# Study on theΥ(1S)→BcM Weak Decays

Authors: Junfeng Sun; Lili Chen; Na Wang; Qin Chang; Jinshu Huang; Yueling Yang
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

Motivated by the prospects of the potentialΥ(1S) particle at high-luminosity heavy-flavor experiments, we studied the Υ(1S)→BcM weak decays, where M = π, ρ, K(∗). The nonfactorizable contributions to hadronic matrix elements are taken into consideration with the QCDF approach. It is found that the CKM-favored Υ(1S)→Bcρ decay has branching ratio of O(10-10), which might be measured promisingly by the future experiments.


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## Body

## 1. Introduction

The first evidence for upsilons, the bound states ofbb¯, 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, and s light quarks, the relatively large mass of the heavy b quark implies a nonnegligible coupling to the Higgs bosons, making upsilons 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.TheΥ(1S) particle is the ground state of the vector bottomonia with quantum number of IGJPC=0-1-- [4]. The mass of the Υ(1S) particle, mΥ1S=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 IGJPC). 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 αs 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, Qb2=1/9, is less than that of the c quark charge, Qc2=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, ΓΥ1S=54.02±1.25keV, is less than that of the J/ψ particle, ΓJ/ψ=92.9±2.8keV [4]. Furthermore, one of the outstanding properties of all upsilons below BB¯ threshold is their narrow decay width of tens of keV [4].TheΥ(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 [5–7], so electromagnetic interactions and radiative transitions 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, that is, 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)Γll, 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 mΥ(1S) and Γll 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)→γηb(1S), is very challenging to experimental physicist due to the very soft photon and pollution from other processes, such as Υ(1S)→π0X→γγX [11]. The experimental signal for Υ(1S)→γηb(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 or b¯ quark decay is tiny, about 2/τBΓΥ1S~10-8 [4]. In this paper, we will estimate the branching ratios for the flavor-changing nonleptonic Υ(1S)→BcM weak decays with the QCD factorization (QCDF) approach [12, 13], where M=π,ρ,K, and K∗. The motivation is listed as follows.From the experimental point of view,(1) there is plenty of upsilons at the high-luminosity dedicated bottomonia factories, for example, over 108Υ(1S) at Belle (see Table 1). Upsilons are also observed by the on-duty ALICE [17], ATLAS [18], CMS [19], and 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. (2) For the two-body Υ(1S)→BcM decay, final states with opposite charges have definite energies and momenta in the center-of-mass frame of the Υ(1S) particle. Particularly, identification of a single charged Bc meson in the final state, which is free from inefficiently double tagging of the bottomed hadron pairs occurring above the BB¯ threshold, would provide an unambiguous signature of the Υ(1S) weak decay. Of course, small branching ratios make the observation of the Υ(1S) weak decays extremely difficult, and evidences of an abnormally large production rate of single Bc mesons in the Υ(1S) decay might be a hint of new physics beyond the standard model.Table 1
Summary of data samples (in the unit of106) 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) particles, 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), π0π0Υ(1S) and Υ(3S)→π+π-Υ(1S), π0π0Υ(1S) are (17.85±0.26)%, (8.6±0.4)%, (4.37±0.08)%, and (2.20±0.13)%, respectively [4].
Υ
(
1
S
)
Υ
(
2
S
)
Υ
(
3
S
)
TotalΥ(1S)
Belle [14]
102
158
11
145
BaBar [15]
121.8
98.6
39
CLEO [16]
22.78
9.45
8.89
26From the theoretical point of view, the bottom-changing upsilon weak decay could permit overconstraining parameters obtained fromB meson decay, but few studies are devoted to the nonleptonic upsilon weak decay in the past. For the Υ(1S)→BcM decay, the amplitude is usually treated as the factorizable product of two independent factors: one describing the transition between heavy quarkonium Υ(1S) and Bc and the other depicting the production of the M state from the vacuum. Previous works, such as [21] based on the spin symmetry and nonrecoil approximation, [22] based on the heavy quark effective theory, and [23] based on the Bauer-Stech-Wirbel (BSW) model [24], concentrated mainly upon the Υ(1S)→Bc transition form factors which are related to the space integrals of the meson wave functions. As is well known, there exist hierarchical scales with nonrelativistic quantum chromodynamics (NRQCD) [25–27] which is an approach to deal with the heavy quarkonium; that is, M2≫(Mv)2≫(Mv2)2, where M is the mass of heavy quark with typical velocities v~αs≪1. The physical contributions at scales of Mv2~ΛQCD are absorbed into wave functions of the Υ(1S) and Bc particles; thus the Υ(1S)→Bc transition should be dominated by the nonperturbative dynamics. The nonfactorizable contributions above scales of Mv2 have not been taken seriously in previous works. About 2000, Beneke et al. proposed the QCDF 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)→BcM weak decays by considering nonfactorizable contributions with the attractive QCDF approach.This paper is organized as follows. In Section2, we will present the theoretical framework and the amplitudes for the nonleptonic two-body Υ(1S)→BcM weak decays with the QCDF approach. Section 3 is devoted to numerical results and discussion. The last section is our summary.


