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Over past five decades, heavy quarkonium spectra have been established due to a collective of theorists and experimentalists working. More than 40 heavy quark-antiquark bound states, also known as charmonium and bottomonium, are observed with masses ranging from 2.9 GeV to 4.7 GeV or from 9.3 GeV to 11.1 GeV, respectively [1]. The heavy quarkonia below the open-flavor production mass-threshold are relatively well understood, providing an ideal laboratory to test perturbative and nonperturbative quantum chromodynamics (QCD) [2, 3].
Charmonium decay is usually a focused research topic that is significant and useful for understanding the fundamental characteristics of charmonium. J/ψ and
ψ(2S) are two observed charmonium states with a wealth of experimental decay information, as recently listed by the Particle Data Group (PDG) [1]. Compared to J/ψ andψ(2S) , the experimental measurement pertinent to the decay of spin singlets, such as the P-wave statehc and S-wave stateηc , and spin triplets, the P-wave stateχcJ is significantly lower. Therefore, there are currently more active theoretical and experimental investigations into such charmonia decays.Experimental measurements of charmonium radiative decay to light-quark vector mesons would help us understand the QCD and QED mechanism between charmonium and light vector mesons by the strong and electromagnetic interactions. The previous theoretical study on radiative decays of charmonium into light vector mesons (the processes
χcJ→γV ) was based on numerical calculations of the quark-gluon loop diagrams in the perturbative QCD (pQCD) frame and nonrelativistic quantum chromodynamics (NRQCD) [4, 5]. This provides a useful starting point to investigate the interatctions between quarks and gluons in OZI suppressed processes. Furthermore, its predicted branching ratios of the decaysχcJ→γV were tested by latere+e− collider experiments.Measurements of the branching ratios of the P-wave charmonia
χcJ→γV were presented by the CLEO-c and BESIII collaborations in 2008 [6] and 2011 [7], respectively. However, there are still some significant discrepancies between the experimental results and the theoretical predictions, as shown in Table 1. To resolve these discrepancies, a phenomenological model with a hadronic loop mechanism was employed [8].Decay Mode CLEO-c [6] BESIII [7] pQCD [4] NRQCD [5] NRQCD+QED [5] χc0→γρ0 
<9.6 < 10.5 1.2 3.2 2.0 χc1→γρ0 
243±19±22 228±13±22 14 41 42 χc2→γρ0 
<50 <20.8 4.4 13 38 χc0→γϕ 
<6.4 <16.2 0.46 1.3 0.03 χc1→γϕ 
<26 25.8±5.2±2.3 3.6 11 11 χc2→γϕ 
<13 <8.1 1.1 3.3 6.5 χc0→γω 
<8.8 <12.9 0.13 0.35 0.22 χc1→γω 
83±15±12 69.7±7.2±6.6 1.6 4.6 4.7 χc2→γω 
<7.0 <6.1 0.5 1.5 4.2 Table 1. Comparison of experimental measurement results (CLEO-c, BESIII) and theoretical calculations (pQCD, NRQCD, NRQCD+QED) for the branching ratio of
χcJ→ γV(V=ρ0,ϕ,ω) [in units of 10−6].The Sokolov-Ternov effect induces self-polarization in high-energy
e+e− beams, which allows them to naturally become transversely polarized in a storage ring [9]. Based on the(2712±14)×106ψ(2S) data samples collected by the BESIII detector in 2009, 2012, and 2021, it is feasible to precisely measure the polarized parameters and impact of transversely polarized beams on them, enabling a thorough examination of these theoretical models and aiding in a better understanding of the properties of P-wave charmonium radiation decays [10]. Despite the measurement of the branching ratio ofχcJ→γV , for the accurate measurement of their polarized parameters, the theoretical calculation must be validated.In this paper, we present a helicity amplitude formula for the process
e+e−→ψ(2S)→γ1χcJ→γ1γ2V(V=ρ0,ϕ,ω) and construct the spin density matrix forχcJ and light vector mesons. Expressions of the joint angular distributions are obtained, and some polarization observables are given for the measurement bye+e− collider. The statistical sensitivities for the relative magnitudes of transverse to longitudinal polarization amplitude of light vector mesons are also discussed with Monte Carlo (MC) simulation results in the paper. By considering the transverse polarization ofe+e− pairs, the angular distribution parameters will be measured with a high accuracy as well as other decay parameters. -
The helicity mechanism can be used to effectively build the dynamic information of entire decays [11, 12]. The decay planes and helicity angles are visually and clearly depicted in Fig. 1. In
ψ(2S)→γ1χcJ decay, the helicity angleθ1 is the polar angle between the directions of the momenta ofe+ andγ1 in thee+e− center-of-mass (CM) frame. InχcJ→γ2V decay, the helicity angleθ2 is chosen as the angle between the direction of momentum ofγ2 in theγ1 rest frame and the direction of momentum ofγ1 frome+e− collision, andϕ2 is the azimuthal angle between theχcJ production plane and its decay plane. Inρ0→π+π− andϕ→K+K− decays, there are two helicity angles: polar angleθ3 and azimuthal angleϕ3 ; meanwhile, in a three-body decayω→π+π−π0 , we use the Euler angles(α,β,γ) to describe its coordinate system rotating process. Specifically, theγ1 rest frame is rotated to theγ2 rest frame by γ aroundz3 , β aroundy3 , and finally α aroundz3 , where β is the angle between the momentum direction ofγ2 and cross product direction of the momenta ofπ+ andπ− in the ω rest frame [13].
Figure 1. (color online) Definition of helicity angles at
e+e− collider experiment.These helicity angles can be constructed by the momenta of final particles. Importantly, the experimentally obtained laboratory-frame momentum must be transformed to the rest frame of the decaying parent particle for calculation.
The helicity angles and amplitudes for sequential decays are defined in Table 2. The total amplitude
M for the sequential decayψ(2S)→γ1χcJ→γ1γ2ρ0(ϕ)→γ1γ2π+π−(K+K−) can be expressed asDecay Mode Solid Angle Helicity Amplitude ψ(2S)(λ0)→γ(λ1)χcJ(λ2) 
Ω1 =(
θ1,ϕ1 )
BJλ1,λ2 
χcJ(λ′2)→γ(λ3)V(λ4) 
Ω2 =(
θ2,ϕ2 )
AJcλ3,λ4 
V(λ′4)→PP(ρ→π+π−,ϕ→K+K−) 
Ω3 =(
θ3,ϕ3 )
F1 
V(λ′4)→PPP(ω→π+π−π0) 
Ω3 =(
α,β,γ )
F10 
Table 2. Definition of helicity angles and amplitudes, where
λi indicates the helicity and energy for the corresponding particles.M(ϕ1,θ1,ϕ2,θ2,ϕ3,θ3;J,M,Jc,λ1,λ2,λ3,λ4)=BJλ1λ2DJ⋆M,λ1−λ2(ϕ1,θ1)AJcλ3λ4DJc⋆λ2,λ3−λ4(ϕ2,θ2)×F1D1⋆λ4,0(ϕ3,θ3),
(1) where
BJλ1,λ2 ("before" theχcJ ) is the helicity amplitude ofψ(2S)→γχcJ decay, the superscript J is the spin ofψ(2S) withJ=1 ,AJcλ3,λ4 ("after" theχcJ ) is the helicity amplitude of the processχcJ→γρ0(χcJ→γϕ) , the superscriptJc is the spin ofχcJ , andF1 is the helicity amplitude of the decayρ0→π+π− orϕ→K+K− . The helicity amplitudes are subscripted by the helicityλ1=±1 of the radiative photon fromψ(2S)→γχcJ decay, the helicityλ2 (λ2=0 forχc0 ,λ2=0,±1 forχc1 , andλ2=0,±1,±2 forχc2 ) ofχcJ , the helicityλ3=±1 of the radiative photon fromχcJ→γV(V=ρ0,ϕ) , and the helicityλ4=0,±1 of the vector meson (ρ0 or ϕ). The indices for the helicity of the daughters (psedudoscalar mesons) fromρ0 or ϕ decay can be omitted because they are zero. The subscriptM=±1 for the first Wigner D-function is the z-component of spin J ofψ(2S) .The total amplitude
M for the decayψ(2S)→γ1χcJ→γ1γ2ω→γ1γ2π+π−π0 isM(ϕ1,θ1,ϕ2,θ2,ϕ3,θ3;J,M,Jc,λ1,λ2,λ3,λ4)=BJλ1λ2DJ⋆M,λ1−λ2(ϕ1,θ1)AJcλ3λ4DJc⋆λ2,λ3−λ4(ϕ2,θ2)×F1μD1⋆λ4,μ(α,β,γ),
(2) where
F1μ is the helicity amplitude of the decayω→π+π−π0 . μ is the z-component of the spin angular momentumJ of ω, while the normal to the ω decay plane is taken as the z-axis. The symmetry relation requires only one amplitudeF1μ(μ=0) [14].The decay rates Γ for the cascade decays
ψ(2S)→γ1χcJ→γ1γ2ρ0(ϕ)→γ1γ2π+π−(K+K−) andψ(2S)→γ1χcJ→γ1γ2ω→γ1γ2π+π−π0 are proportional to∑M,λ1,λ2,λ3,λ4|M(ϕ1,θ1,ϕ2,θ2,ϕ3,θ3;J,M,Jc,λ1,λ2,λ3λ4)|2.
