The ${\Upsilon}(1S)$ ${\to}$ $B_{c}{\rho}$ decay with perturbative QCD approach

0.16 MB·

by  Junfeng Sun

The Υ(1S) B ρ decay with perturbative QCD approach
c
→
Junfeng Sun,1 Yueling Yang,1 Qingxia Li,1 Gongru Lu,1 Jinshu Huang,2 and Qin Chang1
1Institute of Particle and Nuclear Physics,
Henan Normal University, Xinxiang 453007, China
2College of Physics and Electronic Engineering,
6
1 Nanyang Normal University, Nanyang 473061, China
0
2
Abstract

y
a
With the potential prospects of the Υ(1S) data samples at the running LHC and upcoming
M
SuperKEKB, the Υ(1S) B ρ weak decay is studied with the pQCD approach. It is found that
8 → c
1
(1) thelion’s shareof branchingratio comes from the longitudinal polarization helicity amplitudes;

]
h (2) branching ratio for the Υ(1S) B ρ decay can reach up to (10−9), which might behopefully
c
p → O
-
measurable.
p
e
h
[

2
v
9
5
2
0
0
.
1
0
6
1
:
v
i
X
r
a
1
I. INTRODUCTION
¯
The Υ(1S) meson consists of the bottom quark and antiquark pair bb, carries the defi-
nitely established quantum numbers of IGJPC = 0−1−− [1], and lies below the kinematic
BB¯ threshold. The Υ(1S) meson decay mainly through the strong interaction, the electro-
magnetic interaction and radiative transition. Besides, the Υ(1S) meson can also decay via
the weak interactions within the standard model. More than 108 Υ(1S) data samples have
been accumulated at Belle [2]. More and more upsilon data samples with high precision are
promisingly expected at the running LHC and the forthcoming SuperKEKB. Although the
branching ratio for the Υ(1S) weak decay is tiny, it seems to exist a realistic possibility to
search for the signals of the Υ(1S) weak decay at future experiments. In this paper, we will
study the Υ(1S) B ρ weak decay with the perturbative QCD (pQCD) approach [3–5].
c
→
Experimentally, there is no report on the Υ(1S) B ρ weak decay so far. The signals for
c
→
the Υ(1S) B ρ weak decay should, in principle, be easily identified, due to the facts that
c
→
the final states have different electric charges, have definite momentum and energy, and are
back-to-back in the rest frame of the Υ(1S) meson. In addition, the identification of a single
flavored B meson could be used to effectively enhance signal-to-background ratio. Another
c
important and fashionable motivation is that evidences of an abnormally large branching
ratio for the Υ(1S) weak decay might be a hint of new physics.
Theoretically, the Υ(1S) B ρ weak decay belongs to the external W emission topog-
c
→
raphy, and is favored by the Cabibbo-Kabayashi-Maskawa (CKM) matrix elements V V∗ .
| cb ud|
So it should have relatively large branching ratio among the Υ(1S) weak decays, which has
been studied with the naive factorization (NF) approximation [6, 7]. Recently, some attrac-
tive methods have been developed, such as the pQCD approach [3–5], the QCD factorization
approach [8–10], soft and collinear effective theory [11–14], and applied widely to accommo-
date measurements on the B meson weak decays. The Υ(1S) B ρ decay permit one to
c
→
cross check parameters obtained from the B meson decay, to test the practical applicability
of various phenomenological models in the vector meson weak decays, and to further explore
the underlying dynamical mechanism of the heavy quark weak decay. In addition, as it is
well known, the B meson carries two explicit heavy flavors and has extremely abundant de-
c
cay modes, but its hadronic production is suppressed compared with that for hidden-flavor
quarkonia and heavy-light mesons, due to higher order in QCD coupling constants α and
s
2
the presence of additional heavy quarks [15, 16]. The Υ(1S) B ρ decay offers another
c
→
platform to study the B meson production at high energy colliders.
c
This paper is organized as follows. In section II, we present the theoretical framework
and the amplitudes for the Υ(1S) B ρ decay with the pQCD approach. Section III is
c
→
devoted to numerical results and discussion. The last section is our summary.
