Study of ${\Upsilon}(nS)$ ${\to}$ $B_{c}P$ decays with perturbative QCD approach

0.2 MB·

by  Yueling Yang

Study of Υ(nS) B P decays with perturbative QCD approach
c
→
Yueling Yang,1 Junfeng Sun,1 Yan Guo,1 Qingxia Li,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,
7 Nanyang Normal University, Nanyang 473061, China
1
0
Abstract
2

n
With the potential prospects of the Υ(nS) at high-luminosity dedicated heavy-flavor factories,
a
J
the color-favored Υ(nS) B π, B K weak decays are studied with the pQCD approach. It is
7 → c c
1
found that branching ratios for the Υ(nS) B π decay are as large as the order of (10−11),
c
→ O
]
h which might be measured promisingly by the future experiments.
p
-
p
e
h
[

1
v
3
9
5
4
0
.
1
0
7
1
:
v
i
X
r
a
1
I. INTRODUCTION
Since the discovery of upsilons in proton-nucleus collisions at Fermilab in 1977 [1, 2], re-
markable achievements have been made in the understanding of the nature of bottomonium
(bound state of b¯b). The upsilon Υ(nS) is the spin-triplet S-wave state n3S of bottomo-
1
nium with the well established quantum number of IGJPC = 0−1−− [3]. The spectroscopy,
production and decay mechanisms of the bottomonium resemble those of charmonium. The
upsilonΥ(nS)belowtheBB¯ thresholdwiththeradialquantum number n=1, 2and3(note
that for simplicity, the notation Υ(nS) will denote all Υ(1S), Υ(2S) and Υ(3S) mesons in
the following content if not specified explicitly), in a close analogy with J/ψ, decay primarily
¯
through the annihilation of the bb pairs into three gluons, followed by the evolution of gluons
into hadrons, glueballs, multiquark and other exotic states. The strong Υ(nS) decay offers
an ideal plaza to glean the properties of the invisible gluons and of the quark-gluon coupling
[4]. The Υ(nS) strong decays are suppressed by the Okubo-Zweig-Iizuka rules [5–7], which
enable electromagnetic and radiative transitions to become competitive1. Besides, the up-
silon weak decay is also legitimate within the standard model, although the branching ratio
is tiny, about 2/τ Γ (10−8) [3]. In this paper, we will estimate the branching ratios
B Υ
∼ O
for the bottom-changing nonleptonic Υ(nS) B P weak decays with perturbative QCD
c
→
(pQCD) approach [9–11], where P denotes pseudoscalar π and K mesons. The motivation
is listed as follows.
From the experimental point of view, (1) over 108 Υ(nS) samples have been accumulated
at Belle and Babar collaborations due to their outstanding performance [12] (see Table.I). It
is hopefully expected that more than 1011 b¯b quark pairs would be available per fb−1 data at
LHCb [13]. Much more upsilons could be collected with great precision at the forthcoming
SuperKEKB and the running upgraded LHC, which provide a golden opportunity to search
for the Υ(nS) weak decays that in some cases might be detectable. Theoretical studies of
the Υ(nS) weak decays are very necessary to offer a ready reference. (2) For the two-body
Υ(nS) B π, B K decays, the back-to-back final states with opposite charges have definite
c c
→
energies and momenta in the center-of-mass frame of upsilons. Additionally, identification
of a single flavored B meson is free from inefficiently double tagging of flavored hadron
c
1 Because of G-parity conservation, there also exist dipion transitions Υ(3S) ππΥ(2S), ππΥ(1S) and
→
Υ(2S) ππΥ(1S), and hadronic transitions Υ(3S,2S) ηΥ(1S) [3, 8].
→ →
2
pairs produced via conventional decays occurring above the BB¯ threshold [14], and can also
provide a conclusive evidence of the upsilon weak decay. Of course, small branching ratios
make the observation of the upsilon weak decays extremely challenging, and the observation
of an abnormally large production rate of single B mesons in the Υ(nS) decay might be a
c
hint of new physics [14].
