$B_{s} \to (ρ, ω, φ) η^{(\prime)}$ Decays in the Perturbative QCD Approach

0.39 MB·English

by  Xin-fen Chen

NJNU-TH-07-02
B (ρ,ω,φ)η() Decays in the Perturbative QCD Approach
s ′
→
Xin-fen Chen,∗ Dong-qin Guo,† and Zhen-jun Xiao‡
Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210097, P.R.China
(Dated: February 2, 2008)
Abstract
7 In this paper, we calculate the branching ratios and CP-violating asymmetries for Bs
→
0 (ρ0,ω,φ)η(′) decays in the perturbative QCD (pQCD) factorization approach. Numerically we
0
2 found that (a) the pQCD predictions for the CP-averaged branching ratios are Br(Bs ρ0η)
→ ≈
0.07 10−6 , Br(B ρ0η′) 0.10 10−6 , Br(B ωη) 0.02 10−6, Br(B ωη′)
n s s s
× → ≈ × → ≈ × → ≈
a 0.13 10−6, Br(B φη) 2.7 10−5 and Br(B φη′) 2.0 10−5; (b) the gluonic
s s
J × → ≈ × → ≈ ×
contributions are small in size: less than 3% for B (ρ,ω,φ)η decays, and about 10% for
8 →
1 B (ρ,ω,φ)η′ decays; and (c) the pQCD predictions for the CP-violating asymmetries of the
→

considered decays are generally not large in magnitude. The above predictions can be tested in
1
v the forthcoming LHC-b experiments at CERN.
6
4
PACS numbers: 13.25.Hw, 12.38.Bx,14.40.Nd
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∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
1
I. INTRODUCTION
The theoretical calculations and experimental measurements of the rare B meson de-
cays play an important role in testing the standard model (SM), probing CP violation
of B meson system and searching for possible new physics beyond the SM. At present,
about 1000 million events of B meson pair productions and decays have been collected by
BaBar and Belle collaborations. In the forthcoming LHC experiments, a huge amount of
B meson events, say around 1011 1012, are expected, and therefore the rare B meson
∼
decays with a branching ratio around 10−7 can be observed with good precision. Another
advantage is that the heavier B and B mesons and b-baryons, besides the B and B ,
s c u d
can also be produced and studied at LHC [1].
By employing the generalized factorization approach[2, 3] or the QCD factorization
(QCDF) approach [4], about forty B h h (h stand for light pseudo-scalar or vector
s 1 2 i
→
mesons ) decay modes have been studied in the framework of SM [5, 6, 7, 8] or in some
new physics models beyond the SM [9]. In this paper, we will study the B ρ0η(′) ,
s
→
ωη(′) and φη(′) decays in the pQCD factorization approach. In principle, the physics for
the B two-body hadronic decays is very similar to that for the B meson except that the
s d
spectator d quark is replaced by the s quark.
For B (ρ0,ω,φ)η(′) decays, the B meson is heavy, setting at rest and decaying
s s
→
into two light mesons (i.e. ρ0 and η(′) ) with large momenta. Therefore the light final
state mesons are moving very fast in the rest frame of B meson. In this case, the short
s
distance hard process dominates the decay amplitude. We assume that the soft final state
interaction is not important for such decays. The smallness of FSI effects for B meson
decays into two light final state mesons has been put forward by Bjorken [10] based
on the color transparency argument [11], and also supported by further renormalization
group analysis of soft gluon exchanges among initial and final state mesons [12]. With the
Sudakov resummation, wecaninclude theleading double logarithmsfor allloopdiagrams,
in association with the soft contribution. Unlike the usual factorization approach, the
hard part of the pQCD approach consists of six quarks rather than four. We thus call it
six-quark operators or six-quark effective theory. Applying the six-quark effective theory
to B meson decays, we need meson wave functions for the hadronization of quarks into
s
mesons. All the collinear dynamics are included in the meson wave functions.
This paper is organized as follows. In Sec. II, we give a brief review for the PQCD
factorization approach. In Sec. III, we calculate analytically the related Feynman dia-
grams and present the decay amplitudes for the studied decay modes. In Sec. IV, we
show the numerical results for the branching ratios and CP asymmetries ofB ρ0η(′)
s
→
, ωη(′) and φη(′) decays and comparing them with the results obtained in the other two
methods mentioned above. The summary and some discussions are included in the final
section.
