# Associated quarkonia production in a single boson e+e−superscript𝑒superscript𝑒e^{+}e^{-}italic_e start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT annihilation
**Authors**: I. N. Belov, A. V. Berezhnoy, E. A. Leshchenko
> INFN, Sezione di Genova, Italy
> SINP MSU, Moscow, Russia
> Physics department of MSU, Moscow, Russia
\setcaptionwidth \newcaptionstyle
nonumber \usecaptionmargin \onelinecaption \captiontext \captiontext
Abstract
The production cross sections of charmonia, charmonium-bottomonium and bottomonia 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 such as ILC and FCC. One loop QCD corrections to QCD and EW contributions as well as their interference are considered. The both intermediate bosons $\gamma$ and $Z$ are taken into account.
1 Introduction
Heavy quark physics have been remaining attractive for theorists and experimentalists throughout its long history. Nearly every year is now marked with discoveries in this field as a result of various experiments such as LHC, BELLE-II and the BES-III. The production of quarkonium pairs is a popular topic of discussions. One of the most intriguing researches is the observation of $J/\psi\,\eta_{c}$ pairs in the $e^{+}e^{-}$ annihilation where the experimental yield measured at BELLE and BaBar Abe et al. (2004); Aubert et al. (2005) was underestimated by the theoretical predictions Braaten and Lee (2003) by the order of magnitude. This event prompted the countless investigations Dong et al. (2012); Li and Wang (2014); Feng et al. (2019); Zhang et al. (2006); Gong and Wang (2008); Bondar and Chernyak (2005); Braguta et al. (2005); Berezhnoy and Likhoded (2007); Braguta et al. (2006); Bodwin et al. (2006); Ebert and Martynenko (2006); Berezhnoy (2008); Ebert et al. (2009); Braguta et al. (2008); Braguta (2009); Sun et al. (2010); Braguta et al. (2012); Sun et al. (2018), the results of which led to a decent level of data agreement. Another surge of interest to this topic occurred in 2020, when the LHCb Collaboration published article Aaij et al. (2020) about the observation of the structure in the $J/\psi\leavevmode\nobreak\ J/\psi$ spectrum at large statistics.
All these results have motivated us to study the processes of paired quarkonium production, specifically the production of $J/\psi\,\eta_{c}$ and $J/\psi\,J/\psi$ pairs and also $\Upsilon\,\eta_{b}$ and $\Upsilon\,\Upsilon$ pairs in the $e^{+}e^{-}$ annihilation. Proper observation of such processes is still impossible in the currently existing experiments due to low achieved collision energy. Nevertheless, current work may be of practical interest in terms of several discussed future projects, such as ILC and FCC with announced energy ranges as high as $\sqrt{s}\leavevmode\nobreak\ =\leavevmode\nobreak\ 90\div 400\leavevmode%
\nobreak\ \text{GeV}$ Koratzinos (2016) and $\sqrt{s}\leavevmode\nobreak\ =\leavevmode\nobreak\ 250\leavevmode\nobreak\ %
\text{GeV}$ Desch et al. (2019) correspondingly, aimed to investigations at the energies of the order of $Z$ -boson’s mass. As it was emphasised in the Conceptual Design Report Abada (2019), it is planned to obtain as much as $5× 10^{12}$ decays of $Z$ bosons, with energies of about $\sqrt{s}\leavevmode\nobreak\ ≈\leavevmode\nobreak\ 91\leavevmode\nobreak%
\ \text{GeV}$ , which will result into the outstanding luminosity on the facility. And such enormous amount of statistics to be obtained on FCC makes it one of the most perspective successor of B-factories for heavy hadron research field, including charmonia and bottomonia investigations in a wide energy range. We also consider studied processes interesting in terms of the $Z$ -boson decays to the charmonia and the bottomonia, which may have a certain potential for the experiments at the LHC, see Sirunyan et al. (2019).
Our previous studies involved the investigation of the $B_{c}$ pair production Berezhnoy et al. (2017), the $J/\psi\,J/\psi$ and the $J/\psi\,\eta_{c}$ pair production Berezhnoy et al. (2021) around the $Z$ mass within the NLO approximation considering only QCD contribution, as well as the production of $J/\psi\,\eta_{b}$ and $\Upsilon\,\eta_{c}$ pairs considering QCD contribution within NLO approximation and EW contribution within LO accuracy Lesh (2021). The substantial conclusion from the listed researches is that the loop corrections for investigated processes essentially affect the cross section values. This result is consistent with findings by other research groups investigating the paired quarkonium production in the $e^{+}e^{-}$ annihilation. As an extension of this study we investigate the electroweak contribution in NLO approximation to these processes and implement the derived calculation technique for bottomonia case: the $\Upsilon\,\Upsilon$ and the $\Upsilon\,\eta_{b}$ pair production.
