category: literaturenote citekey: zhangstringtensioninitialtemperature2025 title: On string tension and initial temperature from transverse momentum spectra of heavy quarkonia in proton-proton collisions at the LHC authors: "Zhang, Peng-Cheng; Zhu, Hailong; Liu, Fu-Hu; Olimov, Khusniddin K." year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.10702v1" zotero_key: EZ7RCAWK zotero_storage: E4I7M7A6 collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Zhang, Peng-Cheng"

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On string tension and initial temperature from transverse momentum spectra of heavy quarkonia in proton-proton collisions at the LHC

Peng-Cheng Zhang1,^∗ , Hailong Zhu1,† , Fu-Hu Liu1,‡ , Khusniddin K. Olimov2,3,§ 1 Institute of Theoretical Physics & College of Physics and Electronic Engineering, State Key Laboratory of Quantum Optics Technologies and Devices & Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China ²Laboratory of High Energy Physics, Physical-Technical Institute of Uzbekistan Academy of Sciences, Chingiz Aytmatov Str. 2b, Tashkent 100084, Uzbekistan ³Department of Natural Sciences, National University of Science and Technology MISIS (NUST MISIS), Almalyk Branch, Almalyk 110105, Uzbekistan

Abstract: The string tension (κ) in the Schwinger mechanism and the effective temperature (T ) in Bose-Einstein statistics are extracted from the transverse momentum (p^T ) spectra of heavy quarkonia produced in proton-proton (p+p) collisions at the Large Hadron Collider (LHC). Here, T derived from the heavy quarkonium p^T spectra also represents the initial temperature of the collision system, as transverse flow has not yet formed when heavy quarkonia are produced. The related parameters (κ and T ) are obtained by fitting the experimental p^T spectra of J/ψ and Υ(nS) (n = 1, 2, and 3) within various rapidity intervals, produced in p+p collisions at center-of-mass energies of √ s = 13 and 8 TeV, as measured by the LHCb Collaboration. It is found that the multi-component distribution structured within the framework of the Schwinger mechanism or Bose-Einstein statistics can effectively describe the heavy quarkonium p^T spectra in small collision systems. With decreasing rapidity in the forward region, both κ and T increase, indicating a directly proportional relationship between them. Based on κ, the average minimum strong force radius of participant quarks is determined.

Keywords: Schwinger mechanism; Bose-Einstein statistics; small collision system; string tension; initial temperature; linear relationship

PACS numbers: 12.40.Ee, 13.85.Hd, 13.85.Ni, 25.75.Ag, 25.75.Dw

I. INTRODUCTION

Current investigations reveal that small collision systems, such as proton-proton (p+p) or proton-nucleus collisions, also exhibit collectivity [1 –4]. This phenomenon suggests that the temperature concept used in large collision systems, such as nucleus-nucleus collisions, can also be applied to small systems [5 –7]. Generally, the final temperature of the collision system can be extracted from the yields or transverse momentum (p^T ) spectra of identified light hadrons, which are produced in the thermal process and appear during the final kinetic freeze-out stage. In contrast, the initial temperature of the collision system can be extracted from the distribution properties of heavy quarkonia, which are produced at the initial stage [8 –11]. Thus, temperatures extracted from the p^T spectra of different particles reflect the excitation degrees of the collision system at different stages.

^∗ 202312602003@email.sxu.edu.cn

^† Correspondence: zhuhl@sxu.edu.cn

^‡ Correspondence: fuhuliu@163.com; fuhuliu@sxu.edu.cn

^§ Correspondence: khkolimov@gmail.com; kh.olimov@uzsci.net

Typically, the temperature parameter used in p^T distributions refers to the so-called effective temperature (T ), rather than the real thermal temperature, unless the influence of radial or transverse flow [12 –15] is removed from the total amount. However, T extracted from the p^T spectra of heavy quarkonia may represent the real temperature because transverse flow does not form at the initial stage. In other words, if the p^T distribution with the parameter T in Bose-Einstein statistics is used to describe the production of heavy quarkonia [16 –18], T indeed reflects the initial temperature. Although the initial temperature has received relatively little research attention in the community compared to the final temperature (chemical or kinetic freeze-out temperature), which has been extensively studied, it remains an important quantity for understanding the early dynamics of the collision system.