II. THEORETICAL FRAMEWORK
The pQCD factorization approach has been developed and applied in the non-leptonic
B meson decays for some time [11, 13, 14, 15]. This approach is based on k factor-
T
ization scheme, where three energy scales are involved [13]. In this approach, the decay
amplitude is factorized into the convolution of the meson’s light-cone wave functions, the
2
hard scattering kernels and the Wilson coefficients, which stand for the soft (Φ), hard(H),
and harder(C) dynamics respectively. The hard dynamics (H) describes the four quark
operator and the spectator quark connected by a hard gluon. This hard part is charac-
terized by ΛM , and can be calculated perturbatively in pQCD approach. The harder
Bs
dynamics (C)is from m scale to m scale described by renormalization group equation
p W Bs
for the four quark operators. The dynamics below ΛM is soft, which is described by
Bs
the meson wave functions (Φ). While the function (H) depends on the processes consid-
p
ered, the wave function is independent of the specific processes. Using the wave functions
determined from other well measured processes, one can make quantitative predictions
here. Based on this factorization, the decay amplitude can be written as the following
(B M M ) d4k d4k d4k Tr[C(t)Φ (k )Φ (k )Φ (k )H(k ,k ,k ,t)], (1)
A s → 1 2 ∼ 1 2 3 Bs 1 M1 2 M2 3 1 2 3
Z
where k ’s are momenta of light quarks included in each mesons, and Tr denotes the trace
i
over Dirac and color indices. C(t) is Wilson coefficient of the four quark operator which
results from the radiative corrections at short distance. The functions Φ and H are
M
meson wave functions and the hard part respectively.
For the B meson decays, since the b quark is rather heavy we consider the B meson
s s
at rest for simplicity. It is convenient to use light-cone coordinate (p+,p−,p ) to describe
T
the meson’s momenta,
1
p± = (p0 p3), and p = (p1,p2). (2)
T
√2 ±
Using these coordinates the B meson and the two final state meson momenta can be
s
written as:
M M M
P = Bs(1,1,0 ), P = Bs(1,r2,0 ), P = Bs(0,1 r2,0 ), (3)
Bs √2 T ρ √2 ρ T η √2 − ρ T
respectively, where r = m /M and the light pseudoscalar meson masses have been
ρ ρ Bs
neglected (here we just take the decay B ρ0η as an example). Putting the light (anti-)
s
→
quark momenta in B , ρ0 and η mesons as k , k , and k , respectively, we can choose
s 1 2 3
k = (x P+,0,k ), k = (x P+,0,k ), k = (0,x P−,k ). (4)
1 1 1 1T 2 2 2 2T 3 3 3 3T
Then the integration over k−, k−, and k+ in eq.(1) will lead to
1 2 3
(B ρ0η) dx dx dx b db b db b db
s 1 2 3 1 1 2 2 3 3
A → ∼
Z
Tr C(t)Φ (x ,b )Φ (x ,b )Φ (x ,b )H(x ,b ,t)S (x )e−S(t) , (5)
· Bs 1 1 ρ 2 2 η 3 3 i i t i
where b is the conjug(cid:2)ate space coordinate of k , and t is the largest energy(cid:3)scale in
i iT
function H(x ,b ,t), as a function in terms of x and b . The large logarithms (lnm /t)
i i i i W
coming fromQCDradiative corrections tofour quark operatorsareincluded inthe Wilson
coefficients C(t). The large double logarithms (ln2x ) on the longitudinal direction are
i
summedbythethresholdresummation [15], andtheyleadtoS (x )whichsmearstheend-
t i
point singularities on x . The last term, e−S(t), is the Sudakov form factor resulting from
i
3
overlap of soft and collinear divergences, which suppresses the soft dynamics effectively.
Thus it makes the perturbative calculation of the hard part H applicable at intermediate
scale, i.e., M scale. We will calculate analytically the function H(x ,b ,t) for our six
Bs i i
decays in the first order in α expansion and give the convoluted amplitudes in next
s
section.