Thus, the following processes are studied in this paper: ${e^{+}e^{-}\xrightarrow{\gamma^{*},\ Z^{*}}\leavevmode\nobreak\ J/\psi\,\eta_{%
c}}$ , ${e^{+}e^{-}\xrightarrow{Z^{*}}\leavevmode\nobreak\ J/\psi\,J/\psi}$ , ${e^{+}e^{-}\xrightarrow{\gamma^{*},\ Z^{*}}\leavevmode\nobreak\ \Upsilon\,\eta%
_{b}}$ and ${e^{+}e^{-}\xrightarrow{Z^{*}}\leavevmode\nobreak\ \Upsilon\,\Upsilon}$ .
2 Methods
The production features of quarkonia pairs in a single boson $e^{+}e^{-}$ annihilation are determined by the certain set of selection rules:
- The productions of both vector-vector pairs (VV) and pseudoscalar-pseudoscalar (PP) pairs through the intermediate photon and the vector part of $Z$ vertex are prohibited due to the charge parity conservation.
- The productions of vector-pseudoscalar (VP) pairs via the axial part of $Z$ vertex is prohibited for the very same reason.
- The PP pairs production via the axial part of $Z$ vertex is prohibited due to the combined $CP$ parity conservation.
These selection rules acted as additional verification criteria of calculations.
There are two production mechanisms for the investigated processes. The first one is the single gluon exchange, for which the tree level contribution is ${\cal O}(\alpha^{2}\alpha_{s}^{2})$ . The second production mechanism is the single photon or $Z$ boson exchange. In that case the tree level contribution is of order ${\cal O}(\alpha^{4})$ . In text we refer to this contributions as QCD LO and EW LO contributions correspondingly.
We also take into account QCD one-loop correction to both QCD LO and EW LO contributions, which are of orders ${\cal O}(\alpha^{2}\alpha_{s}^{3})$ and ${\cal O}(\alpha^{4}\alpha_{s})$ correspondingly. We refer to them as QCD NLO and EW NLO contributions.
Thus, when studying these processes, one should take into account 7 contributions to the total cross sections:
$$
|\mathcal{A}|^{2}=|\mathcal{A}^{LO}_{QCD}|^{2}+|\mathcal{A}^{LO}_{EW}|^{2}+2Re%
(\mathcal{A}^{LO}_{QCD}\mathcal{A}^{LO*}_{EW})+\\
2Re(\mathcal{A}^{NLO}_{QCD}\mathcal{A}^{LO*}_{QCD})+2Re(\mathcal{A}^{NLO}_{QCD%
}\mathcal{A}^{LO*}_{EW})+\\
2Re(\mathcal{A}^{NLO}_{EW}\mathcal{A}^{LO*}_{QCD})+2Re(\mathcal{A}^{NLO}_{EW}%
\mathcal{A}^{LO*}_{EW})+\dots\ . \tag{1}
$$
To describe the double heavy quarkonia the Nonrelativistic QCD (NRQCD) Bodwin et al. (1995) is used. The NRQCD is based on the hierarchy of scales for the quarkonia: $m_{q}>>m_{q}v,m_{q}v^{2},\Lambda_{QCD}$ , where $m_{q}$ is the mass of the heavy quark and $v$ is the heavy quark velocity in the quarkonium. Such formalism allows to divide the investigated process into the hard subprocess of heavy quarks production and the soft fusion of heavy quarks into quarkonia.
In order to construct the bound states we put $v=0$ and replace the spinor products $v(p_{\bar{q}})\bar{u}(p_{q})$ by the appropriate covariant projectors for color-singlet spin-singlet and spin-triplet states:
$$
\displaystyle\Pi_{P}(Q_{q},m_{P})=\frac{\not{Q}-2m_{P}}{2\sqrt{2}}\gamma^{5}%
\otimes\frac{\bm{1}}{\sqrt{N_{c}}}, \displaystyle\Pi_{V}(P_{q},m_{V})=\frac{\not{P}-2m_{V}}{2\sqrt{2}}\ \not{%
\epsilon}^{V}\otimes\frac{\bm{1}}{\sqrt{N_{c}}}, \tag{2}
$$
where $Q_{q}$ , $m_{P}$ and $P_{q}$ , $m_{V}$ are momenta and masses of the pseudoscalar and vector final states correspondingly. Polarization $\epsilon^{V}$ of the vector meson satisfy the following constraints: $\epsilon^{V}\leavevmode\nobreak\ ·\leavevmode\nobreak\ {\epsilon^{V}}^{*}%
\leavevmode\nobreak\ =\leavevmode\nobreak\ -1$ , $\epsilon^{V}\leavevmode\nobreak\ ·\leavevmode\nobreak\ P_{q}\leavevmode%
\nobreak\ =\leavevmode\nobreak\ 0$ . These operators enclose the fermion lines into traces.