It is well understood that the mechanism of heavy quarkonium production in high-energy collisions involves the synergistic effects of perturbative and non-perturbative quantum chromodynamics (QCD) [19 –21]. Despite heavy quarkonia not being purely produced via a thermal process, statistical results from a large number of events exhibit similar laws to thermal processes [22 –25]. Therefore, the statistical laws applicable to thermal processes can also be applied to the production of heavy quarkonia. Due to differing interaction mechanisms, one might consider T from heavy quarkonium spectra as an extensive temperature. For consistency, however, we prefer to refer to T from heavy quarkonium spectra as the initial temperature in this work. This temperature perfectly reflects the statistical process and is unaffected by transverse flow.

Quantum statistics (Bose-Einstein and Fermi-Dirac) [26 –28] have been widely applied in particle and nuclear physics, although classical (Boltzmann-Gibbs) statistics serve as a suitable approximation in most cases [29 –31]. Of course, a single-component distribution from quantum statistics cannot adequately describe the p^T spectra. Instead, two- or multi-component distributions play a significant role. As products generated at the initial stage, the behavior characteristics of heavy quarkonia, such as J/ψ and Υ(nS) (n = 1, 2, and 3), reflect the strength of initial interactions among participant partons. The multi-component distribution derived from Bose-Einstein statistics (or the multi-component Bose-Einstein distribution) can describe the p^T spectra of J/ψ and Υ(nS) and extract the initial temperature T of collision systems.

Alternatively, the Schwinger mechanism distribution (or Schwinger distribution) can be utilized to describe the interaction strength between two participant partons in high-energy collisions, albeit this application remains relatively uncommon [32 –35]. Naturally, single-, two-, or multi-component Schwinger distributions can be employed depending on the specific scenarios. Compared with proton-nucleus and nucleus-nucleus collisions, p+p collisions exhibit particular advantages due to the minimal influence of spectator partons. Here, the participant-spectator model [36 –39] widely used in proton-nucleus and nucleus-nucleus collisions is extended to p+p collisions, where participants and spectators are considered partons when studying heavy quarkonium production.

The Schwinger mechanism describes particle production at the parton level, while Bose-Einstein statistics characterizes the behavior of bosons. We aim to extract the string tension (κ) from the Schwinger mechanism [32 –35] and the temperature T from Bose-Einstein statistics [29 –31], subsequently investigating the relationship between κ and T . In this study, multi-component Schwinger and Bose-Einstein distributions are employed to describe the p^T spectra of J/ψ and Υ(nS) produced in p+p collisions at center-of-mass energies of √ s = 13 and 8 TeV, as measured by the LHCb Collaboration at the Large Hadron Collider (LHC) [40 –43]. The parameters κ and T are extracted, and their relationship is determined. Additionally, based on κ, the average minimum strong force radius is also obtained.

The remainder of this article is structured as follows: Section 2 outlines the formalism associated with the Schwinger mechanism and Bose-Einstein statistics. Results and discussions are presented in Section 3. Finally, Section 4 provides a summary and conclusions.

II. FORMALISM FROM SCHWINGER MECHANISM AND BOSE-EINSTEIN STATISTICS

Based on the Schwinger mechanism [32 –35], a multi-component distribution can be formulated. Each component i arises from the contribution of two participant partons, where each parton j contributes transverse momentum (ptj ) via a Gaussian-type probability density function fij (ptj ). This is expressed as:

$$f{ij}(p{tj}) = C_0(\kappai) \exp\left[-\frac{\pi(p{tj}^2 + m_0^2)}{\kappa_i}\right]$$

$$= C_0(\kappa_i) \exp\left(-\frac{\pi m_0^2}{\kappai}\right) \exp\left(-\frac{\pi p{tj}^2}{\kappa_i}\right)$$

$$= \frac{1}{\sqrt{\kappai}} \exp\left(-\frac{\pi p{tj}^2}{\kappa_i}\right), \tag{1}$$

where m⁰ represents the rest mass of the participant parton, κⁱ denotes the string tension, and C0(κi) is the normalization constant.

The final p^T of a particle in component i results from the contributions of two participant partons, following the folding of their individual contributions. The p^T distribution in component i is given by:

$$f{i}(p{T}) = \int{0}^{p{T}} f{i1}(p{t1}) f{i2}(p{T} - p{t1}) dp{t1}$$

$$= \int{0}^{p{T}} f{i2}(p{t2}) f{i1}(p{T} - p{t2}) dp{t2}$$

$$= \frac{1}{\kappa{i}} \int{0}^{p{T}} \exp\left{-\frac{\pi \left[p{t1}^{2} + (p{T} - p{t1})^{2}\right]}{\kappa{i}}\right} dp{t1}$$

$$= \frac{1}{\kappa{i}} \int{0}^{p{T}} \exp\left{-\frac{\pi \left[p{t2}^{2} + (p{T} - p{t2})^{2}\right]}{\kappa{i}}\right} dp{t2}, \tag{2}$$

which is normalized to unity. Let kⁱ denote the fraction contributed by component i. The final p^T distribution from the multi-component system can then be written as:

$$f(p_T) = \sum_i k_i f_i(p_T),\tag{3}$$

with P i kⁱ = 1 due to normalization. The average string tension weighted by kⁱ in the multi-component Schwinger distribution is κ = P i kiκⁱ .

