-
The Feynman diagrams for the production of a pair of higher excited heavy quarkonia via
e−(p1)e+(p2)→Z0→|(QˉQ′)[n]⟩(q1)+|(Q′ˉQ)[n′]⟩(q2) are depicted in Fig. 1. According to the QCD factorization formula, its differential cross sections can be factored into the short-distance coefficients and the long-distance matrix elementsFigure 1. Feynman diagrams for
e−(p1)e+(p2)→Z0→|(QˉQ′)[n]⟩ (q1)+|(Q′ˉQ)[n′]⟩(q2) , whereQ,Q′=c -, b-quarks, and[n] represents the color-singlet[n1S0],[n3S1],[n1P1] , and[n3PJ] (n=1,2,3;J=0,1,2 ) heavy quarkonia.dσ=∑ndˆσ((Q′ˉQ)[n]+(Q¯Q′)[n′])⟨O[n]⟩⟨O[n′]⟩.
(1) The short-distance coefficients
ˆσ((Q′ˉQ)[n]+(Q¯Q′)[n′]) can be calculated perturbatively, which depicts the short-distance production of two Fock states(Q′ˉQ)[n] and(Q¯Q′)[n′] (Q(′)=b - or c-quarks) in the spin, color, and angular momentum states[n(′)] . Here,[n(′)] represents the color-singlet states[n(′)1S0],[n(′)3S1],[n(′)1P1] , and[n(′)3PJ] (n=1,2,3;J=0,1,2 ).The long-distance matrix element
⟨O[n]⟩ is a non-perturbative parameter that describes the hadronization of the heavy quark pair(Q¯Q′)[n] with quantum number n into the heavy quarkonium|(Q¯Q′)[n]⟩ . In the NRQCD framework, we have contributions from both color-singlet and color-octect states. In this paper, we consider only the color-singlet state, i.e., the intermediate heavy quark pair(Q¯Q′)[n] and the final heavy quarkonium|(Q¯Q′)[n]⟩ have the same n. The color-singlet matrix elements⟨O([n])⟩ in Eq. (1) can be related to the Schrödinger wave function at the originΨ∣(Q¯Q′)[nS]⟩(0) fornS -wave heavy quarkonia or the first derivative of the wave function at the originΨ′∣(Q¯Q′)[nP]⟩(0) fornP -wave heavy quarkonia:⟨O([n1S0])⟩≃⟨O([n3S1])⟩≃|Ψ∣(Q¯Q′)[nS]⟩(0)|2,⟨O([n1P0])⟩≃⟨O([n3PJ])⟩≃|Ψ′∣(Q¯Q′)[nP]⟩(0)|2.
(2) Because the spin-splitting effect at the same n-th level is small, the same values of wave function for both the spin-triplet and spin-singlet Fock states are adopted in our calculation. The Schrödinger wave function at the origin
Ψ|Q¯Q′)[nS]⟩(0) and its first derivative at the originΨ′|(Q¯Q′)[nP]⟩(0) can be further relevant to the radial wave function at the originR|(Q¯Q′)[nS]⟩(0) and its first derivative of the radial wave function at the originR′|(Q¯Q′)[nP]⟩(0) , respectively [2]:Ψ|(Q¯Q′)[nS]⟩(0)=√1/4πR|(Q¯Q′)[nS]⟩(0),Ψ′|(Q¯Q′)[nP]⟩(0)=√3/4πR′|(Q¯Q′)[nP]⟩(0).
(3) The differential cross section
dˆσ can be calculated perturbatively, which can be formulated asdˆσ=14√(p1⋅p2)2−m4e¯∑|M([n],[n′])|2dΦ2,
(4) where
¯∑ is the average over the spin of the initial positron and electron and sum over the color and spin of the final higher excited heavy quarkonia. The two-body phase space in thee−e+ center-of-momentum (CM) rest frame can be simplified asdΦ2=(2π)4δ4(p1+p2−2∑f=1qf)2∏f=1d3→qf(2π)32q0f=∣→q1∣8π√sd(cosθ).
(5) Here,
s=(p1+p2)2 is the squared CM energy. The3 -momentum of the heavy quarkonium|(QˉQ′)[n]⟩ is|→q1|=√λ[s,M21,M22]/2√s , in whichλ[a,b,c]=(a−b−c)2−4bc , andM1 andM2 are the masses of two higher excited heavy quarkonia. θ is the angle between the momentum (→p1 ) of the electron and that (→q1 ) of the heavy quarkonium|(QˉQ′)[n]⟩ .The hard scattering amplitude
M([n],[n′]) in Eq. (4) can be read directly from the Feynman diagrams in Fig. 1, which can be expressed asiM([n],[n′])=4∑k=1ˉvs′(p2)Lσus(p1)DσρAρk.
