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The
$ \varUpsilon(1S) $ meson serves as a reliable probe in heavy-ion collisions, as the regeneration process in the quark-gluon plasma (QGP) is negligible compared to$ J/\psi $ . Therefore, the distribution of$ \varUpsilon(1S) $ in the hadron gas provides valuable information about the QGP. Consequently, its study holds great significance. The distribution in the hadron gas is influenced by flow, quantum, and strong interaction effects. Previous models have predominantly focused on one or two of these effects while neglecting the others, resulting in the inclusion of unconsidered effects in the fitted parameters. In this paper, we aim to comprehensively examine all three effects simultaneously from a novel fractal perspective through physical calculations, rather than relying solely on data fitting. Close to the critical temperature, the combined action of the three effects leads to the formation of a two-meson structure comprising$ \varUpsilon(1S) $ and its nearest neighboring meson. However, with the evolution of the system, most of these states undergo disintegration. To describe this physical process, we establish a two-particle fractal (TPF) model. Our model proposes that, under the influence of the three effects near the critical temperature, a self-similarity structure emerges, involving a$ \varUpsilon(1S) $ -π two-meson state and a$ \varUpsilon(1S) $ -π two-quark state. As the system evolves, the two-meson structure gradually disintegrates. We introduce an influencing factor,$ q_{{\rm{fqs}}} $ , to account for the flow, quantum, and strong interaction effects, as well as an escort factor,$ q_2 $ , to represent the binding force between b and$ \bar{b} $ and the combined impact of the three effects. By solving the probability and entropy equations, we derive the values of$ q_{{\rm{fqs}}} $ and$ q_2 $ at various collision energies. Substituting the value of$ q_{{\rm{fqs}}} $ into the distribution function, we successfully obtain the transverse momentum spectrum of low-$ p_{\rm{T}} $ $ \varUpsilon(1S) $ , which demonstrates good agreement with experimental data. Additionally, we analyze the evolution of$ q_{{\rm{fqs}}} $ with temperature. Interestingly, we observe that$ q_{{\rm{fqs}}} $ is greater than 1 and decreases as the temperature decreases. This behavior arises from the fact that the three effects reduce the number of microstates, leading to$ q_{{\rm{fqs}}}>1 $ . The decrease in$ q_{{\rm{fqs}}} $ with system evolution aligns with the understanding that the influence of the three effects diminishes as the system expands. In the future, the TPF model can be employed to investigate other mesons and resonance states.-
Keywords:
- transverse momentum spectrum /
- fractal theory /
- escort probability /
- Tsallis entropy
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[2] Van Hove L 1982 Phys. Lett. B 118 138
Google Scholar
[3] Krämer M 2001 Prog. Part. Nucl. Phys. 47 141
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[4] Lansberg J P 2009 Eur. Phys. J. C 61 693
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Google Scholar