(3) The electromagnetic transitions
ψ(2S)→γχcJ are dominated by electric dipole (E1) transitions, and their helicity amplitudesBJλ1λ2 satisfy the E1 transition relations [15]:B11,1=B11,0,forψ(2S)→γ1χc1decay,B11,2=√2B11,1=√6B11,0,forψ(2S)→γ1χc2decay.
(4) Parity conservation gives the relation
B1λ1,λ2=B1−λ1,−λ2ηψ(2S)ηγηχcJ(−1)sψ(2S)−sγ−sχcJ , where η and s represents the parity and spin of the particle, respectively. Thus, we haveB1−1,0=B11,0 for the decayψ(2S)→γχc0 ,B1−1,−1=−B11,1 ,B1−1,0=−B11,0 for the decayψ(2S)→γχc1 , andB1−1,−2=B11,2 ,B1−1,−1=−B11,1 ,B1−1,0=−B11,0 for the decayψ(2S)→γχc2 . Angular momentum conservation|λ1−λ2|≤1 requires that these amplitudes (B11,−1,B1−1,1 ) do not exist for the decayψ(2S)→γχc1 and thatB11,−1,B1−1,1, B1−1,2,B11,−2 do not exist for the decayψ(2S)→γχc2 . The matrix of helicity amplitude can be written asB=[B1−1,0B10,0B11,0]=[B11,00B11,0],forψ(2S)→γχc0decay,
(5) B=[B1−1,−1B1−1,00B10,−1B10,0B10,10B11,0B11,1]=[−B11,1−B11,000000B11,0B11,1]=[−B11,0−B11,000000B11,0B11,0],forψ(2S)→γχc1decay,
(6) B=[B1−1,−2B1−1,−1B1−1,0B1−1,10B10,−2B10,−1B10,0B10,1B10,20B11,−1B11,0B11,1B11,2]=[B11,2B11,1B11,0000000000B11,0B11,1B11,2]=[√6B11,0√3B11,0B11,0000000000B11,0√3B11,0√6B11,0],forψ(2S)→γχc2decay.
(7) For
χcJ→γV(ρ0,ϕ,ω) decays, only one independent helicity amplitude forχc0 decays (A01,1 orA0−1,−1 , the "so-called" transverse polarization amplitudeA⊥0V in Ref. [16]), and there are more than one independent helicity amplitudes forχc1 andχc2 decays. The helicity amplitudes can be expressed asAJcλ3,λ4=aλ3,λ4∗eiζλ3,λ4 , whereJc is the spin of the mother particle,λi is the helicity value of daughter particles, and a and ζ are the magnitude and phase angle of helicity amplitude A, respectively. Parity conservation leads to the following symmetry relations for helicity amplitudes:A01,1=A0−1,−1,forχc0decay,A1−1,−1=−A11,1,A1−1,0=−A11,0,forχc1decay,A2−1,−1=A21,1,A2−1,0=A21,0,A2−1,1=A21,−1,forχc2decay.
(8) We define a parameter for polarization observables to describe the relative magnitudes of two helicity amplitudes:
x≡|A11,1||A11,0|=|A⊥1V||A||1V|
(9) for
χc1→γV(ρ0,ϕ,ω) decays. We define two independent parametersx≡|A21,1||A21,0|=|A⊥2V||A||2V|,y≡|A21,−1||A21,0|=|T⊥2V||A||2V|
(10) for
χc2→γV(ρ,ϕ,ω) decays, where A or T with subscript⊥ (transverse polarization) or|| (longitudinal polarization) is the amplitude defined in Ref. [16]. These parameters describe the ratio of amplitudes that characterize transverse to longitudinal polarization.The phase differences between independent amplitudes are defined as
Δ1=ζ1,0−ζ1,1
(11) for
χc1 andΔ1=ζ1,0−ζ1,1,Δ2=ζ1,−1−ζ1,1
(12) for
χc2 . -
As a method to obtain the decay angular distributions, the spin density matrix (SDM) carries the dynamical information of particle decay, and its different parameterizations can clearly explain various physical phenomena. For example, when expressed in multipole parameter form
rLM (L-rank index ranges from 1 to 2J, M is taken from−L to L), it can provide information about particle polarization [17, 18].The spin density matrix of
ψ(2S) from polarizede+e− beams can is written asρψ(2S)=12[(1−Pz)(1+ˉPz)0P2T000P2T0(1+Pz)(1−ˉPz)],
(13) where
Pz/ˉPz is the degree of longitudinal polarization ofe+/e− , andPT is the degree of transverse polarization ofe+/e− [19].By taking the direction of the photon's momentum as the positive z-axis direction, as shown in Fig. 1(a), the SDM of
χc0 can be written asρχc0=12b21,0[1+cos2(θ1)+P2Tsin2(θ1)cos(2ϕ1)],
(14) while the matrix elements of
ρχc1 andρχc2 are listed in Appendix A, whereb1,0 is the magnitude of helicity amplitudeB1,0 inψ(2S)→γχcJ decays. In these SDM calculations, we have used the E1 transition relations in theψ(2S)→γχcJ decay [20].Here, we take the multipole parameters
rLM to describe the SDM of vectors (V=ρ0,ϕ,ω ), which is expressed asρV=r003[r20+√3r10+1√32(−ir1−1+r11−ir2−1+r21)√3(r22−ir2−2)√32(ir1−1+r11+ir2−1+r21)1−2r20√32(−ir1−1+r11+ir2−1−r21)√3(r22+ir2−2)√32(ir1−1+r11−ir2−1−r21)r20−√3r10+1]. (15) In
χcJ→γV(ρ,ϕ,ω) decays, the real multipole parametersrLM have different expressions forJ=0,1,2 , as listed in Appendix B. -
To compare the experimental results from the electron-positron collider, the joint angular distribution between the production and decay of
χcJ is required. The expressions of joint angular distribution can be easily obtained using the SDMs of decay particles. Here, we construct the joint angular distribution expressions for each decay level, which can be used to experimentally verify the measured results. The joint angular distributions for the processesψ(2S)→γ1χcJ ,ψ(2S)→γ1χcJ,χcJ→γ2V(V=ρ0,ϕ,ω) , andψ(2S)→γ1χcJ,χcJ→γ2V(V=ρ0,ϕ,ω),V→final states areW(Ω1) ∝ Tr[ρχcJ],
(16) W(Ω1,Ω2)∝Tr[ρV]=r00,
(17) W(Ω1,Ω2,Ω3)∝∑λ4,λ′4ρVλ4,λ′4D1∗λ4,0(Ω3)D1λ′4,0(Ω3)|F10|2∝−16r00[2√3sin2θ3(r2−2sin2ϕ3+r22cos2ϕ3)+2√3sin2θ3(r2−1sinϕ3+r21cosϕ3)+3r20cos2θ3+r20−2],
(18) respectively.
Considering the experimental polarization observables, we provide the distribution formulae for polar angle
θi(i=1,2,3) after integrating other polar and azimuthal angles.For
χc0 ,dNdcosθ1∝1+cos2θ1,
(19) dNdcosθ3∝1−cos2θ3,
(20) dNdcosθ1dϕ1∝1+cos2(θ1)+Pt2(1−cos2(θ1))cos(2ϕ1)
(21) The angular distribution
dN/dcosθ2 is trivial because the projection of the cosine polar anglecosθ2 is flat.For
χc1 ,dNdcosθ1∝1−13cos2θ1,
(22) dNdcosθ2∝1+2x2−12x2+3cos2θ2,
(23) dNdcosθ3∝1+2−x2x2cos2θ3,
(24) dNdcosθ1dϕ1∝1−13cos2(θ1)−13Pt2(1−cos2(θ1))cos(2ϕ1).
