Fig. 1 to Fig. 3 with the different parameter values of CKM matrix elements. The dash line, dot
line and solid line corresponds to the maximum, middle, and minimum CKM matrix element for the decay channel
of B¯0 ρ0(ω)ρ0(ω) π+π π+π , respectively. We find the results are not sensitive to the values of ρ, η, λ and
s → → − −
A. In Fig. 1, we give the plot of CP violating asymmetry as a function of √s. From the Fig. 1, one can see the CP
violationparameteris dependent on√s andchangesrapidlydue to ρ ω mixing whenthe invariantmassofπ+π is
−
−
inthe vicinity ofthe ω resonance. Fromthe numericalresults,itis foundthat the maximumCP violating parameter
reaches 28.64% in the case of (ρ , η ).
mini mini
0.3
0.2
Acp 0.1
0.0
-0.1
0.74 0.76 0.78 0.80 0.82
s
FIG. 1: The CP violating asymmetry, A , as a function of √s for different CKM matrix elements. The dash line,
cp
dot line and solid line corresponds to the maximum, middle, and minimum CKM matrix element for the decay
channel of B¯0 ρ0(ω)ρ0(ω) π+π π+π , respectively.
s → → − −
From Eq.(14), one can see that the CP violating parameter depend on both sinδ and r. The plots of sinδ and r
as a function of √s are shown in Fig. 2, and Fig. 3, respectively. It can be seen that sinδ (sinδ and sinδ ) vary
0 +
−
sharply at the range of the resonance in Fig. 2. One can see that r change largely in the vicinity of the ω resonance.
9
1.0
0.5
∆n 0.0
si
-0.5
-1.0
0.74 0.76 0.78 0.80 0.82
s
FIG. 2: sinδ as a function of √s corresponding to central parameter values of CKM matrix elements for
B¯0 ρ0(ω)ρ0(ω) π+π π+π . The dash line, dot line and solid line corresponds to sinδ , sinδ and sinδ ,
s → → − − respectively. 0 + −
4
3
r 2
1
0
0.74 0.76 0.78 0.80 0.82
s
FIG. 3: Plot of r as a function of √s corresponding to central parameter values of CKM matrix elements for
B¯0 ρ0(ω)ρ0(ω) π+π π+π . The dash line, dot line and solid line corresponds to r , r and r , respectively.
s → → − − 0 + −
VI. SUMMARY AND CONCLUSION
In this paper, we study the CP violation for the pure annihilation type decay process of B¯0 π+π π+π in
s → − −
perturbative QCD. It has been found that the CP violation can be enhanced greatly at the area of ρ ω resonance.
−
The maximum CP violation value can reach 28.64% due to double ρ and ω resonance.
The theoretical errors are large which follows to the uncertainties of results. Generally, power corrections beyond
the heavy quark limit give the major theoretical uncertainties. This implies the necessity of introducing 1/m power
b
corrections. Unfortunately, there are many possible 1/m powersuppressed effects and they are generally nonpertur-
b
bative in nature and hence not calculable by the perturbative method. There are more uncertainties in this scheme.
The first error refers to the variation of the CKM parameters, which are given in Eq.(34). The second error comes
from the hadronic parameters: the shape parameters,form factors, decay constants,and the wave function of the B
s
meson. The third errorcorrespondsto the choice of the hardscales, which vary from 0.75tto 1.25t,which character-
10
izing the size of next-to-leading order QCD contributions. Therefore, the results for CP violating asymmetrie of the
decay process B¯0 π+π π+π is given as following:
s → − −
A (B¯0 π+π π+π )=28.43+0.21+0.25+5.62%, (37)
CP s → − − −0.25−0.16−3.98
wherethefirstuncertaintyiscorrespondingtotheCKMparameters,thesecondcomesfromthehadronicparameters,
and the third is associated with the hard scales. The LHC experiment may detect the large CP violation for the
decay process B¯0 π+π π+π in the region of the ω resonance.
s → − −
VII. APPENDIX: RELATED FUNCTIONS DEFINED IN THE TEXT
In this appendix we present explicit expressions of the factorizable and non-factorizable amplitudes with Pertur-
bative QCD in Eq.(23) and Eq.(26) [10, 11, 34, 35]. The factorizable amplitudes FLL,i(a ), and FSP,i(a ) (i=L,N,T)
ann i ann i
are written as
f FLL,N(a ) = f FLR,N(a ) (38)
Bs ann i Bs ann i
1
f FLL,N(a ) = 8πC M4 f r r dx dx ∞b db b db E (t )a (t )h (x ,1 x ,b ,b ))
Bs ann i − F Bs Bs 2 3 2 3 2 2 3 3 a c i c a 2 − 3 2 3
[(2 x )(φv(x )φv(xZ0)+φa(xZ)0φa(x ))+x (φnv(x )φa(x )+φa(x )φv(x ))]
− 3 2 2 3 3 2 2 3 3 3 2 2 3 3 2 2 3 3
h (1 x ,x ,b ,b )[(1+x )(φv(x )φv(x )+φa(x )φa(x ))
− a − 3 2 3 2 2 2 2 3 3 2 2 3 3
(1 x )(φv(x )φa(x )+φa(x )φv(x ))]E (t )a (t ) . (39)
− − 2 2 2 3 3 2 2 3 3 a ′c i ′c
o
f FLL,T(a ) = f FLR,T(a ) (40)
Bs ann i − Bs ann i
1
f FLL,T(a ) = 16πC M4 f r r dx dx ∞b db b db [x (φv(x )φv(x )+φa(x )φa(x ))
Bs ann i − F Bs Bs 2 3 2 3 2 2 3 3 3 2 2 3 3 2 2 3 3
+(2 x )(φv(x )φa(xZ0)+φa(x Z)φ0v(x ))]E (t )na (t )h (x ,1 x ,b ,b )
− 3 2 2 3 3 2 2 3 3 a c i c a 2 − 3 2 3
+h (1 x ,x ,b ,b )[(1 x )(φv(x )φv(x )+φa(x )φa(x ))
a − 3 2 3 2 − 2 2 2 3 3 2 2 3 3
−(1+x2)(φv2(x2)φa3(x3)+φa2(x2)φv3(x3))]Ea(t′c)ai(t′c) . (41)
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