TABLE I: Summary of the mass, decay width and data samples of upsilon Υ(1S,2S,3S).
properties [3] data samples (106) [12]
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
± ± ± ±
From the theoretical point of view, the bottom-changing upsilon weak decays permit
one to reexamine parameters obtained from B meson decay, test various phenomenologi-
cal models and improve our understanding on the strong interactions and the mechanism
responsible for heavy meson weak decay. The Υ(nS) B P decays are monopolized by
c
→
tree contributions and favored by the Cabibbo-Kobayashi-Maskawa (CKM) matrix element
V , so they should have relatively large branching ratios among nonleptonic upsilon weak
cb
decays. The Υ(1S) B π, B K decays have been studied with the naive factorization (NF)
c c
→
approximation in previous works [14–16]. One obvious deficiency of NF approach is the dis-
appearance of strong phases and the renormalization scale from hadronic matrix elements
(HME). Recently, several attractive methods have been developed to reevaluate HME, such
as pQCD [9–11], the QCD factorization (QCDF) [17–19] and soft and collinear effective
theory [20–23]. These methods have been widely used and could explain reasonably many
measurements on nonleptonic B decays. But, few works devote to the nonleptonic upsilon
u,d
weak decays with these new phenomenological approaches. In this paper, we will study the
Υ(nS) B π, B K weak decays with the pQCD approach.
c c
→
This paper is organized as follows. In section II, we present the theoretical framework
and the amplitudes for the Υ(nS) B π, B K decays with pQCD approach. Section III is
c c
→
devoted to numerical results and discussion. The last section is our summary.
3
II. THEORETICAL FRAMEWORK
A. The effective Hamiltonian
The effective Hamiltonian for the Υ(nS) B π, B K decays is written as [24]
c c
→
G
= F V V∗ C (µ)Q (µ)+C (µ)Q (µ) +h.c., (1)
Heff √2 X cb uqn 1 1 2 2 o
q=d,s
where G is the Fermi coupling constant; the CKM factors are expanded as a power series
F
in the small Wolfenstein parameter λ 0.2 [3],
∼
1 1
V V∗ = Aλ2 Aλ4 Aλ6 + (λ8), (2)
cb ud − 2 − 8 O
V V∗ = Aλ3 + (λ8). (3)
cb us O
The Wilson coefficients C (µ) summarize the physical contributions above scales of µ, and
1,2
have been properly calculated to the NLO order with the renormalization group improved
perturbation theory. The local operators are defined as follows.
Q = [c¯ γ (1 γ )b ][q¯ γµ(1 γ )u ], (4)
1 α µ 5 α β 5 β
− −
Q = [c¯ γ (1 γ )b ][q¯ γµ(1 γ )u ], (5)
2 α µ 5 β β 5 α
− −
where α and β are color indices and the sum over repeated indices is understood.
B. Hadronic matrix elements
To obtain the decay amplitudes, one has to calculate the hadronic matrix elements of
local operators. Analogous to the common applications of hard exclusive processes in per-
turbative QCD proposed by Lepage and Brodsky [25], HME could be expressed as the
convolution of hard scattering subamplitudes containing perturbative contributions with
the universal wave functions reflecting the nonperturbative contributions. However, some-
times, the high-order corrections to HME produce collinear and/or soft logarithms based
on collinear factorization approximation, for example, the spectator scattering amplitudes
within the QCDF framework [19]. The pQCD approach advocates that [9–11] this problem
could be settled down by retaining the transverse momentum of quarks and introducing
the Sudakov factor. The decay amplitudes could be factorized into three parts: the soft
4
effects uniting with the universal wave functions Φ, the process-dependent subamplitude H,
the hard effects incorporated into the Wilson coefficients C . For a particular topology, the
i
decay amplitudes could be written as
dx db C (t)H (t ,x ,b )Φ (x ,b )e−S, (6)
Ai ∝ YZ j j i i i j j j j j
j
where t is a typical scale, x is the longitudinal momentum fraction of the valence quark,
i j
b is the conjugate variable of the transverse momentum, e−S is the Sudakov factor, and j
j
denotes participating particles.