For the considered B ρ0η(′),ωη(′) and φη(′) decays, the low energy weak effective
s
→
Hamiltonian H for the b q transition with q = (d,s) can be written as [16]:
eff
→
10
G
= F V V∗ (C (µ)Ou(µ)+C (µ)Ou(µ)) V V∗ C (µ)O (µ) . (6)
Heff √2 " ub uq 1 1 2 2 − tb tq i i #
i=3
X
We specify below the operators in for b d transition:
eff
H →
Ou = d¯ γµLu u¯ γ Lb , Ou = d¯ γµLu u¯ γ Lb ,
O1 = d¯αγµLbβ · β µq¯′γαLq′ , O2 = d¯αγµLbα · β µq¯′γβLq′ ,
3 α α · q′ β µ β 4 α β · q′ β µ α
O = d¯ γµLb q¯′γ Rq′ , O = d¯ γµLb q¯′γ Rq′ , (7)
5 α α ·Pq′ β µ β 6 α β ·Pq′ β µ α
O7 = 23d¯αγµLbαP· q′eq′q¯β′γµRqβ′ , O8 = 32d¯αγµLbβ P· q′eq′q¯β′γµRqα′ ,
O9 = 32d¯αγµLbα ·Pq′eq′q¯β′γµLqβ′ , O10 = 32d¯αγµLbβ ·Pq′eq′q¯β′γµLqα′ ,
where α and β are the SUP(3) color indices; L,R = (1 γ ) arePthe left- and right-handed
5
∓
projection operators. The sum over q′ runs over the quark fields that are active at the
scale µ = O(m ). Since we here work at the leading twist approximation and leading
b
double logarithm summation, we will also use the leading order (LO) expressions of the
Wilson coefficients C (µ) (i = 1,...,10), although the next-to-leading order C (µ) already
i i
exist in the literature [16]. This is the consistent way to cancel the explicit µ dependence
in the theoretical formulae.
For the renormalization group evolution of the Wilson coefficients from higher scale to
lower scale, we use the formulae as given in Ref. [14] directly. At the high m scale, the
W
leading order Wilson coefficients C (M ) are simple and can be found easily in Ref. [16].
i W
In pQCD approach, the scale ‘t’ may be larger or smaller than the m scale. For the case
b
of m < t < m , we evaluate the Wilson coefficients at t scale using leading logarithm
b W
running equations, as given in Eq.(C1) of Ref. [14]. For the case of t < m , we then
b
evaluate the Wilson coefficients at t scale by using C (m ) as input and the formulae
i b
given in Appendix D of Ref. [14].
For the wave function of the heavy B meson, we take
s
1
ΦBs = √2N (p/Bs +MBs)γ5φBs(k1). (8)
c
Here only the contribution of Lorentz structure φBs(k1) is taken into account, since the
¯
contributionofthesecondLorentzstructureφ isnumericallysmallandcanbeneglected.
Bs
The distribution amplitude φ in Eq. (8) will be given lately in Eq. (47).
Bs
For B Vη(′) decays, the vector meson V = (ρ,φ,ω) is longitudinally polarized. The
→
relevant longitudinal polarized component of the wave function for ρ meson, for example,
is given as [17],
1
Φ = /ǫ m φ (x)+p/ φt(x) +m φs(x) , (9)
ρ √2N ρ ρ ρ ρ ρ ρ
c
(cid:8) (cid:2) (cid:3) (cid:9)
4
where the first term is the leading twist wave function (twist-2), while the second and
third terms are sub-leading twist (twist-3) wave functions. For the case of V = ω and φ,
their wave functions arethe same in structure as that defined inEq. (9), but with different
distributionamplitudes. Onecanfindthedistribution amplitudes φ andφt,s (x)innext
ω,φ ω,φ
section.
For η and η′ meson, the wave function for dd¯components of η(′) meson are given as
[18]
iγ
Φ (P,x,ζ) 5 p/φA (x)+mηdd¯φP (x)+ζmηdd¯(/vn/ v n)φT (x) , (10)
ηdd¯ ≡ √2N ηdd¯ 0 ηdd¯ 0 − · ηdd¯
c
h i
where P and x are the momentum and the momentum fraction of η respectively, while
dd¯
φA , φP and φT represent the axial vector, pseudoscalar and tensor components of the
ηdd¯ ηdd¯ ηdd¯
wave functionrespectively, andwill begiven explicitly innext section. Following Ref.[18],
we here also assume that the wave function of η is the same as π wave function based on
dd¯
SU(3) flavor symmetry. The parameter ζ is either +1 or 1 depending on the assignment
−
of the momentum fraction x.