(6) Here, k is the number of Feynman diagrams, and s and
s′ are the spins of the initial electron and positron, respectively. The vertexLσ and propagatorDσρ forZ0 propagated processes have the following forms:Lσ=−ig4cosθWγσ(1−4eQsin2θW−γ5),Dσρ=−igσρp2−m2Z+imZΓZ.
(7) Here,
p=p1+p2 represents the momentum of the propagator; e denotes the unit of electric charge,eQ=1 for the positron and electron,eQ=−1/3 for the b-quark andeQ=2/3 for the c-quark; g is the weak interaction coupling constant;θW is the Weinberg angle, andΓZ andmZ are the total decay width and mass of theZ0 boson, respectively.The concrete expressions of the Dirac γ matrix chains
Aσk in Eq. (6) for thenS -wave spin-singletn1S0 and spin-tripletn3S1 (n=1,2,3 ) can be expressed asAσ(S,L=0)1=iTr[Π(S,L=0)q1γα(⧸q1+⧸q21)+mQ1[(q1+q21)2−m2Q1](q12+q21)2LσΠ(S,L=0)q2γα]q=0,Aσ(S,L=0)2=iTr[Π(S,L=0)q1Lσ−(⧸q2+⧸q12)+mQ2[(q2+q12)2−m2Q2](q12+q21)2γαΠ(S,L=0)q2γα]q=,Aσ(S,L=0)3=iTr[Π(S,L=0)q1γαΠ(S,L=0)q2Lσ−(⧸q1+⧸q22)+mQ′1[(q1+q22)2−m2Q′1](q11+q22)2γα]q=0,Aσ(S,L=0)4=iTr[Π(S,L=0)q1γαΠ(S,L=0)q2γα(⧸q2+⧸q11)+mQ′2[(q2+q11)2−m2Q′2](q11+q22)2Lσ]q=0. (8) Here, S and L represent the quantum number of spin and orbit angular momentums of the heavy quarkonium, respectively.
q11=mQ1q1/M1 andq12=mQ′1q1/M1 are the momenta of the two constituent quarks of the heavy quarkonium|(QˉQ′)[n]⟩(q1) , withM1=mQ1+mQ′1 .q21=mQ′2q2/M2+q andq22=mQ2q2/M2−q are the momenta of the two heavy constituent quarks of the heavy quarkonium|(Q′ˉQ)[n′]⟩(q2) , withM2=mQ′2+mQ2 . q respresents the relative momentum between the heavy quarksQ′2 andQ2 . We introduce the relative momentum q for the quarkonium|(Q′ˉQ)[n′]⟩(q2) because it might be either annS -wave ornP -wave state. For annS -wave state, the relative momentum q is set to zero directly. For annP -wave state, we should first perform the derivative of the amplitude over q and then setq=0 . The two projectorsΠ(S,L=0)qk (k=1,2 ) in Eq. (8) have the following forms:Π(S,L=0)q1=ϵa(q1)−√M14mQ1mQ′1(⧸q12−mQ′1)γa(⧸q11+mQ1)⊗δij√Nc,
Π(S,L=0)q2=ϵa(q2)−√M24mQ2mQ′2(⧸q22−mQ2)γa(⧸q21+mQ′2)⊗δij√Nc.
(9) Here,
ϵa=1 andγa=γ5 for the spin-singlet1S0 states (S=0,L=0 ), andϵa=εμ andγa=γμ for spin-triplet3S1 state (S=1,L=0 ), where μ is the Lorentz vector index.δij/√Nc denotes the color operator for a color-singlet projector withNc=3 .When
|(Q′ˉQ)[n′]⟩(q2) is annP -wave state, the expressions ofAσk(k=1,2,3,4) for thenP -wave spin-singlet statesn1P1(S=0,L=1) and spin-triplet statesn3PJ(S=1,L=1) with total angular momentum (J=0,1,2 ) can be formulated as the first derivative of S-wave amplitudes for the relative momentum q and then we can setq=0 :Aσ(S=0,L=1)k=εν(q2)ddqνAσ(S=0,L=0)k|q=0,Aσ(S=1,L=1)k=εJμν(q2)ddqνAσ(S=1,L=0)k|q=0.
(10) Here,
εν(q2) denotes the polarization vector of an1P1 state,εJμν(q2) is the polarization tensor for ann3PJ state (J=0,1,2 ). The derivative over the relative momentumqν will result in complicated and lengthy amplitudes. Thus, fornP -wave states, obtaining the squared amplitudes|M([n],[n′])|2 using the traditional method is very time-consuming. We continue using the "improved trace technique" to solve the amplitudesM([n],[n′]) . In this prescription, the compact analytical expressions of the complicatednP -wave can be obtained, and the efficiency of numerical evaluation can also be improved. To keep this paper short, we do not further describe the "improved trace technique" in details. For complete techniques and typical examples, please refer to Refs. [25, 29−33].When solving the squared amplitudes
|M([n],[n′])|2 , we must also sum the polarization vectors or tensors of the heavy quarkonia. The polarization sum for the spin-tripletn3S1 or spin-singletn1P1 states with momentum p is given by [3]∑Jzεμεμ′=Πμμ′≡−gμμ′+pμpμ′p2,
(11) where
Jz=Sz andLz denoten3S1 andn1P1 states, respectively. The sum over the polarization tensors ofn3PJ states can be obtained by [3]ε(0)μνε(0)∗μ′ν′=13ΠμνΠμ′ν′,∑Jzε(1)μνε(1)∗μ′ν′=12(Πμμ′Πνν′−Πμν′Πμ′ν),∑Jzε(2)μνε(2)∗μ′ν′=12(Πμμ′Πνν′+Πμν′Πμ′ν)−13ΠμνΠμ′ν′.