## 2. Theoretical Framework

### 2.1. The Effective Hamiltonian

The low energy effective Hamiltonian responsible for theΥ(1S)→BcM decays is [28](1)Heff=GF2∑q=d,sVcbVuq∗C1μQ1μ+C2μQ2μ+h.c.,where the Fermi coupling constant GF=1.166×10-5GeV-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],(2)VcbVud∗=Aλ2-12Aλ4-18Aλ6+Oλ8,VcbVus∗=Aλ3+Oλ8.The Wilson coefficients C1,2(μ) summarize the physical contributions above scales of μ. The Wilson coefficients Ci are calculable with the perturbation theory and have properly been evaluated to the next-to-leading order (NLO) with the renormalization group (RG) equation. The numerical values of the Wilson coefficients C1,2 at scales of μ~O(mb) in naive dimensional regularization scheme are listed in Table 2. The local tree four-quark operators are defined as follows:(3)Q1=c¯αγμ1-γ5bαq¯βγμ1-γ5uβ,Q2=c¯αγμ1-γ5bβq¯βγμ1-γ5uα,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.Table 2
The numerical values of the Wilson coefficientsC1,2 and a1 for the Υ(1S)→Bcπ decay at different scales, where mb=4.78 GeV [4].
μ
NLO
LO
QCDF
C
1
C
2
C
1
C
2
Re(a1)
Im(a1)
0
.
5
m
b
1.128
-
0.269
1.168
-
0.337
1.076
-
0.027
m
b
1.076
-
0.173
1.110
-
0.235
1.054
-
0.015
1.5
m
b
1.054
-
0.128
1.085
-
0.188
1.043
-
0.011
2.0
m
b
1.041
-
0.100
1.070
-
0.159
1.036
-
0.008