The renormalization procedure must be applied to the one-loop contribution. The so-called „On-shell“ scheme has been used for renormalization of masses and spinors and $\overline{MS}$ scheme has been adopted for coupling constant renormalization:
$$
\displaystyle Z_{m}^{OS}=1-\frac{\alpha_{s}}{4\pi}C_{F}C_{\epsilon}\left[\frac%
{3}{\epsilon_{UV}}+4\right]+\mathcal{O}\left(\alpha_{s}^{2}\right), \displaystyle Z_{2}^{OS}=1-\frac{\alpha_{s}}{4\pi}C_{F}C_{\epsilon}\left[\frac%
{1}{\epsilon_{UV}}+\frac{2}{\epsilon_{IR}}+4\right]+\mathcal{O}\left(\alpha_{s%
}^{2}\right), \displaystyle Z_{g}^{\overline{MS}}=1-\frac{\beta_{0}}{2}\frac{\alpha_{s}}{4%
\pi}\left[\frac{1}{\epsilon_{UV}}-\gamma_{E}+\ln{4\pi}\right]+\mathcal{O}\left%
(\alpha_{s}^{2}\right), \tag{3}
$$
where $C_{\epsilon}=\left(\frac{4\pi\mu^{2}}{m^{2}}e^{-\gamma_{E}}\right)^{\epsilon}$ and $\gamma_{E}$ is the Euler constant.
The counter-terms are obtained from the leading order diagrams. The isolated singularities are then cancelled with the singular parts of the calculated counter-terms.
The FeynArts -package Hahn (2001) in Wolfram Mathematica is used to generate the diagrams and the accompanying analytical amplitudes.
In total 6 nonzero tree level EW, 10 nonzero one loop EW and 4 nonzero tree level QCD diagrams contribute to PV pair production ( $e^{+}e^{-}\leavevmode\nobreak\ \xrightarrow{\gamma^{*},Z^{*}}\leavevmode%
\nobreak\ J/\psi\,\eta_{c}$ and $e^{+}e^{-}\leavevmode\nobreak\ \xrightarrow{\gamma^{*},Z^{*}}\leavevmode%
\nobreak\ \Upsilon\,\eta_{b}$ ). The number of nonzero one-loop QCD diagrams depends on the intermediate boson type: 80 in case of virtual photon and 92 in case of virtual $Z$ boson. The VV pair production subprocesses $e^{+}e^{-}\leavevmode\nobreak\ \xrightarrow{Z^{*}}\leavevmode\nobreak\ J/\psi%
\,J/\psi$ and $e^{+}e^{-}\leavevmode\nobreak\ \xrightarrow{Z^{*}}\leavevmode\nobreak\ %
\Upsilon\,\Upsilon$ are described by 8 nonzero tree level EW diagrams, 4 nonzero tree QCD diagrams, 20 nonzero one loop EW diagrams and 86 nonzero one loop QCD diagrams.
To calculate the tree level amplitudes we use FeynArts Hahn (2001) and FeynCalc Shtabovenko et al. (2020) packages in Wolfram Mathematica, while the computation of loop amplitudes demands the following toolchain: $\texttt{FeynArts}→\texttt{FeynCalc}(\texttt{TIDL})\xrightarrow{}%
\texttt{Apart\leavevmode\nobreak\ \cite[cite]{\@@bibref{Authors Phrase1YearPhr%
ase2}{Feng:2012iq}{\@@citephrase{(}}{\@@citephrase{)}}}}→\texttt{%
FIRE\leavevmode\nobreak\ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{%
Smirnov:2008iw}{\@@citephrase{(}}{\@@citephrase{)}}}}→\texttt{X%
\leavevmode\nobreak\ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Patel:2%
016fam}{\@@citephrase{(}}{\@@citephrase{)}}}}$ .
The FeynCalc package performs all required algebraic operations with Dirac and color matrices, in particular the trace evaluation. The Passarino-Veltman reduction is carried out with the implementation of TIDL library included in FeynCalc package. The Apart function performs the additional simplification providing the partial fractioning of IR-divergent integrals. The integrals produced in the above described phases are completely reduced to master integrals by the FIRE package. At last, by using the X -package the master integrals are evaluated by substituting their analytical expressions.
The NLO amplitudes are computed using the conventional dimensional regularization (CDR) approach with a $D$ -dimensional loop and external momenta.