In terms of Bose-Einstein statistics [29 –31], for component i with effective temperature Tⁱ , the invariant yield or boson momentum (p) distribution is expressed as [29]:

$$E\frac{d^3N_i}{d^3p} = \frac{gV_i}{(2\pi)^3} E\left[\exp\left(\frac{E-\mu}{T_i}\right) - 1\right]^{-1},\tag{4}$$

where Nⁱ is the number of bosons, g = 1 is the degeneracy factor for bosons, µ is the chemical potential (close to zero at high energy), E = p p ² + m′² ⁰ = m^T cosh y is the energy, m′ 0 is the rest mass, m^T = p p 2 ^T ⁺ ^m′² is the transverse mass, y = (1/2) ln[(E + pz)/(E − pz)] is the rapidity, p^z is the longitudinal momentum of the considered boson, and Vi is the volume of the collision system.

The probability density function of p^T in component i is:

$$f{i}(p{T}) = \frac{1}{N{i}} \frac{dN{i}}{dp_{T}}$$

$$= \frac{1}{N{i}} \frac{gV{i}}{(2\pi)^{2}} p{T} \sqrt{p{T}^{2} + m{0}^{\prime 2}} \int{y{\min}}^{y{\max}} \cosh y$$

$$\times \left[ \exp\left(\frac{\sqrt{p{T}^{2} + m{0}^{\prime 2}} \cosh y - \mu}{T_{i}}\right) - 1 \right]^{-1} dy, \tag{5}$$

where ymin and ymax are the minimum and maximum rapidities, respectively, within the experimental rapidity interval [ymin, ymax]. The multi-component distribution is expressed as:

$$f(p_T) = \frac{1}{N} \frac{dN}{dpT} = \sum{i} k_i \frac{1}{N_i} \frac{dN_i}{dpT} = \sum{i} k_i f_i(p_T),$$ (6)

which is consistent with Eq. (3). The average temperature from the multi-component Bose-Einstein distribution is given by T = P kiTⁱ

In the multi-component Schwinger (Bose-Einstein) distribution, the functional forms of various components remain consistent, despite differences in parameter values due to observed commonalities, similarities [44 –47], and universality [48 –51] in high-energy collisions. Variations in parameter values arise from differing strengths or degrees of analogous processes.

III. RESULTS AND DISCUSSION

Figure 1 illustrates the double differential cross section, d ²σ/(dydp^T ), of (a) prompt J/ψ and (b) J/ψ originating from b-quarks in p+p collisions at √ s = 13 TeV, where σ denotes the cross section. Figure 2 depicts the double differential cross section, d ²σ/(dydp^T ), of (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) produced in p+p collisions at √ s = 13 TeV. Similarly, Figure 3 displays the double differential cross section, d ²σ/(dydp^T ), of (a) prompt J/ψ and (b) J/ψ from b-quarks in p+p collisions at √ s = 8 TeV, while Figure 4 presents the double differential cross section, d ²σ/(dydp^T ), of (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) produced in p+p collisions at √ s = 8 TeV. In Figures 1–4, different symbols represent experimental data measured by the LHCb Collaboration in various y intervals within the forward rapidity region at the LHC [40 –43]. Solid and dashed curves correspond to our results fitted using three-component Schwinger and Bose-Einstein distributions, respectively. To enhance clarity, experimental data and fitted results are rescaled by factors indicated in the panels. Using the least square method, optimal parameter values are obtained during fitting. It is evident that the experimental double differential cross sections of J/ψ and Υ(nS) across different y intervals in 13 and 8 TeV p+p collisions, as measured by the LHCb Collaboration, can be accurately described by three-component Schwinger and Bose-Einstein distributions.

Figure 1. The double differential cross section, d ²σ/(dydp^T ), for (a) prompt J/ψ and (b) J/ψ from b produced in p+p collisions at √ s = 13 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [40], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using threecomponent Schwinger and Bose-Einstein distributions, respectively.