(12) -
For numerical evaluations, the masses of the constituent charm and bottom quarks for the heavy quarkonia
|(Q¯Q′)[n]⟩ and|(Q′ˉQ)[n′]⟩ are shown in Table 1. The mass of the higher excited heavy quarkonium is set to be the sum of the masses of its constituent quarks at the same n-th order. This is assured by the gauge invariance of amplitudes within the NRQCD framework. The radial wave functions at the originR|(Q¯Q′)[nS]⟩(0) and their first derivatives at the originR′|(QˉQ′)[nP]⟩(0) for heavy quarkonia|(Q¯Q′)[n]⟩ were calculated under five different potential models in our previous study [26]. Because the Buchmüller and Tye potential (BT-potential) model [34, 35] has the correct two-loop short-distance behavior in the QCD, we adopt the results of the BT-potential in this paper, which are shown in Table 1. The uncertainties ofR|(QˉQ)[nS]⟩(0) andR′|(QˉQ)[nP]⟩(0) are induced by the corresponding varying constituent quark masses. They are considered when we discuss the uncertainties of the total cross sections caused by the varying quark masses in Sec. III.C. The LO running strong coupling constantαs=0.26 is adopted for|(cˉc)[n]⟩ and|(bˉc)[n]⟩ , andαs=0.18 for|(bˉb)[n]⟩ . Other parameters are derived from the PDG [36].mc ,|R|(cˉc)[nS]⟩(0)|2 mc ,|R′|(cˉc)[nP]⟩(0)|2 n=1 1.48 ± 0.1,2.458+0.227−0.327 1.75 ± 0.1,0.322+0.077−0.068 n=2 1.82 ± 0.1,1.671+0.115−0.107 1.96 ± 0.1,0.224+0.012−0.012 n=3 1.92 ± 0.1,0.969+0.063−0.057 2.12 ± 0.1,0.387+0.045−0.042 mb ,|R|(bˉb)[nS]⟩(0)|2 mb ,|R′|(bˉb)[nP]⟩(0)|2 n=1 4.71 ± 0.2,16.12+1.28−1.23 4.94 ± 0.2,5.874+0.728−0.675 n=2 5.01 ± 0.2,6.746+0.598−0.580 5.12 ± 0.2,2.827+0.492−0.432 n=3 5.17 ± 0.2,2.172+0.178−0.155 5.20 ± 0.2,2.578+0.187−0.186 mc /mb ,|R|(cˉb)[nS]⟩(0)|2 mc /mb ,|R′|(cˉb)[nP]⟩(0)|2 n=1 1.45 ± 0.1 /4.85± 0.2,3.848+0.238−0.225 1.75 ± 0.1 /4.93± 0.2,0.518+0.122−0.105 n=2 1.82 ± 0.1 /5.03± 0.2,1.987+0.116−0.118 1.96 ± 0.1 /5.13± 0.2,0.500+0.036−0.036 n=3 1.96 ± 0.1 /5.15± 0.2,1.347+0.079−0.082 2.15 ± 0.1 /5.25± 0.2,0.729+0.080−0.075 Table 1. Masses (units: GeV) of the constituent heavy quarks, the radial wave functions at the origin
|R|(QˉQ′)[nS]⟩(0)|2 (units: GeV3 ), and their first derivatives at the origin|R′|(QˉQ′)[nP]⟩(0)|2 (units: GeV5 ) under the BT-potential model [26]. Note: the uncertainties ofR|(QˉQ)[nS]⟩(0) andR′|(QˉQ)[nP]⟩(0) are induced by the corresponding varying constituent quark masses. -
The total cross sections for the production of the higher excited heavy quarkonium pair in
e−e+→Z0→|(Q¯Q′)[n]⟩+|(Q′ˉQ)[n′]⟩ (Q,Q′=c - or b-quarks) at CM energy√s=91.1876 GeV are listed in Tables 2−4 for double charmonium, double bottomonium, andBc pairs, respectively. The uncertainties are caused by the varying quark masses. The charm quark massmc has the variation±0.1 GeV, and the bottom quark massmb has the variation±0.2 GeV. In Tables 2−4, the top three rankings are marked in bold. For the double charmonium channels, we always haveσ(|(cˉc)[n3S1]⟩+|(cˉc)[n′1P1]⟩)>σ(|(cˉc)[n1S0]⟩+|(cˉc)[n′3P2]⟩)>σ(|(cˉc)[n1S0]⟩+|(cˉc)[n′3P0]⟩) at the same nth level. For double bottomonium channels, the largest cross sections can either be|(bˉb)[n1S0]⟩+|(bˉb)[n′3S1]⟩ or|(bˉb)[n3S1]⟩+|(bˉb)[n′3P2]⟩ . ForBc pair channels, the largest cross section can either be|(cˉb)[n3S1]⟩+|(bˉc)[n′3S1]⟩ or|(cˉb)[n3S1]⟩+|(bˉc)[n′3P2]⟩ . Note that in Tables 2−4, we have redundant data in the row ofσ(|(QˉQ′)[n3S1]⟩+|(QˉQ′)[n′3S1]⟩) , i.e., the cross sections are the same when n andn′ are exchanged. This is also a check for our extensive calculations.