The ${\Upsilon}(nS)$ ${\to}$ $B_{c}D_{s}$, $B_{c}D_{d}$ decays with perturbative QCD approach

0.23 MB·

by  Junfeng Sun

The Υ(nS) B D , B D decays with perturbative QCD approach
c s c d
→
Junfeng Sun,1 Yueling Yang,1 Qingxia Li,1 Haiyan Li,1 Qin Chang,1 and Jinshu Huang2
1Institute of Particle and Nuclear Physics,
Henan Normal University, Xinxiang 453007, China
2College of Physics and Electronic Engineering,
7 Nanyang Normal University, Nanyang 473061, China
1
0
Abstract
2

n
The Υ(nS) B D , B D weak decays are studied with the pQCD approach firstly. It is found
a → c s c d
J
that branching ratios r(Υ(nS) B D ) (10−10) and r(Υ(nS) B D ) (10−11), which
7 B → c s ∼ O B → c d ∼ O
1
might be measurable in the future experiments.

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I. INTRODUCTION
Since the discovery of bottomonium (the bound states of the bottom quark b and the
¯ ¯
corresponding antiquark b, i.e., bb) at Fermilab in 1977 [1, 2], remarkable achievements have
been made in the understanding of the properties of bottomonium, thanks to the endeavor
from the experiment groups of CLEO, BaBar, Belle, CDF, D0, LHCb, ATLAS, and so on
[3]. The upsilon, Υ(nS), is theS-wave spin-triplet state, n3S , of bottomoniumwith thewell
1
established quantum number of IGJPC = 0−1−− [4]. The typical total widths of the upsilons
below the kinematical open-bottom threshold (where the radial quantum number n = 1, 2
and 3) are a few tens of keV (see Table I), at least two orders of magnitude lower less than
thoseofbottomoniumabovetheBB¯ threshold. (notethatforsimplicity, thenotationΥ(nS)
will denote the Υ(1S), Υ(2S) and Υ(3S) mesons in the following content if not specified
explicitly) Asitiswell known, theΥ(nS)mesondecays primarily throughtheannihilationof
¯
the bb pairs into three gluons, which are suppressed by the phenomenological Okubo-Zweig-
Iizuka rule [5–7]. The allowed G-parity conserving transitions, Υ(nS) ππΥ(mS) and
→
Υ(nS) ηΥ(mS) where 3 n > m 1, are greatly limited by the compact phase spaces,
→ ≥ ≥
because the mass difference m m is just slightly larger than 2m , and m
Υ(3S) Υ(2S) π Υ(2S)
−
m is just slightly larger than m . The coupling strengths of the electromagnetic
Υ(1S) η
−
and radiative interactions are proportional to the electric charge of the bottom quark, Q =
b
1/3 in the unit of e . Besides, the Υ(nS) meson can also decay via the weak interactions
− | |
within the standard model, although the branching ratio is small, about 2/τ Γ (10−8)
B Υ
∼ O
[4], where τ and Γ are the lifetime of the B meson and the total width of the Υ(nS)
B Υ u,d,s
meson, respectively. In this paper, we will study the Υ(nS) B D , B D weak decays
c s c d
→
with the perturbative QCD (pQCD) approach [8–10]. The motivation is listed as follows.
TABLE I: Summary of the mass, total width and data samples of the Υ(1S,2S,3S) mesons.
properties [4] data samples (106) [11]
meson mass (MeV) width (keV) Belle BaBar
Υ(1S) 9460.30 0.26 54.02 1.25 102 2 ...
± ± ±
Υ(2S) 10023.26 0.31 31.98 2.63 158 4 98.3 0.9
± ± ± ±
Υ(3S) 10355.2 0.5 20.32 1.85 11 0.3 121.3 1.2
± ± ± ±
2
From the experimental point of view, (1) over 108 Υ(nS) data samples have been accu-
mulated by the Belle detector at the KEKB and the BaBar detector at the PEP-II e+e−
asymmetric energy colliders [11] (see Table I). It is hopefully expected that more and more
upsilons will be collected with great precision at the running upgraded LHC and the forth-
coming SuperKEKB. An abundant data samples offer a realistic possibility to search for the
Υ(nS) weak decays which in some cases might be detectable. (2) The signals for the Υ(nS)
B D weak decays should be clear and easily distinguishable from background, because
c s,d
→
the back-to-back final states with opposite electric charges have definite momentums and
energies in the center-of-mass frame of the Υ(nS) meson. In addition, the identification of
either a single flavored D or B meson can be used not only to avoid the low double-
s,d c
tagging efficiency [12], but also to provide an unambiguous evidence of the Υ(nS) weak
decay. It should be noticed that on one hand, the Υ(nS) weak decays are very challenging
to be observed experimentally due to their small branching ratios, on the other hand, any
evidences of an abnormally large production rate of either a single charmed or bottomed
meson might be a hint of new physics beyond the standard model [12].