The so-called „naive“ prescription for $\gamma^{5}$ was implemented: $\gamma^{5}$ anticommutes with all other $\gamma$ matrices, the remaining $\gamma^{5}$ in traces with an odd number of $\gamma^{5}$ matrices is shifted to the right and then replaced by
$$
\gamma^{5}=-\frac{i}{24}\varepsilon_{\alpha\beta\sigma\rho}\gamma^{\alpha}%
\gamma^{\beta}\gamma^{\sigma}\gamma^{\rho}.
$$
The strong coupling constant was treated within the two loops accuracy:
$$
\alpha_{S}(\mu)=\frac{4\pi}{\beta_{0}L}\left(1-\frac{\beta_{1}\ln{L}}{\beta_{0%
}^{2}L}\right),
$$
where $L=\ln{\mu^{2}/\Lambda^{2}}$ , $\beta_{0}=11-\frac{2}{3}N_{f}$ , $\beta_{1}=10-\frac{38}{3}N_{f}$ , and $\alpha_{S}(M_{Z})=0.1179$ . The renormalization and coupling constant scales are set to be equal: $\mu_{R}=\mu$ . The fine structure constant is fixed at the Thomson limit: $\alpha=1/137$ . $u$ -, $d$ - and $s$ -quarks are considered massless. The numerical values of other parameters are outlined in Table 1.
3 Results
Analytical results
In this subsection we present some analytical results obtained for the discussed the amplitudes and cross sections.
The asymptotic behaviour of the amplitudes are presented below:
$$
\displaystyle\mathcal{A}^{LO}_{QCD}\sim\frac{1}{s^{2}}; \displaystyle\ \ \ \ \ \frac{\mathcal{A}^{NLO}_{QCD}}{\mathcal{A}^{LO}_{QCD}}%
\sim\alpha_{s}\left(c^{QCD}_{2}\ln^{2}\left(\frac{s}{m^{2}_{q}}\right)+c^{QCD}%
_{1}\ln\left(\frac{s}{m^{2}_{q}}\right)+c^{QCD}_{0}+c^{QCD}_{\mu}\ln\left(%
\frac{\mu}{m_{q}}\right)\right); \displaystyle\mathcal{A}^{LO}_{EW}\sim\frac{1}{s}; \displaystyle\ \ \ \ \ \frac{\mathcal{A}^{NLO}_{EW}}{\mathcal{A}^{LO}_{EW}}%
\sim\alpha_{s}\left(c^{EW}_{1}\ln\left(\frac{s}{m^{2}_{q}}\right)+c^{EW}_{0}%
\right), \tag{4}
$$
where $c^{QCD}_{i}$ and $c^{EW}_{i}$ are coefficients independent of the collision energy, the quark masses and the chosen scale.
Since the tree-level amplitudes for the investigated processes have a trivial Lorentz structure, the analytical expressions for the appropriate cross sections are quite simple:
$$
\displaystyle\sigma^{LO}_{QCD}\left(VP\right) \displaystyle=\frac{131072\ \pi^{3}\alpha^{2}\alpha_{s}^{2}e_{q}^{2}{\cal O}_{%
V}{\cal O}_{P}\left(s-16m_{q}^{2}\right)^{3/2}}{243\ s^{11/2}}\ \left(1+q_{%
\gamma Z}+q_{Z}\right), \displaystyle\sigma^{LO}_{EW}\left(VP\right) \displaystyle=\frac{32\ \pi^{3}\alpha^{4}e_{q}^{6}{\cal O}_{V}{\cal O}_{P}%
\left(s-16m_{q}^{2}\right)^{3/2}\left(s+\frac{16}{3}m_{q}^{2}\right)^{2}}{3\ m%
_{q}^{4}s^{11/2}}\left(1+q_{\gamma Z}+q_{Z}\right), \displaystyle\sigma^{LO}_{QCD}\left(V_{1}V_{2}\right) \displaystyle=\frac{1}{2!}\frac{512\ \pi^{3}\alpha^{2}\alpha_{s}^{2}{\cal O}_{%
V}^{2}\left(s-16m_{q}^{2}\right)^{5/2}}{243\ s^{9/2}}\frac{\sec^{4}{\theta_{w}%
}\left(\csc^{4}{\theta_{w}}-4\csc^{2}{\theta_{w}}+8\right)}{\left(M_{Z}^{2}-s%
\right)^{2}+\Gamma^{2}M_{Z}^{2}}, \displaystyle\sigma^{LO}_{EW}\left(V_{1}V_{2}\right) \displaystyle=\frac{1}{2!}\frac{\pi^{3}\alpha^{4}e_{q}^{4}{\cal O}_{V}^{2}%
\left(s-16m_{q}^{2}\right)^{5/2}\left(s+\frac{8}{3}m_{q}^{2}\right)^{2}}{6\ m_%
{q}^{4}s^{9/2}}\frac{\sec^{4}{\theta_{w}}\left(\csc^{4}{\theta_{w}}-4\csc^{2}{%
\theta_{w}}+8\right)}{\left(M_{Z}^{2}-s\right)^{2}+\Gamma^{2}M_{Z}^{2}}, \tag{6}
$$
where $q_{\gamma Z}$ in (6) and (7) corresponds to the interference of photonic and $Z$ bosonic amplitudes, $q_{Z}$ corresponds to the $Z$ boson annihilation amplitude squared, and $1$ stands for the photon annihilation amplitude squared:
$$