[n]+[n′] 1+1 1+2 1+3 2+1 2+2 2+3 3+1 3+2 3+3 σ([n1S0]+[n′3S1]) 18.95+3.64−4.70 11.32+1.98−2.20 6.632+1.121−1.250 13.91+2.24−2.55 8.176+1.19−1.04 4.770+0.672−0.579 8.594+1.308−1.506 5.032+0.694−0.598 2.933+0.390−0.332 σ([n3S1]+[n′3S1]) 64.16+12.29−15.87 42.55+7.09−8.00 25.63+4.08−4.64 42.55+7.09−8.00 27.62+4.01−3.49 16.55+2.29−1.98 25.63+4.08−4.64 16.55+2.29−1.98 9.900+1.308−1.099 σ([n1S0]+[n′1P1]) 60.76+4.49−7.35 18.71+1.35−0.57 25.47+0.16−1.04 20.42+1.32−1.43 10.07+1.04−0.80 13.69+0.26−0.19 11.63+0.72−0.76 5.729+0.572−0.468 7.792+0.184−0.128 σ([n3S1]+[n′1P1]) 454.0+39.1−59.2 227.1+4.4−14.0 312.4+1.8−16.2 247.4+19.0−19.7 123.8+10.2−8.4 170.6+0.6−0.2 141.3+10.4−10.7 70.77+6.18−5.00 97.48+1.11−0.41 σ([n1S0]+[n′3P0]) 137.0+9.9−16.4 68.03+2.17−5.01 93.16+0.70−3.71 72.27+4.44−4.85 35.87+3.50−2.93 49.10+1.04−0.79 40.87+2.47−2.65 20.29+2.09−1.71 27.76+0.73−0.53 σ([n1S0]+[n′3P1]) 18.87+3.98−4.27 10.56+0.41−0.90 15.74+1.69−2.28 12.37+2.30−2.10 6.835+0.136−0.117 10.11+0.89−0.82 7.432+1.327−1.213 4.093+0.058−0.038 6.039+0.496−0.448 σ([n1S0]+[n′3P2]) 284.2+22.2−35.3 140.9+3.8−9.7 192.4+0.5−8.5 153.2+10.5−11.2 75.96+5.51−6.91 103.8+1.0−1.6 87.22+5.69−5.97 43.26+3.39−4.16 59.13+0.75−1.18 σ([n3S1]+[n′3P0]) 44.03+4.60−6.38 21.89+0.95−0.01 30.20+0.80−2.2 25.92+2.31−2.33 12.76+0.85−0.69 17.50+0.27−0.17 15.15+1.28−1.27 7.437+0.542−0.431 10.18+0.12−0.04 σ([n3S1]+[n′3P1]) 17.62+3.57−3.88 10.79+0.29−0.80 17.07+1.59−2.26 9.625+1.822−1.663 5.909+0.105−0.086 9.354+0.774−0.711 5.492+1.019−0.926 3.374+0.048−0.032 5.344+0.423−0.380 σ([n3S1]+[n′3P2]) 57.51+8.25−10.02 24.83+0.06−1.18 38.15+2.66−4.20 22.09+3.70−3.42 13.26+0.08−0.01 20.55+1.33−1.19 12.46+2.08−1.92 7.521+0.015−0.069 11.69+0.73−0.64 Table 2. Total cross sections (units:
×10−5fb ) for the production of higher excited charmonium pairs ine+e−→Z0→ |(cˉc)[n]⟩+|(cˉc)[n′]⟩ at√s=91.1876 GeV. The uncertainties are caused by the charm quark mass varying by 0.1 GeV, where effects of the uncertainties of theR|(cˉc)[nS]⟩(0) andR′|(cˉc)[nP]⟩(0) induced by varying masses are also considered (see Table 1 for explicit values). The top three rankings are marked in bold.[n]+[n′] 1+1 1+2 1+3 2+1 2+2 2+3 3+1 3+2 3+3 σ([n1S0]+|[n′3S1]) 428.6+67.9−60.9 173.3+29.4−26.9 54.82+8.92−7.61 184.3+30.7−27.6 74.40+13.23−11.87 23.53+4.03−3.45 60.18+9.54−8.15 24.28+4.12−3.53 7.675+1.25−1.02 σ([n3S1]+[n′3S1]) 119.9+18.4−16.6 49.84+8.14−7.08 16.00+2.47−2.16 49.84+8.14−7.08 20.69+3.58−3.23 6.634+1.097−0.946 16.00+2.47−2.16 6.634+1.097−0.946 2.127+0.336−0.274 σ([n1S0]+[n′1P1]) 53.27+1.46−1.51 22.83+1.77−1.68 19.80+0.25−0.28 21.08+0.82−0.68 9.036+0.809−0.778 7.835+0.024−0.007 6.601+0.225−0.119 2.829+0.238−0.206 2.453+0.033−0.019 σ([n3S1]+[n′1P1]) 61.46+4.22−4.05 26.91+3.25−2.95 23.55+0.60−0.63 24.90+1.99−1.72 10.90+1.44−1.32 9.543+0.347−0.374 7.891+0.586−0.434 3.454+0.437−0.373 3.024+0.093−0.075 σ([n1S0]+|([n′3P0]) 9.215+0.141−0.160 4.010+0.262−0.256 3.501+0.081−0.088 3.495+0.087−0.067 1.520+0.114−0.113 1.327+0.015−0.021 1.068+0.020−0.004 0.465+0.032−0.028 0.405+0.011−0.009 σ([n1S0]+[n′3P1]) 14.31+1.60−1.60 6.331+1.041−0.925 5.568+0.362−0.368 6.049+0.725−0.635 2.672+0.463−0.412 2.349+0.172−0.176 1.959+0.220−0.176 0.865+0.143−0.121 0.760+0.050−0.045 σ([n1S0]+|[n′3P2]) 28.15+1.38−1.35 