From the theoretical point of view, the Υ(nS) weak decays permit one to cross check
parameters obtained from the B meson decays, to further explore the underlying dynamical
mechanism of the heavy quark weak decay, to test various theoretical approaches and to
improve our understanding on the factorization properties. Phenomenologically, the Υ(nS)
B D , B D weak decays are favored by the color factor due to the external W emission
c s c d
→
topological structure, and by the Cabibbo-Kabayashi-Maskawa (CKM) elements V due to
cb
| |
the b c transition, so usually their branching ratio should not be too small. In addition,
→
these two decay modes are the U-spin partners with each other, so the flavor symmetry
breakingeffectscanbeinvestigated. However, asfarasweknow,thereisnostudyconcerning
on the Υ(nS) B D weak decays theoretically and experimentally at the moment. We
c s,d
→
wish this paper can provide a ready reference to the future experimental searches. Recently,
manyattractivemethodshavebeenfullydeveloped toevaluatethehadronicmatrixelements
(HME) where the local quark-level operators are sandwiched between the initial and final
hadron states, such as the pQCD approach [8–10], the QCD factorization [13–15] and the
soft and collinear effective theory [16–19], which could give an appropriate explanation for
many measurements on the nonleptonic B decays. In this paper, we will estimate the
u,d
branching ratios for the Υ(nS) B D weak decays with the pQCD approach to offer a
c s,d
→
3
possibility of searching for these processes at the future experiments.
This paper is organized as follows. Section II is devoted to the theoretical framework
and the amplitudes for the Υ(nS) B D decays. We present the numerical results and
c s,d
→
discussion in section III, and conclude with a summary in the last section.
II. THEORETICAL FRAMEWORK
A. The effective Hamiltonian
Using the operator product expansion and renormalization group equation, the effective
Hamiltonian responsible for the Υ(nS) B D weak decays is written as [20]
c s,d
→
G 2 10
= F V V∗ C (µ)Qq(µ) V V∗ C (µ)Qq(µ) +H.c., (1)
Heff √2 cb cp i i − tb tp j j
qX=d,sn Xi=1 jX=3 o
where G = 1.166 10−5GeV−2 [4] is the Fermi coupling constant; the CKM factors are
F
×
expressed as a power series in the Wolfenstein parameter λ 0.2 [4],
∼
1 1
V V∗ = +Aλ2 Aλ4 Aλ6(1+4A2)+ (λ7), (2)
cb cs − 2 − 8 O
V V∗ = V V∗ Aλ4(ρ iη)+ (λ7), (3)
tb ts − cb cs − − O
for the Υ(nS) B D decays, and
c s
→
V V∗ = Aλ3 + (λ7), (4)
cb cd − O
1
V V∗ = +Aλ3(1 ρ+iη)+ Aλ5(ρ iη)+ (λ7). (5)
tb td − 2 − O
for the Υ(nS) B D decays. The Wilson coefficients C (µ) summarize the physical con-
c d i
→
tributions above the scale of µ, and have been reliably calculated to the next-to-leading
order with the renormalization group assisted perturbation theory. The local operators are
defined as follows.