[12] Karsch F, Mehr M T, Satz H 1988 Z. Phys. C: Part. Fields 37 617
Google Scholar
[13] Guo Y, Dong L, Pan J, Moldes M R 2019 Phys. Rev. D 100 036011
Google Scholar
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[15] Digal S, Petreczky P, Satz H 2001 Phys. Rev. D 64 094015
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[16] Burnier Y, Rothkopf A 2017 Phys. Rev. D 95 054511
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[18] Akamatsu Y, Rothkopf A 2012 Phys. Rev. D 85 105011
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[27] Cleymans J, Satz H 1993 Z. Phys. C: Part. Fields 57 135
Google Scholar
[28] Andronic A, Braun-Munzinger P, Redlich K, Stachel J 2007 Nucl. Phys. A 789 334
Google Scholar
[29] Mandelbrot B 1967 Science 156 636
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Google Scholar
[31] Pathria R, Beale D P 2022 Formulation of Quantum Statistics (London:Elsevier) pp127–128
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 强子气体中b夸克和
$ \bar{b} $ 反夸克在不同层次的自相似结构 (a)真空中的自由$ \varUpsilon(1 S) $ 介子; (b)介子层次强子气体中的$ \varUpsilon(1 S) $ 介子; (c)真空中的自由b夸克和$ \bar{b} $ 反夸克; (d)夸克层次强子气体中的b夸克和$ \bar{b} $ 反夸克Figure 1. Self-similarity structure of b quark and
$ \bar{b} $ anti-quark in hadron gas: (a) Free$ \varUpsilon(1 S) $ in vacuum; (b)$ \varUpsilon(1 S) $ in hadron gas from meson aspect; (c) free b and$ \bar{b} $ in vacuum; (d)$ \varUpsilon(1 S) $ in hadron gas from quark aspect
图 2 Pb-Pb在不同碰撞能量以及0—100%碰撞对心度下, 低横动量
$ \varUpsilon(1 S) $ 介子在中心快度区间$ |y|<2.4 $ 的横动量谱. 实验数据来源于LHC[46,47]Figure 2. Transverse momentum spectrum of low-
$ p_{\rm{T}}$ $ \varUpsilon(1 S) $ in Pb-Pb at different collision energies for 0–100% centrality, in mid-rapidity region$ |y| <2.4 $ . The experimental data are taken from LHC[46,47]
图 3 不同固定温度T下的强子气体影响因子
$q_{{\rm{fqs}}}$ Figure 3. Hadron gas influencing factor
$q_{{\rm{fqs}}}$ at different fixed temperature T表 1 Pb-Pb在不同碰撞能量以及0—100%碰撞对心度下, 中心快度区
$|y|<2.4$ 内的$\varUpsilon(1 S)$ 介子运动空间半径$r_0$ 的值Table 1. In mid-rapidity region
$|y|<2.4$ , radius$r_0$ of$\varUpsilon(1 S)$ motion space under different collision energies and 0–100% centrality for Pb-Pb碰撞能量($\sqrt{s_{\rm{NN}}}$) 2.76/TeV 5.02/TeV $r_0$/fm 2.92 3.20 
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表 2 Pb-Pb在不同碰撞能量以及0—100%碰撞对心度下, 中心快度区
$|y|<2.4$ 内影响因子$q_{{\rm{fqs}}}$ 和$q_2$ 的数值Table 2. In mid-rapidity region
$|y|<2.4$ , values of$q_{{\rm{fqs}}}$ and$q_2$ for Pb-Pb in 0–100% centrality at different collision energies碰撞能量 ($\sqrt{s_{\rm{NN}}}$ ) 2.76/TeV 5.02/TeV $q_{{\rm{fqs}}}$ 1.0732 1.1051 $q_{2}$ 1.4249 1.4144
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[1] Lee K S, Heinz U, Schnedermann E 1990 Z. Phys. C: Part. Fields 48 525
Google Scholar
[2] Van Hove L 1982 Phys. Lett. B 118 138
Google Scholar
[3] Krämer M 2001 Prog. Part. Nucl. Phys. 47 141
Google Scholar
[4] Lansberg J P 2009 Eur. Phys. J. C 61 693
Google Scholar
[5] Brambilla N 2011 Eur. Phys. J. C 71 1534
Google Scholar
[6] Andronic A 2016 Eur. Phys. J. C 76 107
Google Scholar
[7] Mócsy Á, Petreczky P, Strickland M 2013 Int. J. Mod. Phys. A 28 1340012
Google Scholar