(25) For
χc2 ,dNdcosθ1∝1+113cos2θ1,
(26) dNdcosθ2∝1+−6x2+6y2−310x2+6y2+9cos2θ2,
(27) dNdcosθ3∝1+2−x2−y2x2+y2cos2θ3,
(28) dNdcosθ1dϕ1∝1+113cos2θ1+113Pt2(1−cos2(θ1))cos(2ϕ1)
(29) To validate the above angular distribution functions, MC simulation was performed by modeling the amplitude sampling of phase space events in Eqs. (3)−(12). The phase differences were naively set to
Δ1=π/3 inχc1 decays and toΔ1=π/3 andΔ2=π/4 inχc2 decays. The degree of beam polarization was simply set asPT=0.24 for allχcJ decays. The parameter x was set as 0.43 forχc1→γρ0 decay, 0.63 forχc1→γϕ decay, and 0.57 forχc1→γω decay, as obtained from BESIII measurement in 2011 [7]. Referring to Ref. [20], the parameters forχc2→γV decays were arbitrarily chosen to bex=1.55 ,y=2.06 forχc2→γρ0 decay,x=1.55 ,y=2.13 forχc2→γϕ decay, andx=0 ,y=1 forχc2→γω decay. 500000 pseudo experiments were generated and fitted for each decay mode, using a probability density function derived from the full angular distributions shown in Eqs. (19)−(29).Here, we list some fitting results; the others can be found in Table C1, C2 and Figs. C1−Fig. C6 of Appendix C. We use the function
1+αcos2θ to fit the angular distributions ofψ(2S)→γχcJ(J=0,1,2) in Eqs. (19), (22), (26) andρ0→π+π− ,ϕ→K+K− ,ω→π+π−π0 fromχc0→γV(V=ρ0,ϕ,ω) decays in Eq. (21). In this case, the fitted α values are0.992±0.009 for the decayψ(2S)→γχc0 and−0.999±0.001 for the decayϕ→K+K− fromχc0→γϕ decay in Fig. 2(a) and Fig. 2(b), respectively, which are consistent with the default values in Eqs. (19) and (21) within the standard deviation. With the measured angular distributions ofχcJ→γV(J=0,1,2) , we can extract the parameterˆx(x,y,PT) inχcJ decays by fitting these equations of joint angular distributions to data. Given an example of fitting the angular distributions ofχcJ→γϕ(J=1,2) based on MC simulation samples, the fitted x andPT values inχc1→γϕ decays are0.630±0.002 and0.26±0.02 , as shown in Fig. 3(b), Fig. 3(c), and Fig. 3(d), which are in agreement with the inputsx=0.63 andPT=0.24 . Inχc2→γϕ decays, the fitted parameters of x and y arex=1.558±0.344 ,y=2.137±0.251 , as shown in Fig. 4(b) and Fig. 4(c), respectively, which are consistant with the inputsx=1.55,y=2.13 .
Figure 2. (color online) Fits to the angular distributions of
cosθ1 ,ϕ1 inψ(2S)→γχc0 andcosθ3 inϕ→K+K− fromχc0→γϕ decays. Dots with error bars represent MC events, and the blue solid curve represents the fit.
Figure 3. (color online) Fits to the angular distributions of
cosθi(i=1,2,3) andϕ1 inψ(2S)→γχc1 ,χc1→γϕ , andϕ→K+K− decays. Dots with error bars represent MC events, and the blue solid curve represents the fit.
Figure 4. (color online) Fits to the angular distributions of
cosθi(i=1,2,3) andϕ1 inψ(2S)→γχc2 ,χc2→γϕ , andϕ→K+K− decays. Dots with error bars represent MC events, and the blue solid curve represents the fit.Decay mode Default α value Fitted α value Figure ψ(2S)→γχc0 
1.00 1.00±0.01 
C1(a) ω→π+π−π0 
−1.00 −1.00±0.01 
C1(b) ψ(2S)→γχc0 
1.00 1.02±0.01 
C2(a) ρ0→π+π− 
−1.00 −1.00±0.01 
C2(b) ψ(2S)→γχc1 (
χc1→γρ )
−0.33 −0.34±0.01 
C3(a) ψ(2S)→γχc1 (
χc1→γω )
-0.33 −0.33±0.01 
C4(a) ψ(2S)→γχc2 (
χc2→γρ )
0.08 0.07±0.01 
C5(a) ψ(2S)→γχc2 (
χc2→γω )
0.08 0.07±0.01 
C6(a) Table C1. Monte Carlo simulation and fitting results of the projection of the polar angles in the processes
ψ(2S)→γχc0 andψ(2S)→γχc1,2 Decay mode Input value Fit value Figure χc0→γρ 
PT=0.24 
PT=0.24±0.01 
C2(c) χc0→γω 
PT=0.24 
PT=0.25±0.01 
C1(c) χc1→γρ 
x=0.43,PT=0.24 
x=0.43±0.01,PT=0.25±0.02 
C3(b),C3(c),C3(d) χc1→γω 
x=0.57,PT=0.24 
x=0.56±0.01,PT=0.23±0.02 
C4(b),C4(c) C4(d) χc2→γρ 
x=1.55,y=2.06,PT=0.24 
x=1.58±0.79,y=2.09±0.77,PT=0.29±0.07 
C5(b),C5(c),C5(d) χc2→γω 
x=0,y=1,PT=0.24 
x=0.00±0.02,y=1.000±0.02,PT=0.22±0.01 
C(b),C6(c),C6(d) Table C2. Monte Carlo simulation and fitting results in the decays
χc1,2→γV -
By investigating the statistical sensitivity of specific parameters, precise measurement is beneficial for evaluating the potential impact of uncertainties on the final results. The estimation aims to enhance the understanding of experimental uncertainties associated with these parameters and observed signal yields. It is necessary to estimate sensitivity to guide the data acquisition plan for current large-scale
e+e− experimental devices, such as BEPCII, as well as those to be built in the future, such as Super-Tau Charm Facility (STCF) [21, 22]. Through rigorous statistical analysis and simulations, the research provides valuable insights into the sensitivity of the key parameters, enabling researchers to make informed decisions and draw meaningful conclusions from experimental data.Assuming that the parameters will be obtained by fitting to the maximum likelihood function event-by-event, the normalized joint angular distribution can be defined as
˜W(Ω1,Ω2,Ω3,ˆx)=W(Ω1,Ω2,Ω3,ˆx)∫⋅⋅⋅∫W(⋅⋅⋅)∏3i=1dcosθi∏3j=1dϕj,
(30) where
ˆx is a set of parameters containingPT forχc0 ,x,PT forχc1 , andx,y,PT forχc2 . The maximum likelihood function isL=N∏i=1˜W(Ω1,Ω2,Ω3,ˆx),
(31) where N is the number of observed events [23]. The estimated statistical sensitivity
δxi for parameterˆx(xi) is obtained byδxi=√Vxi,xi|xi|×100%
(32) where the covariance matrix gives
V−1xi,xj=E[−∂2lnL∂xi∂xj]=N∫−˜W⋅(∂2ln˜W∂xi∂xj)3∏k=1dcosθk3∏l=1dϕl.
(33) By taking the phase difference
\Delta_1= {\pi}/{3} in\chi_{c1} decays, the dependence of the statistical sensitivity for a set of different x values is plotted in Fig. 6(a). In\chi_{c2} decays, the phase differences\Delta_1 and\Delta_2 are set to{\pi}/{3} and{\pi}/{4} , respectively. The parametery(x) is set to1 when plotting the dependence of the statistical sensitivity for a set of differentx(y) , as shown in Fig. 7(a) and Fig. 7(b).
Figure 6. (color online) (a) Sensitivity of x (in
\chi_{c1} decay) for different x values relative to the observed events N. (b) Sensitivity ofP_T (in\chi_{c1} decay) for differentP_T values relative to the observed events N.
Figure 7. (color online) (a) Snsitivity of x (in
\chi_{c2} decay) for different x values relative to the observed events N. (b) Sensitivity of y (in\chi_{c2} decay) for different y values relative to the observed events N. (c) Sensitivity ofP_T (in\chi_{c2} decay) for differentP_T values relative to the observed events N.From Fig. 5, Fig. 6(b), and Fig. 7(c), we can see that as polarization increases, the number of events required under the same statistical significance decreases. In Fig. 6(a), we can see that the ratio of the two independent helicity amplitude moduli x in
\chi_{c1} decays can be measured at a statistical sensitivity of an order of 1% with at least 20,000 observed signal yields, where background, detector acceptance, and other experimental effects are not taken into account, if assumingx=1 . In comparison to the decay of\chi_{c1} , the decay of\chi_{c2} involves two independent parameters x and y, necessitating a higher number of observed events to achieve the same statistical sensitivity in the measurement of these two parameters.