Qq = [c¯ γ (1 γ )b ][q¯ γµ(1 γ )c ], (6)
1 α µ − 5 α β − 5 β
Qq = [c¯ γ (1 γ )b ][q¯ γµ(1 γ )c ], (7)
2 α µ − 5 β β − 5 α
Qq = [q¯ γ (1 γ )b ][q¯′γµ(1 γ )q′], (8)
3 α µ − 5 α β − 5 β
q′
X
4
Qq = [q¯ γ (1 γ )b ][q¯′γµ(1 γ )q′ ], (9)
4 α µ − 5 β β − 5 α
q′
X
Qq = [q¯ γ (1 γ )b ][q¯′γµ(1+γ )q′], (10)
5 α µ − 5 α β 5 β
q′
X
Qq = [q¯ γ (1 γ )b ][q¯′γµ(1+γ )q′ ], (11)
6 α µ − 5 β β 5 α
q′
X
3
Qq7 = 2Qq′ [q¯αγµ(1−γ5)bα][q¯β′γµ(1+γ5)qβ′], (12)
q′
X
3
Qq8 = 2Qq′ [q¯αγµ(1−γ5)bβ][q¯β′γµ(1+γ5)qα′ ], (13)
q′
X
3
Qq9 = 2Qq′ [q¯αγµ(1−γ5)bα][q¯β′γµ(1−γ5)qβ′], (14)
q′
X
3
Qq10 = 2Qq′ [q¯αγµ(1−γ5)bβ][q¯β′γµ(1−γ5)qα′ ], (15)
q′
X
where Qq , Qq , and Qq are usually called as the tree operators, QCD penguin oper-
1,2 3,···,6 7,···,10
ators, and electroweak penguin operators, respectively; α and β are color indices; q′ denotes
all the active quarks at the scale of µ (mb), i.e., q′ = u, d, s, c, b; and Qq′ is the electric
∼ O
charge of the q′ quark in the unit of e .
| |
B. Hadronic matrix elements
Theoretically, to obtain the decay amplitudes, the remaining essential work and also the
most complex part is the calculation of the hadronic matrix elements of local operators as
accurate as possible. Combining the k factorization theorem [21] with the collinear factor-
T
ization hypothesis, and based on the Lepage-Brodsky approach for exclusive processes [22],
the HME can be written as the convolution of universal wave functions reflecting the non-
perturbative contributions with hard scattering subamplitudes containing the perturbative
contributions within the pQCD framework, where the transverse momentums of quarks are
retained and the Sudakov factors are introduced, in order to regulate the endpoint singular-
ities and provide a naturally dynamical cutoff on the nonperturbative contributions [8–10].
Generally, the decay amplitude can be separated into three parts: the Wilson coefficients C
i
incorporatingthe hardcontributions abovethetypical scale oft, theprocess-dependent scat-
tering amplitudes T accounting for the heavy quark decay, and the universal wave functions
5
Φ including the soft and long-distance contributions, i.e.,
dxdbC (t)T(t,x,b)Φ(x,b)e−S, (16)
i
Z
where x is the longitudinal momentum fraction of valence quarks, b is the conjugate variable
of the transverse momentum k , and e−S is the Sudakov factor.
T
C. Kinematic variables
In the center-of-mass frame of the Υ(nS) mesons, the light cone kinematic variables are
defined as follows.
m
1
p = p = (1,1,0), (17)
Υ 1
√2
p = p = (p+,p−,0), (18)
Bc 2 2 2
p = p = (p−,p+,0), (19)
Ds,d 3 3 3
~
k = x p +(0,0,k ), (20)
i i i iT
1
k
ǫ = (1, 1,0), (21)
Υ √2 −
p± = (E p)/√2, (22)
i i±
s = 2p p , (23)
2 3
·
t = 2p p = 2m E , (24)
1 2 1 2
·
u = 2p p = 2m E , (25)
1 3 1 3
·
[m2 (m +m )2][m2 (m m )2]
p = 1 − 2 3 1 − 2 − 3 , (26)
q 2m
1
~
where x is the longitudinal momentum fraction; k is the transverse momentum; p is the
i iT
k
common momentum of final states; ǫ is the longitudinal polarization vector of the Υ(nS)
Υ
meson; m = m , m = m and m = m denote the masses of the Υ(nS), B and
1 Υ(nS) 2 Bc 3 Ds,d c
D mesons, respectively. The notation of momentum is displayed in Fig.2(a).
s,d
D. Wave functions
With the notation of Refs. [23, 24], the HME of diquark operators squeezed between the
vacuum and the Υ(nS), B , D mesons are defined as follows.
c q
1
0 b (z)¯b (0) Υ(p ,ǫ ) = f dk e−ik1·z ǫ m φv(k ) p φt (k ) , (27)
h | i j | 1 k i 4 Υ 1 6 k 1 Υ 1 −6 1 Υ 1 ji
Z n h io
6
i
B+(p ) c¯(z)b (0) 0 = f dk eik2·z γ p φa (k )+m φp (k ) , (28)
h c 2 | i j | i 4 Bc 2 5 6 2 Bc 2 2 Bc 2 ji
Z n h io
i 1
D−(p ) q¯(z)c (0) 0 = f dk eik3·z γ p φa (k )+m φp (k ) , (29)
h q 3 | i j | i 4 DqZ 0 3 n 5h6 3 Dq 3 3 Dq 3 ioji
where f , f , f are decay constants.