Before we proceed to do the perturbative calculations, we firstly give a brief discussion
about the φ ω mixing , as well as the η η′ mixing and the gluonic component of the
− −
η(′) mesons.
Forthe vector φ ω meson system, we choose the ideal mixing scheme between φ(1020)
−
and ω(782)
1
¯
ω = uu¯+dd , φ = ss¯, (11)
√2 −
(cid:0) (cid:1)
since the current data support this ideal mixing scheme [19]. The quark contents of ρ0
meson is chosen as ρ0 = ( uu¯+dd¯)/√2 [19].
−
For the η η′ system, there exist two popular mixing basis: the octet-singlet basis and
−
the quark-flavor basis [20, 21]. Here we use the quark-flavor basis [20] and define
η = (uu¯+dd¯)/√2, η = ss¯. (12)
q s
The physical states η and η′ are related to η and η through a single mixing angle φ,
q s
η η cosφ sinφ η
= U(φ) q = − q . (13)
η′ η sinφ cosφ η
s s
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
The corresponding decay constants f ,f ,fq,s and fq,s have been defined in Ref. [20] as
q s η η′
i
< 0 q¯γµγ q η (P) > = f Pµ,
5 q q
| | −√2
< 0 s¯γµγ s η (P) > = if Pµ , (14)
5 s s
| | −
i
< 0 q¯γµγ q η(′)(P) > = fq Pµ ,
| 5 | −√2 η(′)
< 0 s¯γµγ s η(′)(P) > = ifs Pµ , (15)
| 5 | − η(′)
5
while thedecay constants fq,s andfq,s arerelatedtof andf viathesame mixing matrix,
η η′ q s
fq fs f 0
η η = U(φ) q . (16)
fq fs 0 f
(cid:18) η′ η′ (cid:19) (cid:18) s (cid:19)
The three input parameters f , f and φ in the quark-flavor basis have been extracted
q s
from various related experiments [20, 21]
f = (1.07 0.02)f , f = (1.34 0.06)f , φ = 39.3◦ 1.0◦, (17)
q π s π
± ± ±
where f = 130 MeV. In the numerical calculations, we will use these mixing parameters
π
as inputs.
As shown in Eq. (13), the physical states η and η′ are generally considered as a linear
combination of light quark pairs uu¯,dd¯ and ss¯. But it should be noted that the η′
meson may have a gluonic component. Following Ref. [22], we also estimated the possible
gluonic contributions to B (ρ,ω,φ)η(′) decays induced by the gluonic corrections to
→
the B η(′) transition form factors [22] and found that these corrections to both the
→
branching ratios and CP violating asymmetries are indeed small: less than 10%.
Frankly speaking, on the other hand, we currently still do not know how to calculate
reliably the gluonic contributions to the B meson decays involving η′ meson as final state
particle. For the studied decay modes in this paper, we firstly consider only the dominant
contributions from the quark contents of η(′) meson, and then take the subdominant
contribution from the possible gluonic content of η(′) meson as a source of theoretical
uncertainties.
III. PERTURBATIVE CALCULATIONS
Inthissection, wewillcalculatethehardpartH(t)fortheconsidereddecays. Following
the same procedure as being used in Refs. [23, 24, 25, 26] and taking B ρ0η(′) decays
→
as an example, we will calculate and show the analytical results for all decay amplitudes,
and then extend the results of B ρ0η(′) decays to other decay modes under study.
→
As illustrated in Fig. 1, there are eight type Feynman diagrams contributing to the
B ρ0η(′) decays. We first calculate the usual factorizable diagrams (a) and (b). Opera-
s
→
tors O , O , O , O , O , and O are (V A)(V A) currents, the sum of their amplitudes
1 2 3 4 9 10
− −
can be written as:
1 ∞
F = 8πC M4 f dx dx b db b db φ (x ,b )
eη − F Bs ρ 1 3 1 1 3 3 Bs 1 1
Z0 Z0
(1+x )φA(x ,b )+r (1 2x )(φP(x ,b )+φT(x ,b ))
· 3 η 3 3 η − 3 η 3 3 η 3 3
α (t1)h (x ,x ,b ,b )exp[ S (t1)]
(cid:8)s(cid:2) e e 1 3 1 3 − ab e (cid:3)
+2r φP(x ,b )α (t2)h (x ,x ,b ,b )exp[ S (t2)] . (18)
η η 3 3 s e e 3 1 3 1 − ab e
where C = 4/3 is a color factor and r = mηss/M or r = mηdd/M(cid:9) since r depends
F η 0 Bs η 0 Bs η
on the quark components in η. The function hi, the scales ti and the Sudakov factors S
e e ab
are displayed in the appendix. From diagrams Fig. 1(a) and 1(b), one can also extract
out the form factor FBs→ηss.