C. Kinematic variables
In the Υ(nS) rest frame, the light cone kinematic variables are defined as
m
1
p = p = (1,1,0), (7)
Υ 1
√2
p = p = (p+,p−,0), (8)
Bc 2 2 2
p = p = (p−,p+,0), (9)
π(K) 3 3 3
~
k = x p +(0,0,k ), (10)
i i i i⊥
1
k
ǫ = (1, 1,0), (11)
Υ √2 −
n = (1,0,0), n = (0,1,0), (12)
+ −
E p
p± = i± , (13)
i √2
[m2 (m +m )2][m2 (m m )2]
p = q 1 − 2 3 1 − 2 − 3 , (14)
2m
1
s = 2p p = m2 m2 m2, (15)
2· 3 1 − 2 − 3
t = 2p p = m2 +m2 m2 = 2m E , (16)
1· 2 1 2 − 3 1 2
u = 2p p = m2 m2 +m2 = 2m E , (17)
1· 3 1 − 2 3 1 3
t+u s = m2 +m2 +m2, (18)
− 1 2 3
~
where x and k are the longitudinal momentum fraction and transverse momentum of the
i i⊥
k
light valence quark, respectively; ǫ is the longitudinal polarization vector of the Υ(nS)
Υ
particle; n and n are positive and negative null vectors, respectively; p is the common
+ −
momentum of final states; m , m and m denote the masses of the Υ(nS), B and π(K)
1 2 3 c
mesons, respectively. The notation of momentum is displayed in Fig.2(a).
5
D. Wave functions
With the notation in [26–28], the definitions of matrix elements of diquark operators
sandwiched between vacuumandthelongitudinallypolarizedΥ(nS), thedouble-heavy pseu-
doscalar B , the light pseudoscalar P are
c
1
0 b (z)¯b (0) Υ(p ,ǫ ) = f dx e−ix1p1·z ǫ m φv(x ) p φt (x ) , (19)
h | i j | 1 k i 4 ΥZ 1 n6 kh 1 Υ 1 −6 1 Υ 1 ioji
i
B+(p ) c¯(z)b (0) 0 = f dx eix2p2·z γ p φa (x )+m φp (x ) , (20)
h c 2 | i j | i 4 BcZ 2 n 5h6 2 Bc 2 2 Bc 2 ioji
P(p ) u (z)q¯(0) 0
3 i j
h | | i
i
= f dx eix3p3·z γ p φa(x )+µ φp(x ) µ (n n 1)φt (x ) , (21)
4 PZ 3 n 5h6 3 P 3 P P 3 − P 6 −6 +− P 3 ioji
where f , f , f are decay constants, µ = m2/(m +m ) and q = d(s) for π(K) meson.
Υ Bc P P 3 u q
The twist-2 distribution amplitudes of light pseudoscalar π, K mesons are defined as [28]:
∞
φa(x) = 6xx¯ 1+ aP C3/2(x x¯) , (22)
P n X n n − o
n=1
where x¯ = 1 x; aP and C3/2(z) are Gegenbauer moment and polynomials, respectively;
− n n
aπ = 0 for i = 1, 3, 5, due to the explicit G-parity of pion.
i ···
BothΥ(nS) andB systems arenearlynonrelativistic, duetom 2m andm m +
c Υ ≃ b Bc ≃ b
m . Nonrelativistic quantum chromodynamics (NRQCD) [29–31] and Schro¨dinger equation
c
can be used to describe their spectrum. The radial wave functions with isotropic harmonic
oscillator potential are written as
φ (~k) e−~k2/2β2, (23)
1S
∼
φ (~k) e−~k2/2β2(2~k2 3β2), (24)
2S
∼ −
φ (~k) e−~k2/2β2(4~k4 20~k2β2 +15β4), (25)
3S
∼ −
where the parameter β determines the average transverse momentum, i.e., 1S ~k2 1S =
h | ⊥| i
β2. According to the NRQCD power counting rules [29], the characteristic magnitude of
the momentum of heavy quark is order of Mv, where M is the mass of the heavy quark
with typical velocity v α (M). So, value of β = Mα (M) is taken in our calculation.