To explore the relationship between κ and T , Figures 5(a) and 5(b) depict results derived from the spectra of heavy quarkonia J/ψ and Υ(nS) produced in p+p collisions at √ s = 13 and 8 TeV, respectively. For each case — i) prompt J/ψ, ii) J/ψ from b, iii) Υ(1S), iv) Υ(2S), and v) Υ(3S) — the result corresponding to 2.0 < y < 2.5 is positioned at the higher edge, whereas the result for 4.0 < y < 4.5 is located at the lower edge. Values gradually increase from

Figure 2. The double differential cross section, d ²σ/(dydp^T ), for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) produced in p+p collisions at √ s = 13 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [41], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using threecomponent Schwinger and Bose-Einstein distributions, respectively.

4.0 < y < 4.5 to 2.0 < y < 2.5. It is observed that the relationship between κ and T is approximately linear for each case. However, considering all five cases collectively, the relationship between κ and T exhibits complexity.

The values of κ and T extracted from the spectra of prompt J/ψ, J/ψ from b, Υ(1S), Υ(2S), and Υ(3S) increase in an orderly manner. This reflects the interaction strength between two participant partons as well as the magnitude of the average transverse momenta of the bosons themselves. Given the initial temperature (^∼ ⁰.75–2.15 GeV at ^√ s = 13 TeV and ^∼ ⁰.00–0.00 GeV at ^√ s = 8 TeV), T is significantly larger than the chemical freeze-out temperature (∼ 0.16 GeV) determined by the statistical (thermal) model and hydrodynamical description [52 –55] and the kinetic freeze-out temperature (∼ 0.10–0.16 GeV) derived from the blast wave model [56 –59]. While this relative size appears to be a natural outcome, direct comparisons among the three temperatures are not feasible due to their differing definitions.

Based on κ, the average minimum distance between the two participant quarks [cc¯ for J/ψ or b ¯b for Υ(nS)] can be calculated. According to the Schwinger mechanism [32 –35], the average minimum distance is given by 2m0/κ. This distance also corresponds to twice the average minimum radius Rmin of the strong interaction force between the two quarks, where Rmin = m0/κ. The dependence of Rmin on y, extracted from the spectra of heavy quarkonia produced in p+p collisions at √ s = 13 and 8 TeV, is illustrated in Figures 6(a) and 6(b), respectively. It is evident that Rmin

Figure 3. The double differential cross section, d ²σ/(dydp^T ), for (a) prompt J/ψ and (b) J/ψ from b produced in p+p collisions at √ s = 8 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [42], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using three-component Schwinger and Bose-Einstein distributions, respectively.

decreases as y decreases from 4.0 < y < 4.5 to 2.0 < y < 2.5 in the forward rapidity region. Extremely small values of Rmin (0.03 ∼ 0.09 fm) are obtained, which are one tenth of various radii (hard-core, charge, mass, axial) of nucleons (0.3 ∼ 1.0 fm) [60 –62].

Comparing relevant parameters across different distributions holds great significance in high-energy collisions. In our previous and recent studies [63–67], we have investigated the parameters associated with string tension, effective temperature, and average transverse momentum extracted from the transverse momentum spectra of particles produced in both large and small collision systems. Approximate linear correlations under specific restrictive conditions have been identified. The present work further validates these observations by comparing the string tension in the Schwinger mechanism with the effective temperature in Bose-Einstein statistics in p+p collisions at the LHC. This study provides an important perspective for researchers to deepen their understanding of system evolution and interaction mechanisms.

The string tension characterizes the energy density per unit length of the color flux tube in QCD, representing the gradient of quark confinement strength and the strong interaction potential. Its numerical value directly influences the critical energy threshold for string breaking during hadronization and determines the transverse momentum distribution of particles. The effective temperature reflects the degree of thermalization in high-temperature and high-density systems and is directly linked to transport properties such as sound velocity and viscosity coefficients. Larger values result in a broader transverse momentum distribution of particles.

The experimental determination of string tension can verify the consistency between the QCD string model and lattice calculation predictions, providing a benchmark for describing strong interactions beyond perturbation theory. Simultaneously, when coupled with non-equilibrium dynamics, string tension may induce the formation of strange hadron states (e.g., color superconducting states). Abnormal fluctuations in the effective temperature may indicate the critical region in the QCD phase diagram, verifying the phase transition paths predicted by lattice QCD. Furthermore, the correlation between the effective temperature and boson occupancy can reveal collective excitations dominated by boson modes during the QGP de-confinement process.

