Associated charmonium-bottomonium production in a single boson $e^+e^-$ annihilation

The production cross sections of $J/\psi~\eta_b$, $\Upsilon\;\eta_c$ pairs in a single boson $e^+e^-$ annihilation have been studied in a wide range of energies, which will be achieved at future $e^+e^-$ colliders. The main color singlet contributions to the production processes are taken into account, including the one loop QCD contribution.

In the project of muon collider, it is planned to implement the µ + µ − collisions at energies from 3 TeV to 14 TeV [9], which are beyond the energy range studied in this work.
It is worth to note that the decays of Z-boson to the charmonium and the bottomonium may be of some interest for the experiments at the LHC, see [10].
In our previous studies we already have investigated the paired B c production [11], as well as the J/ψ J/ψ and the J/ψ η c pair production around the Z mass within the NLO approximation [12]. We have found that for these processes the loop corrections essentially contribute to the cross section values. This result is in agreement with the studies of other research groups investigated the paired quarkonium production in the e + e − annihilation. Both cases of J/ψ η b and Υ η c production are special, because the tree level diagrams with the single gluon exchange can not contribute to the production of cc and bb pairs in the color singlet states, and therefore the lowest order QCD contribution to these processes contains loops. This one-loop contribution is of the order of O(α 2 α 4 s ), that is why it makes sense to investigate it with the purely electromagnetic J/ψ η b and Υ η c production which is of the order of O(α 4 ), since α 2 s ∼ α.
When studying these processes, one cannot ignore the discussion of the role of color octet contributions. Indeed, from one side the paired color octet production is suppressed as v 2 cc v 2 bb ∼ 0.03, where v cc and v bb are velocities of quarks inside charmonium and botommonium, correspondingly. From the other side, it has no additional suppression by α 2 s , going through the tree level QCD diagrams. Therefore a color octet mechanism could essentially contribute to the discussed quarkonium-pair production. It deserves a separate detailed consideration, which goes beyond the scope of the current study.
Another problem that is outside the scope of this study is the relativistic effects caused by the use of relativistic or relativized wave functions. It is known that accounting for such effects can crucially change the predictions, especially for the charmonium-pair production in the e + e − annihilation, where accounting for the relativistic effects decreases the virtuality of the intermediate gluon, and therefore essentially increases the cross section value (see for example [5]).
Another examples of influence of the relativistic effects on the paired quarkonium production could be found in [13]. For the processes discussed in this study we do not expect such a huge change of the cross section due to relativistic effects, as for the paired charmonium production, but, of course, it does not eliminate the problem, and such effects should be thoroughly investigated.
It is worth to mention that the one loop QCD diagrams do not contribute to the production of J/ψ Υ-pair in a single boson e + e − -annihilation. In leading order such a process goes via the electroweak Z boson decay only. As we planned the current work as a continuation of our previous studies on the one loop QCD corrections, in a sense, this process is slightly out of our interest. However, we keep it in our consideration for comparison with J/ψ η b and Υ η c production. Thus, the following processes are considered in this study: (1)

II. CALCULATION TECHNIQUE
The production of the pair of charmonium and bottomonium in a single boson annihilation is constrained by several selection rules, which are discussed below.
The production of the J/ψ Υ pairs is not allowed in the photon exchange, as well as it does not go via the vector part of Z vertex due to the charge parity conservation. The J/ψ Υ pairs are produced via the interaction with the axial part of Z vertex only. As concerned the η b η c pair production, it goes neither via the photon exchange, nor via the Z-boson exchange: the photon decay to the η b η c pair is forbidden due to the charge parity conservation, and the Z decay to the η b η c pair is forbidden due to the combined CP parity conservation. At the same time the vector-pseudoscalar (VP) pairs: J/ψ η b and Υ η c are produced via the exchange of both the photon and the Z-boson. These selection rules were reproduced directly in our calculations.
As we already pointed out in the Introduction the tree level diagrams describing the single gluon exchange do not contribute to the discussed processes, and the lowest order QCD contribution to such processes contains loops. This one-loop QCD contribution is of the order of O(α 2 α 4 s ) and therefore it could be comparable with the pure electroweak tree level contribution, which is of the order of O(α 4 ) ∼ O(α 2 α 4 s ). Thus, when studying these processes, one should take into account the electroweak contribution (EW) of the order of O(α 4 ), the one loop QCD contribution of the order of O(α 2 α 4 s ), and the interference between them of the order of O(α 3 α 2 s ): For the more detailed study of the processes we consider the amplitudes with different intermediate bosons separately: The production of double heavy quarkonia is described in the framework of nonrelativistic QCD (NRQCD). This formalism allows to factor out the perturbative degrees of freedom and therefore separate the production mechanism into hard and soft subprocesses, using the hierarchy of scales for the quarkonia, which is m q >> m q v, m q v 2 , Λ QCD , where m q is the heavy quark mass and v is the velocity of heavy quark in the quarkonium. The hard subprocess corresponds to the perturbative production of qq-pair, while the soft subprocess corresponds to the fusion of quarks into the bound state.
To compute the matrix elements for the studied processes, we start from the matrix element for e + e − → c(p c )c(pc)b(p b )b(pb) with heavy quarks and antiquarks on their mass shells: p 2 c = p 2 c = m 2 c and p 2 b = p 2 b = m 2 b . As we put v = 0 before the projection onto the bound states, the momentum P of charmonium and the momentum Q of bottomonium are related with the heavy quark momenta as follows: To construct the bound states we replace the spinor products v(pq)ū(p q ) by the appropriate covariant projectors for color-singlet spin-singlet and spin-triplet states: where J/ψ and Υ are the polarizations of the J/ψ and Υ mesons, satisfying the following constraints: J/ψ · J/ψ * = −1, J/ψ · P = 0, Υ · Υ * = −1 and Υ · Q = 0.