s s
∼
Employing the substitution ansatz [32],
1 ~k2 +m2
~k2 i⊥ qi, (26)
→ 4 Xi xi
6
~
where x , k , m are the longitudinal momentum fraction, transverse momentum, mass of
i i⊥ qi
~
the light valence quark, respectively, with the relations x = 1 and k = 0. Integrating
i i⊥
P P
~
out k and combining with their asymptotic forms, one can obtain
i⊥
x¯m2 +xm2
φa (x) = Axx¯exp c b , (27)
Bc n− 8β2xx¯ o
2
x¯m2 +xm2
φp (x) = Bexp c b , (28)
Bc n− 8β2xx¯ o
2
m2
φv (x) = Cxx¯exp b , (29)
Υ(1S) n− 8β2xx¯o
1
m2
φt (x) = D(x x¯)2exp b , (30)
Υ(1S) − n− 8β2xx¯o
1
m2
φt,v (x) = Eφt,v (x) 1+ b , (31)
Υ(2S) Υ(1S) n 2β2xx¯o
1
m2 2
φt,v (x) = F φt,v (x) 1 b +6 , (32)
Υ(3S) Υ(1S) n(cid:16) − 2β2xx¯(cid:17) o
1
where β = ξ α (ξ ) with ξ = m /2; parameters A, B, C, D, E, F are the normalization
i i s i i i
coefficients satisfying the conditions
1 1
dxφa,p(x) = 1, dxφv,t(x) = 1. (33)
Z Bc Z Υ
0 0
The shape lines of the normalized distribution amplitudes of φa,p(x) and φv,t (x) are
Bc Υ(nS)
displayed in Fig.1. Here we would like to point out that the relativistic corrections of (v2)
O
are left out. According to the arguments in Ref. [29], the (v2) corrections could bring
O
about 10 30% errors, and it is expected that such error could be reduced systematically by
∼
including new interactions in principle, which is beyond the scope of this paper.
4 ΦUvH3SLHxL 4 ΦUt H3SLHxL
3 ΦUvH2SLHxL 3 ΦUt H2SLHxL
2 ΦUvH1SLHxL 2 ΦUt H1SLHxL
1 ΦaBcHxL 1 ΦBpcHxL
0 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)
FIG. 1: The distribution amplitudes of φa,p(x) and φv,t (x).
Bc Υ(nS)
7
E. Decay amplitudes
The Feynman diagrams for the Υ(nS) B π decay within the pQCD framework are
c
→
shown in Fig.2, where (a) and (b) are factorizable topology; (c) and (d) are nonfactorizable
topology.
p
3
π− π− π− π−
d(k¯3) u¯(k3) d u¯ d u¯ d u¯
b(k ) c(k )
1 2
b c b c b c
p p
1 Υ G Bc+ 2 Υ G Bc+ Υ G Bc+ Υ G Bc+
¯b ¯b ¯b ¯b ¯b ¯b ¯b ¯b
(a) (b) (c) (d)
FIG. 2: Feynman diagrams for the Υ(nS) B π decay with the pQCD approach.
c
→
The decay amplitudes of Υ(nS) B P decay can be written as
c
→
πC
(Υ(nS) B P) = √2G FV V∗ m3 pf f f , (34)
A → c F N cb uq Υ Υ Bc P X AFig.2(i)
i=a,b,c,d
where C = 4/3 and the color number N = 3.