12.15+1.22−1.13 10.57+0.071−0.087 11.36+0.69−0.59 4.903+0.55−0.51 4.265+0.077−0.091 3.591+0.199−0.136 1.550+0.166−0.142 1.349+0.017−0.010 σ([n3S1]+[n′3P0]) 68.37+6.17−5.83 30.69+4.38−3.92 27.16+1.24−1.26 27.39+2.78−2.41 12.29+1.90−1.70 10.88+0.61−0.63 8.629+0.824−0.639 3.871+0.574−0.486 3.425+0.173−0.150 σ([n3S1]+[n′3P1]) 117.5+13.6−15.0 54.03+9.03−7.98 48.32+3.24−3.25 46.76+5.93−5.16 21.50+3.85−3.40 19.23+1.50−1.51 14.68+1.75−1.44 6.748+1.166−0.981 6.034+0.436−0.391 σ([n3S1]+[n′3P2]) 230.6+23.6−22.2 104.8+16.2−14.4 93.29+5.3−10.5 91.99+10.5−9.08 41.81+6.97−6.19 37.20+2.49−2.53 28.90+3.12−46 13.13+2.11−1.78 11.69+0.72−0.63 Table 3. Total cross sections (units:
×10−4fb ) for the production of higher excited bottomonium pairs ine+e−→Z0→ |(bˉb)[n]⟩+|(bˉb)[n′]⟩ at√s=91.1876 GeV. The uncertainties are caused by the bottom quark mass varying by 0.2 GeV, where effects of the uncertainties of theR|(bˉb)[nS]⟩(0) andR′|(bˉb)[nP]⟩(0) induced by varying masses are also considered (see Table 1 for explicit values). The top three rankings are marked in bold.[n]+[n′] 1+1 1+2 1+3 2+1 2+2 2+3 3+1 3+2 3+3 σ([n1S0]+[n′3S1]) 634.6+47.8−45.8 212.9+5.7−3.9 127.9+2.3−1.6 220.3+5.9−41 72.63+0.30−0.29 43.41+0.47−0.57 133.9+2.4−1.7 43.89+0.47−0.67 26.19+1.34−0.58 σ([n3S1]+[n′3S1]) 1150+87−84 397.2+10.5−7.4 240.5+4.3−3.1 397.2+10.5−7.4 136.3+0.6−0.1 82.33+0.72−0.94 240.5+4.3−3.1 82.33+0.72−0.94 49.69+0.75−1.00 σ([n1S0]+[n′1P1]) 4.810+0.399−0.406 2.134+0.154−0.141 1.405+0.033−0.045 2.747+0.021−0.024 1.357+0.148−0.129 1.042+0.035−0.034 1.840+0.016−0.022 0.926+0.100−0.089 0.729+0.027−0.026 σ([n3S1]+[n′1P1]) 11.77+1.10−1.12 8.984+0.985−0.784 6.849+0.344−0.256 2.878+0.182−0.174 2.122+0.128−0.114 2.510+0.013−0.009 1.586+0.1080.105− 1.153+0.057−0.053 1.355+0.008−0.010 σ([n1S0]+[n′3P0]) 15.27+1.10−1.12 12.01+1.67−1.35 14.96+1.37−1.12 4.430+0.205−0.226 3.500+0.328−0.300 4.378+0.217−0.204 2.536+0.140−0.157 2.006+0.163−0.155 2.512+0.095−0.097 σ([n1S0]+[n′3P1]) 6.162+0.701−0.448 4.751+0.443−0.352 5.659+0.242−0.174 1.451+0.151−0.145 1.161+0.041−0.038 1.419+0.014−0.0137 0.780+0.093−0.090 0.632+0.008−0.013 0.781+0.017−0.019 σ([n1S0]+[n′3P2]) 3.093+0.253−0.236 1.326+0.214−0.163 1.802+0208−0.159 0.541+0.050−0.049 0.472+0.025−0.023 0.628+0.009−0.007 0.266+0.030−0.0296 0.236+0.007−0.007 0.319+0.001−0.002 σ([n3S1]+[n′3P0]) 454.2+15.7−30.8 239.3+56.8−42.4 213.1+33.5−26.1 153.3+12.5−10.78 79.95+16.38−12.90 70.46+8.89−7.40 91.71+6.72−5.99 47.74+9.32−7.48 41.98+4.91−4.20 σ([n3S1]+[n′3P1]) 270.1+16.2−15.8 180.1+25.2−20.0 194.7+15.7−12.5 84.10+2.73−2.56 55.80+5.52−4.81 60.11+2.53−2.27 49.46+1.83−1.98 32.77+2.88−2.60 35.27+1.11−1.08 σ([n3S1]+[n′3P2]) 1133+16−22 394.1+89.4−67.1 595.3+80.1−62.6 376.9+22.5−19.1 207.9+37.3−29.8 193.6+19.9−16.7 224.8+11.5−10.2 123.7+21.0−17.1 114.9+20.7−9.3 Table 4. Total cross sections (units:
×10−3fb ) for the production of higher excitedBc pairs ine+e−→Z0→|(cˉb)[n]⟩+|(bˉc)[n′]⟩ at√s=91.1876 GeV. The uncertainties are caused by the charm quark mass varying by 0.1 GeV and bottom quark mass by 0.2 GeV, where effects of the uncertainties of theR|(cˉb)[nS]⟩(0) andR′|(cˉb)[nP]⟩(0) caused by varying masses are also considered (see Table 1 for explicit values). The top three rankings are marked in bold.As shown in Tables 2−4, compared with the ''