0
6
ρ0 ρ0
B0 η B0 η
s s
(a) (b)
ρ0 ρ0
B0 η B0 η
s s
(c) (d)
ρ0 ρ0
B0 B0
s s
η η
(e) (f)
ρ0 ρ0
B0 B0
s s
η η
(g) (h)
FIG. 1: Diagrams contributing to the B ρ0η decay (diagram (a) and (b) contribute to the
s
→
B η form factor FBs→ηss).
s → ss 0
The operators O , O , O , and O have a structure of (V A)(V +A). In some decay
5 6 7 8
−
channels, some of these operators contribute to the decay amplitude in a factorizable way.
Since only the vector part of (V A) current contribute to the scaler meson production,
±
ρ V +A B η V A 0 = ρ V A B η V A 0 , (19)
h | | ih | − | i h | − | ih | − | i
the result of these operators is the same as Eq. (18), and therefore we find easily that
FP1 = F . (20)
eη eη
In some other cases, one needs to do Fierz transformation for these operators to get right
flavor and color structure for factorization to work. In this case, we get (S +P)(S P)
−
operators from (V A)(V +A) ones. For these (S +P)(S P) operators, Fig. 1(a) and
− −
1(b) will give
FP2 = 0 . (21)
eη
For the non-factorizable diagrams 1(c) and 1(d), all three meson wave functions are
involved. The integration of b can be performed by using δ function δ(b b ), leaving
3 3 1
−
7
only integration of b and b . For the (V A)(V A) operators, the corresponding decay
1 2
− −
amplitude is:
32 1 ∞
M = πC M4 dx dx dx b db b db
eη √6 F Bs 1 2 3 1 1 2 2
Z0 Z0
φ (x ,b )φ (x ,b ) x φA(x ,b ) 2r φT(x ,b )
· Bs 1 1 ρ 2 2 { 3 η 3 1 − η η 3 1
α (t )h (x ,x ,x ,b ,b )exp[ S (t )] . (22)
· s f f 1 2 3 1 2(cid:2) − cd f } (cid:3)
For the (S A)(V +A) operators, the decay amplitudes read
−
64 1 ∞
MP1 = πC M4 r dx dx dx b db b db φ (x ,b )
eη √6 F Bs ρ 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
x φA(x ,b ) φs(x ,b ) φt(x ,b ) +r (x +x ) φP(x ,b )
· 2 η 3 1 ρ 2 2 − ρ 2 2 η 2 3 η 3 1
φs(x ,b )+φT(x ,b )φt(x ,b ) +(x x )(φP(x ,b )φt(x ,b )
·(cid:8)(cid:2)ρ 2 2 η(cid:0) 3 1 ρ 2 2 (cid:1)3 − 2(cid:0) η 3 1(cid:0) ρ 2 2
+φT(x ,b )φs(x ,b ) α (t )h (x ,x ,x ,b ,b )exp[ S (t )] , (23)
η 3 1 ρ 2 2 s f (cid:1)f 1 2 3 1 2 − cd f }
MP2 = M . (24)
eη − eη (cid:1)(cid:1)(cid:3)
For the non-factorizable annihilation diagrams 1(e) and 1(f), again all three wave func-
tions are involved. Here we have all three kinds of contributions. M is the contribution
aη
containing operator type (V A)(V A), while MP1 and MP2 is the contribution con-
− − aη aη
