Study on the ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays

Motivated by the prospects of the potential ${\Upsilon}(1S)$ particle at high-luminosity heavy-flavor experiments, we studied the ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays, where $M$ $=$ ${\pi}$, ${\rho}$, $K^{(\ast)}$. The nonfactorizable contributions to hadronic matrix elements are taken into consideration with the QCDF approach. It is found that the CKM-favored ${\Upsilon}(1S)$ ${\to}$ $B_{c}{\rho}$ decay has branching ratio of ${\cal O}(10^{-10})$, which might be measured promisingly by the future experiments.


I. INTRODUCTION
The first evidence for upsilons, the bound states of bb, was observed in collisions of protons with a stationary nuclear target at Fermilab in 1977 [1,2]. From that moment on, bottomonia have been a subject of intensive experimental and theoretical research. Some of the salient features of upsilons are as follows [3]: (1) In the upsilons rest frame, the relative motion of the b quark is sufficiently slow. Nonrelativistic Schrödinger equation can be used to describe the spectrum of bottomonia states and thus one can learn about the interquark binding forces. (2) The Υ(nS) particles, with the radial quantum number n = 1, 2 and 3, decay primarily via the annihilation of the bb quark pairs into three gluons.
Thus the properties of the invisible gluons and of the gluon-quark coupling can be gleaned through the study of the upsilons decay. (3) Compared with the u, d, s light quarks, the relatively large mass of the heavy b quark implies a nonnegligible coupling to the Higgs bosons, making upsilons to be one of the best hunting grounds for light Higgs particles. By now, our knowledge of the properties of bottomonia comes primarily from e + e − annihilation.
The Υ(1S) particle is the ground state of the vector bottomonia with quantum number of I G J P C = 0 − 1 −− [4]. The mass of the Υ(1S) particle, m Υ(1S) = 9460.30±0.26 [4], is about three times heavier than the mass of the J/ψ particle (the ground state of charmonia with the same quantum number of I G J P C ). On one hand, compared with the J/ψ decay, much richer decay channels could be accessed by the Υ(1S) particle. On the other hand, the coupling constant α s for the Υ(1S) decay is smaller than that for the J/ψ decay due to the QCD nature of asymptotic freedom, which results in hadronic partial width Γ(Υ(1S)→ggg) < Γ(J/ψ→ggg), although the possible phase space in the Υ(1S) decay is larger than that in the J/ψ decay. In addition, the squared value of the b quark charge, Q 2 b = 1/9, is less than that of the c quark charge, Q 2 c = 4/9, which results in electromagnetic partial width Γ(Υ(1S)→γ * ) < Γ(J/ψ→γ * ). So the full decay width of the Υ(1S) particle, Γ Υ(1S) = 54.02±1.25 keV, is less than that of the J/ψ particle, Γ J/ψ = 92.9±2.8 keV [4]. Furthermore, one of the outstanding properties of all upsilons below BB threshold is their narrow decay width of tens of keV [4].
The Υ(1S) and J/ψ particles share the similar decay mechanism. As is the case for the J/ψ particle, strong decays of the Υ(1S) particle are suppressed by the phenomenological OZI (Okubo-Zweig-Iizuka) rules [5][6][7], so electromagnetic interactions and radiative transi-tions become competitive. It is expected that, at the lowest order approximation, the decay modes of the Υ(1S) particle could be subdivided into four types: (1) The lion's share of the decay width is the hadronic decay via the annihilation of the bb quark pairs into three gluons, i.e., some (81.7±0.7)% via Υ(1S) → ggg [4]. (2) The partial width of the electromagnetic decay via the annihilation of the bb quark pairs into a virtual photon could be written approximately as (3 + R)Γ ℓℓ , where the value of R is the ratio of inclusive production of hadrons to the µ + µ − pair production rate at the energy scale of m Υ(1S) , and Γ ℓℓ is the partial width of decay into dileptons. (3) Branching ratio of the radiative decay is about Br(Υ(1S)→γgg) ≃ (2.2±0.6)% [4]. The up-to-date research for light Higgs bosons in the Υ(1S) radiative decay has been performed by CLEO [8], Belle [9] and BaBar [10] Collaborations. (4) The magnetic dipole transition decay, Υ(1S) → γη b (1S), is very challenging to experimental physicist due to the very soft photon and pollution from other processes, such as Υ(1S) → π 0 X → γγX [11]. The experimental signal for Υ(1S) → γη b (1S) has not been discovered until now. Besides, the Υ(1S) particle could also decay via the weak interactions, although the branching ratio for a single b orb quark decay is tiny, about 2/τ B Γ Υ(1S) ∼ 10 −8 [4]. In this paper, we will estimate the branching ratios for the flavor-changing nonleptonic Υ(1S) → B c M weak decays with the QCD factorization (QCDF) approach [12,13], where M = π, ρ, K and K * . The motivation is listed as follows.
TABLE I: Summary of data samples (in the unit of 10 6 ) of the Υ(nS) particles below BB threshold collected by Belle, BaBar and CLEO Collaborations. The data in the 5th column correspond to total number of the Υ(1S) particle, including events from hadronic dipion transitions between upsilons Υ(2S, 3S) → ππΥ(1S), where branching ratios for the dipion decays Υ(2S) → π + π − Υ(1S), π 0 π 0 Υ(1S) and Υ(3S) → π + π − Υ(1S), π 0 π 0 Υ(1S) are ( From the experimental point of view, (1) there is plenty of upsilons at the high-luminosity dedicated bottomonia factories, for example, over 10 8 Υ(1S) at Belle (see Table I). Upsilons are also observed by the on-duty ALICE [17], ATLAS [18], CMS [19], LHCb [20] experiments at LHC. It is hopefully expected that more upsilons could be accumulated with great precision at the running upgraded LHC and forthcoming SuperKEKB. The huge Υ(1S) data samples will provide good opportunities to search for the Υ(1S) weak decays which in some cases might be detectable. Theoretical studies on the Υ(1S) weak decays are just necessary to offer a ready reference. approach [12,13], where nonfactorizable contributions could be estimated systematically with the perturbation theory based on collinear factorization approximation and power countering rules in the heavy quark limit [13], and the QCDF approach has been widely applied to nonleptonic B meson decays. So it should be very interesting to study the Υ(1S) → B c M weak decays by considering nonfactorizable contributions with the attractive QCDF approach.
This paper is organized as follows. In section II, we will present the theoretical framework and the amplitudes for the nonleptonic two-body Υ(1S) → B c M weak decays with the QCDF approach. Section III is devoted to numerical results and discussion. The last section is our summary.