II. THEORETICAL FRAMEWORK
A. The effective Hamiltonian
The effective Hamiltonian responsible for the Υ(1S) B ρ weak decay is [17]
c
→
G
= F V V∗ C (µ)Q (µ)+C (µ)Q (µ) +H.c., (1)
Heff √2 cb udn 1 1 2 2 o
whereG 1.166 10−5GeV−2 [1]istheFermi couplingconstant; theCKMfactoriswritten
F
≃ ×
as a power series in the Wolfenstein parameter λ 0.2 [1],
≃
1 1
V V∗ = Aλ2 Aλ4 Aλ6 + (λ8). (2)
cb ud − 2 − 8 O
The local operators are defined as follows:
Q = [c¯ γ (1 γ )b ][q¯ γµ(1 γ )u ], (3)
1 α µ 5 α β 5 β
− −
Q = [c¯ γ (1 γ )b ][q¯ γµ(1 γ )u ], (4)
2 α µ 5 β β 5 α
− −
where α and β are color indices.
From Eq.(1), it is clearly seen that only the tree operators contribute to the concerned
process, and there is no pollution from penguin and annihilation contributions. As it is
well known, degrees of freedom with mass scales above µ are integrated out into the Wilson
coefficients C (µ) typically using the renormalization group assisted perturbation theory.
1,2
The physical contributions below the scale of µ are included in the hadronic matrix elements
(HME) where the local operators sandwiched between initial and final hadron states. The
most complicated part is the treatment on HME, where the perturbative and nonperturba-
tive effects entangle with each other. To obtain the decay amplitudes, the remaining work
is to calculate HME properly.
3
B. Hadronic matrix elements
With the Lepage-Brodsky approach for exclusive processes [18], HME could be expressed
as the convolution of hard scattering subamplitudes containing perturbative contributions
with the universal wave functions reflecting the nonperturbative contributions. To eliminate
the endpoint singularities appearing in the collinear factorization approximation, the pQCD
approach suggests [3–5] retaining the transverse momentum of quarks and introducing the
Sudakov factor. Finally, the decay amplitudes could be factorized into three parts [4, 5]: the
hard effects enclosed by the Wilson coefficients C , the heavy quark decay subamplitudes ,
i
H
and the universal wave functions Φ,
dkC (t) (t,k)Φ(k)e−S, (5)
Z i H
where t is a typical scale, k is the momentum of the valence quarks, and the Sudakov factor
e−S can effectively suppress the long-distance contributions and make the hard scattering
more perturbative.
C. Kinematic variables
The light cone kinematic variables in the Υ(1S) rest frame are defined as follows:
m
1
p = p = (1,1,0), (6)
Υ 1
√2
p = p = (p+,p−,0), (7)
Bc 2 2 2
p = p = (p−,p+,0), (8)
ρ 3 3 3
~
k = x p +(0,0,k ), (9)
i i i i⊥
p m
k i i
ǫ = n , (10)
i m − p n +
i i +
·
ǫ⊥ = (0,0,~1), (11)
i
n = (1,0,0), (12)
+
p± = (E p)/√2, (13)
i i±
s = 2p p , (14)
2 3
·
t = 2p p = 2m E , (15)
1 2 1 2
·
4
u = 2p p = 2m E , (16)
1 3 1 3
·
[m2 (m +m )2][m2 (m m )2]
p = q 1 − 2 3 1 − 2 − 3 , (17)
2m
1
~
where x and k are the longitudinal momentum fraction and transverse momentum of
i i⊥
the valence quark, respectively; ǫk and ǫ⊥ are the longitudinal and transverse polarization
i i
vectors, respectively, satisfying with the relations ǫ2 = 1 and ǫ p = 0; the subscript i
i − i· i
k,⊥
= 1, 2, 3 on variables (p , E , m and ǫ ) correspond to the Υ(1S), B and ρ mesons,
i i i i c
respectively; n is the null vector; s, t and u are the Lorentz-invariant variables; p is the
+
common momentum of final states. The notation of momentum is displayed in Fig.1(a).
