The ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$, $B_{c}^{\ast}K$ decays with perturbative QCD approach

Besides the traditional strong and electromagnetic decay modes, ${\Upsilon}(nS)$ meson can also decay through the weak interactions within the standard model of elementary particle. With anticipation of copious ${\Upsilon}(nS)$ data samples at the running LHC and coming SuperKEKB experiments, the two-body nonleptonic bottom-changing ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$, $B_{c}^{\ast}K$ decays ($n$ $=$ 1, 2, 3) are investigated with perturbative QCD approach firstly. The absolute branching ratios for ${\Upsilon}(nS)$ ${\to}$ $B_{c}^{\ast}{\pi}$ and $B_{c}^{\ast}K$ decays are estimated to reach up to about $10^{-10}$ and $10^{-11}$, respectively, which might possibly be measured by the future experiments.


I. INTRODUCTION
The upsilon Υ(nS) meson is the spin-triplet S-wave state of bottomonium (bound state consisting of bottom quark b and anti-bottom quarkb) with well-established quantum number of I G J P C = 0 − 1 −− [1]. The characteristic narrow decay widths of Υ(nS) mesons for n = 1, 2 and 3 provide insight into the study of strong interactions. [see Table. I, and note that for simplicity, Υ(nS) will denote Υ(1S), Υ(2S) and Υ(3S) mesons in the following content if not specified definitely.] The mass of Υ(nS) meson is below the B meson pair threshold.
The Υ(nS) meson decays into bottomed hadrons through strong and electromagnetic interactions are forbidden by the law of conservation of flavor number. The bottom-changing Υ(nS) decays can occur only via the weak interactions within the standard model, although with tiny incidence probability. Both constituent quarks of upsilons can decay individually, which provide an alternative system for investigating the weak decay of heavy-flavored hadrons. In this paper, we will study the nonleptonic Υ(nS) → B * c P (P = π and K) weak decays with perturbative QCD (pQCD) approach [2][3][4].
TABLE I: Summary of mass, decay width, on(off)-peak luminosity and numbers of Υ(nS).
properties [1] luminosity (f b −1 ) [5] numbers ( Experimentally, (1) over 10 8 Υ(nS) data samples have been accumulated at Belle and BaBar experiments [5]. More and more upsilon data samples will be collected at the running hadron collider LHC and the forthcoming e + e − collider SuperKEKB a . There seems to exist a realistic possibility to explore Υ(nS) weak decay at future experiments. (2) Signals of the Υ(nS) → B * c π, B * c K decays should be easily distinguished with "charge tag" technique, due to the facts that the back-to-back final states with different electric charges have definite momentum and energy in the rest frame of Υ(nS) meson. (3) The B * c meson has not been observed experimentally by now. The B * c meson production via the strong interaction are a The SuperKEKB has started commissioning test run (http://www.kek.jp/en/NewsRoom/Release).
suppressed due to the simultaneous presence of two heavy quarks with different flavors and higher order in QCD coupling constant α s . The Υ(nS) → B * c π, B * c K decays provide a novel pattern to study the B * c meson production. The identification of a single explicitly flavored B * c meson could be used as an effective selection criterion to detect upsilon weak decays. Moreover, the radiative decay of B * c meson provide a useful extra signal and a powerful constraint b . Of course, any discernible evidences of an anomalous production rate of single bottomed meson from upsilon decays might be a hint of new physics.
Theoretically, many attractive QCD-inspired methods have been developed recently to describe the exclusive nonleptonic decay of heavy-flavored mesons, such as the pQCD approach [2][3][4], the QCD factorization approach [7][8][9], soft and collinear effective theory [10][11][12][13], and have been applied widely to vindicate measurements on B meson decays. The upsilon weak decay permits one to further constrain parameters obtained from B meson decay, and cross comparisons provide an opportunity to test various phenomenological models. The upsilon weak decay possess a unique structure due to the Cabibbo-Kobayashi-Maskawa (CKM) matrix properties which predicts the channels with one B ( * ) c meson are dominant. The Υ(nS) → B * c P decay belongs to the favorable b → c transition, which should, in principle, have relatively large branching ratio among upsilon weak decays. However, there is still no theoretical study devoted to the Υ(nS) → B * c P decay for the moment. In this paper, we will present a phenomenological investigation on Υ(nS) → B * c P weak decay with the pQCD approach to supply a ready reference for the future experiments. This paper is organized as follows. Section II focus on theoretical framework and decay amplitudes for Υ(nS) → B * c π, B * c K weak decays. Section III is devoted to numerical results and discussion. The last section is a summary.