### 2.2. 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)→BcM decay [22, 23] were based on NF approach.As is well known, NF’s defect is the disappearance of the renormalization scale dependence, the strong phases, and the nonfactorizable contributions from hadronic matrix elements, resulting in nonphysical amplitudes and noCP violating asymmetries. To remedy NF’s deficiencies, Beneke et al. proposed that hadronic matrix elements could be written as the convolution integrals of hard scattering kernels and light cone distribution amplitudes with the QCDF approach [12, 13].For theΥ(1S)→BcM 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(4)BcMQiΥ1S=∑iFiΥ→Bc∫dxHixϕMx,where transition form factor FiΥ→Bc and light cone distribution amplitudes ϕM(x) of the emitted meson M are nonperturbative input parameters and hard scattering kernels Hi(x) are computable order by order with the perturbation theory in principle.The leading twist two-valence-particle distribution amplitudes of pseudoscalar and longitudinally polarized vector meson are defined in terms of Gegenbauer polynomials [32, 33]:(5)ϕMx=6xx¯∑n=0∞anMCn3/2x-x¯,where x¯=1-x, anM is the Gegenbauer moment, and a0M≡1.After calculation, the decay amplitudes could be written as(6)AΥ1S⟶BcM=BcMHeffΥ1S=GF2VcbVuq∗a1MJμ0BcJμΥ1S.The coefficienta1 in (6), including nonfactorizable contributions from QCD radiative vertex corrections, is written as [34](7)a1=C1NLO+1NcC2NLO+αs4πCFNcC2LOV.For the transversely polarized vector meson, the factorV 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](8)V=3log⁡mb2μ2+3log⁡mc2μ2-18+∫01dxTxϕMx,where(9)Tx=ca1-calog⁡ca-4cb1-cblog⁡cb+cd1-cdlog⁡cd-4cc1-cclog⁡cc-rcca1-ca2log⁡ca+11-ca-rc-1cd1-cd2log⁡cd+11-cd+fca-fcb-fcc+fcd+2log⁡rc2log⁡ca-log⁡cb,fc=2Li2c-1c-log2⁡c-2c1-clog⁡c,and the relations are as follows:(10)rc=mcmb,ca=x1-rc2,cb=x¯1-rc2,cc=-carc2,cd=-cbrc2.The numerical values of coefficienta1 at scales of μ~O(mb) are listed in Table 2.


### 2.3. Decay Constants and Form Factors

The matrix elements of current operators are defined as follows:(11)PpAμ0=-ifPpμ,Vp,ϵVμ0=fVmVϵV,μ∗,where fP and fV are the decay constants of pseudoscalar and vector mesons, respectively, and mV and ϵV denote the mass and polarization of vector meson, respectively.The transition form factors are defined as follows [22–24]:(12)Bcp2Vμ-AμΥp1,ϵ=ϵμναβϵΥνqαp1+p2βVΥ→Bcq2mΥ+mBc-i2mΥϵΥ·qq2qμA0Υ→Bcq2-iϵΥ,μmΥ+mBcA1Υ→Bcq2-iϵΥ·qmΥ+mBcp1+p2μA2Υ→Bcq2+i2mΥϵΥ·qq2qμA3Υ→Bcq2,where q=p1-p2 and A0(0)=A3(0) is required compulsorily to cancel singularities at the pole q2=0. There is a relation among these form factors:(13)2mΥA3q2=mΥ+mBcA1q2+mΥ-mBcA2q2.It is clearly seen that there are only three independent form factors,A0,1(0) and V(0), at the pole q2=0 for the Υ(1S)→BcM decays. The form factors at the pole q2=0 could be written as the overlap integrals of wave functions of mesons [24]; that is,(14)A0Υ→Bc0=∫dk→⊥∫01dx·ΦΥk⊥,x,1,0σzΦBck⊥,x,0,0,A1Υ→Bc0=mb+mcmΥ1S+mBcIΥ→Bc,VΥ→Bc0=mb-mcmΥ1S-mBcIΥ→Bc,IΥ→Bc=2∫dk→⊥∫01dxx·ΦΥk⊥,x,1,-1iσyΦBck⊥,x,0,0,where σy,z is a Pauli matrix acting on the spin indices of the decaying bottom quark and 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(15)Φk→⊥,x,j,jz=ϕk→⊥,xs,sz,s1,s2,with the normalization condition,(16)∑s1,s2∫dk→⊥∫01dxΦk→⊥,x,j,jz2=1,where s1,2 denote the spin of valence quark in meson; s→=s→1+s→2; s=1 and 0 for the Υ(1S) and Bc particles, respectively.For the ground states of heavy quarkoniaΥ(1S) and Bc particles, we will take the solution of the Schödinger equation with nonrelativistic three-dimensional scalar harmonic oscillator potential,(17)ϕk→~∫dr→ϕr→e-ik→·r→~∫dr→e-β2r2/2e-ik→·r→~e-k→2/2β2,where the parameter β determines the average transverse quark momentum; that is, 〈k→⊥2〉=β2. According to the power counting rules of NRQCD [25], the characteristic magnitude of the moment is order of Mv and v~αs. So we will take β=Mαs in our calculation. Employing the substitution ansatz [35],(18)k→2⟶14k→⊥2+m12x1+k→⊥2+m22x2+m12-m222k→⊥2+m12/x1+k→⊥2+m22/x2,where x1+x2=1 and m1,2 is the mass of valence quark. Setting x1=x, we can obtain(19)ϕk→⊥,x=Nexp⁡-18β2k→⊥2+x¯m12+xm22xx¯+m12-m222xx¯k→⊥2+x¯m12+xm22,where N is a normalization factor.Using the aforementioned convention, we get(20)A0Υ→Bc0=A3Υ→Bc0=0.81±0.01,A1Υ→Bc0=0.83±0.01,A2Υ→Bc0=0.73±0.08,VΥ→Bc0=1.98±0.06,where the uncertainties come from variation of valence quark mass mb,c. In addition, according to the NRQCD argument, the relativistic corrections and higher-twist effects might give uncertainties of O(v2), about 10%~30%. Values at q2≠0 could, in principle, be extrapolated by assuming the form factors dominated by a proper pole which is unknown or calculated with other methods, 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]. Here, we will follow the common practice for nonleptonic B decays with the QCDF approach. Values of form factors at q2=0 are taken to offer an order of magnitude estimation, because both Υ and Bc are heavy quarkonium and the recoil effects might be not so significant.