taining operator type (V A)(V +A) and (S P)(S +P), respectively
− −
32 1 ∞
M = πC M4 dx dx dx b db b db φ (x ,b )
aη −√6 F Bs 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
x φ (x ,b )φA(x ,b )+r r (x x ) φP(x ,b )φt(x ,b )
· − 2 ρ 2 2 η 3 2 ρ η 2 − 3 η 3 2 ρ 2 2
+φT(x ,b )φs(x ,b ) +(2+x +x )φP(x ,b )φs(x ,b )+( 2+
(cid:8) η(cid:2) 3 2 ρ 2 2 2 (cid:0) 3 η 3 (cid:0)2 ρ 2 2 −
x +x )φT(x ,b )φt(x ,b ) α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2)]
2 3 η 3 2 ρ(cid:1) 2 2 s f f 1 2 3 1 2 − ef f
+ x φ (x ,b )φA(x ,b )+r r (x x ) φP(x ,b )φt(x ,b )+
3 ρ 2 2 η 3 2 (cid:1)ρ(cid:3)η 3 − 2 η 3 2 ρ 2 2
φT(x ,b )φs(x ,b ) +(x +x ) φP(x ,b )φs(x ,b )+φT(x ,b )
η(cid:2) 3 2 ρ 2 2 2 3(cid:0) η 3 2(cid:0) ρ 2 2 η 3 2
φt(x ,b ) α (t1)h1(x ,x ,x ,b ,b )exp[ S (t1)] . (25)
· ρ 2 2 s f(cid:1) f 1 2 3(cid:0) 1 2 − ef f }
(cid:1)(cid:1)(cid:3)
32 1 ∞
MP1 = πC M4 dx dx dx b db b db φ (x ,b )
aη −√6 F Bs 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
r (2 x )φA(x ,b ) φt(x ,b )+φs(x ,b ) +r (x 2)φ (x ,b )
· ρ − 2 η 3 2 ρ 2 2 ρ 2 2 η 3 − ρ 2 2
φP(x ,b )+φT(x ,b ) α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2)]
·(cid:8)(cid:2)η 3 2 η 3 2(cid:0) s f f 1 2 3 (cid:1)1 2 − ef f
+ x r φA(x ,b ) φt(x ,b )+φs(x ,b ) x r φ (x ,b )
(cid:0) 2 ρ η 3 2 ρ 2 (cid:1)(cid:3)2 ρ 2 2 − 3 η ρ 2 2
φP(x ,b )+φT(x ,b ) α (t1)h1(x ,x ,x ,b ,b )exp[ S (t1)] . (26)
· (cid:2) η 3 2 η(cid:0) 3 2 s f f 1 (cid:1)2 3 1 2 − ef f }
(cid:0) (cid:1)(cid:3)
8
32 1 ∞
MP2 = πC M4 dx dx dx b db b db φ (x ,b )
aη −√6 F Bs 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
x φ (x ,b )φA(x ,b )+r r φP(x ,b ) (x +x +2)φs(x ,b )
· 3 ρ 2 2 η 3 2 ρ η η 3 2 2 3 ρ 2 2
(x x )φt(x ,b ) +r r φT(x ,b ) (x x )φs(x ,b )+(x
−(cid:8)(cid:2) 2 − 3 ρ 2 2 ρ η η 3 2 (cid:0)3 − 2 ρ 2 2 2
+x 2)φt(x ,b ) α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2)]
3 − ρ 2 2 (cid:1) s f f 1 2 (cid:0)3 1 2 − ef f
+ ( x )φ (x ,b )φA(x ,b )+r r (x x ) φP(x ,b )φt(x ,b )
− 2 ρ 2 2 (cid:1)η(cid:3) 3 2 ρ η 3 − 2 η 3 2 ρ 2 2
+φT(x ,b )φs(x ,b ) (x +x ) φP(x ,b )φs(x ,b )+φT(x ,b )
(cid:2)η 3 2 ρ 2 2 − 2 3 (cid:0)η 3 2 (cid:0)ρ 2 2 η 3 2
φt(x ,b ) α (t1)h1(x ,x ,x ,b ,b )exp[ S (t1)] . (27)
ρ 2 2 s f (cid:1)f 1 2 3 1(cid:0) 2 − ef f }
The factorizable annih(cid:1)il(cid:1)a(cid:3)tion diagrams 1(g) and 1(h) involve only ρ and η wave func-
tions. Again decay amplitude F is for (V A)(V A) type operators, while the FP1