\displaystyle q_{\gamma Z} \displaystyle=\frac{\tan^{2}{\theta_{w}}\left(\csc^{2}{\theta_{w}}-4\right)%
\left(\csc^{2}{\theta_{w}}-4|e_{q}|\right)}{8\ |e_{q}|}\ \frac{s\left(s-M_{Z}^%
{2}\right)}{(M_{Z}^{2}-s)^{2}+\Gamma^{2}M_{Z}^{2}}, \displaystyle q_{Z} \displaystyle=\frac{\tan^{4}{\theta_{w}}\left(\csc^{4}{\theta_{w}}-4\csc^{2}{%
\theta_{w}}+8\right)\left(\csc^{2}{\theta_{w}}-4|e_{q}|\right)^{2}}{128\ e_{q}%
^{2}}\ \frac{s^{2}}{\left(M_{Z}^{2}-s\right)^{2}+\Gamma^{2}M_{Z}^{2}}. \tag{10}
$$
The analytical expressions for the one-loop cross sections are cumbersome, and this is why we do not present them in the text.
Numerical results
The numerical results are encapsulated in the Table 2.
As it is seen in Fig. 1, $J/\psi\,\eta_{c}$ pairs production smaller than $J/\psi\,J/\psi$ only near the $Z$ -pole, while $\Upsilon\,\eta_{b}$ pairs production greater than $\Upsilon\,\Upsilon$ at all investigated energies.
The relative QCD and EW contributions to the total cross sections are shown in Fig. 2. It is worth to note that in the charmonia production the QCD contribution is dominant up to energies of $\sim 25\leavevmode\nobreak\ \text{GeV}$ , while at higher energies the EW contribution dominates. As for the associated bottomonia production, at low energies the QCD mechanism appears to be dominant, while at energies above the $Z$ -boson threshold both contributions are comparable.
K-factors
In order to estimate the contribution of NLO corrections to the cross section the K-factors were numerically calculated for each of the production mechanisms.
The K-factors dependencies on interaction energy are shown in Fig. 3 and Fig. 4. All K-factors increase with energy as expected from (4) and (5). The K-factors for the QCD mechanism are consimilar for PV and VV pairs production, whereas K-factors for the EW mechanism are sufficiently depend on quarkonia quantum numbers. The NLO corrections to QCD mechanism contribute positively to the cross section at all investigated energies. The NLO corrections to the EW charmonia production are negative at low energies and positive at high energies. The NLO corrections to the EW bottomonia production are negative at all energies.
The obtained results highlight the significance of NLO corrections to the both discussed mechanisms.
4 Conclusions
The exclusive production of charmonia and bottomonia pairs ( $J/\psi\,\eta_{c}$ , $J/\psi\,J/\psi$ , $\Upsilon\,\eta_{b}$ and $\Upsilon\,\Upsilon$ ) in a single boson $e^{+}e^{-}$ annihilation has been investigated at interaction energies from the production threshold to $2M_{Z}$ within the QCD NLO accuracy.
It has been shown that the both studied mechanisms, namely QCD and EW ones, significantly contribute to the cross section in a wide energy range. Analytical expressions for QCD and EW amplitudes and cross sections have been obtained and numerical results for cross sections have been presented. Also, the relative contributions of production mechanisms to the total cross section have been analysed. We believe that the presented results may be of considerable interest for experiments at future $e^{+}e^{-}$ colliders.
Authors are grateful to the organizing committee of ICPPA2022 for the opportunity to make this report. This study was conducted within the scientific program of the National Center for Physics and Mathematics, section #5 „Particle Physics and Cosmology“. Stage 2023-2025. I. Belov acknowledges the support from „BASIS“ Foundation, grant No. 20-2-2-2-1.