1+1 '' configuration, the cross sections for the production of the higher excited heavy quarkonium pair are sizable. Because most of higher excited heavy quarkonia will decay to ground states forn=1 , their contribution must be considered carefully when we study the production rates of the ground states.● For the top three double charmonium channels
|(cˉc)[n3S1]⟩+|(cˉc)[n′1P1]⟩ ,|(cˉc)[n1S0]⟩+|(cˉc)[n′3P2]⟩ , and|(cˉc)[n1S0]⟩+|(cˉc)[n′3P0]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 50%, 69%, 54%, 27%, 38%, 31%, 16%, and 21% of the cross section of the[1]+[1] configuration, respectively. An interesting observation is that the top three double charmonium channels for other configurations have very similar ratios to that of the[1]+[1] configuration.● For the double bottomonium channel
|(bˉb)[n1S0]⟩+|(bˉb)[n′3S1]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 40%, 13%, 43%, 17%, 5.5%, 14%, 5.7%, and 1.8% of the cross section of the[1]+[1] configuration, respectively.For the double bottomonium channel
|(bˉb)[n3S1]⟩+|(bˉb)[n′3P1]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 46%, 41%, 40%, 18%, 16%, 12%, 5.7%, and 5.1% of the cross section of the[1]+[1] configuration, respectively.For the double bottomonium channel
|(bˉb)[n3S1]⟩+|(bˉb)[n′3P2]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 45%, 40%, 40%, 18%, 16%, 13%, 5.7%, and 5.1% of the cross section of the[1]+[1] configuration, respectively.● For the
Bc pair channel|(cˉb)[n1S0]⟩+|(bˉc)[n′3S1]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 34%, 20%, 35%, 11%, 6.8%, 21%, 6.9%, and 4.1% of the cross section of the[1]+[1] configuration, respectively.For the
Bc pair channel|(cˉb)[n3S1]⟩+|(bˉc)[n′3S1]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 35%, 21%, 35%, 18%, 7.2%, 21%, 7.2%, and 4.3% of the cross section of the[1]+[1] configuration, respectively.For the
Bc pair channel|(cˉb)[n3S1]⟩+|(bˉc)[n′3P2]⟩ , the total cross sections of the[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] ,[3]+[1] ,[3]+[2] , and[3]+[3] configurations are about 35%, 53%, 33%, 18%, 17%, 20%, 11%, and 10% of the cross section of the[1]+[1] configuration, respectively.When CEPC is running in the Z factory operation mode, its designed integrated luminosity with two interaction points can reach as high as
16ab−1 [1]. Thus, we can estimate the events of the production of double higher excited heavy quarkonia. We show the events in Table 5 for the top three channels in Tables 2−4 as an illustration.[n]+[n′] 1+1 1+2 1+3 2+1 2+2 2+3 3+1 3+2 3+3 |(cˉc)[n3S1]⟩+|(cˉc)[n′1P1]⟩ 73 37 50 40 20 27 23 11 16 |(cˉc)[n1S0]⟩+|(cˉc)[n′3P2]⟩ 45 22 30 24 12 16 14 7 9 |(cˉc)[n1S0]⟩+|(cˉc)[n′3P0]⟩ 22 11 15 12 6 8 7 3 4 |(bˉb)[n1S0]⟩+|(bˉb)[n′3S1]⟩ 686 277 88 295 119 38 96 39 12 |(bˉb)[n3S1]⟩+|(bˉb)[n′3P2]⟩ 369 168 149 147 67 60 46 21 19 |(bˉb)[n3S1]⟩+|(bˉb)[n′3S1]⟩ 192 80 26 80 33 11 26 11 3 |(cˉb)[n3S1]⟩+|(bˉc)[n′3S1]⟩ 1.84×104 6.36×103 3.85×103 6.36×103 2.18×103 1.32×103 3.85×103 1.32×104 7.95×102 |(cˉb)[n3S1]⟩+|(bˉc)[n′3P0]⟩ 1.81×104 6.30×103 9.51×103 6.02×103 3.32×103 3.09×103 3.59×103 1.98×103 1.84×103 |(cˉb)[n1S0]⟩+|(bˉc)[n′3S1]⟩ 1.02×104 3.42×103 2.06×103 3.42×103 1.17×103 6.98×102 2.06×103 6.98×102 4.21×102 In Figs. 2−4, we depict the distribution dσ/dcosθ at