A. The effective Hamiltonian
Theoretically, the Υ(nS) → B * c π, B * c K weak decays are described by an effective bottomchanging Hamiltonian based on operator product expansion [14]: where G F ≃ 1.166×10 −5 GeV −2 [1] is the Fermi coupling constant; the CKM factors V cb V * ud and V cb V * us correspond to Υ(nS) → B * c π and B * c K decays, respectively; with the Wolfenstein parameterization, the CKM factors are expanded as a power series in a small Wolfenstein parameter λ ∼ 0.2 [1]: The local tree operators Q 1,2 are defined as: where α and β are color indices and the sum over repeated indices is understood.
The scale µ factorizes physics contributions into short-and long-distance dynamics. The Wilson coefficients C i (µ) summarize the physics contributions at scale higher than µ, and are calculable with the renormalization group improved perturbation theory. The hadronic matrix elements (HME), where the local operators are inserted between initial and final hadron states, embrace the physics contributions below scale of µ. To obtain decay amplitudes, the remaining work is to calculate HME properly by separating from perturbative and nonperturbative contributions.

B. Hadronic matrix elements
Based on Lepage-Brodsky approach for exclusive processes [15], HME is commonly expressed as a convolution integral of hard scattering subamplitudes containing perturbative contributions with universal wave functions reflecting nonperturbative contributions. In order to effectively regulate endpoint singularities and provide a naturally dynamical cutoff on nonperturbative contributions, transverse momentum of valence quarks is retained and the Sudakov factor is introduced within the pQCD framework [2][3][4]. Phenomenologically, the pQCD's decay amplitude could be divided into three parts: the Wilson coefficients C i incorporating the hard contributions above typical scale of t, process-dependent rescattering subamplitudes T accounting for the heavy quark decay, and wave functions Φ of all participating hadrons, which is expressed as where k is the momentum of valence quarks, and e −S is the Sudakov factor.

C. Kinematic variables
The light cone kinematic variables in the Υ(nS) rest frame are defined as follows.
where x i and k i⊥ are the longitudinal momentum fraction and transverse momentum of valence quarks, respectively; ǫ i and ǫ ⊥ i are the longitudinal and transverse polarization vectors, respectively, and satisfy relations ǫ 2 i = −1 and ǫ i ·p i = 0; the subscript i on variables p i , E i , m i , ǫ i corresponds to participating hadrons, namely, i = 1 for Υ(nS) meson, i = 2 for the recoiled B * c meson, i = 3 for the emitted pseudoscalar meson; n + and n − are positive and negative null vectors, respectively; 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. 2(a).

D. Wave functions
With the notation in [16,17], wave functions are defined as where it might assume that the motion of heavy valence quarks in Υ(nS) and B * c mesons is nearly nonrelativistic. The wave functions of Υ(nS) and B * c mesons could be approximately described with nonrelativistic quantum chromodynamics (NRQCD) [18][19][20] and time-independent Schrödinger equation.
For an isotropic harmonic oscillator potential, the eigenfunctions of stationary state with quantum numbers nL are written as [21] where parameter β determines the average transverse momentum, i.e., nS|k 2 ⊥ |nS ∼ β 2 . Employing the substitution ansatz [22], where x i and m q i are the longitudinal momentum fraction and mass of valence quark, respectively, then integrating out k ⊥ and combining with their asymptotic forms, the distribution amplitudes (DAs) for Υ(nS) and B * c mesons can be written as [21], According to NRQCD power counting rules [18], The shape lines of normalized DAs for Υ(nS) and B * c mesons are showed in Fig. 1. It is clearly seen that (1) DAs for Υ(nS) and B * c mesons fall quickly down to zero at endpoint x,x Our study shows that only the leading twist (twist-2) DAs of the emitted light pseudoscalar meson P is involved in decay amplitudes (see Appendix A). The twist-2 DAs has the expansion [16]: and are normalized as where C and each term corresponds to a nonperturbative Gegenbauer moment a i ; note that a 0 = 1 due to the normalization condition Eq.(40); the G-parity invariance of the pion DAs requires Gegenbauer moment a i = 0 for i = 1, 3, 5 · · ·.