In the extraction of string tension, there exists a challenge regarding insufficient real-time detection accuracy in non-perturbative processes. Future research could focus on developing case reconstruction algorithms based on deep learning to address this issue. Similarly, in the extraction of effective temperature, measurement model distortion

Figure 4. The double differential cross section, d ²σ/(dydp^T ), for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) produced in p+p collisions at √ s = 8 TeV. Different symbols represent experimental data in various y intervals measured by the LHCb Collaboration [43], with the data rescaled by different factors for clarity. The solid and dashed curves correspond to our results fitted using three-component Schwinger and Bose-Einstein distributions, respectively.

caused by multi-body correlation effects poses a significant problem. This can potentially be resolved by constructing thermodynamic evolution equations that incorporate quantum entanglement. As these are fundamental research tasks, accurately extracting their values is essential for meaningful comparisons and understanding their relationships.

Based on the aforementioned discussion, it can be inferred that through synergistic analysis of string tension and effective temperature, researchers can reconstruct the collision dynamics of the complete spatiotemporal chain, from the initial energy deposition of the color field to the evolution of the thermalization medium. This approach allows for comparing prediction differences between the string model and the thermal/statistical model for the same observation, enabling cross-validation of the models. Furthermore, it explores the possibility of discovering strong interaction corrections or Bose condensation phases beyond the standard model in the TeV energy region. In short, the cross-study of these two parameters provides a unique perspective for uncovering the deep structure of matter and the early evolution laws of the universe.

Figure 5. Relationship between string tension κ and initial temperature T derived from the spectra of heavy quarkonia J/ψ and Υ(nS) produced in p+p collisions at (a) 13 TeV and (b) 8 TeV. Results corresponding to 2.0 < y < 2.5 (4.0 < y < 4.5) are located at the higher (lower) value edge for each case. Five cases are illustrated in the panels.

Figure 6. Dependence of the average minimum strong force radius Rmin of participant c or b quarks on the rapidity y of quarkonium extracted from p+p collisions at (a) 13 TeV and (b) 8 TeV. Five cases are illustrated in the panels.

IV. SUMMARY AND CONCLUSIONS

The transverse momentum spectra (double differential cross sections) of J/ψ and Υ(nS) produced in p+p collisions at √ s = 13 and 8 TeV within different rapidity intervals are analyzed using the Schwinger mechanism and Bose-Einstein statistics. The multi-component distribution successfully fits the experimental results measured by the LHCb Collaboration. Our study demonstrates that both the string tension and initial temperature increase as rapidity decreases in the forward rapidity region, indicating a positive correlation between these parameters.

At higher collision energies, the string tension and initial temperature correspond to larger values, reflecting greater deposited energy in the collisions. For the production of i) prompt J/ψ, ii) J/ψ from b, iii) Υ(1S), iv) Υ(2S), and v) Υ(3S), both parameters increase sequentially. This suggests that more collision energy is required for Υ(nS) production due to the larger mass of participating quarks (b ¯b). Based on the Schwinger mechanism, the minimum strong force radii of c and b quarks in the formation of J/ψ and Υ(nS) are approximately 0.03 ∼ 0.09 fm. Author Contributions: The authors contributed to the paper in this way: conceptualization, H.L.Z., F.H.L. and K.K.O.; methodology, H.L.Z., F.H.L. and K.K.O.; software, P.C.Z.; validation, H.L.Z., F.H.L. and K.K.O.; formal analysis, P.C.Z.; investigation, P.C.Z.; resources, P.C.Z.; data curation, P.C.Z.; writing—original draft preparation, P.C.Z.; writing—review and editing, H.L.Z., F.H.L. and K.K.O.; visualization, P.C.Z.; supervision, H.L.Z., F.H.L. and K.K.O.; project administration, H.L.Z. and F.H.L.; funding acquisition, H.L.Z., F.H.L. and K.K.O. All authors have read and agreed to the published version of the manuscript.

Funding: The work of Shanxi group was supported by National Natural Science Foundation of China under Grant No. 12147215, Fundamental Research Program of Shanxi Province under Grant No. 202303021221071, Shanxi Scholarship Council of China and the Fund for Shanxi "1331 Project" Key Subjects Construction. The work of K.K.O. was supported by the Ministry of Innovative Development of the Republic of Uzbekistan within the fundamental project No. F3-20200929146 on analysis of open data on heavy-ion collisions at RHIC and LHC.

Data Availability Statement: The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper. The funders had no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.

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