### 2.4. Decay Amplitudes

With the aforementioned definition of hadronic matrix elements, the decay amplitudes ofΥ(1S)→BcM decays can be written as(21)AΥ⟶Bc+π-=2GFVcbVud∗a1fπmΥϵΥ·pπ·A0Υ→Bc,AΥ⟶Bc+K-=2GFVcbVus∗a1fKmΥϵΥ·pK·A0Υ→Bc,AΥ⟶Bc+ρ-=-iGF2·VcbVud∗a1fρmρϵΥ·ϵρ∗mΥ+mBcA1Υ→Bc+ϵΥ·pρϵρ∗·pΥ2A2Υ→BcmΥ+mBc-iϵμναβϵΥμϵρ∗νpΥαpρβ2VΥ→BcmΥ+mBc,AΥ⟶Bc+K∗-=-iGF2·VcbVus∗a1fK∗mK∗ϵΥ·ϵK∗∗mΥ+mBcA1Υ→Bc+ϵΥ·pK∗ϵK∗∗·pΥ2A2Υ→BcmΥ+mBc-iϵμναβϵΥμϵK∗∗νpΥαpK∗β2VΥ→BcmΥ+mBc.For theΥ(1S)→BcV decays, the hadronic matrix elements in (6) can also be expressed as [37](22)Hλ=VJμ0BcJμΥ1S=ϵV∗μϵΥνagμν+bmΥmVpΥ+pBcμpVν+icmΥmVϵμναβpVαpΥ+pBcβ.The definition of helicity amplitudes is(23)H0=-ay-2by2-1,H±=a±2cy2-1,where invariant amplitudes a, b, and c and variable y are(24)a=-ifVmVmΥ+mBcA1Υ→Bcq2,b=-ifVmΥmV2A2Υ→Bcq2mΥ+mBc,c=+ifVmΥmV2VΥ→Bcq2mΥ+mBc,y=pΥ·pVmΥmV=mΥ2-mBc2+mV22mΥmV.The scalar amplitudes a, b, and c describe the s, d, and p wave contributions, respectively. Clearly, compared with the s wave amplitude, the p and d wave amplitudes are suppressed by a factor mV/mΥ.