Figure 5. (color online) Sensitivity of
P_T (in\chi_{c0} decay) for differentP_T values relative to the observed events N.The expected number of observed signal yields for the processes
\psi(2S) \to \gamma_1 \chi_{c1,2} \to \gamma_1 \gamma_2 \rho^0 \to \gamma_1 \gamma_2 \pi^+ \pi^- ,\psi(2S) \to \gamma_1 \chi_{c1,2} \to \gamma_1 \gamma_2 \phi \to \gamma_1 \gamma_2 K^+ K^- , and\psi(2S) \to \gamma_1 \chi_{cJ} \to \gamma_1 \gamma_2 \omega \to \gamma_1 \gamma_2 \pi^+ \pi^- \pi^0 are calculated by the following equation:N_{sig} = N_{\Psi(2S)} \times {\rm Br}_{\psi(2S) \to \gamma_1 \chi_{cJ}} \times {\rm Br}_{\chi_{cJ}\to\gamma_2 V} \times {\rm Br}_{V \to\rm final\; states}\times \epsilon ,
(34) where
N_{\rm sig} represents the expected number of observed signal events,N_{\Psi(2S)} represents the total number of\psi(2S) data samples at BESIII or STCF, and\epsilon is the expected experimental reconstruction efficiency. Additinally,{\rm Br}_{\psi(2S) \to \gamma_1 \chi_{cJ}} ,{\rm Br}_{\chi_{cJ}\to\gamma_2 V} , andBr_{V \to\rm final\; states} denote the branching ratios of\psi(2S) \to \gamma_1 \chi_{cJ}\; (J=0,1,2) ,\chi_{cJ}\to\gamma_2 V (V=\rho^0,\; \phi,\; \omega) , and\rho^0 \to \pi^+ \pi^- |\phi\to K^+ K^- |\omega \to \pi^+ \pi^- \pi^0 , respectively. BESIII collected approximately 2.71 billion (B)\psi(2S) events in 2009, 2012, and 2021 and plans to collect 3 billion (B)\psi(2S) events [21]. The future high-luminositye^+e^- collider STCF will collect approximately 640 billion (B)\psi(2S) data samples each year [22]. Table 3 lists the expected numbers of observed signal yields for the processes\psi(2S) \to \gamma_1 \chi_{c0,1,2},\; \chi_{c0,1,2} \to \gamma_2 V (V=\rho^0,\; \phi,\; \omega) based on 2.71 B\psi(2S) data samples from BESIII and 640 B\psi(2S) data samples from STCF. With the expected signal yields, we can estimate the statistical sensitivity of the relative magnitudes of the transverse to longitudinal polarization amplitude and the degree of transverse polarization ofe^++e^- beams,\delta_x (\delta_y, \delta_{P_T} ) for the processes\psi(2S) \to \gamma_1 \chi_{c0,1,2} \to \gamma_1 \gamma_2 V (V=\rho^0,\; \phi,\; \omega) based on 2.71 B and 640 B\psi(2S) events, as shown in Table 4. The results of\delta_{P_T} indicate that the degree of beam transverse polarization has a large statistical uncertainty based on current data. The statistical sensitivity\delta_x is 1.4%−4.3% for\chi_{c1} \to \gamma V decays with 2.71 B\psi(2S) data samples in the current BESIII experiment, which reaches at least 5-fold improvement over the BESIII measurement in 2011 with(1.06\pm 0.04)\times10^8\; \psi(2S) data samples [7]. The processes\chi_{c2} \to \gamma V is promising to be observed at BESIII, and its statistical sensitivities\delta_x and\delta_y are conservatively estimated to be up to the levels of 10%−20% based on 2.71 B\psi(2S) data samples. The STCF experiment is expected to further improve the sensitivity\delta_x for\chi_{c1} decays and\delta_x ,\delta_y for\chi_{c2} decays, with an impressive precision of less than or equal to 1% based on 640 B\psi(2S) data samples, presenting an improvement of 1 order of magnitude compared to the BESIII experiment with 2.71 B\psi(2S) data samples. The sensitivity\delta_{P_T} for\chi_{c1} and\chi_{c2} decays ranges from 2% to 20% at the STCF experiment.N_{sig} 
N_{\Psi(2S)} 
Br_{\psi(2S) \to \gamma_1 \chi_{cJ}} (%) [1]
Br_{\chi_{cJ}\to\gamma_2 V} (
10^{-5} ) [1]
{\rm Br}_{V \to\rm final~ states} (%) [1]
\epsilon (%) [7]
537 2.71\times 10^9 (BESIII)
9.77 ( \psi(2S) \to \gamma \chi_{c0}) 
<0.9 ( \chi_{c0}\to\gamma \rho^0 )
100 ( \rho^0 \to \pi^+ \pi^- )
22.6 253 <0.6 ( \chi_{c0}\to\gamma \phi )
49.1 ( \phi\to K^+ K^- )
32.4 337 <0.8 ( \chi_{c0}\to\gamma \omega )
89.2 ( \omega \to \pi^+ \pi^- \pi^0 )
18.6 11072 9.75 ( \psi(2S) \to \gamma \chi_{c1}) 
21.6 ( \chi_{c1}\to\gamma \rho^0 )
100 ( \rho^0 \to \pi^+ \pi^- )
19.4 1080 2.4 ( \chi_{c1}\to\gamma \phi )
49.1 ( \phi\to K^+ K^- )
34.6 3493 6.8 ( \chi_{c1}\to\gamma \omega )
89.2 ( \omega \to \pi^+ \pi^- \pi^0 )
22.7 788 9.36 ( \psi(2S) \to \gamma \chi_{c2} )
<1.9 ( \chi_{c2}\to\gamma \rho^0 )
100 ( \rho^0 \to \pi^+ \pi^- )
15.7 339 <0.8 ( \chi_{c2}\to\gamma \phi )
49.1 ( \phi\to K^+ K^- )
32.6 261 <0.6 ( \chi_{c2}\to\gamma \omega )
89.2 ( \omega \to \pi^+ \pi^- \pi^0 )
19.2 1.27\times10^5 
6.4\times 10^{11} (STCF)
9.77 ( \psi(2S) \to \gamma \chi_{c0}) 
<0.9 ( \chi_{c0}\to\gamma \rho^0 )
100 ( \rho^0 \to \pi^+ \pi^- )
22.6 5.97\times10^4 
<0.6 ( \chi_{c0}\to\gamma \phi )
49.1 ( \phi\to K^+ K^- )
32.4 7.95\times10^4 
<0.8 ( \chi_{c0}\to\gamma \omega )
89.2 ( \omega \to \pi^+ \pi^- \pi^0 )
18.6 2.61\times10^6 
9.75 ( \psi(2S) \to \gamma \chi_{c1}) 
21.6 ( \chi_{c1}\to\gamma \rho^0 )
100 ( \rho^0 \to \pi^+ \pi^- )
19.4 2.55\times10^5 
2.4 ( \chi_{c1}\to\gamma \phi )
49.1 ( \phi\to K^+ K^- )
34.6 8.25\times10^5 
6.8 ( \chi_{c1}\to\gamma \omega )
89.2 ( \omega \to \pi^+ \pi^- \pi^0 )
22.7 1.86\times10^5 
9.36 ( \psi(2S) \to \gamma \chi_{c2} )
<1.9 ( \chi_{c2}\to\gamma \rho^0 )
100 ( \rho^0 \to \pi^+ \pi^- )
15.7 8.01\times10^4 
<0.8 ( \chi_{c2}\to\gamma \phi )
49.1 ( \phi\to K^+ K^- )
32.6 6.16\times10^4 
<0.6 ( \chi_{c2}\to\gamma \omega )
89.2 ( \omega \to \pi^+ \pi^- \pi^0 )
19.2 Table 3. Expected numbers of observed signal events and related parameters for calculating the expected signal yields for the processes
\psi(2S) \to \gamma_1 \chi_{c0,1,2} \to \gamma_1 \gamma_2 V (V=\rho^0,\; \phi,\; \omega) at the BESIII and STCF experiments. The upper limits for\chi_{c2}\to\gamma V are at 90% C.L..Decay Mode Parameter Input value \delta_x (2.71B)
\delta_y (2.71B)
\delta_{P_T} (2.71B)
\delta_x (640B)
\delta_y (640B)
\delta_{P_T} (640B)
\chi_{c0}\to\gamma \rho^0 
P_T 
0.2 \backslash 
− − 7.69% − − \chi_{c0}\to\gamma \phi 
P_T 
0.2 \backslash 
− − 7.27% − − \chi_{c0}\to\gamma \omega 
P_T 
0.2 \backslash 
− − 8.27% − − \chi_{c1}\to\gamma \rho^0 
x 1 1.4% − 32.5% 0.1% − 2.1% P_T 
0.2 (fixed) \Delta_1 
\dfrac{\pi}{3} (fixed)
\chi_{c1}\to\gamma \phi 
x 1 4.3% − \backslash 
0.3% − 6.8% P_T 
0.2 (fixed) \Delta_1 
\dfrac{\pi}{3} (fixed)
\chi_{c1}\to\gamma \omega 
x 1 2.4% − 59.1% 0.2% − 3.8% P_T 
0.2 (fixed) \Delta_1 
\dfrac{\pi}{3} (fixed)
\chi_{c2}\to\gamma \rho^0 
x 1 8.6% 8.4% \backslash 
0.6% 0.5% 9.0% y 1 P_T 
0.2 (fixed) \Delta_1 
\dfrac{\pi}{3} (fixed)
\Delta_2 
\dfrac{\pi}{4} (fixed)
\chi_{c2}\to\gamma \phi 
x 1 13.7% 13.3% \backslash 
0.9% 0.9% 14.2% y 1 P_T 
0.2 (fixed) \Delta_1 
\dfrac{\pi}{3} (fixed)
\Delta_2 
\dfrac{\pi}{4} (fixed)
\chi_{c2}\to\gamma \omega 
x 1 15.6% 15.1% \backslash 
1.0% 1.0% 16.2% y 1 P_T 
0.2 (fixed) \Delta_1 
\dfrac{\pi}{3} (fixed)
\Delta_2 
\dfrac{\pi}{4} (fixed)
Table 4. Statistical sensitivity
\delta_x (\delta_y ,\delta_{P_T} ) for the processes\psi(2S) \to \gamma_1 \chi_{c0,1,2} \to \gamma_1 \gamma_2 V (V=\rho^0,\; \phi,\; \omega) based on 2.71 billion\Psi(2S) events at BESIII and 640 billion\Psi(2S) events at STCF. -
To better understand the radiative decays of
