Research on u-d quark stars and their tidal deformaions under new mass scaling

XU Jianfeng; WANG Jingtao; XIA Chengjun

doi:10.7498/aps.74.20250535

Abstract

Strange quark matter (SQM) is considered to be the true ground state of the strong interactions, but recent studies have shown that ordinary quark matter (u-d quark matter, u-d QM) may also be the ground state of the strong interactions. By inserting an attenuation factor of Woods-Saxon potential type into the quark mass scaling, the resulting calculations of equation of state of u-d QM based on equiv-particle model show that the stability window of model parameters for stable u-d QM can be significantly enlarged with proper model parameters, which can be seen in the following figure. In this figure, the red solid and dashed lines represent the curves of $ \sqrt{D} $ versus C with and without attenuation factor, respectively, when the minimum value of the average energy per baryon is set to 930 MeV; the blue solid and dashed lines represent the curves of $ \sqrt{D} $ versus C with and without attenuation factor, respectively, when $ m_\mathrm{u}=0 $. Thereby, the red and blue shaded areas are the absolute stable regions of u-d QM without and with attenuation factor in mass scaling. It is obvious that with the attenuation factor and proper model parameters, the absolute stable region (blue shaded area) for u-d QM can be much larger than that without the attenuation factor (red shaded area). The introduction of the attenuation factor allows the maximum mass of ordinary quark star (u-d quark star, u-d QS) to be larger than twice the solar mass, while the tidal deformability satisfies $ \varLambda_{1.4} \in [70,580] $, which is consistent with the current astronomical observations. Therefore, the pulsars may be essentially the u-d QSs. This result provides a possibility for understanding the nature of pulsars, and it also further deepens the understanding of the strong interactions.

Keywords:

Authors and contacts

1.
School of Methmatics and Physics, Suqian University, Suqian 223800, China
2.
School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100084, China
3.
Center for Gravitation and Cosmology, School of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China

Corresponding author: XU Jianfeng, xujf@squ.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12005005, 12275234) and the Scientifc Research Start-up Fund of Talent Introduction of Suqian University, China (Grant No. Xiao2022XRC061).

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References

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Cited By

图 1 u-d QM的稳定窗口, 红色区域表示在没有考虑衰减因子时的u-d QM的稳定区域, 而蓝色区域表示衰减因子引入后u-d QM的稳定区域. 在图(a)–(c)中, 红色实线与红色虚线分别表示包含衰减因子与不包含衰减因子且当平均重子能量最小值等于930 MeV时$ \sqrt{D} $随C的变化曲线. 蓝色实线与蓝色虚线分别表示包含衰减因子与不包含衰减因子且当$ m_\mathrm{u}=0 $时$ \sqrt{D} $随C的变化曲线. 图(a)–(c)中的$ (w/\mathrm{fm}^{-3} $, $ n_{\mathrm{a}}/\mathrm{fm}^{-3}) $参数取值从上往下分别为(4.0, 0.6), (1.0, 0.6) 和(1.0, 2.0)

Figure 1. Stability window for u-d QM, the red and blue areas are the absolute stable regions for u-d QM without and with attenuation factor in mass scaling. In the panels (a)–(c), the red solid and dashed lines represent the curves of $ \sqrt{D} $ versus C with and without attenuation factor, respectively, when the minimum value of the average energy per baryon is 930 MeV. The blue solid and dashed lines represent the curves of $ \sqrt{D} $ versus C with and without attenuation factor, respectively, when $ m_\mathrm{u}=0 $. The values of parameters $ (w/\mathrm{fm}^{-3}, n_{\mathrm{a}}/\mathrm{fm}^{-3}) $ in the panels (a)–(c) are (4.0, 0.6), (1.0, 0.6), and (1.0, 2.0) from top to bottom.

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图 2 衰减因子随密度的变化曲线

Figure 2. Curves of the attenuation factor as function of baryon number density.

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图 3 平均重子能量与压强随密度的变化曲线

Figure 3. Curves of energy per baryon and pressure as functions of baryon number density.

DownLoad: Full-Size Img PowerPoint

图 4 潮汐形变Λ和QS质量M随QS中心密度$ n_0 $的变化曲线, 各分图中的蓝色实线为Λ随$ n_0 $的变化曲线, 其值对应于左纵轴; 红色实线为M随$ n_0 $的变化曲线, 其值对应于右纵轴. (a)—(c)中的参数值与图1(a)—(c)中的参数值相同

Figure 4. Curves of tidal deformability Λ and QS mass M as functions of central density $ n_0 $ of quark star, the blue solid curves corresponding to the left axis represent Λ versus $ n_0 $; the red solid curves corresponding to right axis represent M versus $ n_0 $. The parameters of panels (a)—(c) are the same with that in Fig. 1(a)—(c)

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图 5 u-d QS的质量-半径关系. 图中黑色、红色和蓝色曲线的参数分别与图1(a)—(c)中的参数相同

Figure 5. Mass-radius relations of u-d QSs. The parameter values for the black, red and blue curves are corresponding to that given in the three sub-figures in Fig. 1(a)—(c).

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表 1 当衰减因子参数$ (w/\mathrm{fm}^{-3}, n_{\mathrm{a}}/\mathrm{fm}^{-3}) $分别取$ (4.0, 0.6), (1.0, 0.6) $和$ (1.0, 2.0) $时u-d QS的最大质量, 最大质量u-d QS的半径, 以及与潮汐形变范围$ \varLambda_{1.4} \in $$ [70, 580] $相对应的u-d QS中心密度范围和质量范围

Table 1. Maximum masses and corresponding radii of u-d QSs under $ (w/\mathrm{fm}^{-3}, n_{\mathrm{a}}/\mathrm{fm}^{-3})= (4.0, 0.6),\; (1.0, 0.6), $$ (1.0, 2.0)$, as well as the central density range and mass range of u-d quark stars for the tidal deformability range $ \varLambda_{1.4} \in [70, 580] $.