## 2.1. The Effective Hamiltonian

The low energy effective Hamiltonian responsible for theΥ(1S)→BcM decays is [28](1)Heff=GF2∑q=d,sVcbVuq∗C1μQ1μ+C2μQ2μ+h.c.,where the Fermi coupling constant GF=1.166×10-5GeV-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],(2)VcbVud∗=Aλ2-12Aλ4-18Aλ6+Oλ8,VcbVus∗=Aλ3+Oλ8.The Wilson coefficients C1,2(μ) summarize the physical contributions above scales of μ. The Wilson coefficients Ci are calculable with the perturbation theory and have properly been evaluated to the next-to-leading order (NLO) with the renormalization group (RG) equation. The numerical values of the Wilson coefficients C1,2 at scales of μ~O(mb) in naive dimensional regularization scheme are listed in Table 2. The local tree four-quark operators are defined as follows:(3)Q1=c¯αγμ1-γ5bαq¯βγμ1-γ5uβ,Q2=c¯αγμ1-γ5bβq¯βγμ1-γ5uα,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.Table 2
The numerical values of the Wilson coefficientsC1,2 and a1 for the Υ(1S)→Bcπ decay at different scales, where mb=4.78 GeV [4].
μ
NLO
LO
QCDF
C
1
C
2
C
1
C
2
Re(a1)
Im(a1)
0
.
5
m
b
1.128
-
0.269
1.168
-
0.337
1.076
-
0.027
m
b
1.076
-
0.173
1.110
-
0.235
1.054
-
0.015
1.5
m
b
1.054
-
0.128
1.085
-
0.188
1.043
-
0.011
2.0
m
b
1.041
-
0.100
1.070
-
0.159
1.036
-
0.008


## 2.2. 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)→BcM decay [22, 23] were based on NF approach.As is well known, NF’s defect is the disappearance of the renormalization scale dependence, the strong phases, and the nonfactorizable contributions from hadronic matrix elements, resulting in nonphysical amplitudes and noCP violating asymmetries. To remedy NF’s deficiencies, Beneke et al. proposed that hadronic matrix elements could be written as the convolution integrals of hard scattering kernels and light cone distribution amplitudes with the QCDF approach [12, 13].For theΥ(1S)→BcM 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(4)BcMQiΥ1S=∑iFiΥ→Bc∫dxHixϕMx,where transition form factor FiΥ→Bc and light cone distribution amplitudes ϕM(x) of the emitted meson M are nonperturbative input parameters and hard scattering kernels Hi(x) are computable order by order with the perturbation theory in principle.The leading twist two-valence-particle distribution amplitudes of pseudoscalar and longitudinally polarized vector meson are defined in terms of Gegenbauer polynomials [32, 33]:(5)ϕMx=6xx¯∑n=0∞anMCn3/2x-x¯,where x¯=1-x, anM is the Gegenbauer moment, and a0M≡1.After calculation, the decay amplitudes could be written as(6)AΥ1S⟶BcM=BcMHeffΥ1S=GF2VcbVuq∗a1MJμ0BcJμΥ1S.The coefficienta1 in (6), including nonfactorizable contributions from QCD radiative vertex corrections, is written as [34](7)a1=C1NLO+1NcC2NLO+αs4πCFNcC2LOV.For the transversely polarized vector meson, the factorV 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](8)V=3log⁡mb2μ2+3log⁡mc2μ2-18+∫01dxTxϕMx,where(9)Tx=ca1-calog⁡ca-4cb1-cblog⁡cb+cd1-cdlog⁡cd-4cc1-cclog⁡cc-rcca1-ca2log⁡ca+11-ca-rc-1cd1-cd2log⁡cd+11-cd+fca-fcb-fcc+fcd+2log⁡rc2log⁡ca-log⁡cb,fc=2Li2c-1c-log2⁡c-2c1-clog⁡c,and the relations are as follows:(10)rc=mcmb,ca=x1-rc2,cb=x¯1-rc2,cc=-carc2,cd=-cbrc2.The numerical values of coefficienta1 at scales of μ~O(mb) are listed in Table 2.