E. Decay amplitudes
The Feynman diagrams for Υ(nS) → B * c π weak decay are shown in Fig. 2. There are two types. One is factorizable emission topology where gluon attaches to quarks in the same meson, and the other is nonfactorizable emission topology where gluon connects to quarks between different mesons.
With the pQCD master formula Eq.(6), the amplitude for Υ(nS) → B * c P decay can be expressed as [23], which is conventionally written as the helicity amplitudes [23], where C F = 4/3 and the color number N c = 3; the subscript i on A j i corresponds to three different helicity amplitudes, i.e., i = L, N, T ; the superscript j on A j i denotes to indices of Fig. 2. The explicit expressions of building blocks A j i are collected in Appendix A.

III. NUMERICAL RESULTS AND DISCUSSION
In the center-of-mass of Υ(nS) meson, branching ratio Br for Υ(nS) → B * c P decay are defined as The input parameters are listed in Table I and II. If not specified explicitly, we will take their central values as the default inputs. Our numerical results are collected in Table. III, where the first uncertainty comes from scale (1±0.1)t i and the expression of t i is given in Eq.(A20) and Eq.(A21); the second uncertainty is from mass m b and m c ; the third uncertainty is from hadronic parameters including decay constants and Gegenbauer moments; the fourth uncertainty is from CKM parameters. The followings are some comments.   (1) Branching ratio for Υ(nS) → B * c π decay is about O(10 −10 ) with pQCD approach, which is well within the measurement potential of LHC and SuperKEKB. For example, experimental studies have showed that production cross sections for Υ(nS) meson in pp and p-Pb collisions are a few µb at the LHCb [27,28] and ALICE [29,30] detectors.
Consequently, there will be more than 10 12 Υ(nS) data samples per ab −1 data collected by c The decay constant f B * c cannot be extracted from the experimental data because of no measurement on B * c weak decay at the present time. Theoretically, the value of f B * c has been estimated, for example, in Ref. [25] with the QCD sum rules. From Table. 3 of Ref. [25], one can see that the value of f B * c are model-dependent. In our calculation, we will take the latest value given by the lattice QCD approach [26] just to offer an order of magnitude estimation on branching ratio for Υ(nS) → B * c P decays.
the LHCb and ALICE, corresponding to a few hundreds of Υ(nS) → B * c π events. Branching ratio for Υ(nS) → B * c K decay, O(10 −11 ), is generally less than that for Υ(nS) → B * c π decay by one order of magnitude due to the CKM suppression, |V * us /V * ud | 2 ∼ λ 2 . (2) As it is well known, due to the large mass of B * c , the momentum transition in the Υ(nS) → B * c P decay may be not large enough. One might naturally wonder whether the pQCD approach is applicable and whether the perturbative calculation is reliable. Therefore, it is necessary to check what percentage of the contributions comes from the perturbative region. The contributions to branching ratio for Υ(nS) → B * c π decay from different α s /π region are showed in Fig. 3. It can be clearly seen that more than 93% (97%) contributions come from the α s /π ≤ 0.2 (0.3) region, implying that the Υ(nS) → B * c π decay is computable with the pQCD approach. As the discussion in [2][3][4], there are many factors for this, for example, the choice of the typical scale, retaining the quark transverse moment and introducing the Sudakov factor to suppress the nonperturbative contributions, which deserve much attention and further investigation.  Table. III are beyond such expectation.
(4) Besides the uncertainties listed in Table III, other factors, such as the models of wave functions, contributions of higher order corrections to HME, relativistic effects, and so on, deserve the dedicated study. Our results just provide an order of magnitude estimation.

IV. SUMMARY
The Υ(nS) decay via the weak interaction, as a complementary to strong and electromagnetic decay mechanism, is allowable within the standard model. Based on the potential prospects of Υ(nS) physics at high-luminosity collider experiment, Υ(nS) decay into B * c π and B * c K final states is investigated with the pQCD approach firstly. It is found that (1) the dominant contributions come from perturbative regions α s /π ≤ 0.3, which might imply that the pQCD calculation is practicable and workable; (2) there is a promiseful possibility of searching for Υ(nS) → B * c π (B * c K) decay with branching ratio about 10 −10 (10 −11 ) at the future experiments.

Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant Nos. 11547014, 11475055, U1332103 and 11275057). The building blocks A j i , where the superscript j corresponds to indices of Fig. 2 and the subscript i relates with different helicity amplitudes, are expressed as follows.