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

Inspired by the potential prospects of ${\Upsilon}(nS)$ data samples ($n$ $=$ $1$, $2$, $3$) at LHC and SuperKEKB, ${\Upsilon}(nS)$ ${\to}$ $B_{c}{\rho}$, $B_{c}K^{\ast}$ decays are studied phenomenologically with pQCD approach. Branching ratios for ${\Upsilon}(nS)$ ${\to}$ $B_{c}{\rho}$ and $B_{c}K^{\ast}$ decays are estimated to reach up to ${\cal O}(10^{-11})$ and ${\cal O}(10^{-12})$, respectively. Given the identification and detection efficiency of final states, searching for these weak decay modes should be fairly challenging experimentally in the future.

They all lie below the open bottom threshold, and carry the same quantum numbers of I G J P C = 0 − 1 −− [1]. For each of them, the mass is ten times as large as proton, but the full decay width is very narrow, only a few keV. Based on the above-mentioned facts, here we will use a notation Υ(nS) to represent special Υ(1S), Υ(2S), and Υ(3S) mesons for simplicity if it is not specified explicitly. Thanks to the unremitting endeavor and splendid performance from experimental groups of CLEO, CDF, D0, BaBar, Belle, LHCb, ATLAS, and so on, great achievements have been made in understanding of bottomonium properties [2]. The Υ(nS) decays through the strong interaction, electromagnetic interaction and radiative transition, have been extensively studied. The rapid accumulation of Υ(nS) data samples with high precision will enable a realistic possibility to search for Υ(1S) weak decay at the LHC and SuperKEKB. In this paper, we will study the Υ(nS) → B c V weak decays (V = ρ, K * ) with perturbative QCD (pQCD) approach [3][4][5] to offer a ready reference for the future experimental research.
Both b andb quarks in Υ(nS) meson can decay individually via the weak interaction. It is well known that a clear hierarchy of the quark-mixing Cabibbo-Kabayashi-Maskawa (CKM) matrix elements opts favorably for the b → c transition, so Υ(nS) weak decay into final states containing abc or bc bound state should have a relatively large branching fraction.

A. The effective Hamiltonian
Phenomenologically, assisted with the operator product expansion and renormalization group (RG) technique, the effective weak Hamiltonian accounting for Υ(nS) → B c V decay has the following structure [14], where is the Fermi constant. Using the Wolfenstein parameterization, the CKM factors are written approximately in term of A and λ, i.e., for Υ(nS) → B c ρ decay, and for Υ(nS) → B c K * decay. The local operators are expressed as where α and β are color indices, and q denotes d and s.
In Eq.(1), the auxiliary scale µ factorizes physical contributions into two parts. The physical contributions above µ are integrated into the Wilson coefficients C 1,2 , which has been reliably calculated to the next-to-leading order with the RG-improved perturbation theory [14]. The physical contributions below µ are embodied in hadronic matrix elements (HME), where the local operators are sandwiched between initial and final hadron states. The incorporation of long distance contributions make HME very challenging and complicated to evaluate. HME is not yet fully understood so far. However, to obtain decay amplitudes, one has to treat HME with certain comprehensible approximation or assumptions, which result in a number of uncertainties.

B. Hadronic matrix elements
Based on factorization ansatz [15][16][17] and hard-scattering approach [18][19][20][21][22], HME has a simple structure, and is commonly expressed as a convolution of hard scattering kernel func-tion T with distribution amplitudes (DAs). Only DAs are nonperturbative inputs, which, on the other hand, are process independent, i.e., DAs determined by nonperturbative methods or extracted from experimental data can be employed to make predictions. With the collinear approximation, hard scattering kernels for annihilation contributions and spectator interactions can not provide sufficient endpoint suppression [23][24][25]. In order to admit a perturbative treatment for HME, the intrinsic transverse momentum of valence quarks is kept explicitly and a Sudakov factor for each DAs is introduced with pQCD approach [3][4][5]. Finally, a pQCD amplitude is written as a convolution integral of three parts: Wilson coefficients C i , hard scattering kernel T and wave functions Φ, where t is a typical scale, k is the momentum of valence quarks and e −S is a Sudakov factor.