√s=91.1876 GeV for the production of double higher excited charmonium, double higher excited bottomonium, and higher excitedBc pairs, respectively. The distribution dσ/dcosθ can be obtained easily using the differential phase space in Eq. (5). Note that θ is the angle between the momentum→p1 of the electron and the momentum→q1 of the heavy quarkonium. Here, we only show parts of the configurations[n]+[n′] of top three ranking channels. We show that the largest differential cross section dσ/dcosθ can be obtained atθ=90∘ for the configurations depicted in Fig. 2, and the minimum is achieved atθ=90∘ for the configurations in Fig. 3. In Fig. 4, for doubleBc pair production, we obtain the largest differential cross section dσ/dcosθ near (but not precisely at)θ=90∘ in the left and right plots and obtain the minimum near (but not precisely at)θ=90∘ in the middle plot. The plots, particularly the middle and right ones in Fig. 4, does not indicate the symmetry of θ as those in Figs. 2 and 3.Figure 2. (color online) Differential angle distributions of cross sections dσ/dcosθ at
√s=91.1876 GeV fore−e+→Z0→ |(cˉc)[n]⟩+|(cˉc)[n′]⟩ . The dash-dotted black, dotted blue, dashed green, solid red, diamond cyan, and cross magenta lines are for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] configurations of the|(cˉc)[n3S1]⟩+|(cˉc)[n′1P1]⟩ channel (left),|(cˉc)[n1S0]⟩+ |(cˉc)[n′3P2]⟩ channel (middle), and|(cˉc)[n1S0]⟩+|(cˉc)[n′3P0]⟩ channel (right), respectively.Figure 3. (color online) Differential angle distributions of cross sections dσ/dcosθ at
√s=91.1876 GeV for the channele−e+→Z0→|(bˉb)[n]⟩+|(bˉb)[n′]⟩ . The dash-dotted black, dotted blue, dashed green, solid red, diamond cyan, and cross magenta lines are for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] configurations of the|(bˉb)[n1S0]⟩+|(bˉb)[n′3S1]⟩ channel (left) and|(bˉb)[n3S1]⟩+|(bˉb)[n′3P2]⟩ channel (middle) but for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[2] ,[2]+[3] ,[3]+[3] configurations of the|(bˉb)[n3S1]⟩+|(bˉb)[n′3S1]⟩ channel (right), respectively.Figure 4. (color online) Differential angle distributions of cross sections dσ/dcosθ at
√s=91.1876 GeV for the channele−e+→Z0→|(cˉb)[n]⟩+|(bˉc)[n′]⟩ . The dash-dotted black, dotted blue, dashed green, solid red, diamond cyan, and cross magenta lines are for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[2] ,[2]+[3] ,[3]+[3] configurations of the|(cˉb)[n3S1]⟩+|(bˉc)[n′3S1]⟩ channel (left) but for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] configurations of the|(cˉb)[n3S1]⟩+|(bˉc)[n′3P2]⟩ channel (middle) and|(cˉb)[n1S0]⟩+|(bˉc)[n′3S1]⟩ channel (right), respectively. -
To obtain reliable results in LO calculation, we should consider the main uncertainty sources of the cross sections. For values of the input parameters, the fine-structure constant α, Weinberg angle
θW , Fermi constantGF , and mass and width of theZ0 boson are relatively precise. The non-perturbative matrix elements are an overall factor when the quark mass is fixed. In the following, we probe the uncertainties caused by the masses of constituent heavy quarks, and the running coupling constantαs(μ) , which is related to the renormalization scale μ.The uncertainties of total cross sections caused by the varying quark masses are presented in Tables 2−4 for the production of double higher excited charmonium, double higher excited bottomonium, and higher excited