\psi(2S) \to \gamma \chi_{cJ}, \chi_{cJ} \to \gamma V(\rho^0, \phi, \omega) , we presented formulae for helicity amplitude analysis and derived the joint angular distribution for these decay chains. Furthermore, we provided observables for experimentally measuring the polarization of the vector mesons in\chi_{cJ} decays, performed a Monte Carlo simulation, and fitted the angular distributions to validate the theoretical calculations. Finally, we investigated the statistical sensitivity of the degree of transverse polarizationP_T ofe^+ e^- beams and the modulus ratio of helicity amplitudes in the decays of\chi_{c1} and\chi_{c2} , and we predicted the expected number of signal events required to achieve the relevant statistical precision in experimental measurements. Based on 3 billion planned\psi(2S) data samples at the BESIII experiment, the ratio of transverse to longitudinal polarization amplitude for the process\chi_{cJ} \to \gamma V(\rho, \phi, \omega) can be measured in the near future [21]. The fomalism in the work can be used in the near future in high-energy physics experiments, such as STCF with 640 billion\psi(2S) data samples per year [22]. Analogous to the radiative decay of a charmonium state with the same spin-parity quantum number, it also provides a reference for future measurements by the super-B factory to study the polarization effect of P-wave bottonia\chi_{bJ} \to \gamma V(\rho^0, \phi, \omega) [24]. -
\begin{aligned}[b] \rho^{\chi_{c1}}_{1,1} &=-\frac{1}{2} b_{1,0}^2 \sin ^2\left(\theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right),\\ \rho^{\chi_{c1}}_{1,0} &= \frac{b_{1,0}^2 \sin \left(\theta _1\right) \left(\cos \left(\theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)-{\rm i} P_T^2 \sin \left(2 \phi _1\right)\right)}{2 \sqrt{2}},\\ \rho^{\chi_{c1}}_{0,1} &= \frac{b_{1,0}^2 \sin \left(\theta _1\right) \left(\cos \left(\theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)+{\rm i} P_T^2 \sin \left(2 \phi _1\right)\right)}{2 \sqrt{2}},\\ \rho^{\chi_{c1}}_{0,0} &= \frac{1}{4} b_{1,0}^2 \left(\cos \left(2 \theta _1\right)+2 P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right)+3\right),\\ \rho^{\chi_{c1}}_{0,-1} &= \frac{b_{1,0}^2 \sin \left(\theta _1\right) \left(\cos \left(\theta _1\right) \left(1-P_T^2 \cos \left(2 \phi _1\right)\right)+{\rm i} P_T^2 \sin \left(2 \phi _1\right)\right)}{2 \sqrt{2}},\\ \rho^{\chi_{c1}}_{-1,0} &= -\frac{b_{1,0}^2 \sin \left(\theta _1\right) \left(\cos \left(\theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)+{\rm i} P_T^2 \sin \left(2 \phi _1\right)\right)}{2 \sqrt{2}},\\ \rho^{\chi_{c1}}_{-1,-1} &= -\frac{1}{2} b_{1,0}^2 \sin ^2\left(\theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right),\\ \rho^{\chi_{c1}}_{1,-1} &= \rho^{\chi_{c1}}_{-1,1} = 0. \end{aligned}
(A1) -
\begin{aligned}\rho_{2,2}^{\chi_{c2}} & =\frac{3}{4}b_{1,0}^2\left(\cos\left(2\theta_1\right)+2P_T^2\sin^2\left(\theta_1\right)\cos\left(2\phi_1\right)+3\right), \\ \rho_{2,1}^{\chi_{c2}} & =\frac{3}{2}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(1-P_T^2\cos\left(2\phi_1\right)\right)+{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{2,0}^{\chi_{c2}} & =\frac{1}{4}\sqrt{\frac{3}{2}}b_{1,0}^2\left(2\sin^2\left(\theta_1\right)+P_T^2\left(\left(\cos\left(2\theta_1\right)+3\right)\cos\left(2\phi_1\right)-4{\rm i}\cos\left(\theta_1\right)\sin\left(2\phi_1\right)\right)\right), \\ \rho_{1,2}^{\chi_{c2}} & =\frac{1}{2}(-3)b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right)+{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{1,1}^{\chi_{c2}} & =\frac{1}{2}(-3)b_{1,0}^2\sin^2\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right), \\ \rho_{1,0}^{\chi_{c2}} & =\frac{1}{2}\sqrt{\frac{3}{2}}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right)-{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{0,2}^{\chi_{c2}} & =\frac{1}{4}\sqrt{\frac{3}{2}}b_{1,0}^2\left(2\sin^2\left(\theta_1\right)+P_T^2\left(\left(\cos\left(2\theta_1\right)+3\right)\cos\left(2\phi_1\right)+4{\rm i}\cos\left(\theta_1\right)\sin\left(2\phi_1\right)\right)\right), \\ \rho_{0,1}^{\chi_{c2}} & =\frac{1}{2}\sqrt{\frac{3}{2}}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right)+{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{0,0}^{\chi_{c2}} & =\frac{1}{4}b_{1,0}^2\left(\cos\left(2\theta_1\right)+2P_T^2\sin^2\left(\theta_1\right)\cos\left(2\phi_1\right)+3\right),\\\rho_{0,-1}^{\chi_{c2}} & =\frac{1}{2}\sqrt{\frac{3}{2}}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(1-P_T^2\cos\left(2\phi_1\right)\right)+{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{0,-2}^{\chi_{c2}} & =\frac{1}{4}\sqrt{\frac{3}{2}}b_{1,0}^2\left(2\sin^2\left(\theta_1\right)+P_T^2\left(\left(\cos\left(2\theta_1\right)+3\right)\cos\left(2\phi_1\right)-4{\rm i}\cos\left(\theta_1\right)\sin\left(2\phi_1\right)\right)\right), \end{aligned}
\begin{aligned}[b] \rho_{-1,0}^{\chi_{c2}} & =-\frac{1}{2}\sqrt{\frac{3}{2}}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right)+{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{-1,-1}^{\chi_{c2}} & =\frac{1}{2}(-3)b_{1,0}^2\sin^2\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right), \\ \rho_{-1,-2}^{\chi_{c2}} & =\frac{3}{2}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right)-{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{-2,0}^{\chi_{c2}} & =\frac{1}{4}\sqrt{\frac{3}{2}}b_{1,0}^2\left(2\sin^2\left(\theta_1\right)+P_T^2\left(\left(\cos\left(2\theta_1\right)+3\right)\cos\left(2\phi_1\right)+4{\rm i}\cos\left(\theta_1\right)\sin\left(2\phi_1\right)\right)\right), \\ \rho_{-2,-1}^{\chi_{c2}} & =\frac{3}{2}b_{1,0}^2\sin\left(\theta_1\right)\left(\cos\left(\theta_1\right)\left(P_T^2\cos\left(2\phi_1\right)-1\right)+{\rm i}P_T^2\sin\left(2\phi_1\right)\right), \\ \rho_{-2,-2}^{\chi_{c2}} & =\frac{3}{4}b_{1,0}^2\left(\cos\left(2\theta_1\right)+2P_T^2\sin^2\left(\theta_1\right)\cos\left(2\phi_1\right)+3\right), \\ \rho_{2,-1}^{\chi_{c2}} & =\; \rho_{2,-2}^{\chi_{c2}}=\; \rho_{1,-1}^{\chi_{c2}}=\; \rho_{1,-2}^{\chi_{c2}}=\; \rho_{-1,2}^{\chi_{c2}}=\; \rho_{-1,1}^{\chi_{c2}}=\; \rho_{-2,1}^{\chi_{c2}}=\; \rho_{-2,1}^{\chi_{c2}}=0\\\end{aligned}
(A2) -
The multipole parameters
r^L_M for\chi_{c0} are expressed as\begin{aligned} r^0_0 =a_{1,1}^2 b_{1,0}^2 \left(1+\cos ^2\left(\theta _1\right)+P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right)\right), r^0_0 r^2_0 = \frac{1}{2} a_{1,1}^2 b_{1,0}^2 \left(1+\cos ^2\left(\theta _1\right)+P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right)\right). \end{aligned}
(B1) The other unlisted
r^L_M are equal to zero. -
The multipole parameters