F
The explicit expressions of are
AFig.2(i)
1 ∞ 1 ∞
= dx b db dx b db α (t )a (t )E (t )
AFig.2(a) Z 1Z 1 1Z 2Z 2 2 s a 1 a a a
0 0 0 0
H (x ,x ,b ,b )φv(x ) φa (x )(x +r2x¯ )+φp (x )r r , (35)
× a 1 2 1 2 Υ 1 n Bc 2 2 3 2 Bc 2 2 bo
1 ∞ 1 ∞
= dx b db dx b db α (t )a (t )E (t )
AFig.2(b) Z 1Z 1 1Z 2Z 2 2 s b 1 b b b
0 0 0 0
H (x ,x ,b ,b ) 2r φp (x ) r φv(x )+x φt (x )
× b 1 2 2 1 n 2 Bc 2 h c Υ 1 1 Υ 1 i
φa (x ) φv(x )(r2x +r2x¯ )+φt (x )r , (36)
− Bc 2 h Υ 1 2 1 3 1 Υ 1 cio
1 ∞ 1 ∞ 1 ∞
= dx db dx b db dx b db δ(b b )α (t )
AFig.2(c) Z 1Z 1Z 2Z 2 2Z 3Z 3 3 1 − 2 s c
0 0 0 0 0 0
C (t )
2 c E (t )H (x ,x ,x ,b ,b )φa(x ) φp (x )φt (x )r (x x )
× N c c c 1 2 3 2 3 P 3 n Bc 2 Υ 1 2 2 − 1
+ φa (x )φv(x ) (1+r2 r2)(x x )+2r2(x x ) , (37)
Bc 2 Υ 1 h 2 − 3 1 − 3 2 3 − 2 io
1 ∞ 1 ∞ 1 ∞
= dx db dx b db dx b db δ(b b )α (t )
AFig.2(d) Z 1Z 1Z 2Z 2 2Z 3Z 3 3 1 − 2 s d
0 0 0 0 0 0
C (t )
2 d E (t )H (x ,x ,x ,b ,b )φa(x ) φp (x )φt (x )r (x x )
× N d d d 1 2 3 2 3 P 3 n Bc 2 Υ 1 2 2 − 1
+ φa (x )φv(x )(1 r2 r2)(x¯ x ) , (38)
Bc 2 Υ 1 − 2 − 3 2 − 3 o
8
where α is the QCD coupling; a = C + C /N; C is the Wilson coefficients; r = m /m .
s 1 1 2 1,2 i i 1
Itcanbeeasilyseenthat(1)thenonfactorizablecontributions arecolor-suppressed
AFig.2(c,d)
with respect to the factorizable contributions ; (2) The twist-3 distribution ampli-
AFig.2(a,b)
tudes φp,t have no contribution to decay amplitudes.
P
The typical scales t and the Sudakov factor E are defined as
i i
t = max( α , β ,1/b ,1/b ), (39)
a(b) g a(b) 1 2
q− q−
t = max( α , β ,1/b ,1/b ,1/b ), (40)
c(d) g c(d) 1 2 3
q− q| |
E (t) = exp S (t) , (41)
a(b) {− Bc }
E (t) = exp S (t) S (t) , (42)
c(d) {− Bc − P }
α = x¯2m2 +x¯2m2 x¯ x¯ t, (43)
g 1 1 2 2 − 1 2
β = m2 m2 +x¯2m2 x¯ t, (44)
a 1 − b 2 2 − 2
β = m2 m2 +x¯2m2 x¯ t, (45)
b 2 − c 1 1 − 1
β = x2m2 +x2m2 +x2m2 x x t x x u+x x s, (46)
c 1 1 2 2 3 3 − 1 2 − 1 3 2 3
β = x¯2m2 +x¯2m2 +x2m2 x¯ x¯ t x¯ x u+x¯ x s, (47)
d 1 1 2 2 3 3 − 1 2 − 1 3 2 3
t dµ
S (t) = s(x ,p+,1/b )+2 γ , (48)
Bc 2 2 2 Z µ q
1/b2
t dµ
S (t) = s(x ,p+,1/b )+s(x¯ ,p+,1/b )+2 γ , (49)
P 3 3 3 3 3 3 Z µ q
1/b3
whereα andβ arethevirtualityoftheinternalgluonandquark, respectively; γ = α /π is
g i q s
−
the quark anomalous dimension; the expression of s(x,Q,1/b) can be found in the appendix
of Ref.[9].