aη − − aη
and FP2 come from the (V A)(V +A) and (S P)(S+P) type operators, respectively
aη − −
1 ∞
F = 8πC f M4 dx dx b db b db
aη F Bs Bs 2 3 2 2 3 3
Z0 Z0
x φ (x ,b )φA(x ,b )+2r r φs(x ,b ) (1+x )φP(x ,b )
· 3 ρ 2 2 η 3 3 ρ η ρ 2 2 3 η 3 3
+(x 1)φT(x ,b ) α (t3)h (x ,x ,b ,b )exp[ S (t3)]
(cid:8)(cid:2) 3 − η 3 3 s e a 2 3 2 3(cid:0) − gh e
x φ (x ,b )φA(x ,b )+2r r φP(x ,b ) (1+x )φs(x ,b )
− 2 ρ 2 2 η 3(cid:1)(cid:3)3 ρ η η 3 3 2 ρ 2 2
+(x 1)φt(x ,b ) α (t4)h (x ,x ,b ,b )exp[ S (t4)] , (28)
(cid:2) 2 − ρ 2 2 s e a 3 2 3 2(cid:0) − gh e
(cid:1)(cid:3) (cid:9)
FP1 = F , (29)
aη − aη
1 ∞
FP2 = 16πC f M4 dx dx b db b db
aη − F Bs Bs 2 3 2 2 3 3
Z0 Z0
2r φs(x ,b )φA(x ,b )+r x φ (x ,b ) φP(x ,b )
· ρ ρ 2 2 η 3 3 η 3 ρ 2 2 η 3 3
φT(x ,b ) α (t3)h (x ,x ,b ,b )exp[ S (t3)]
−(cid:8)(cid:2)η 3 3 s e a 2 3 2 3 −(cid:0)gh e
+ x r φA(x ,b ) φs(x ,b ) φt(x ,b ) +2r φ (x ,b )
2 ρ η (cid:1)3(cid:3) 3 ρ 2 2 − ρ 2 2 η ρ 2 2
φP(x ,b ) α (t4)h (x ,x ,b ,b )exp[ S (t4)] . (30)
· (cid:2)η 3 3 s e(cid:0) a 3 2 3 2 −(cid:1) gh e
When we exchange the posi(cid:3)tion of ρ and η(′) mesons in Fig. 1, t(cid:9)hen only 4 annihila-
tion diagrams 1(e)-1(h) can contribute to B ρ0η(′) decays. The corresponding decay
→
amplitudes from the non-factorizable annihilation diagrams 1(e) and 1(f), where it is the
ρ0 meson who picks up the spectator s quark, can be written as
32 1 ∞
M = πC M4 dx dx dx b db b db φ (x ,b )
aρ −√6 F Bs 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
x φ (x ,b )φA(x ,b )+r r (x x ) φP(x ,b )φt(x ,b )
· 3 ρ 3 2 η 2 2 ρ η 3 − 2 η 2 2 ρ 3 2
+φT(x ,b )φs(x ,b ) +(x +x ) φP(x ,b )φs(x ,b )+φT(x ,b )
(cid:8)(cid:2)η 2 2 ρ 3 2 2 (cid:0)3 η 2 (cid:0)2 ρ 3 2 η 2 2
φt(x ,b ) α (t1)h1(x ,x ,x ,b ,b )exp[ S (t1)] [x φ (x ,b )
· ρ 3 2 s f (cid:1)f 1 2 3 (cid:0)1 2 − ef f − 2 ρ 3 2
φA(x ,b )+r r (x x ) φP(x ,b )φt(x ,b )+φT(x ,b )
· η 2 2 (cid:1)(cid:1)(cid:3)ρ η 2 − 3 η 2 2 ρ 3 2 η 2 2
φs(x ,b ) +r r (2+x +x )φP(x ,b )φs(x ,b ) (2 x x )
· ρ 3 2 ρ(cid:0)η 2(cid:0) 3 η 2 2 ρ 3 2 − − 2 − 3
φT(x ,b )φt(x ,b ) α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2)] , (31)
· η 2 2(cid:1)(cid:1)ρ 3 2(cid:0) s f f 1 2 3 1 2 − ef f }
(cid:1)(cid:3)
9
32 1 ∞
MP1 = πC M4 dx dx dx b db b db φ (x ,b )
aρ √6 F Bs 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
x r φ (x ,b ) φP(x ,b )+φT(x ,b ) x r φs(x ,b )