r^L_M for\chi_{c1} are expressed as\begin{aligned}[b] r^0_0 =\;& \frac{1}{12} a_{1,0}^2 b_{1,0}^2 (-3 \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) \cos \left(\phi _2\right) +6 P_T^2 \left(1-2 x^2\right) \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) \\ &+ 6 P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right) \left(\left(2 x^2-1\right) \cos \left(2 \theta _2\right)-1\right) +3 \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) \cos \left(\phi _2\right) \left(P_T^2 \left(1-2 x^2\right) \cos \left(2 \phi _1\right)+2 x^2\right) \\ &+2 \left(2 x^2-1\right) \cos ^2\left(\theta _1\right) \left(3 \cos \left(2 \theta _2\right)+1\right)-2 \left(x^2+1\right) \cos \left(2 \theta _1\right)+10 x^2+10),\\ r^0_0 r^1_{-1} =\;& \frac{1}{4} \sqrt{3} x a_{1,0}^2 b_{1,0}^2 \sin \left(\Delta _1\right) (2 \sin \left(2 \theta _2\right) \left(\cos ^2\left(\theta _1\right)+P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right)\right) \\ &+\cos \left(2 \theta _2\right) \left(2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) +\sin \left(2 \theta _1\right) \cos \left(\phi _2\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)\right)),\\ r^0_0 r^1_1 =\;&\frac{1}{4} \sqrt{3} x a_{1,0}^2 b_{1,0}^2 \sin \left(\Delta _1\right) \cos \left(\theta _2\right) (2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \cos \left(\phi _2\right) +\sin \left(2 \theta _1\right) \sin \left(\phi _2\right) \left(1-P_T^2 \cos \left(2 \phi _1\right)\right)),\\ r^0_0 r^2_{-1} =\;& \frac{1}{4} \sqrt{3} x a_{1,0}^2 b_{1,0}^2 \cos \left(\Delta _1\right) \cos \left(\theta _2\right) (\sin \left(2 \theta _1\right) \sin \left(\phi _2\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right) -2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \cos \left(\phi _2\right)),\\ r^0_0 r^2_{0} =\;& \frac{1}{12} a_{1,0}^2 b_{1,0}^2 (3 \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) \cos \left(\phi _2\right) -6 P_T^2 \left(x^2+1\right) \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) \\ &+6 P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right) \left(\left(x^2+1\right) \cos \left(2 \theta _2\right)+1\right) -3 \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) \cos \left(\phi _2\right) \left(P_T^2 \left(x^2+1\right) \cos \left(2 \phi _1\right)-x^2\right) \\ &+2 \left(x^2+1\right) \cos ^2\left(\theta _1\right) \left(3 \cos \left(2 \theta _2\right)+1\right)-\left(x^2-2\right) \cos \left(2 \theta _1\right)+5 x^2-10), \\ r^0_0 r^2_{1} =\;& \frac{1}{4} \sqrt{3} x a_{1,0}^2 b_{1,0}^2 \cos \left(\Delta _1\right) (2 \sin \left(2 \theta _2\right) \left(\cos ^2\left(\theta _1\right)+P_T^2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _1\right)\right) \\ &+\cos \left(2 \theta _2\right) (2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) +\sin \left(2 \theta _1\right) \cos \left(\phi _2\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right))) .\\ \end{aligned}
(B2) The other unlisted
r^L_M are equal to zero. -
The multipole parameters
r^L_M for\chi_{c2} are expressed as\begin{aligned}[b] r^0_0 =\;& \frac{1}{64} \{4 x^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2+54 y^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 +108 x^2 \cos \left(4 \theta _2\right) \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 \\ &+ 18 y^2 \cos \left(4 \theta _2\right) \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 -72 \cos \left(4 \theta _2\right) \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2+72 x^2 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) P_T^2 \\ &+108 y^2 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) P_T^2-144 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) P_T^2 \\ &+48 x^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) P_T^2-72 y^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) P_T^2 \\ &+48 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) P_T^2-72 x^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) P_T^2\\ &-12 y^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) P_T^2+48 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) P_T^2\\ &+96 x^2 \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2-144 y^2 \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &+96 \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2-144 x^2 \sin \left(\theta _1\right) \sin \left(4 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &-24 y^2 \sin \left(\theta _1\right) \sin \left(4 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2+96 \sin \left(\theta _1\right) \sin \left(4 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &+48 \left(2 x^2+3 y^2-4\right) \cos \left(\theta _1\right) \sin ^2\left(\theta _2\right) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right) P_T^2+48 x^2 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right) \\ &+72 y^2 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right)-96 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right)+18 x^2 \cos \left(4 \theta _2\right) +3 y^2 \cos \left(4 \theta _2\right) \\ &+\cos \left(2 \theta _1\right) \{2 x^2+27 y^2+12 P_T^2 \left(2 x^2+3 y^2-4\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) +9 \left(6 x^2+y^2-4\right) \cos \left(4 \theta _2\right) \\ &+ 12 \cos \left(2 \theta _2\right) [-2 x^2+5 y^2 +P_T^2 \left(6 x^2+y^2-4\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right)-4]-12\}-48 x^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) \\ &+72 y^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right)+3 [50 x^2 + 67 y^2 - 8 P_T^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) - 4 \cos \left(4 \theta _2\right) - 16 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) + 52] \\ &+72 x^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) +12 y^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right)-48 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right)\\ &+12 \cos \left(2 \theta _2\right) [\cos \left(2 \phi _1\right) \left(3 \left(6 x^2+y^2-4\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right)-2 \left(2 x^2-5 y^2+4\right) \sin ^2\left(\theta _1\right)\right) P_T^2 -6 x^2+7 y^2\\ &+2 \left(6 x^2+y^2-4\right) \sin ^2\left(\theta _2\right) (2 \cos \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right) P_T^2 +\cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right))-4]\} a_{1,0}^2 b_{1,0}^2,\\r^0_0 r^1_{-1} &= \frac{3}{64} a_{1,0}^2 b_{1,0}^2 \{2 x \sin \Delta _1 \{\sin \left(2 \theta _2\right) [4 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _2\right)+\cos \left(2 \theta _1\right) \left(4 P_T^2 \cos \left(2 \phi _1\right) \cos ^2\left(\phi _2\right)-2\right) \\ &+2 P_T^2 \cos \left(2 \phi _1\right) \left(3 \cos \left(2 \phi _2\right)-1\right)-6] +18 P_T^2 \sin \left(4 \theta _2\right) \sin ^2\left(\phi _2\right) \cos \left(2 \phi _1\right)-3 \sin \left(4 \theta _2\right) [2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _2\right)+\cos \left(2 \theta _1\right) \\ &\times \left(P_T^2 \cos \left(2 \phi _1\right) \left(\cos \left(2 \phi _2\right)+3\right)-3\right)-1]-8 P_T^2 \sin \left(\theta _1\right) [\cos \left(2 \theta _2\right) -3 \cos \left(4 \theta _2\right)] \sin \left(2 \phi _1\right) \sin \left(\phi _2\right)\}\\ &+ 8 \cos \left(\theta _1\right) \{x \sin \Delta _1 [P_T^2 (2 \sin \left(2 \theta _2\right)-3 \sin \left(4 \theta _2\right)) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right)\\ &-2 \sin \left(\theta _1\right) (\cos \left(2 \theta _2\right) -3 \cos \left(4 \theta _2\right)) \cos \left(\phi _2\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)]\\ &+\sqrt{6} y \sin \left(\Delta _1-\Delta _2\right) [\sin \left(\theta _1\right) \left(3 \cos \left(2 \theta _2\right)+\cos \left(4 \theta _2\right)\right) \cos \left(\phi _2\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)\\ &-4 P_T^2 \sin \left(\theta _2\right) \cos ^3\left(\theta _2\right) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right)]\} \\ &+\sqrt{6} y \sin \left(\Delta _1-\Delta _2\right) \{6 P_T^2 \sin \left(4 \theta _2\right) \sin ^2\left(\phi _2\right) \cos \left(2 \phi _1\right)-2 \sin \left(2 \theta _2\right) [2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _2\right)\\ &+\cos \left(2 \theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right) \left(\cos \left(2 \phi _2\right)+5\right)-5\right)+P_T^2 \cos \left(2 \phi _1\right) \left(3 \cos \left(2 \phi _2\right)-5\right)-7] -\sin \left(4 \theta _2\right) (2 \sin ^2\left(\theta _1\right) \cos \left(2 \phi _2\right)\\ &+\cos \left(2 \theta _1\right) \left(P_T^2 \cos \left(2 \phi _1\right) \left(\cos \left(2 \phi _2\right)+3\right)-3\right)-1) +8 P_T^2 \sin \left(\theta _1\right) \left(3 \cos \left(2 \theta _2\right)+\cos \left(4 \theta _2\right)\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right)\}\},\\ r^0_0 r^1_1 =\;&\frac{3}{224} a_{1,0}^2 b_{1,0}^2 \{7 \sin \left(\theta _2\right) [\sin \left(2 \phi _2\right) \left(2 \sin ^2\left(\theta _1\right)+P_T^2 \left(\cos \left(2 \theta _1\right)+3\right) \cos \left(2 \phi _1\right)\right)\\ &-4 P_T^2 \cos \left(\theta _1\right) \sin \left(2 \phi _1\right) \cos \left(2 \phi _2\right)] [2 x \sin \Delta _1 \left(3 \cos \left(2 \theta _2\right)+1\right) \end{aligned}
\begin{aligned} &+\sqrt{6} y \sin \left(\Delta _1-\Delta _2\right) \left(\cos \left(2 \theta _2\right)+3\right)] +28 \sin \left(\theta _1\right) \cos \left(\theta _2\right) [\cos \left(\theta _1\right) \sin \left(\phi _2\right) \left(1-P_T^2 \cos \left(2 \phi _1\right)\right) \\&+P_T^2 \sin \left(2 \phi _1\right) \cos \left(\phi _2\right)] [2 x \sin \Delta _1 \left(3 \cos \left(2 \theta _2\right)-1\right) +\sqrt{6} y \sin \left(\Delta _1-\Delta _2\right) \left(\cos \left(2 \theta _2\right)+3\right)]\}, \\ r^0_0 r^2_{-2} =\;& -\frac{3 x y }{8 \sqrt{2}}a_{1,0}^2 b_{1,0}^2 \cos \left(\Delta _2\right) \{-\cos \left(\theta _2\right) \left(3 \cos \left(2 \theta _2\right)+1\right) \sin \left(2 \phi _2\right) [2 \sin ^2\left(\theta _1\right) +P_T^2 \left(\cos \left(2 \theta _1\right)+3\right) \cos \left(2 \phi _1\right)] \\ &+2 P_T^2 \sin \left(2 \phi _1\right) [\cos \left(\theta _1\right) \left(5 \cos \left(\theta _2\right)+3 \cos \left(3 \theta _2\right)\right) \cos \left(2 \phi _2\right) +\sin \left(\theta _1\right) \left(7 \sin \left(\theta _2\right)+3 \sin \left(3 \theta _2\right)\right) \cos \left(\phi _2\right)] \\ &-2 \sin \left(2 \theta _1\right) \sin \left(\theta _2\right) \left(3 \cos \left(2 \theta _2\right)+5\right) \sin \left(\phi _2\right)\left(P_T^2 \cos \left(2 \phi _1\right)-1\right)\}\\ r^0_0 r^2_{-1} =\;&-\frac{3}{224} a_{1,0}^2 b_{1,0}^2 \{2 x \cos \Delta _1 \{14 \cos \left(\theta _2\right) \left(3 \cos \left(2 \theta _2\right)-1\right) (2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \cos \left(\phi _2\right) \\ &+\sin \left(2 \theta _1\right) \sin \left(\phi _2\right) \left(1-P_T^2 \cos \left(2 \phi _1\right)\right)) -7 \sin \left(\theta _2\right) \left(3 \cos \left(2 \theta _2\right)+1\right) [4 P_T^2 \cos \left(\theta _1\right) \sin \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \\ &-\sin \left(2 \phi _2\right) \left(2 \sin ^2\left(\theta _1\right)+P_T^2 \left(\cos \left(2 \theta _1\right)+3\right) \cos \left(2 \phi _1\right)\right)]\}\\&+7 \sqrt{6} y \cos \left(\Delta _1-\Delta _2\right) \{\sin \left(\theta _2\right) \left(\cos \left(2 \theta _2\right)+3\right) \sin \left(2 \phi _2\right) \left(2 \sin ^2\left(\theta _1\right)+P_T^2 \left(\cos \left(2 \theta _1\right)+3\right) \cos \left(2 \phi _1\right)\right)\\ &-2 P_T^2 \left(5 \sin \left(\theta _2\right)+\sin \left(3 \theta _2\right)\right) \cos \left(\theta _1\right) \sin \left(2 \phi _1\right) \cos \left(2 \phi _2\right) +\sin \left(2 \theta _1\right) \cos \left(3 \theta _2\right) \sin \left(\phi _2\right) \left(1-P_T^2 \cos \left(2 \phi _1\right)\right) \\ &+\cos \left(\theta _2\right) \left(4 P_T^2 \sin \left(\theta _1\right) \left(\cos \left(2 \theta _2\right)+3\right) \sin \left(2 \phi _1\right) \cos \left(\phi _2\right)-7 \sin \left(2 \theta _1\right) \sin \left(\phi _2\right) \left(P_T^2 \cos \left(2 \phi _1\right)-1\right)\right)\}\},\\ r^0_0 r^2_0 =\;& \frac{1}{128} \{4 x^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2+54 y^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 + 108 x^2 \cos \left(4 \theta _2\right) \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 \\ &+18 y^2 \cos \left(4 \theta _2\right) \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 +144 \cos \left(4 \theta _2\right) \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2+48 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) P_T^2 \\ &+72 x^2 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) P_T^2+108 y^2 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) P_T^2 +288 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) P_T^2 \\ &+48 x^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) P_T^2 -72 y^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) P_T^2\\ &-96 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) P_T^2 -72 x^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) P_T^2\\ &-12 y^2 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) P_T^2 -96 \cos \left(2 \phi _1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) P_T^2 \\ &+96 x^2 \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 -144 y^2 \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2\\ &-192 \sin \left(\theta _1\right) \sin \left(2 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 -144 x^2 \sin \left(\theta _1\right) \sin \left(4 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2\\ &-24 y^2 \sin \left(\theta _1\right) \sin \left(4 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 -192 \sin \left(\theta _1\right) \sin \left(4 \theta _2\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &+48 \left(2 x^2+3 y^2+8\right) \cos \left(\theta _1\right) \sin ^2\left(\theta _2\right) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right) P_T^2+150 x^2+201 y^2 +48 x^2 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right)\\ &+72 y^2 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right) +192 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right)+18 x^2 \cos \left(4 \theta _2\right)+3 y^2 \cos \left(4 \theta _2\right)+24 \cos \left(4 \theta _2\right) \\ &+\cos \left(2 \theta _1\right) [2 x^2+27 y^2+12 P_T^2 \left(2 x^2+3 y^2+8\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right) +9 \left(6 x^2+y^2+8\right) \cos \left(4 \theta _2\right)\\ &+12 \cos \left(2 \theta _2\right) [-2 x^2+5 y^2 +P_T^2 \left(6 x^2+y^2+8\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right)+8]+24] -48 x^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right)\\ &+72 y^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right) +96 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(2 \theta _2\right)+72 x^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) \\ &+12 y^2 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right)+96 \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \sin \left(4 \theta _2\right) +12 \cos \left(2 \theta _2\right) \{\cos \left(2 \phi _1\right) [2 \left(-2 x^2+5 y^2+8\right) \sin ^2\left(\theta _1\right) \\ &+3 \left(6 x^2+y^2+8\right) \cos \left(2 \phi _2\right) \sin ^2\left(\theta _2\right)] P_T^2-6 x^2+7 y^2 +2 \left(6 x^2+y^2+8\right) \sin ^2\left(\theta _2\right) (2 \cos \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right) P_T^2\\ &+\cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right))+8\} -312\} a_{1,0}^2 b_{1,0}^2, \\ r^0_0 r^2_1 =\;& \frac{3}{224} \{7 \sqrt{\frac{3}{2}} y \cos \left(\Delta _1-\Delta _2\right) \{-32 P_T^2 \cos \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(2 \phi _1\right) \sin \left(2 \phi _2\right) \cos ^3\left(\theta _2\right) \\ &+20 P_T^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) \sin \left(2 \theta _2\right)-4 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin \left(2 \theta _2\right)+2 [5 \cos \left(2 \theta _1\right)+\cos \left(2 \theta _2\right) +7] \sin \left(2 \theta _2\right) \\ &-6 P_T^2 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(2 \theta _2\right)-2 P_T^2 \cos \left(2 \theta _1\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(2 \theta _2\right) +6 P_T^2 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) \sin \left(4 \theta _2\right) \end{aligned}
\begin{aligned} &-2 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin \left(4 \theta _2\right)+3 \cos \left(2 \theta _1\right) \sin \left(4 \theta _2\right) -3 P_T^2 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(4 \theta _2\right)\\ &-P_T^2 \cos \left(2 \theta _1\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(4 \theta _2\right) +12 \cos \left(2 \theta _2\right) (2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &+\left(P_T^2 \cos \left(2 \phi _1\right)-1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right)) +4 \cos \left(4 \theta _2\right) (2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &+\left(P_T^2 \cos \left(2 \phi _1\right)-1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right))\} -7 x \cos \Delta _1 \{4 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) \sin \left(2 \theta _2\right) P_T^2 \\ &-2 \cos \left(2 \theta _1\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(2 \theta _2\right) P_T^2-6 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(2 \theta _2\right) P_T^2 -18 \cos \left(2 \phi _1\right) \sin ^2\left(\theta _1\right) \sin \left(4 \theta _2\right) P_T^2\\ &+3 \cos \left(2 \theta _1\right) \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(4 \theta _2\right) P_T^2 +9 \cos \left(2 \phi _1\right) \cos \left(2 \phi _2\right) \sin \left(4 \theta _2\right) P_T^2+4 \cos \left(\theta _1\right) (3 \sin \left(4 \theta _2\right)\\&-2 \sin \left(2 \theta _2\right)) \sin (2 \phi _1) \sin (2 \phi _2) P_T^2 -4 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin \left(2 \theta _2\right)+2 \left(\cos \left(2 \theta _1\right)+3\right) \sin \left(2 \theta _2\right)\\&+6 \cos \left(2 \phi _2\right) \sin ^2\left(\theta _1\right) \sin \left(4 \theta _2\right) -9 \cos \left(2 \theta _1\right) \sin \left(4 \theta _2\right)-3 \sin \left(4 \theta _2\right)+4 \cos \left(2 \theta _2\right) [2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right) P_T^2 \\ &+\left(P_T^2 \cos \left(2 \phi _1\right)-1\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right)]+12 \cos \left(4 \theta _2\right) [\left(1-P_T^2 \cos \left(2 \phi _1\right)\right) \cos \left(\phi _2\right) \sin \left(2 \theta _1\right) \\ &-2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right)]\}\} a_{1,0}^2 b_{1,0}^2,\\ r^0_0 r^2_2=\; & \frac{3 x y}{32 \sqrt{2}} a_{1,0}^2 b_{1,0}^2 \cos \left(\Delta _2\right) \{2 \sin ^2\left(\theta _1\right) \left(4 \cos \left(2 \theta _2\right)+3 \cos \left(4 \theta _2\right)+9\right) \cos \left(2 \phi _2\right) +4 \sin ^2\left(\theta _2\right) (3 \left(\cos \left(2 \theta _2\right)+3\right) \\&+\cos \left(2 \theta _1\right) \left(9 \cos \left(2 \theta _2\right)+11\right)) +P_T^2 \cos \left(2 \phi _1\right) [\left(\cos \left(2 \theta _1\right)+3\right) \left(4 \cos \left(2 \theta _2\right)+3 \cos \left(4 \theta _2\right)+9\right) \cos \left(2 \phi _2\right) \\&+4 \sin \left(2 \theta _1\right) \left(2 \sin \left(2 \theta _2\right)+3 \sin \left(4 \theta _2\right)\right) \cos \left(\phi _2\right)+8 \sin ^2\left(\theta _1\right) \sin ^2\left(\theta _2\right) \left(9 \cos \left(2 \theta _2\right)+11\right)] +4 P_T^2 \cos \left(\theta _1\right) (4 \cos \left(2 \theta _2\right)\\ &+3 \cos \left(4 \theta _2\right)+9) \sin (2 \phi _1) \sin (2 \phi _2) -4 \left(2 \sin \left(2 \theta _2\right)+3 \sin \left(4 \theta _2\right)\right) \left(\sin \left(2 \theta _1\right) \cos \left(\phi _2\right)-2 P_T^2 \sin \left(\theta _1\right) \sin \left(2 \phi _1\right) \sin \left(\phi _2\right)\right)\}. \end{aligned}
(B3) -
Figure C1. (color online)
{\mathrm{d}N}/{\mathrm{d}\cos\theta_1} ,{\mathrm{d}N}/{\mathrm{d}\cos\theta_3} , and{\mathrm{d}N}/{\mathrm{d}\phi_1} distributions versus\cos\theta_1 ,\cos\theta_3 , and\phi_1 in\psi(2S)\to \gamma \chi_{c0} and\omega \to \pi^+ \pi^- \pi^0 from\chi_{c0} \to \gamma \omega decays. Dots with error bars are filled with MC events, and the blue solid curve represents the fit.
Figure C2. (color online)
{\mathrm{d}N}/{\mathrm{d}\cos\theta_1} ,{\mathrm{d}N}/{\mathrm{d}\cos\theta_3} , and{\mathrm{d}N}/{\mathrm{d}\phi_1} distributions versus\cos\theta_1 ,\cos\theta_3 , and\phi_1 in\psi(2S)\to \gamma \chi_{c0} and\rho^0 \to \pi^+ \pi^- from\chi_{c0} \to \gamma \rho decays. Dots with error bars are filled with MC events, and the blue solid curve represents the fit.
Figure C3. (color online)
{\mathrm{d}N}/{\mathrm{d}\cos\theta_i}, i=1,2,3 , and{\mathrm{d}N}/{\mathrm{d}\phi_1} distributions versus\cos\theta_i and\phi_1 in\psi(2S)\to \gamma \chi_{c1} ,\chi_{c1} \to \gamma \rho , and\rho^0 \to \pi^+ \pi^- decays. Dots with error bars are filled with MC events, and the blue solid curve represents the fit.
Figure C4. (color online)
{\mathrm{d}N}/{\mathrm{d}\cos\theta_i}, i=1,2,3 , and{\mathrm{d}N}/{\mathrm{d}\phi_1} distributions versus\cos\theta_i and\phi_1 in\psi(2S)\to \gamma \chi_{c1} ,\chi_{c1} \to \gamma \omega , and\omega \to \pi^+ \pi^- \pi^0 decays. Dots with error bars are filled with MC events, and the blue solid curve represents the fit.
Figure C5. (color online) Fits to the angular distributions of
\cos\theta_i (i=1,2,3) and{\mathrm{d}N}/{\mathrm{d}\phi_1} versus\cos\theta_i and\phi_1 in\psi(2S)\to \gamma \chi_{c2} ,\chi_{c2} \to \gamma \rho , and\rho^0 \to \pi^+ \pi^- decays. Dots with error bars represent MC events, and the blue solid curve represents the fit.
Figure C6. (color online)
{\mathrm{d}N}/{\mathrm{d}\cos\theta_i}, i=1,2,3 , and{\mathrm{d}N}/{\mathrm{d}\phi_1} distributions versus\cos\theta_i and\phi_1 in\psi(2S)\to \gamma \chi_{c2} ,\chi_{c2} \to \gamma \omega , and\omega \to \pi^+ \pi^- \pi^0 decays. Dots with error bars are filled with MC events, and the blue solid curve represents the fit.
Polarization study of P-wave charmonium radiative decay into a light vector meson at e+e− collider experiment
- Received Date: 2024-06-25
- Available Online: 2025-01-15
Abstract: In this paper, a formalism is presented for the helicity amplitude analysis of the decays
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