Υ Bc Dq
Because of the relations, m 2m , m m + m , and m m + m (see Table
Υ(nS) ≃ b Bc ≃ b c Dq ≃ c q
II), it might assume that the motion of the valence quarks in the considered mesons is nearly
nonrelativistic. The wave functions of the Υ(nS), B , D mesons could be approximately
c q
described with the nonrelativistic quantum chromodynamics [25–27] and Schro¨dinger equa-
tion. The wave functions of a nonrelativistic three-dimensional isotropic harmonic oscillator
potential are given in Ref. [28],
m2
φv (x) = Axx¯exp b , (30)
Υ(1S) − 8β2xx¯
1
n o
m2
φt (x) = B(x x¯)2exp b , (31)
Υ(1S) − − 8β2xx¯
1
n o
m2
φt,v (x) = Cφt,v (x) 1+ b , (32)
Υ(2S) Υ(1S) 2β2xx¯
1
n o
m2 2
φt,v (x) = Dφt,v (x) 1 b +6 , (33)
Υ(3S) Υ(1S) − 2β2xx¯
1
n(cid:16) (cid:17) o
x¯m2 +xm2
φa (x) = Exx¯exp c b , (34)
Bc − 8β2xx¯
2
n o
x¯m2 +xm2
φp (x) = F exp c b , (35)
Bc − 8β2xx¯
2
n o
x¯m2 +xm2
φa (x) = Gxx¯exp q c , (36)
Dq − 8β2xx¯
3
n o
x¯m2 +xm2
φp (x) = Hexp q c , (37)
Dq − 8β2xx¯
3
n o
whereβ =ξ α (ξ )withξ =m /2; parametersA,B,C,D,E,F,G,H arethenormalization
i i s i i i
coefficients satisfying the following conditions
1 1 1
dxφv,t (x) = 1, dxφa,p(x) = 1, dxφa,p(x) = 1. (38)
Υ(nS) Bc Dq
Z 0 Z 0 Z 0
The shape lines of the distribution amplitudes φv,t (x) and φa,p(x) have been displayed
Υ(nS) Bc
in Ref. [28], which are basically consistent with the physical picture that the valence quarks
share momentums according to their masses.
7
Here, one may question the nonrelativistic treatment on the wave functions of the D
s,d
mesons, because the motion of the light valence quark in D meson is commonly assumed
to be relativistic, and the behavior of the light valence quark in the heavy-light charmed
D mesons should be different from that in the heavy-heavy B and Υ(nS) mesons. In
s,d c
addition, there are several phenomenological models for the D meson wave functions, for
s,d
example, Eq.(30) in Ref. [29]. The D wave function, which is widely used within the pQCD
framework, and is also favored by Ref. [29] via fitting with measurements on the B DP
→
decays, is written as
1
φ (x,b) = 6xx¯ 1+C (1 2x) exp w2b2 , (39)
D D
− − 2
n o n o
where C = 0.5 and w = 0.1 GeV for the D meson; C = 0.4 and w = 0.2 GeV for
D d D
the D meson; the exponential function represents the k distribution. The same model of
s T
Eq.(39) is usually taken as the twist-2 and twist-3 distribution amplitudes in many practical
applications [29].
3.0 3.0 3.0
2.5 ΦDdHx,0L 2.5 ΦDsHx,0L 2.5 ΦDdHx,0L
ΦDdHx,1L ΦDsHx,1L ΦDsHx,0L
2.0 ΦDdHx,2L 2.0 ΦDsHx,2L 2.0
1.5 1.5 1.5
001...050 ΦΦDaDpddHHxxLL 001...050 ΦΦDaDpssHHxxLL 001...050 ΦΦDaDadsHHxxLL
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(a) (b) (c)
FIG. 1: Thedistributions of theD meson wave functions in (a), the distributionsof the D meson
d s
wave functions in (b), and the distributions of the D meson wave functions in (c), where φa (x),
s,d Dq
φp (x), and φ (x,b) correspond to the Eq.(36), Eq.(37), and Eq.(39), respectively.