[8] Rapp R, Blaschke D, Crochet P 2010 Prog. Part. Nucl. Phys. 65 209
Google Scholar
[9] Rothkopf A 2020 Phys. Rep. 858 1
Google Scholar
[10] Zhao J, Zhou K, Chen S, Zhuang P 2020 Prog. Part. Nucl. Phys. 114 103801
Google Scholar
[11] Mócsy Á 2009 Eur. Phys. J. C 61 705
Google Scholar
[12] Karsch F, Mehr M T, Satz H 1988 Z. Phys. C: Part. Fields 37 617
Google Scholar
[13] Guo Y, Dong L, Pan J, Moldes M R 2019 Phys. Rev. D 100 036011
Google Scholar
[14] Kogut J B 1983 Rev. Mod. Phys. 55 775
Google Scholar
[15] Digal S, Petreczky P, Satz H 2001 Phys. Rev. D 64 094015
Google Scholar
[16] Burnier Y, Rothkopf A 2017 Phys. Rev. D 95 054511
Google Scholar
[17] Young C, Dusling K 2013 Phys. Rev. C 87 065206
Google Scholar
[18] Akamatsu Y, Rothkopf A 2012 Phys. Rev. D 85 105011
Google Scholar
[19] Zhou K, Xu N, Zhuang P 2014 Nucl. Phys. A 931 654
Google Scholar
[20] Herrmann N, Wessels J P, Wienold T 1999 Annu. Rev. Nucl. Part. Sci. 49 581
Google Scholar
[21] Schnedermann E, Sollfrank J, Heinz U 1993 Phys. Rev. C 48 2462
Google Scholar
[22] Wong C Y 2002 Phys. Rev. C 65 034902
Google Scholar
[23] Lin Z, Ko C M 2001 Phys. Lett. B 503 104
Google Scholar
[24] Abreu L M, Navarra F S, Nielsen M 2020 Phys. Rev. C 101 014906
Google Scholar
[25] Tang Z, Xu Y, Ruan L, van Buren G, Wang F, Xu Z 2009 Phys. Rev. C 79 051901
Google Scholar
[26] Reygers K, Schmah A, Berdnikova A, Sun X 2020 Phys. Rev. C 101 064905
Google Scholar
[27] Cleymans J, Satz H 1993 Z. Phys. C: Part. Fields 57 135
Google Scholar
[28] Andronic A, Braun-Munzinger P, Redlich K, Stachel J 2007 Nucl. Phys. A 789 334
Google Scholar
[29] Mandelbrot B 1967 Science 156 636
Google Scholar
[30] Li B A, Ko C M 1995 Phys. Rev. C 52 2037
Google Scholar
[31] Pathria R, Beale D P 2022 Formulation of Quantum Statistics (London:Elsevier) pp127–128
[32] Mandelbrot B 1982 The Fractal Geometry of Nature (New York: W. H. Freeman) pp25–74
[33] Dumitru A, Guo Y, Mócsy Á, Strickland M 2009 Phys. Rev. D 79 054019
Google Scholar
[34] Particle Data Group 2022 Prog. Theor. Exp. Phys. 2022 083C01
Google Scholar
[35] Srivastava P K, Chaturvedi O S K, Thakur L 2018 Eur. Phys. J. C 78 440
Google Scholar
[36] Crater H W, Yoon J H, Wong C Y 2009 Phys. Rev. D 79 034011
Google Scholar
[37] Ristea C 2018 Eur. Phys. J. Web Conf. 191 01004
Google Scholar
[38] Cè M, Harris T, Meyer H B, Toniato A, Török C 2021 J. High Energy Phys. 12 215
Google Scholar
[39] Beck C, Schögl F 1995 Thermodynamics of Chaotic Systems (Cambridge: Cambridge University Press) pp88–127
[40] Tél T 1988 Z. Naturforsch., A: Phys. Sci. 43 1154
Google Scholar
[41] Schroeder M 2009 Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (New York: W. H. Freeman and Company) pp103–121
[42] Abe S, Okamoto Y 2001 Nonextensive Statistical Mechanics and Its Applications (Berlin: Springer) pp5–6
[43] Tsallis C 1988 J. Stat. Phys. 52 479
Google Scholar
[44] Cleymans J, Worku D 2012 Eur. Phys. J. A 48 160
Google Scholar
[45] Beck C 2000 Physica A 286 164
Google Scholar
[46] CMS Collaboration 2017 Phys. Lett. B 770 357
Google Scholar
[47] CMS Collaboration 2019 Phys. Lett. B 790 270
Google Scholar
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