A. The effective Hamiltonian
The low energy effective Hamiltonian responsible for the Υ(1S) → B c M decays is [28] H where the Fermi coupling constant G F = 1.166×10 −5 GeV −2 [4]; Using the Wolfenstein parameterization, the Cabibbo-Kobayashi-Maskawa (CKM) factors can be expanded as a power series in the small parameter λ = 0.22537(61) [4], The Wilson coefficients C 1,2 (µ) summarize the physical contributions above scales of µ.
The  Table II. The local tree four-quark operators are defined as follows.
where α and β are color indices and the sum over repeated indices is understood. To obtain the decay amplitudes, the remaining and the most intricate works are to calculate accurately hadronic matrix elements of local operators.

B. Hadronic matrix elements
Phenomenologically, the simplest treatment with hadronic matrix elements of four-quark operators is approximated by the product of two current matrix elements with color transparency ansatz [29] and naive factorization (NF) scheme [30,31], and current matrix elements are further parameterized by decay constants and transition form factors. For example, previous studies on the Υ(1S) → B c M decay [22,23] were based on NF approach.
As is well known, NF's defect is the disappearance of the renormalization scale depen- with the QCDF approach [12,13].
For the Υ(1S) → B c M decay, the spectator quark is the heavy bottom (anti)quark. According to the QCDF's power counting rules [13], contributions from the spectator scattering are power suppressed. With the QCDF master formula, hadronic matrix elements could be written as : where transition form factor F Υ→Bc The leading twist two-valence-particle distribution amplitudes of pseudoscalar and longitudinally polarized vector meson are defined in terms of Gegenbauer polynomials [32,33]: wherex = 1 − x; a M n is the Gegenbauer moment and a M 0 ≡ 1. After calculation, the decay amplitudes could be written as The coefficient a 1 in Eq.(8), including nonfactorizable contributions from QCD radiative vertex corrections, is written as [34]: For the transversely polarized vector meson, the factor V is zero beyond leading twist (twist-2) contributions. For the pseudoscalar and longitudinally polarized vector meson, with the modified minimal subtraction (MS) scheme, the factor V is written as [34]: where and the relations The numerical values of coefficient a 1 at scales of µ ∼ O(m b ) are listed in Table II.