## 2.3. Decay Constants and Form Factors

The matrix elements of current operators are defined as follows:(11)PpAμ0=-ifPpμ,Vp,ϵVμ0=fVmVϵV,μ∗,where fP and fV are the decay constants of pseudoscalar and vector mesons, respectively, and mV and ϵV denote the mass and polarization of vector meson, respectively.The transition form factors are defined as follows [22–24]:(12)Bcp2Vμ-AμΥp1,ϵ=ϵμναβϵΥνqαp1+p2βVΥ→Bcq2mΥ+mBc-i2mΥϵΥ·qq2qμA0Υ→Bcq2-iϵΥ,μmΥ+mBcA1Υ→Bcq2-iϵΥ·qmΥ+mBcp1+p2μA2Υ→Bcq2+i2mΥϵΥ·qq2qμA3Υ→Bcq2,where q=p1-p2 and A0(0)=A3(0) is required compulsorily to cancel singularities at the pole q2=0. There is a relation among these form factors:(13)2mΥA3q2=mΥ+mBcA1q2+mΥ-mBcA2q2.It is clearly seen that there are only three independent form factors,A0,1(0) and V(0), at the pole q2=0 for the Υ(1S)→BcM decays. The form factors at the pole q2=0 could be written as the overlap integrals of wave functions of mesons [24]; that is,(14)A0Υ→Bc0=∫dk→⊥∫01dx·ΦΥk⊥,x,1,0σzΦBck⊥,x,0,0,A1Υ→Bc0=mb+mcmΥ1S+mBcIΥ→Bc,VΥ→Bc0=mb-mcmΥ1S-mBcIΥ→Bc,IΥ→Bc=2∫dk→⊥∫01dxx·ΦΥk⊥,x,1,-1iσyΦBck⊥,x,0,0,where σy,z is a Pauli matrix acting on the spin indices of the decaying bottom quark and 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(15)Φk→⊥,x,j,jz=ϕk→⊥,xs,sz,s1,s2,with the normalization condition,(16)∑s1,s2∫dk→⊥∫01dxΦk→⊥,x,j,jz2=1,where s1,2 denote the spin of valence quark in meson; s→=s→1+s→2; s=1 and 0 for the Υ(1S) and Bc particles, respectively.For the ground states of heavy quarkoniaΥ(1S) and Bc particles, we will take the solution of the Schödinger equation with nonrelativistic three-dimensional scalar harmonic oscillator potential,(17)ϕk→~∫dr→ϕr→e-ik→·r→~∫dr→e-β2r2/2e-ik→·r→~e-k→2/2β2,where the parameter β determines the average transverse quark momentum; that is, 〈k→⊥2〉=β2. According to the power counting rules of NRQCD [25], the characteristic magnitude of the moment is order of Mv and v~αs. So we will take β=Mαs in our calculation. Employing the substitution ansatz [35],(18)k→2⟶14k→⊥2+m12x1+k→⊥2+m22x2+m12-m222k→⊥2+m12/x1+k→⊥2+m22/x2,where x1+x2=1 and m1,2 is the mass of valence quark. Setting x1=x, we can obtain(19)ϕk→⊥,x=Nexp⁡-18β2k→⊥2+x¯m12+xm22xx¯+m12-m222xx¯k→⊥2+x¯m12+xm22,where N is a normalization factor.Using the aforementioned convention, we get(20)A0Υ→Bc0=A3Υ→Bc0=0.81±0.01,A1Υ→Bc0=0.83±0.01,A2Υ→Bc0=0.73±0.08,VΥ→Bc0=1.98±0.06,where the uncertainties come from variation of valence quark mass mb,c. In addition, according to the NRQCD argument, the relativistic corrections and higher-twist effects might give uncertainties of O(v2), about 10%~30%. Values at q2≠0 could, in principle, be extrapolated by assuming the form factors dominated by a proper pole which is unknown or calculated with other methods, 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]. Here, we will follow the common practice for nonleptonic B decays with the QCDF approach. Values of form factors at q2=0 are taken to offer an order of magnitude estimation, because both Υ and Bc are heavy quarkonium and the recoil effects might be not so significant.