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5mm \onelinecaptionsfalse
<details>
<summary>x1.png Details</summary>
### Visual Description
## Chart: Cross Section vs. Center-of-Mass Energy
### Overview
The image is a plot showing the cross section (sigma) of electron-positron annihilation into various final states as a function of the center-of-mass energy (sqrt(s)). The plot includes four different data series, each representing a different final state. The y-axis is on a logarithmic scale.
### Components/Axes
* **Title:** There is no explicit title on the chart.
* **X-axis:**
* Label: "√s, GeV" (Center-of-mass energy in GeV)
* Scale: Linear, ranging from approximately 0 to 150 GeV.
* Ticks: Present at approximately 50, 100, and 150 GeV.
* **Y-axis:**
* Label: "σ(e+e- → Q1 Q2), fb" (Cross section in femtobarns)
* Scale: Logarithmic.
* Ticks: Present at approximately 1, 0.001, and 10^-6.
* **Legend:** Located in the top-right corner.
* Red solid line: "CC, J/ψ ηc"
* Blue solid line: "BB, Υ ηb"
* Red dashed line: "CC, J/ψ J/ψ"
* Blue dashed line: "BB, Υ Υ"
* **Vertical Line:** A vertical line is present at approximately 95-100 GeV.
### Detailed Analysis
* **Red Solid Line (CC, J/ψ ηc):**
* Trend: Starts high on the left, decreases rapidly, then flattens out. Shows a peak around 95-100 GeV.
* Approximate values:
* At x=20 GeV, y ≈ 2 fb
* At x=50 GeV, y ≈ 0.005 fb
* Peak at x=95-100 GeV, y ≈ 0.0015 fb
* At x=150 GeV, y ≈ 1e-6 fb
* **Blue Solid Line (BB, Υ ηb):**
* Trend: Starts at a lower value than the red solid line, increases to a peak around x=40 GeV, then decreases and flattens out. Shows a peak around 95-100 GeV.
* Approximate values:
* At x=20 GeV, y ≈ 0.0002 fb
* Peak at x=40 GeV, y ≈ 0.004 fb
* At x=50 GeV, y ≈ 0.002 fb
* Peak at x=95-100 GeV, y ≈ 0.001 fb
* At x=150 GeV, y ≈ 2e-7 fb
* **Red Dashed Line (CC, J/ψ J/ψ):**
* Trend: Starts low, increases rapidly, then flattens out. Shows a peak around 95-100 GeV.
* Approximate values:
* At x=20 GeV, y ≈ 2e-7 fb
* At x=50 GeV, y ≈ 5e-7 fb
* Peak at x=95-100 GeV, y ≈ 0.0005 fb
* At x=150 GeV, y ≈ 2e-7 fb
* **Blue Dashed Line (BB, Υ Υ):**
* Trend: Starts very low, increases rapidly, then flattens out. Shows a peak around 95-100 GeV.
* Approximate values:
* At x=20 GeV, y ≈ 1e-7 fb
* At x=50 GeV, y ≈ 2e-7 fb
* Peak at x=95-100 GeV, y ≈ 0.0002 fb
* At x=150 GeV, y ≈ 5e-8 fb
### Key Observations
* All four data series show a peak in the cross section around the same center-of-mass energy (95-100 GeV).
* The solid lines (CC, J/ψ ηc and BB, Υ ηb) have higher cross sections at lower center-of-mass energies compared to the dashed lines (CC, J/ψ J/ψ and BB, Υ Υ).
* The red solid line (CC, J/ψ ηc) has the highest cross section at low energies.
* The blue dashed line (BB, Υ Υ) has the lowest cross section across the entire range.
### Interpretation
The plot likely represents the cross sections for different decay channels of a resonance particle with a mass around 95-100 GeV. The peaks in the cross sections indicate that the electron-positron annihilation is resonantly enhanced when the center-of-mass energy matches the mass of the resonance. The different final states (J/ψ ηc, Υ ηb, J/ψ J/ψ, Υ Υ) represent different decay modes of this resonance. The differences in the cross sections for different decay modes reflect the different coupling strengths of the resonance to these final states. The vertical line at approximately 95-100 GeV likely indicates the mass of a known particle, possibly the Z boson.