The scattering functions H in the subamplitudes are defined as
i AFig.2(i)
H (x ,x ,b ,b ) = K ( α b ) θ(b b )K ( βb )I ( βb )+(b b ) , (50)
a(b) 1 2 i j 0 g i i j 0 i 0 j i j
q− n − q− q− ↔ o
π
H (x ,x ,x ,b ,b ) = θ( β)K ( βb )+ θ(β) iJ ( βb ) Y ( βb )
c(d) 1 2 3 2 3 0 3 0 3 0 3
n − q− 2 h q − q io
θ(b b )K ( α b )I ( α b )+(b b ) , (51)
2 3 0 g 2 0 g 3 2 3
× n − q− q− ↔ o
where J and Y (I and K ) are the (modified) Bessel function of the first and second kind,
0 0 0 0
respectively.
9
III. NUMERICAL RESULTS AND DISCUSSION
In the rest frame of the Υ(nS) particle, branching ratio for the Υ(nS) B P weak
c
→
decays can be written as
1 p
r(Υ(nS) B P) = (Υ(nS) B P) 2. (52)
B → c 12π m2Γ |A → c |
Υ Υ
TABLE II: Numerical values of the input parameters
Wolfenstein parameters [3]: A = 0.814+0.023, λ = 0.22537 0.00061;
−0.024 ±
Masses of quarks [3] : m = 1.67 0.07 GeV, m = 4.78 0.06 GeV;
c b
± ±
Gegenbauer moments: aπ (1 GeV) = 0.17 0.08 [33], aπ (1 GeV) = 0.06 0.10 [33],
2 ± 4 ±
aK (1 GeV) = 0.06 0.03 [28], aK (1 GeV) = 0.25 0.15 [28];
1 ± 2 ±
decay constant: f = 130.41 0.20 MeV [3], f = 156.2 0.7 MeV [3],
π K
± ±
f = 489 5 MeV [34], f = (676.4 10.7) MeV,
Bc ± Υ(1S) ±
f = (473.0 23.7) MeV, f = (409.5 29.4) MeV.
Υ(2S) Υ(3S)
± ±
The input parameters are collected in Table. II. As for the decay constant f , one can
Υ
use the definition of decay constant,
0¯bγµb Υ = f m ǫµ. (53)
h | | i Υ Υ Υ
and relate f to the experimentally measurable leptonic branching ratio,
Υ
4π f2 m2 m2
Γ(Υ ℓ+ℓ−) = α2 Υ v1 2 ℓ 1+2 ℓ , (54)
→ 27 QEDmΥuut − m2Υn m2Υo
where α is the fine-structure constant, m is the lepton mass and ℓ = e, µ, τ.
QED ℓ
The values of f determined from measurements are listed in Table.III. One may notice
Υ
that there are some clear hierarchical relations among these decay constant. (1) The decay
constants fe+e− < fµ+µ− < fτ+τ− for the same radial quantum number n. There are two
Υ(nS) Υ(nS) Υ(nS)
reasons. One is that the final phase space for decay into ℓ+ℓ− states is more compact than
i i
that for ℓ+ℓ− decay when the lepton family number i > j. The other is that branching ratio
j j
of ℓ+ℓ− decay is relatively less than that of ℓ+ℓ− decay with the lepton family number i < j.
i i j j
(2) There are also two reasons for the relation among the weighted average f > f
Υ(1S) Υ(2S)
> f . One is that the possible phase space increases with the radial quantum number of
Υ(3S)
10