D. Wave functions
With the notation in [19, 20], the definitions of the diquark operator HME are
f
0 b (z)¯b (0) Υ(p ,ǫk) = Υ d4k e−ik1·z ǫk m Φv (k ) p Φt (k ) , (18)
h | i j | 1 1 i 4 Z 1 n6 1h 1 Υ 1 −6 1 Υ 1 ioji
f
0 b (z)¯b (0) Υ(p ,ǫ⊥) = Υ d4k e−ik1·z ǫ⊥ m ΦV(k ) p ΦT(k ) , (19)
h | i j | 1 1 i 4 Z 1 n6 1h 1 Υ 1 −6 1 Υ 1 ioji
i
B (p ) c¯(z)b (0) 0 = f dx eix2p2·z γ p +m φ (x ) , (20)
h c 2 | i j | i 4 BcZ 2 n 5h6 2 2i Bc 2 oji
1 1
ρ(p ,ǫk) u (0)d¯(z) 0 = dk eik3·z ǫkm Φv(k )+ǫk p Φt(k )+m Φs(k ) , (21)
h 3 3 | i j | i 4Z 0 3 n6 3 3 ρ 3 6 36 3 ρ 3 3 ρ 3 oji
1 1
ρ(p ,ǫ⊥) u (0)d¯(z) 0 = dk eik3·z ǫ⊥m ΦV(k )
h 3 3 | i j | i 4Z 0 3 n6 3 3 ρ 3
im
+ ǫ⊥ p ΦT(k )+ 3 ε γ γµǫ⊥,ν pαnβ ΦA(k ) , (22)
6 3 6 3 ρ 3 p n µναβ 5 3 3 + ρ 3 oji
3 +
·
where f and f are decay constants; the definitions of wave functions Φv,t,s and ΦV,T,A can
Υ Bc ρ ρ
be found in Ref. [19, 20]. In fact, for the ρ meson, only three wave functions Φv and ΦV,A
ρ ρ
are involved in the decay amplitudes (see Appendix A). The twist-2 distribution amplitude
for the longitudinal polarization ρ meson is [19, 20]:
φv(x) = f 6xx¯ ak C3/2(t), (23)
ρ ρ 2i 2i
X
i=0
k 3/2
where f is the decay constant; x¯ = 1 x; t = x¯ x; a and C (t) are the Gegenbauer
ρ − − i i
k
moment and polynomial, respectively; a = 0 for odd i due to the G-parity invariance of
i
5
the ρ distribution amplitudes. As to the twist-3 distribution amplitudes of the transverse
polarization ρ meson, for simplicity, we will take their asymptotic forms [19, 20]:
3
φV(x) = f (1+t2), (24)
ρ ρ 4
3
φA(x) = f ( t). (25)
ρ ρ 2 −
Becauseofm 2m andm m +m , bothΥ(1S)andB systemsarenearlynon-
Υ(1S) ≃ b Bc ≃ b c c
relativistic. Nonrelativistic quantum chromodynamics (NRQCD) [21–23] and Schro¨dinger
equation can be used to describe their spectrum. The eigenfunction of the time-independent
Schro¨dingerequationwithscalarharmonicoscillatorpotentialcorrespondingtothequantum
numbers nL = 1S is written as
φ(~k) e−~k2/2β2, (26)
∼
where parameter β determines the average transverse momentum, i.e., 1S ~k2 1S = β2.
h | ⊥| i
Employing the Brodsky-Huang-Lepage ansatz [24, 25] which has been used to structure
wave functions for light and heavy mesons [26],
1 ~k2 +m2
~k2 i⊥ qi, (27)
→ 4 Xi xi
~
where x , k , m are the longitudinal momentum fraction, transverse momentum, mass of
i i⊥ qi
~
the valence quarks in hadrons, respectively, with the relations x = 1 and k = 0, then
i i⊥
P P
~
integrating out k and combining with their asymptotic forms, one can obtain [19, 28]
i⊥
x¯m2 +xm2
φ (x) = Axx¯exp c b , (28)
Bc n− 8β2xx¯ o
2
m2
φv(x) = φT(x) = Bxx¯exp b , (29)
Υ Υ n− 8β2xx¯o
1
m2
φt (x) = Ct2exp b , (30)
Υ n− 8β2xx¯o
1
m2
φV(x) = D(1+t2)exp b , (31)
Υ n− 8β2xx¯o
1
where the exponential function represents the transverse momentum distribution and can
suppress the end-point singularity; β ξ α (ξ ) with ξ = m /2 based on the NRQCD power
i i s i i i
≃
counting rules [21]; parameters A, B, C, D are the normalization coefficients satisfying the
conditions
1 1 1
dxφ (x) = 1, dxφv,t(x) = dxφV,T(x) = 1. (32)
Z Bc Z Υ Z Υ
0 0 0
6
The shape lines for the normalized distribution amplitudes of φ (x) and φv,t,V,T(x) have
Bc Υ
beendisplayedinFig.1ofRef.[27], fromwhichonecanseethatEqs.(28)-(31)reflectgenerally
the feature that valence quarks of hadrons share momentum fractions according to their
masses.
p
3
ρ− ρ− ρ− ρ−
d(k3) u¯(k¯3) d u¯ d u¯ d u¯
b(k ) c(k )
1 2
b c b c b c
p p
1 2
Υ G Bc+ Υ G Bc+ Υ G Bc+ Υ G Bc+
¯b ¯b ¯b ¯b ¯b ¯b ¯b ¯b
(a) (b) (c) (d)
FIG. 1: Feynman diagrams for the Υ B ρ decay with the pQCD approach, where (a) and (b)
c
→
are factorizable emission diagrams, (c) and (d) are nonfactorizable emission diagrams.