## 2.4. Decay Amplitudes

With the aforementioned definition of hadronic matrix elements, the decay amplitudes ofΥ(1S)→BcM decays can be written as(21)AΥ⟶Bc+π-=2GFVcbVud∗a1fπmΥϵΥ·pπ·A0Υ→Bc,AΥ⟶Bc+K-=2GFVcbVus∗a1fKmΥϵΥ·pK·A0Υ→Bc,AΥ⟶Bc+ρ-=-iGF2·VcbVud∗a1fρmρϵΥ·ϵρ∗mΥ+mBcA1Υ→Bc+ϵΥ·pρϵρ∗·pΥ2A2Υ→BcmΥ+mBc-iϵμναβϵΥμϵρ∗νpΥαpρβ2VΥ→BcmΥ+mBc,AΥ⟶Bc+K∗-=-iGF2·VcbVus∗a1fK∗mK∗ϵΥ·ϵK∗∗mΥ+mBcA1Υ→Bc+ϵΥ·pK∗ϵK∗∗·pΥ2A2Υ→BcmΥ+mBc-iϵμναβϵΥμϵK∗∗νpΥαpK∗β2VΥ→BcmΥ+mBc.For theΥ(1S)→BcV decays, the hadronic matrix elements in (6) can also be expressed as [37](22)Hλ=VJμ0BcJμΥ1S=ϵV∗μϵΥνagμν+bmΥmVpΥ+pBcμpVν+icmΥmVϵμναβpVαpΥ+pBcβ.The definition of helicity amplitudes is(23)H0=-ay-2by2-1,H±=a±2cy2-1,where invariant amplitudes a, b, and c and variable y are(24)a=-ifVmVmΥ+mBcA1Υ→Bcq2,b=-ifVmΥmV2A2Υ→Bcq2mΥ+mBc,c=+ifVmΥmV2VΥ→Bcq2mΥ+mBc,y=pΥ·pVmΥmV=mΥ2-mBc2+mV22mΥmV.The scalar amplitudes a, b, and c describe the s, d, and p wave contributions, respectively. Clearly, compared with the s wave amplitude, the p and d wave amplitudes are suppressed by a factor mV/mΥ.