Dq D
To show that the nonrelativistic description of the D wave functions seems to be ac-
s,d
ceptable, the shape lines of the D wave functions are displayed in Fig.1. It is clearly seen
from Fig.1 that the shape lines of both Eq.(36) and Eq.(37) have a broad peak at the small
x regions, while the distributions of Eq.(39) is nearly symmetric to the variable x. This
fact may imply that although the nonrelativistic model of the D wave functions is crude,
Eq.(36) and Eq.(37) can reflect, at least to some extent, the feature that the light valence
quark might carry less momentums than the charm quark in the D mesons. In addition,
s,d
the flavor asymmetric effects, and the difference between the twist-2 and twist-3 distribution
8
amplitudes are considered at least in part by Eq.(36) and Eq.(37). In the following calcula-
tion, we will use Eq.(36) and Eq.(37) as the twist-2 and twist-3 distribution amplitudes of
the D meson, respectively.
s,d
E. Decay amplitudes
The Feynman diagrams for the Υ(nS) B D decay are shown in Fig.2. There are
c s
→
two types: the emission and annihilation topologies, where diagram with gluon attaching to
quarks in the same meson and between two different mesons are entitled factorizable and
nonfactorizable diagrams, respectively.
p
3
s(k3) Ds− c¯(k¯3) s Ds− c¯ s Ds− c¯ s Ds− c¯
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)
s D− s D− s D− s D−
s s s s
G
c¯ c¯ c¯ c¯
b b b G b
Υ Υ Υ Υ
¯b c ¯b c ¯b c ¯b G c
G
¯b Bc+ ¯b Bc+ ¯b Bc+ ¯b Bc+
(e) (f) (g) (h)
FIG. 2: Feynman diagrams for the Υ(nS) B D decay with the pQCD approach, including the
c s
→
factorizable emission diagrams (a,b), the nonfactorizable emission diagrams (c,d), the nonfactoriz-
able annihilation diagrams (e,f), and the factorizable annihilation diagrams (g,h).
By calculating these diagrams with the pQCD master formula Eq.(16), the decay ampli-
tudes of Υ(nS) B D decays (where q = d, s) can be expressed as:
c q
→
C
(Υ(nS) B D ) = √2G πf f f F m3 (ǫ p )
A → c q F Υ Bc Dq N Υ Υ· Dq
V V∗ LL a + LL C V V∗ LL (a +a )
× cb cq Aa+b 1 Ac+d 2 − tb tq Aa+b 4 10
n h i h
+ SP (a +a )+ LL (C +C )+ SP (C +C )
Aa+b 6 8 Ac+d 3 9 Ac+d 5 7
1 1 1
+ LL (C +C C C )+ LR (C C )
Ae+f 3 4 − 2 9 − 2 10 Ae+f 6 − 2 8
1 1 1
+ LL (a +a a a )+ LR (a a )
Ag+h 3 4 − 2 9 − 2 10 Ag+h 5 − 2 7
9
1
+ SP (C C ) , (40)
Ae+f 5 − 2 7
io
where C = 4/3 and the color number N = 3.
F
The parameters a are defined as follows.
i
a = C +C /N, (i = 1,3,5,7,9); (41)
i i i+1
a = C +C /N, (i = 2,4,5,6,10). (42)
i i i−1
The building blocks , , , denote the contributions of the factorizable
a+b c+d e+f g+h
A A A A
emission diagrams Fig.2(a,b), the nonfactorizable emission diagrams Fig.2(c,d), the nonfac-
torizable annihilation diagrams Fig.2(e,f), the factorizable annihilation diagrams Fig.2(g,h),
respectively. They are defined as
k = k + k, (43)
Ai+j Ai Aj
where the subscripts i and j correspond to the indices of Fig.2; the superscript k refers to
one of the three possible Dirac structures, namely k = LL for (V A) (V A), k = LR
− ⊗ −
for (V A) (V + A), and k = SP for 2(S P) (S + P). The explicit expressions of
− ⊗ − − ⊗
these building blocks are collected in the Appendix A.
III. NUMERICAL RESULTS AND DISCUSSION
In the rest frame of the Υ(nS) meson, the CP-averaged branching ratios for the Υ(nS)
B D weak decays are written as
c s,d
→
1 p
r(Υ(nS) B D ) = (Υ(nS) B D ) 2. (44)
B → c s,d 12π m2Γ |A → c s,d |
Υ Υ
The input parameters are listed in Table I and II. If not specified explicitly, we will
take their central values as the default inputs. The numerical results on the CP-averaged
branching ratios for the Υ(nS) B D weak decays are listed in Table III, where the
c s,d
→
first uncertainties come from the CKM parameters; the second uncertainties are due to
the variation of mass m and m ; the third uncertainties arise from the typical scale µ =
b c
(1 0.1)t and the expressions of t for different topologies are given in Eqs.(A31-A34). The
i i
±
following are some comments.
10