These operators close the fermion lines into traces. The examples of diagrams contributing
to the process e + e − → J/ψ η b are shown in Figure 1.
The factorized matrix elements have the following form: where M µ J/ψ η b , M µ Υ ηc , and M µν J/ψ Υ are the hard production amplitudes of two quark-antiquark pairs projected onto the quark-antiquark states with zero relative velocities and the appropriate quantum numbers by projectors (5) and (6). The NRQCD matrix elements for annihilation decays defined in [14]. The numerical values of these matrix elements can be estimated from the experimental data on decays [1,15,16], or adopted from the potential models, such as [17], using the relation O ≈ Nc 2π |R(0)| 2 , where R(0) is the quarkonium wave function at origin (see Table A

III. WORKFLOW
The diagrams and the corresponding analytic expressions are generated with the FeynArtspackage [18] in Wolfram Mathematica. The electroweak contribution to the production amplitudes is determined by the tree diagrams of type (a) shown in Figure 1, whereas the one-loop diagrams of type (b) and (c) contribute to the QCD amplitudes.
We obtain 4 nonzero electroweak and 6 nonzero QCD diagrams for each subprocesses The associative production of two vector states J/ψ and Υ is described only by the tree electroweak diagrams of type (a). These results, as well as the explicitly obtained zero contribution to the process of η b η c production, are in exact agreement with the earlier discussed selection rules, providing the additional verification of the procedure.
To calculate the tree level amplitudes we use only FeynArts [18] and FeynCalc It is known that γ 5 is poorly defined in D-dimensions. In the current study the so-called naive interpretation of γ 5 was used: γ 5 anticommutes with all other matrices and therefore disappears in traces with an even number of γ 5 . In traces with an odd number of γ 5 the remaining γ 5 is moved to the right and replaced by Since ε αβσρ is contracted after the regularization procedure, we can safely treat it as 4dimensional.

IV. ANALYTICAL FORM OF THE AMPLITUDES
The relative simplicity of the considered processes makes it possible to provide the analytical expressions for the amplitudes right in the text.
The electroweak amplitudes for the processes e + e − → J/ψ η b and e + e − → Υ η c can be written as per Such a simple structure of amplitudes (11) and (12) is explained by the fact that only the vector, i. e. the photon-like part of Zqq vertex contributes to the decays Z * → J/ψ η b and The analytical expressions for the cross sections of the electromagnetic production of J/ψ η b and Υ η c pairs via the photon exchange are very simple and thus might be shown in the text: It should be noted that in the e + e − → J/ψ η b and e + e − → Υ η c processes the virtual photon transforming into the vector meson (γ * → J/ψ or γ * → Υ, see picture (a) in Figure 1 The QCD one loop contributions to the amplitudes of the discussed processes have exactly the same Lorentz structure as the electroweak contributions (11) and (12): where C 0 is the scalar three-point Passarino-Veltman function ScalarC0[s 1 , s 12 , s 2 ; m 0 , m 1 , m 2 ] defined in Package-X.
As already mentioned the virtual photon does not decay to the J/ψ Υ pair due to the charge parity conservation, while the virtual Z-boson does. The Lorentz structure of this amplitude is a little bit more complicated than presented in (11) and (12), as contains the additional polarization vector and consists of two components: . (19) The photonic parts of the electroweak amplitudes (11) and (12) (15) and (16)). The same applies for the photonic parts of the one loop QCD amplitudes (17) and (18).
It is interesting to note, that EW and QCD amplitudes for the VP-pair production have a different asymptotic behaviour: Therefore asymptotically the total cross section σ tot = σ EW + σ int + σ QCD fall off with the increase of the energy as per where the main contribution proceeds from the tree level electroweak amplitude. It is interesting to note that the tree level QCD cross section of J/ψ η c -pair production falls off with energy increasing faster than (21), namely as 1/s 4 , because the latter process is helicity suppressed (see [1] for details).