## 3. Numerical Results and Discussion

In the rest frame ofΥ(1S) particle, branching ratio for nonleptonic Υ(1S)→BcM weak decays can be written as(25)BrΥ1S⟶BcM=112πpcmmΥ2ΓΥAΥ1S⟶BcM2,where the decay width ΓΥ=54.02±1.25keV [4]; the momentum of final states is(26)pcm=mΥ2-mBc+mM2mΥ2-mBc-mM22mΥ.The input parameters, including the CKM Wolfenstein parameters, masses ofb and c quarks, decay constants, and Gegenbauer moments of distribution amplitudes in (5), are collected in Table 3. If not specified explicitly, we will take their central values as the default inputs. Our numerical results on the CP-averaged branching ratios for the Υ(1S)→BcM decays are displayed in Table 4, where theoretical uncertainties of the QCDF results come from the CKM parameters, the renormalization scale μ=(1±0.5)mb, masses of b and c quarks, and hadronic parameters (decay constants and Gegenbauer moments), respectively. For the sake of comparison, previous results of [22, 23] are reevaluated with coefficient a1=1.054, where the scenario of the flavor dependent parameter ω in [23] is taken. The following are some comments.(1)
The QCDF’s results fall in between those of [22, 23], because the form factors A0,1Υ→Bc in our calculation fall in between those of [22, 23].(2)
There is a clear hierarchical relationship,Br(Υ(1S)→Bcρ)>Br(Υ(1S)→Bcπ)>Br(Υ(1S)→BcK∗)>Br(Υ(1S)→BcK). These are two dynamical reasons. One is that the CKM factor VcbVus∗ responsible for the Υ(1S)→BcK(∗) decay is suppressed by a factor of λ relative to the CKM factor VcbVud∗ responsible for the Υ(1S)→Bcπ, Bcρ decays. The other is that the orbital angular momentum LBcP>LBcV.(3)
The CKM-favoreda1 dominated Υ(1S)→Bcρ 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 fb−1 data collected at ALICE, over 1011Υ(1S) particles are in principle available, corresponding to tens of Υ(1S)→Bcρ 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 parameterA. The second uncertainty from the renormalization scale should, in principle, be reduced by inclusion of higher order αs 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, andq2 dependence of form factors, which are not considered in this paper, deserve the dedicated study. Our results just provide an order of magnitude estimation.Table 3
The values of input parameters.
Wolfenstein parameters
λ
=
0.22537
±
0.00061 [4]
A
=
0.81
4
-
0.024
+
0.023 [4]
Masses of quarks
m
c
=
1.67
±
0.07 GeV [4]
m
b
=
4.78
±
0.06 GeV [4]
Decay constants
f
π
=
130.41
±
0.20 MeV [4]
f
K
=
156.2
±
0.7 MeV [4]
f
ρ
=
216
±
3 MeV [32]
f
K
∗
=
220
±
5 MeV [32]
Gegenbauer moments at the scaleμ=1 GeV
a
1
ρ
=
0 [32]
a
2
ρ
=
0.15
±
0.07 [32]
a
1
K
∗
=
-
0.03
±
0.02 [32]
a
2
K
∗
=
0.11
±
0.09 [32]
a
1
π
=
0 [33]
a
2
π
=
0.25
±
0.15 [33]
a
1
K
=
-
0.06
±
0.03 [33]
a
2
K
=
0.25
±
0.15 [33]Table 4
TheCP-averaged branching ratios for the Υ(1S)→BcM decays.
Reference [22]
Reference [23]
This work
1
0
10
×
B
r
Υ
1
S
→
B
c
ρ
1.84
1.2
1.5
3
-
0.10
-
0.04
-
0.04
-
0.04
+
0.11
+
0.10
+
0.03
+
0.04
1
0
11
×
B
r
Υ
1
S
→
B
c
π
6.91
2.8
5.0
3
-
0.34
-
0.14
-
0.11
-
0.02
+
0.36
+
0.34
+
0.09
+
0.02
1
0
12
×
B
r
Υ
1
S
→
B
c
K
∗
10.47
6.2
8.7
5
-
0.64
-
0.20
-
0.22
-
0.39
+
0.68
+
0.55
+
0.16
+
0.40
1
0
12
×
B
r
Υ
1
S
→
B
c
K
5.03
2.3
3.7
3
-
0.27
-
0.10
-
0.08
-
0.03
+
0.29
+
0.25
+
0.07
+
0.03


## 4. Summary

With the sharp increase of theΥ(1S) data sample at high-luminosity dedicated heavy-flavor factories, the bottom-changing Υ(1S)→BcM 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 Υ(1S) weak decays are allowable within the standard model, though their branching ratios are expected to be tiny in comparison to the conventional strong and electromagnetic decays. In this paper, we studied the nonleptonic Υ(1S)→BcM weak decays, which are a1-dominated based on the low energy effective theory and hence should have large branching ratios among weak decay modes. Considering the nonfactorizable contributions to hadronic matrix elements with the QCDF approach, we estimated the branching ratios of the Υ(1S)→BcM weak decays, where transition form factors are obtained by the integrals of wave functions with the nonrelativistic isotropic harmonic oscillator potential. The prediction on branching ratios for the Υ(1S)→BcM decays is the same order as previous works [22, 23]. The CKM-favored Υ(1S)→Bcρ decay has relatively large branching ratio, ~10−10, and might be detectable in future experiments.



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