</details>
\captionstyle normal
Figure 1: The total cross sections dependence on the collision energy for the double charmonia and the double bottomonia production. \setcaptionmargin
5mm \onelinecaptionsfalse $\begin{array}[]{cc}∈cludegraphics[page=1,width=216.81pt]{pictures/sigma/Fig2%
.pdf}&∈cludegraphics[page=2,width=216.81pt]{pictures/sigma/Fig2.pdf}\\
(a)\leavevmode\nobreak\ J/\psi\,\eta_{c}&(b)\leavevmode\nobreak\ J/\psi\,J/%
\psi\\
∈cludegraphics[page=3,width=216.81pt]{pictures/sigma/Fig2.pdf}&%
∈cludegraphics[page=4,width=216.81pt]{pictures/sigma/Fig2.pdf}\\
(c)\leavevmode\nobreak\ \Upsilon\,\eta_{b}&(d)\leavevmode\nobreak\ \Upsilon\,%
\Upsilon\end{array}$ \captionstyle normal
Figure 2: Relative contributions of different different mechanisms to the total cross section values as a function of collision energy: the QCD mechanism (red), the EW mechanism (blue), the interference of QCD and EW production mechanisms (green). \setcaptionmargin
5mm \onelinecaptionsfalse
<details>
<summary>x6.png Details</summary>
### Visual Description
## Line Chart: QCD K-factor vs. √s
### Overview
The image is a line chart comparing the QCD K-factor for different particle combinations as a function of √s (center-of-mass energy) in GeV. Four different particle combinations are plotted: J/ψ ηc, J/ψ J/ψ, Υ ηb, and Υ Υ. The chart shows how the QCD K-factor changes with increasing energy for each combination.
### Components/Axes
* **X-axis:** √s, GeV (square root of s, measured in GeV). The axis ranges from 0 to 150 GeV, with tick marks at intervals of 50 GeV.
* **Y-axis:** QCD K-factor. The axis ranges from 0 to 5, with tick marks at integer intervals.
* **Legend:** Located in the top-left corner, the legend identifies the four data series:
* Red: J/ψ ηc
* Orange: J/ψ J/ψ
* Blue: Υ ηb
* Cyan: Υ Υ
* A vertical line is present at approximately x = 95 GeV.
### Detailed Analysis
* **J/ψ ηc (Red):** The line starts at approximately (0, 1.9) and increases to approximately (150, 4.4). The slope decreases as x increases.
* **J/ψ J/ψ (Orange):** The line starts at approximately (0, 2.0) and increases to approximately (150, 4.6). The slope decreases as x increases.
* **Υ ηb (Blue):** The line starts at approximately (0, 1.6) and increases to approximately (150, 3.0). The slope decreases as x increases.
* **Υ Υ (Cyan):** The line starts at approximately (0, 1.7) and increases to approximately (150, 3.0). The slope decreases as x increases.
### Key Observations
* The J/ψ ηc and J/ψ J/ψ combinations have higher QCD K-factors than the Υ ηb and Υ Υ combinations across the entire range of √s.
* The J/ψ J/ψ combination has the highest QCD K-factor at √s = 150 GeV.
* The Υ ηb and Υ Υ combinations have very similar QCD K-factors across the entire range of √s.
* All four lines show a positive correlation between √s and the QCD K-factor, but the rate of increase diminishes as √s increases.
### Interpretation
The chart illustrates how the QCD K-factor, a measure of the radiative corrections in quantum chromodynamics (QCD), varies with the center-of-mass energy (√s) for different quarkonium states. The higher K-factors for the J/ψ ηc and J/ψ J/ψ combinations compared to the Υ ηb and Υ Υ combinations suggest that the radiative corrections are more significant for the charmonium states (containing charm quarks) than for the bottomonium states (containing bottom quarks) within the energy range considered. The convergence of the lines at higher energies might indicate a saturation effect, where the radiative corrections become less sensitive to the energy scale. The vertical line at approximately 95 GeV does not have an explicit label, but it may represent a specific energy threshold or a point of interest for comparison.
</details>
\captionstyle normal
Figure 3: The total K-factor dependence on the collision energy in case of QCD production. \setcaptionmargin
5mm \onelinecaptionsfalse
<details>
<summary>x7.png Details</summary>
### Visual Description
## Chart: EW K-factor vs. √s, GeV
### Overview
The image is a 2D line chart displaying the relationship between the EW K-factor (Electroweak K-factor) and the square root of s (√s), measured in GeV. There are four distinct data series represented by different colored lines: red, orange, blue, and cyan. A horizontal black line is present at EW K-factor = 1.0. A vertical black line is present at √s = 90 GeV.
### Components/Axes
* **X-axis (Horizontal):** √s, GeV (Square root of s, measured in GeV). The axis ranges from 0 to 150, with tick marks at intervals of 50 (0, 50, 100, 150).
* **Y-axis (Vertical):** EW K-factor (Electroweak K-factor). The axis ranges from 0.6 to 1.4, with tick marks at intervals of 0.2 (0.6, 0.8, 1.0, 1.2, 1.4).