E. Decay amplitudes
The Feynman diagrams for the Υ(1S) B ρ decay are shown in Fig.1, including factor-
c
→
izable emission topologies(a) and(b) where gluonconnects to thequarks inthe samemeson,
and nonfactorizable emission topologies (c) and (d) where gluon attaches to the quarks in
two different mesons.
The amplitude for the Υ(1S) B ρ decay is defined as below [29],
c
→
(Υ(1S) B ρ) = (ǫk,ǫk)+ (ǫ⊥,ǫ⊥)+i ε ǫµǫν pαpβ, (33)
A → c AL 1 3 AN 1 3 AT µναβ 1 3 1 3
which is conventionally written as the helicity amplitudes [29],
= C i (ǫk,ǫk), (34)
A0 − A XAL 1 3
i
= √2C i (ǫ⊥,ǫ⊥), (35)
Ak AXAN 1 3
i
= √2C m p i , (36)
A⊥ A 1 XAT
i
G C
C = i F F πf f V V∗, (37)
A √2 N Υ Bc cb ud
where C = 4/3 and the color number N = 3; the superscript i on i corresponds to the
F AL,N,T
indices of Fig.1. The explicit expressions of building blocks i are collected in Appendix
AL,N,T
A.
7
III. NUMERICAL RESULTS AND DISCUSSION
In the rest frame of the Υ(1S) meson, branching ratio ( r), polarization fractions (f )
0,k,⊥
B
and relative phase between helicity amplitudes (φ ) for the Υ(1S) B ρ weak decay are
k,⊥ c
→
defined as
1 p
r = 2 + 2 + 2 , (38)
B 12π m2Γ n|A0| |Ak| |A⊥| o
Υ Υ
2
0,k,⊥
f = |A | , (39)
0,k,⊥ 2 + 2 + 2
0 k ⊥
|A | |A | |A |
φ = arg( / ), (40)
k,⊥ k,⊥ 0
A A
where mass m = 9460.30 0.26 MeV and decay width Γ = 54.02 1.25 keV [1].
Υ Υ
± ±
The values of other input parameters are listed as follows. If not specified explicitly, we
will take their central values as default inputs.
(1) Wolfenstein parameters [1]: A = 0.814+0.023 and λ = 0.22537 0.00061.
−0.024 ±
(2) Masses of quarks [1]: m = 1.67 0.07 GeV and m = 4.78 0.06 GeV.
c b
± ±
(3) Gegenbauer moments1 ak = 1 and ak = 0.15 0.07 for twist-2 distribution amplitudes
0 2 ±
of the ρ meson [20].
(4) Decay constants: f = (676.4 10.7) MeV [28], f = 489 5 MeV [31], f = 216 3
Υ ± Bc ± ρ ±
MeV [20].
Our numerical results are presented as follows:
r = (8.34+0.47+1.35+0.40+1.44) 10−9, (41)
B −0.69−0.88−0.40−1.26 ×
f = (82.2+0.0+1.1+0.0)%, (42)
0 −0.7−1.3−0.0
f = (15.0+0.6+1.0+0.0)%, (43)
k −0.0−0.8−0.0
f = (2.8+0.1+0.3+0.0)%, (44)
⊥ −0.0−0.3−0.0
φ 0, φ π, (45)
k ⊥
≃ ≃
where the first uncertainty comes from the choice of the typical scale (1 0.1)t , and the
i
±
expression t is given in Eq.(A25) and Eq.(A26); the second uncertainty is from masses m
i b
and m ; the third uncertainty is from hadronic parameters including decay constants and
c
1 ak = 1 is due to the normalization condition 1φv(x)dx = 1. More discussion on the ρ wave functions
0 0 ρ
and Gegenbauer moments ak can be found in tRhe recent references, such as Ref.[30].
2
8
Gegenbauer moments; and the fourth uncertainty of branching ratio comes from the CKM
parameters. The following are some comments.