C. Kinematic variables
In the center-of-mass frame of Υ(nS), kinematic variables are defined as follows.
where x i and k i⊥ are the longitudinal momentum fraction and transverse momentum of valence quark, respectively; ǫ i and ǫ ⊥ i are the longitudinal and transverse polarization vectors, respectively, satisfying relationship ǫ 2 i = −1 and ǫ i ·p i = 0; n + is a positive null vector; the subscript i = 1, 2, 3 on variables (p i , E i , m i and ǫ i ) corresponds to Υ(nS), B c and V mesons, respectively; s, t and u are Lorentz-invariant variables. The notation of momentum is displayed in Fig.2(a).

D. Wave functions
With the notation in [26,27], meson wave functions are defined as where f Υ and f Bc are decay constants; Φ v,T V and Φ a Bc are twist-2; Φ t,s,V,A V and Φ p Bc are twist-3. The expressions of DAs for double heavy Υ(nS) and B c mesons are [7] φ according to nonrelativistic quantum chromodynamics (NRQCD) power counting rules [28][29][30]; parameters A, B, C, D, E, F , G are normalization coefficients satisfying the conditions The shape lines of 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 are basically consistent with a picture that valence quarks share momentum fractions according to their masses; (2) DAs fall quickly down to zero at endpoint x,x → 0 due to suppression from exponential functions, which are bound to offer a natural and effective cutoff for soft contributions. . Their asymptotic forms are [26,27]: After a straightforward calculation, amplitude for Υ(nS) → B c V decay can be decomposed as below, which is conventionally written as helicity amplitudes,

III. NUMERICAL RESULTS AND DISCUSSION
In the rest frame of Υ(nS), decaying into B c and light vector V mesons, branching ratio is defined as where p is the center-of-mass momentum of final states.    Table I. If it is not specified explicitly, their central values will be used as default inputs. Our numerical results are presented in Table   II, where the uncertainties come from scale (1±0.1)t i , m b and m c , and CKM parameters, respectively. The following are some comments. and Γ Υ(3S) < Γ Υ(2S) < Γ Υ(1S) . However, the numbers in Table II are beyond expectation.

The values of input parameters are listed in
Why is it that? In addition to form factors, one of the possible factors is (2) Branching ratio for Υ(nS) → B c ρ decay can reach up to O(10 −11 ). The Υ(nS) production cross section in p-Pb collision is about a few µb at LHCb [32] and ALICE [33].  (3) From Fig.2, the spectator is a heavy bottom quark in the Υ(nS) → B c transition. It is assumed that the bottom quark is near on-shell and the gluon attaching to the spectator might be soft. It is natural to question the validity of perturbative calculation with pQCD approach. So, it is necessary to check how many shares come from the perturbative region.
The contributions to branching ratio Br(Υ(nS)→B c ρ) from different region of α s /π are displayed in Fig.3. It is clearly seen that more than 85% (some 95%) contributions to branching ratio come from α s /π ≤ 0.2 (0.3) regions, which implies that the calculation with pQCD approach is feasible. Compared with contributions from α s /π ∈ [0.1, 0.2] region, one of crucial reasons for a small percentage in the region α s /π ≤ 0.1 is that the absolute values of Wilson coefficients C 1,2 , parameter a 1 and coupling α s decrease along with the increase of renormalization scale.
(4) Besides uncertainties listed in Table II, decay constants f Υ and f Bc can bring some 8%, 12%, 16% uncertainties for Υ(1S), Υ(2S), Υ(3S) decays, respectively. These are two ways to reduce theoretical uncertainty. One is to construct some relative ratios of branching ratios, The other is to consider higher order corrections to HME, relativistic effects on DAs, and so on. Here, our results just provide an order of magnitude estimation. The expressions of A i,j are written as follows.
where x i andx i = 1 − x i are longitudinal momentum fractions of valence quarks; b i is the conjugate variable of the transverse momentum k i⊥ ; a 1 = C 1 + C 2 /N c ; N c = 3 is the color number; C 1,2 are the Wilson coefficients.
The Sudakov factor E f,n and function H f,n are defined as follows, where the subscript f (n) corresponds to (non)factorizable topologies.