* **Legend (Top-Left):**
* Red line: J/ψ ηc
* Orange line: J/ψ J/ψ
* Blue line: Y ηb
* Cyan line: Y Y
* **Horizontal Reference Line:** A black horizontal line is present at EW K-factor = 1.0.
* **Vertical Reference Line:** A black vertical line is present at √s = 90 GeV.
### Detailed Analysis
* **Red Line (J/ψ ηc):** This line starts at approximately 0.7 at √s = 0 and increases rapidly initially, then gradually flattens out to approximately 1.13 at √s = 150.
* (0, 0.7)
* (50, 0.98)
* (100, 1.08)
* (150, 1.13)
* **Orange Line (J/ψ J/ψ):** This line starts at approximately 0.63 at √s = 0 and increases rapidly initially, then gradually flattens out to approximately 1.02 at √s = 150.
* (0, 0.63)
* (50, 0.9)
* (100, 0.98)
* (150, 1.02)
* **Blue Line (Y ηb):** This line starts at approximately 0.56 at √s = 0 and increases gradually, reaching approximately 0.85 at √s = 150.
* (0, 0.56)
* (50, 0.72)
* (100, 0.79)
* (150, 0.85)
* **Cyan Line (Y Y):** This line starts at approximately 0.7 at √s = 0 and increases gradually, reaching approximately 0.95 at √s = 150.
* (0, 0.7)
* (50, 0.83)
* (100, 0.9)
* (150, 0.95)
### Key Observations
* All four lines show an increasing trend as √s increases.
* The red line (J/ψ ηc) has the highest EW K-factor values across the range of √s.
* The blue line (Y ηb) has the lowest EW K-factor values across the range of √s.
* The red and orange lines increase more rapidly at lower values of √s compared to the blue and cyan lines.
* The vertical line at √s = 90 GeV intersects all four data series, providing a reference point for comparison.
* The horizontal line at EW K-factor = 1.0 intersects the red and orange lines.
### Interpretation
The chart illustrates how the EW K-factor varies with the square root of s (√s) for different particle combinations (J/ψ ηc, J/ψ J/ψ, Y ηb, and Y Y). The K-factor is a correction factor used in theoretical calculations, and its dependence on √s provides insights into the underlying physics of these interactions. The fact that all K-factors increase with √s suggests that the corrections become more significant at higher energies. The different behaviors of the lines indicate that the magnitude of these corrections varies depending on the specific particle combination. The reference lines at EW K-factor = 1.0 and √s = 90 GeV provide a basis for comparing the relative magnitudes and trends of the K-factors.
</details>
\captionstyle normal
Figure 4: The total K-factor dependence on the collision energy in case of EW production. \setcaptionmargin
0mm \onelinecaptionstrue \captionstyle flushleft $m_{c}$ = 1.5 GeV $m_{b}$ = 4.7 GeV $M_{Z}$ = 91.2 GeV $\Gamma_{Z}$ = 2.5 GeV $\langle O\rangle_{J/\psi}=\langle O\rangle_{\eta_{c}}=0.523\mbox{ GeV}^{3}$ $\langle O\rangle_{\Upsilon}=\langle O\rangle_{\eta_{b}}=2.797\mbox{ GeV}^{3}$ $\sin^{2}\theta_{w}$ = 0.23
Table 1: The cross section values in fb units at different collision energies. \setcaptionmargin
0mm \onelinecaptionstrue \captionstyle flushleft E, GeV 15 20 30 50 90 180 $\leavevmode\nobreak\ J/\psi\,\eta_{c}\leavevmode\nobreak\$ $1.15· 10^{0}$ $1.38· 10^{-1}$ $9.96· 10^{-3}$ $5.71· 10^{-4}$ $1.04· 10^{-3}$ $2.25· 10^{-6}$ $\leavevmode\nobreak\ J/\psi\,J/\psi\leavevmode\nobreak\$ $5.67· 10^{-5}$ $3.23· 10^{-5}$ $1.92· 10^{-5}$ $2.00· 10^{-5}$ $5.70· 10^{-3}$ $9.37· 10^{-7}$ $\leavevmode\nobreak\ \Upsilon\,\eta_{b}\leavevmode\nobreak\$ $-$ $1.40· 10^{-2}$ $8.30· 10^{-3}$ $2.59· 10^{-4}$ $1.15· 10^{-3}$ $6.47· 10^{-8}$ $\leavevmode\nobreak\ \Upsilon\,\Upsilon\leavevmode\nobreak\$ $-$ $9.84· 10^{-7}$ $2.07· 10^{-5}$ $1.17· 10^{-5}$ $7.65· 10^{-4}$ $3.41· 10^{-8}$
Table 2: The cross section values in fb units at different collision energies.