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

The ${\Upsilon}(nS)$ ${\to}$ $B_{c}D_{s}$, $B_{c}D_{d}$ weak decays are studied with the pQCD approach firstly. It is found that branching ratios ${\cal B}r({\Upsilon}(nS){\to}B_{c}D_{s})$ ${\sim}$ ${\cal O}(10^{-10})$ and ${\cal B}r({\Upsilon}(nS){\to}B_{c}D_{d})$ ${\sim}$ ${\cal O}(10^{-11})$, which might be measurable in the future experiments.


I. INTRODUCTION
Since the discovery of bottomonium (the bound states of the bottom quark b and the corresponding antiquarkb, 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 the S-wave spin-triplet state, n 3 S 1 , of bottomonium with the well established quantum number of I G J P C = 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 those of bottomonium above the BB threshold. (note that for simplicity, the notation Υ(nS) will denote the Υ(1S), Υ(2S) and Υ(3S) mesons in the following content if not specified explicitly) As it is well known, the Υ(nS) meson decays primarily through the annihilation of the bb pairs into three gluons, which are suppressed by the phenomenological Okubo-Zweig-Iizuka rule [5][6][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 Υ(3S) − m Υ(2S) is just slightly larger than 2m π , and m Υ(2S) − m Υ(1S) is just slightly larger than m η . The coupling strengths of the electromagnetic 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 B Γ Υ ∼ O(10 −8 ) [4], where τ B and Γ Υ are the lifetime of the B u,d,s meson and the total width of the Υ(nS) meson, respectively. In this paper, we will study the Υ(nS) → B c D s , B c D d weak decays with the perturbative QCD (pQCD) approach [8][9][10]. The motivation is listed as follows. properties [4] data samples ( From the experimental point of view, (1) over 10 8 Υ(nS) data samples have been accumulated 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 forthcoming 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)  (HME) where the local quark-level operators are sandwiched between the initial and final hadron states, such as the pQCD approach [8][9][10], the QCD factorization [13][14][15] and the soft and collinear effective theory [16][17][18][19], which could give an appropriate explanation for many measurements on the nonleptonic B u,d decays. In this paper, we will estimate the branching ratios for the Υ(nS) → B c D s,d weak decays with the pQCD approach to offer a 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 c D s,d decays. We present the numerical results and discussion in section III, and conclude with a summary in the last section.

A. The effective Hamiltonian
Using the operator product expansion and renormalization group equation, the effective Hamiltonian responsible for the Υ(nS) → B c D s,d weak decays is written as [20] where G F = 1.166×10 −5 GeV −2 [4] is the Fermi coupling constant; the CKM factors are expressed as a power series in the Wolfenstein parameter λ ∼ 0.2 [4], for the Υ(nS) → B c D s decays, and for the Υ(nS) → B c D d decays. The Wilson coefficients C i (µ) summarize the physical contributions 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.
where Q q 1,2 , Q q 3,···,6 , and Q q 7,···,10 are usually called as the tree operators, QCD penguin operators, and electroweak penguin operators, respectively; α and β are color indices; q ′ denotes all the active quarks at the scale of µ ∼ O(m b ), i.e., q ′ = u, d, s, c, b; and Q q ′ is the electric 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 T factorization theorem [21] with the collinear factorization 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 nonperturbative 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 singularities and provide a naturally dynamical cutoff on the nonperturbative contributions [8][9][10].
Generally, the decay amplitude can be separated into three parts: the Wilson coefficients C i incorporating the hard contributions above the typical scale of t, the process-dependent scattering amplitudes T accounting for the heavy quark decay, and the universal wave functions Φ including the soft and long-distance contributions, i.e., where x is the longitudinal momentum fraction of valence quarks, b is the conjugate variable of the transverse momentum k T , and e −S is the Sudakov factor.

C. Kinematic variables
In the center-of-mass frame of the Υ(nS) mesons, the light cone kinematic variables are defined as follows.
where x i is the longitudinal momentum fraction; k iT is the transverse momentum; p is the common momentum of final states; ǫ Υ is the longitudinal polarization vector of the Υ(nS) meson; m 1 = m Υ(nS) , m 2 = m Bc and m 3 = m D s,d denote the masses of the Υ(nS), B c and D s,d mesons, respectively. The notation of momentum is displayed in Fig.2(a).

D. Wave functions
With the notation of Refs. [23,24], the HME of diquark operators squeezed between the vacuum and the Υ(nS), B c , D q mesons are defined as follows.
where f Υ , f Bc , f Dq are decay constants. Table   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 c , D q mesons could be approximately described with the nonrelativistic quantum chromodynamics [25][26][27] and Schrödinger equation. The wave functions of a nonrelativistic three-dimensional isotropic harmonic oscillator potential are given in Ref. [28], H are the normalization coefficients satisfying the following conditions The shape lines of the distribution amplitudes φ v,t Υ(nS) (x) and φ a,p Bc (x) have been displayed in Ref. [28], which are basically consistent with the physical picture that the valence quarks share momentums according to their masses.
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 s,d mesons should be different from that in the heavy-heavy B c and Υ(nS) mesons. In addition, there are several phenomenological models for the D s,d meson wave functions, for 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 applications [29].  By calculating these diagrams with the pQCD master formula Eq.(16), the decay amplitudes of Υ(nS) → B c D q decays (where q = d, s) can be expressed as: +A SP a+b (a 6 + a 8 ) + A LL c+d (C 3 + C 9 ) + A SP c+d (C 5 + C 7 ) where C F = 4/3 and the color number N = 3.
The parameters a i are defined as follows.
a i = C i + C i+1 /N, (i = 1, 3, 5, 7, 9); (41) The building blocks A a+b , A c+d , A e+f , A g+h denote the contributions of the factorizable emission diagrams Fig.2(a,b), the nonfactorizable emission diagrams Fig.2(c,d), the nonfactorizable annihilation diagrams Fig.2(e,f), the factorizable annihilation diagrams Fig.2(g,h), respectively. They are defined as where the subscripts i and j correspond to the indices of

III. NUMERICAL RESULTS AND DISCUSSION
In the rest frame of the Υ(nS) meson, the CP -averaged branching ratios for the Υ(nS) → B c D s,d weak decays are written as The input parameters are listed in Table I The relation between parameters (ρ, η) and (ρ,η) is [4]:    Table III show that branching ratios for the Υ(nS) → B c D q ) weak decays seem to be close to each other, and have almost nothing with the radial quantum number n. The reason may be that the decay amplitudes are proportional to decay constant f Υ(nS) , and hence there is an approximation, for the same radial quantum number n. (5) It is seen that branching ratios for the Υ(nS) → B c D s (B c D d ) decay can reach up to 10 −10 (10 −11 ), which might be accessible at the running LHC and forthcoming SuperKEKB.
For example, the Υ(nS) production cross section in p-Pb collision is a few µb with the LHCb [33] and ALICE [34] detectors at LHC. Over 10 12 Υ(nS) mesons per ab −1 data collected at LHCb and ALICE are in principle available, corresponding to a few hundreds (tens) of the (6) Besides the uncertainties listed in Table III, the decay constants can bring where the mass ratio r i = m i /m 1 ;x i = 1 − x i ; variable x i is the longitudinal momentum fraction of the valence quark; b i is the conjugate variable of the transverse momentum k i⊥ ; and α s (t) is the QCD coupling at the scale of t.