V. CROSS SECTIONS ESTIMATIONS
The numerical values of parameters used in the calculations are presented in Table I. The values of NRQCD matrix elements are adopted from the potential model [17]. The strong coupling constant is used within the two loops accuracy: where L = ln Q 2 /Λ 2 , β 0 = 11 − 2 3 N f and β 1 = 102 − 38 3 N f with N f = 5; the reference value is α S (M Z ) = 0.1179. The appropriate scale Q = √ s is chosen for α s for all investigated energies.
For sake of simplicity the fine structure constant is used in Thomson limit: α = 1/137.
As it is customary in most studies on quarkonia production within NRQCD, the masses of quarks inside the quarkonium are chosen so that their sum is equal to the quarkonium mass. Table I. The parameters used in the calculations. The NRQCD matrix elements are adopted from [17].
The cross sections reference values are given in Table II. In Figures 2,3,4, and 5 the calculated cross sections and there ratios are performed as functions of √ s.
In Figure 2 we compare the QCD and EW contributions at low energies (left) and at energies around a Z-mass (right). In Figure 3 the total cross sections including all discussed contributions are demonstrated. In Figure 4 the ratios between QCD and EW contributions are performed. The relative contribution of Z-boson annihilation to the studied processes is shown in Figure 5.  Υ η c 1.5 · 10 −3 5.2 · 10 −4 9.6 · 10 −5 1.2 · 10 −5 3.7 · 10 −5 8.1 · 10 −8 J/ψ Υ 2.6 · 10 −6 3.4 · 10 −6 4.1 · 10 −6 6.8 · 10 −6 2.3 · 10 −3 4.0 · 10 −7 As it can be concluded from the presented Figures 2,3,4, and 5, the QCD and EW subprocesses contribute differently to the total yield of J/ψ η b and Υ η c . In the J/ψ η b production the EW contribution dominates at all energies. Contrary, in the Υ η c production the EW contribution dominates only at high energies, while at energies less than 20 GeV the main contribution comes from QCD mechanism.
To understand such a strange behaviour, we suggest to look at the problem from another side, and compare the EW contributions with each other, as well as to compare the QCD contributions with each other. As it can be obtained from the expressions (11), (12), (17) and   (18) for EW and QCD amplitudes, the scalar parts of the amplitudes near the Z pole relate as follows: Thus, the ratios between the EW and QCD contributions for the discussed processes near the Z pole are essentially different, and moreover, these ratios are in inverse proportion to each other. Also it can be concluded from Figure 2, that the EW and QCD contributions approximately obey this pattern at all investigated energies: σ EW (J/ψ η b ) is at least by an order of magnitude greater, than σ EW (Υ η c ), whereas σ QCD (J/ψ η b ) is at least by an order of magnitude smaller than σ QCD (Υ η c ). If one keeps this circumstance in mind, then the behavior of the discussed contributions no longer seems so mysterious. Indeed, if at some As seen in Figure 3, both the J/ψ η b -pair and the Υ η c -pair production cross sections have a maximum near the threshold ( √ s max (J/ψ η b ) ≈ 15.6 GeV and √ s max (Υ η c ) ≈ 14.1 GeV).
The cross section ratios near the maximum take the following values: As already mentioned, near the Z pole the discussed cross sections behave completely differently: As shown in Figure 4 if the Υ η c -pair is produced the EW contribution exceeds the QCD one starting with energy about 20 GeV which agrees with (20). It should be mentioned, that the interference between the EW and QCD contributions is strong and positive at all investigated energies. Particularly if the Υ η c -pair is produced it achieves ∼48% of the total cross section when the EW and QCD cross-sections are comparable.
Since J/ψ Υ pair production goes only via the Z boson exchange it is not surprising that such a process is highly suppressed at low energies against the production of J/ψ η b -pairs and Υ η cpairs (see Figure 3). However at energies higher than 70 GeV the production cross section of J/ψ Υ pair becomes greater than the other cross sections: The Z boson exchange obviously dominates at the Z pole, and also essentially contributes to the production cross section around this pole is such a way that the its contribution to the total cross section value is greater than 20% for energies √ s > 60 GeV for the J/ψ η b -pair production process and for energies in the range 70 GeV < √ s < 150 GeV for the Υ η c -pair production process (see Figure 5).

VI. CONCLUSIONS
The exclusive production of the charmonium-bottomonium pairs (J/ψ η b , Υ η c and J/ψ Υ) has been studied in a single boson e + e − annihilation in the interaction energy range from the threshold to 2M Z within the color singlet approximation of NRQCD.
Both J/ψ η b and Υ η c productions essentially differ from the thoroughly investigated J/ψ η c production, since the main QCD contribution to these processes contains loops and occurs to be comparable in magnitude with the purely electromagnetic contribution (O(α 2 α 4 s ) v.s. O(α 4 )). This is why the QCD contribution, the electromagnetic contribution and their interference have been studied together. As concerned the J/ψ Υ-pair production, in the leading order this process goes via the electroweak Z boson exchange only. The rather simple structure of the studied amplitudes allows one to provide the analytical expressions right in the text.
It has been shown in the current study, that the QCD and EW subprocesses contribute differently to the total yield of J/ψ η b and Υ η c . In the J/ψ η b production the EW contribution obviously dominates at all energies. Contrary, in the Υ η c production the EW contribution dominates only at high energies, while at energies less than 20 GeV the main contribution comes from QCD mechanism. The suppression of one-loop QCD cross sections at high energies is explained by the fact that the one-loop QCD amplitude and the electroweak amplitude Ref 0.335 ± 0.024 0.297 ± 0.032 [15] 0