(1) Branching ratio for the Υ(1S) B ρ decay with the pQCD approach is different
c
→
from previous estimation [6, 7] with the NF approximation. Many factors lead to these
differences. For example, as it is showed in Ref. [7], the values of form factors for Υ(1S)
B transition are very sensitive to the choice of wave functions. In addition, form factors
c
→
written as the convolution integral of wave functions in Ref. [7] are usually enhanced by one-
gluon-exchange scattering amplitudes with the pQCD approach. These discrepancy deserve
much dedicated study and should be carefully tested by the future experiments.
(2) Branching ratio for the Υ(1S) B ρ decay can reach up to (10−9), which might
c
→ O
be measurable at the running LHC and forthcoming SuperKEKB. For example, the Υ(1S)
production cross section in p-Pb collision is about a few µb at LHCb [32] and ALICE [33].
Over 1012 Υ(1S) data samples per ab−1 data collected at LHCb and ALICE are in principle
available, corresponding to a few thousands of the Υ(1S) B ρ events.
c
→
(3) There is a hierarchical pattern among the longitudinal f , parallel f , and perpendic-
0 k
ular f polarization fractions, i.e.,
⊥
p p2
f : f : f 1 : : , (46)
0 k ⊥ ≃ √2m 2m2
Υ(1S) Υ(1S)
where p is the common momentum of final state in the rest frame of the Υ(1S) meson. The
relation Eq.(46) is basically agree with previous estimation [7]. It means that the contri-
butions to branching ratio for the Υ(1S) B ρ decay mainly come from the longitudinal
c
→
polarization fractions, because of f > f > f .
0 k ⊥
(4) The relative phase φ is close to zero. The reason is that the factorizable contribu-
k
tions from diagrams Fig.1(a,b) is real and proportional to the large coefficient a , while the
1
nonfactorizable contributions from diagrams Fig.1(c,d) is suppressed by the color factor and
proportional to the small Wilson coefficient C , and the strong phases arise only from the
2
nonfactorizable contributions, which is consistent with the prediction of the QCD factor-
ization approach [8, 9] where the strong phase arising from nonfactorizable contributions is
suppressed bycolor andα forthea -dominatedprocesses. Therelative phases, if they could
s 1
be determined experimentally, will improve our understanding on the strong interactions.
9
IV. SUMMARY
The Υ(1S) weak decay is allowable within the standard model. In this paper, the Υ(1S)
B ρ weak decays are studied with the pQCD approach. It is found that with the nonrel-
c
→
ativistic wave functions for Υ(1S) and B mesons, the longitudinal polarization fraction is
c
the largest one, and branching ratios for the Υ(1S) B ρ decay can reach up to (10−9),
c
→ O
which might be detectable at the future experiments.
Acknowledgments
We thank Professor Dongsheng Du (IHEP@CAS), Professor Caidian Lu¨ (IHEP@CAS)
and Professor Yadong Yang (CCNU) for helpful discussion. The work is supported by the
National Natural Science Foundation of China (Grant Nos. 11547014, 11475055, U1332103
and 11275057).
Appendix A: Building blocks of decay amplitudes
For the sake of simplicity, the amplitude for the Υ(1S) B ρ decay, Eq.(33), is de-
c
→
composed into building blocks i , where the superscript i corresponds to the indices of
AL,N,T
Fig.1. With the pQCD master formula Eq.(5), the explicit expressions of i are written
AL,N,T
as follows:
1 1 ∞ ∞
a = dx dx b db b db φv(x )
AL Z 1Z 2Z 1 1Z 2 2 Υ 1
0 0 0 0
φ (x )E (t )α (t )a (t )H (α ,β ,b ,b )
Bc 2 f a s a 1 a f e a 1 2
m2s+m m u (4m2p2 +m2u)x¯ , (A1)
n 1 2 b − 1 2 2o
1 1 ∞ ∞
a = m m dx dx b db b db φV(x )
AN 1 3Z 1Z 2Z 1 1Z 2 2 Υ 1
0 0 0 0
φ (x )E (t )α (t )a (t )H (α ,β ,b ,b )
Bc 2 f a s a 1 a f e a 1 2
2m2x¯ 2m m t , (A2)
n 2 2 − 2 b − o
1 1 ∞ ∞
a = 2m m dx dx b db b db φV(x )
AT 1 3Z 1Z 2Z 1 1Z 2 2 Υ 1
0 0 0 0
φ (x )E (t )α (t )a (t )H (α ,β ,b ,b ), (A3)
Bc 2 f a s a 1 a f e a 1 2
1 1 ∞ ∞
b = dx dx b db b db
AL Z 1Z 2Z 1 1Z 2 2
0 0 0 0
10