· 2 η ρ 3 2 η 2 2 η 2 2 − 3 ρ ρ 3 2
+φt(x ,b ) φA(x ,b ) α (t1)h1(x ,x ,x ,b ,b )exp[ S (t1)]
(cid:8)(cid:2)ρ 3 2 η (cid:0)2 2 s f f 1 2 (cid:1)3 1 2 (cid:0) − ef f
+ (2 x )r φ (x ,b ) φP(x ,b )+φT(x ,b ) (2 x )r φs(x ,b )
− 2 (cid:1)η ρ 3 2 (cid:3) η 2 2 η 2 2 − − 3 ρ ρ 3 2
+φt(x ,b ) φA(x ,b ) α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2)] , (32)
(cid:2) ρ 3 2 η 2 2 (cid:0) s f f 1 2 3 1 (cid:1)2 − ef f(cid:0) }
(cid:1) (cid:3)
32 1 ∞
MP2 = πC M4 dx dx dx b db b db φ (x ,b )
aρ √6 F Bs 1 2 3 1 1 2 2 Bs 1 1
Z0 Z0
x φ (x ,b )φA(x ,b )+r r (x +x ) φP(x ,b )φs(x ,b )
· 2 ρ 3 2 η 2 2 ρ η 2 3 η 2 2 ρ 3 2
+φT(x ,b )φt(x ,b ) +(x x ) φP(x ,b )φt(x ,b )+φT(x ,b )
(cid:8)(cid:2)η 2 2 ρ 3 2 2 − (cid:0)3 η 2 (cid:0)2 ρ 3 2 η 2 2
φs(x ,b ) α (t1)h1(x ,x ,x ,b ,b )exp[ S (t1)] [x φ (x ,b )
· ρ 3 2 s f (cid:1)f 1 2 3 (cid:0)1 2 − ef f − 3 ρ 3 2
φA(x ,b )+r r (x x ) φP(x ,b )φt(x ,b )+φT(x ,b )φs(x ,b )
· η 2 2 (cid:1)(cid:1)(cid:3) ρ η 3 − 2 η 2 2 ρ 3 2 η 2 2 ρ 3 2
+r r (2+x +x )φP(x ,b )φs(x ,b )+(x +x 2)φT(x ,b )
ρ η 2 (cid:0)3 η 2 2(cid:0) ρ 3 2 2 3 − η 2 2 (cid:1)(cid:1)
φt(x ,b ) α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2)] . (33)
· ρ (cid:0)3 2 s f f 1 2 3 1 2 − ef f }
For the factorizable(cid:1)a(cid:3)nnihilation diagrams 1(g) and 1(h) after the exchange of ρ0 and
η(′) mesons, the corresponding decay amplitudes can be obtained directly through the
links with their counterparts F , F and FP2
aη aη aη
F = F , FP1 = F , FP2 = FP2. (34)
aρ − aη aρ aη aρ aη
Combining the contributions from different diagrams, the total decay amplitude for
B ρ0η decay can be written as:
s
→
1 3 1 3 1
(ρ0η) = F ξ C + C ξ C + C + C + C F (φ)
eη u 1 2 t 7 8 9 10 2
M 3 − 2 2 2 2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
3 3
+M ξ C ξ C + C F (φ)
eη u 2 t 8 10 2
− −2 2
(cid:20) (cid:18) (cid:19)(cid:21)
3 3
+(M +M ) ξ C ξ C F (φ)+ MP2 +MP2 ξ C F (φ)
aη aρ u 2 − t 2 10 1 aη aρ − t 2 8 1
(cid:20) (cid:21) (cid:20) (cid:21)
1 (cid:0) (cid:1)
+(F +F ) ξ C + C
aη aρ u 1 2
3
(cid:20) (cid:18) (cid:19)
3 1 3 1
ξ C C + C + C F (φ), (35)
t 7 8 9 10 1
− −2 − 2 2 2
(cid:18) (cid:19)(cid:21)
where ξ = V∗V , ξ = V∗V , and F (φ) = cosφ/2 and F (φ) = sinφ/√2 are the
u ub ud t tb td 1 2 −
mixing factors. The Wilson coefficients C should be calculated at the appropriate scale
i
t by using the formulas as given in the Appendices of Ref. [27].
10