Bc pairs, respectively. We adopt the mass deviations of 0.1 GeV formc (changing by about 7%) and 0.2 GeV formb (changing by about 4%). Here, the effects of uncertainties of the radial wave function at the originR|(QˉQ)[nS]⟩(0) and its first derivative at the originR′|(QˉQ)[nP]⟩(0) induced by the varying quark masses are also considered. The uncertainties from the radial wave function and its first derivative at the origin induced by varying quark masses were calculated in our previous paper using the BT-potential model [26], which we show explicitly in Table 1. The table shows that a 7% change in the charm quark mass can result in a correction of up to 20% in the total cross section.In Figs. 5−7, we present total cross sections σ as a function of the renormalization scale μ at
√s=91.1876 GeV for the production of the double higher excited charmonium, double higher excited bottomonium, and higher excitedBc pairs, respectively. We observe that all the cross sections decrease as the renormalization scale μ increases. The NLO corrections might improve the μ dependence.Figure 5. (color online) Total cross sections σ as a function of the renormalization scale μ at
√s=91.1876 GeV for the channele−e+→Z0→|(cˉc)[n]⟩+|(cˉc)[n′]⟩ . The dash-dotted black, dotted blue, dashed green, solid red, diamond cyan, and cross magenta lines are for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] configurations of the|(cˉc)[n3S1]⟩+|(cˉc)[n′1P1]⟩ channel (left),|(cˉc)[n1S0]⟩+|(cˉc)[n′3P2]⟩ channel (middle) and|(cˉc)[n1S0]⟩+|(cˉc)[n′3P0]⟩ channel (right), respecttively.Figure 6. (color online) Total cross sections σ as a function of the renormalization scale μ at
√s=91.1876 GeV for the channele−e+→Z0→|(bˉb)[n]⟩+|(bˉb)[n′]⟩ . The dash-dotted black, dotted blue, dashed green, solid red, diamond cyan, and cross magenta lines are for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] configurations of the|(bˉb)[n1S0]⟩+|(bˉb)[n′3S1]⟩ channel (left) and|(bˉb)[n3S1]⟩+|(bˉb)[n′3P2]⟩ channel (middle), but for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[2] ,[2]+[3] ,[3]+[3] configurations of the|(bˉb)[n3S1]⟩+|(bˉb)[n′3S1]⟩ channel (right), respectively.Figure 7. (color online) Total cross sections σ as a function of the renormalization scale μ at
√s=91.1876 GeV for the channele−e+→Z0→|(cˉb)[n]⟩+|(bˉc)[n′]⟩ . The dash-dotted black, dotted blue, dashed green, solid red, diamond cyan, and cross magenta lines are for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[2] ,[2]+[3] ,[3]+[3] configurations of the|(cˉb)[n3S1]⟩+|(bˉc)[n′3S1]⟩ channel (left) but for the[1]+[1] ,[1]+[2] ,[1]+[3] ,[2]+[1] ,[2]+[2] ,[2]+[3] configurations of the|(cˉb)[n3S1]⟩+|(bˉc)[n′3P2]⟩ channel (middle) and|(cˉb)[n1S0]⟩+ |(bˉc)[n′3S1]⟩ channel (right), respectively.
Production of higher excited quarkonium pair at the super Z factory
- Received Date: 2023-12-21
- Available Online: 2024-07-15
Abstract: The heavy constituent quark pair of the heavy quarkonium is produced perturbatively and subsequently undergoes hadronization into the bound state non-perturbatively. The production of the heavy quarkonium is essential to testing our understanding of quantum chromodynamics (QCD) in both perturbative and non-perturbative aspects. The electron-positron collider will provide a suitable platform for the precise study of the heavy quarkonium. The higher excited heavy quarkonium may contribute significantly to the ground states, which should be considered for sound estimation. We study the production rates of the higher excited states quarkonium pair in