C. Decay constants and form factors
The matrix elements of current operators are defined as follows: where f P and f V are the decay constants of pseudoscalar and vector mesons, respectively; m V and ǫ V denote the mass and polarization of vector meson, respectively.
The transition form factors are defined as follows [22][23][24]: where q = p 1 − p 2 ; and A 0 (0) = A 3 (0) is required compulsorily to cancel singularities at the pole q 2 = 0. There is a relation among these form factors It is clearly seen that there are only three independent form factors, A 0,1 (0) and V (0), at the pole q 2 = 0 for the Υ(1S) → B c M decays. The form factors at the pole q 2 = 0 could be written as the overlap integrals of wave functions of mesons [24], i.e., A Υ→Bc where σ y,z is a Pauli matrix acting on the spin indices of the decaying bottom quark; x and k ⊥ denote the fraction of the longitudinal momentum and the transverse momentum carried by the nonspectator quark, respectively.
Using the separation of momentum and spin variables, the wave functions of mesons can be written as with the normalization condition, where s 1,2 denote the spin of valence quark in meson; s = s 1 + s 2 ; s = 1 and 0 for the Υ (1S) and B c particles, respectively.
For the ground states of heavy quarkonia Υ(1S) and B c particles, we will take the solution of the Schödinger equation with nonrelativistic three-dimensional scalar harmonic oscillator potential, where the parameter β determines the average transverse quark momentum, i.e., k 2 ⊥ = β 2 . According to the power counting rules of NRQCD [25], the characteristic magnitude of the moment is order of Mv and v ∼ α s . So we will take β = Mα s in our calculation. Employing the substitution ansatz [35], where x 1 + x 2 = 1; m 1,2 is the mass of valence quark. Setting x 1 = x, we can obtain where N is a normalization factor.
Using the aforementioned convention, we get A Υ→Bc A Υ→Bc 2 (0) = 0.73±0.08, where the uncertainties come from variation of valence quark mass m b,c . In addition, according to the NRQCD argument, the relativistic corrections and higher-twist effects might give uncertainties of O(v 2 ), about 10%∼30%. Values at q 2 = 0 could, in principle, be extrapolated by assuming the form factors dominated by a proper pole which is unknown, or calculated with other method, such as the approach using the Bethe-Salpeter wave functions with the help of the nonrelativistic instantaneous approximation and the potential model based on the Mandelstam formalism [36] and so on. Here, we will follow the common practice for nonleptonic B decays with the QCDF approach. Values of form factors at q 2 = 0 are taken to offer an order of magnitude estimation, because both Υ and B c are heavy quarkonium and the recoil effects might be not so significant.

D. Decay amplitudes
With the aforementioned definition of hadronic matrix elements, the decay amplitudes of Υ(1S) → B c M decays can be written as For the Υ(1S) → B c V decays, the hadronic matrix elements in Eq.(8) can also be expressed as [37] The definition of helicity amplitudes is where invariant amplitudes a, b, c and variable y are The scalar amplitudes a, b, c describe the s, d, p wave contributions, respectively. Clearly, compared with the s wave amplitude, the p and d wave amplitudes are suppressed by a factor m V /m Υ .

III. NUMERICAL RESULTS AND DISCUSSION
In the rest frame of Υ(1S) particle, branching ratio for nonleptonic Υ(1S) → B c M weak decays can be written as where the decay width Γ Υ = 54.02±1.25 keV [4]; the momentum of final states is The input parameters, including the CKM Wolfenstein parameters, masses of b and c quarks, decay constants, and Gegenbauer moments of distribution amplitudes in Eq. (7), are collected in Table III (1) The QCDF's results fall in between those of Ref. [22] and Ref. [23], because the form factors A Υ→Bc 0,1 in our calculation fall in between those of Ref. [22] and Ref. [23].
(2) There is a clear hierarchical relationship, Br(Υ(1S)→B c ρ) > Br(Υ(1S)→B c π) > Br(Υ(1S)→B c K * ) > Br(Υ(1S)→B c K). These are two dynamical reasons. One is that the CKM factor V cb V * us responsible for the Υ(1S) → B c K ( * ) decay is suppressed by a factor of λ relative to the CKM factor V cb V * ud responsible for the Υ(1S) → B c π, B c ρ decays. The other is that the orbital angular momentum L BcP > L BcV .
(3) The CKM-favored a 1 dominated Υ(1S) → B c ρ decay has the largest branching ratio, ∼ 10 −10 , which should be sought for with high priority and firstly observed at the running LHC and forthcoming SuperKEKB. For example, the Υ(1S) production cross section in p-Pb collision can reach up to a few µb with the ALICE detector at LHC [38]. Therefore, per 100 f b −1 data collected at ALICE, over 10 11 Υ(1S) particles are in principle available, corresponding to tens of Υ(1S) → B c ρ events if with about 10% reconstruction efficiency.
(4) There are many uncertainties on the QCDF's results. The first uncertainty, about 7∼8%, from the CKM factors could be lessened with the improvement on the precision of the Wolfenstein parameter A. The second uncertainty from the renormalization scale should, in principle, be reduced by inclusion of higher order α s corrections to hadronic matrix elements.
The third uncertainty is due to the fact that masses of b and c quark affect the shape lines of wave functions, and hence the magnitude of form factors and branching ratios. The fourth uncertainty from hadronic parameters is expected to be reduced with the relative ratio of branching ratios.
(5) Other factors, such as the contributions of higher order corrections to hadronic matrix elements, relativistic effects, q 2 dependence of form factors and so on, which are not considered in this paper, deserve the dedicated study. Our results just provide an order of magnitude estimation.

IV. SUMMARY
With the sharp increase of the Υ(1S) data sample at high-luminosity dedicated heavyflavor factories, the bottom-changing Υ(1S) → B c M weak decays are interesting in exploring the underlying mechanism responsible for transition between heavy quarknoia, investigating perturbative and nonperturbative effects and overconstraining parameters from B decays.
The CKM favored Υ(1S) → B c ρ decay has relatively large branching ratio, ∼10 −10 , and might be detectable in future experiments.