Parameter	$ M_\mathrm{max}/{M}_\odot $	R/km	$ n_0/\mathrm{fm}^{-3} $	$ M/{M}_\odot $
(a)	2.136	11.681	[0.369, 0.560]	[1.449, 1.990]
(b)	2.150	10.927	[0.421, 0.562]	[1.302, 1.869]
(c)	2.090	11.021	[0.410, 0.582]	[1.343, 1.887]

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[1]	Peng G X, Li A, Lombardo U 2008 Phys. Rev. C 77 065807 Google Scholar
[2]	Weissenborn S, Sagert I, Pagliara G, Hempel M, Schaffner-Bielich J 2011 Astrophys. J. 740 L14 Google Scholar
[3]	Clemente F D, Casolino M, Drago A, Lattanzi M, Ratti C 2025 Mon. Not. R. Astron. Soc. 537 1056 Google Scholar
[4]	Li C M, Zheng H R, Zuo S Y, Zhao Y P, Wang F, Huang Y F 2025 Astrophys. J. 980 231 Google Scholar
[5]	Song X Y 2025 Phys. Rev. D 111 063018 Google Scholar
[6]	Zhang C 2020 Phys. Rev. D 101 043003 Google Scholar
[7]	Xu J F, Peng G X, Liu F, Hou D F, Chen L W 2015 Phys. Rev. D 92 025025 Google Scholar
[8]	Xu R X 2003 Astrophys. J. 596 L59 Google Scholar
[9]	Peng G X, Chiang H C, Zou B S, Ning P Z, Luo S J 2000 Phys. Rev. C 62 025801 Google Scholar
[10]	Klähn T, Fischer T 2015 Astrophys. J. 810 134 Google Scholar
[11]	Xia T, He L Y, Zhuang P F 2013 Phys. Rev. D 88 056013 Google Scholar
[12]	Wen X J, Feng Z Q, Li N, Peng G X 2009 J. Phys. G 36 025011 Google Scholar
[13]	Li B L, Cui Z F, Yu Z H, Yan Y, An S, Zong H S 2019 Phys. Rev. D 99 043001 Google Scholar
[14]	Zhang C, Gao Y, Xia C J, Xu R X 2023 Phys. Rev. D 108 063002 Google Scholar
[15]	Xia C J, Peng G X, Chen S W, Lu Z Y, Xu J F 2014 Phys. Rev. D 89 105027 Google Scholar
[16]	Chen S W, Gao L, Peng G X 2012 Chin. Phys. C 36 947 Google Scholar
[17]	Xu J F, Cui L, Lu Z Y, Xia C J, Peng G X 2023 Nucl. Sci. Tech. 34 171 Google Scholar
[18]	Demorest P, Pennucci T, Ransom S M, Roberts M S E, Hessels J W T 2010 Nature 467 1081 Google Scholar
[19]	Antoniadis J, Freire P C C, Wex N, Tauris T M, Lynch R S, Kerkwijk M H V, Kramer M, Bassa C, Dhillon V S, Driebe T, Hessels J W T, Kaspi V M, Kondratiev V I, Langer N, Marsh T R, Mclaughlin M A, Pennucci T T, Ransom S M, Stairs I H, Leeuwen J V, Verbiest P W, Whelan D G 2013 Science 340 1233232 Google Scholar
[20]	Cromartie H T, Fonseca E, Ransom S M, Demorest P B, Arzoumanian Z, Blumer H, Brook P R, DeCesar M E, Dolch T, Ellis J A, Ferdman R D, Ferrara E C, Garver-Daniels N, Gentile P A, Jones M L, Lam M T, Lorimer D R, Lynch R S, McLaughlin M A, Ng C, Nice D J, Pennucci T T, Spiewak R, Stairs I H, Stovall K, Swiggum J K, Zhu W W 2020 Nat. Astron. 4 72 Google Scholar
[21]	Fonseca E, Cromartie H T, Pennucci T T, Ray P S, Kirichenko A Y, Ransom S M, Demorest P B, Stairs I H, Arzoumanian Z, Guillemot L, Parthasarathy A, Kerr M, Cognard I, Baker P T, Blumer H, Brook P R, DeCesar M, Dolch T, Dong F A, Ferrara E C, Fiore W, Garver-Daniels N, Good D C, Jennings R, Jones M L, Kaspi V M, Lam M T, Lorimer D R, Luo J, McEwen A, McKee J W, McLaughlin M A, McMann N, Meyers B W, Naidu A, Ng C, Nice D J, Pol N, Radovan H A, ShapiroAlbert B, Tan C M, Tendulkar S P, Swiggum J K, Wahl H M, Zhu W W 2021 Astrophys. J. Lett. 915 L12 Google Scholar
[22]	Abbott R, Abbott T D, Abraham S, et al, LIGO Scientific and Virgo Collaboration 2020 Astrophys. J. Lett. 896 L44 Google Scholar
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[24]	LIGO Scientific and Virgo Collaboration, Abbott B P, et al. 2017 Astrophys. J. 848 L12 Google Scholar
[25]	LIGO Scientific and Virgo Collaboration, Abbott B P, et al. 2018 Phys. Rev. Lett. 121 161101 Google Scholar
[26]	Witten E 1984 Phys. Rev. D 30 272 Google Scholar
[27]	Farhi E, Jaffe R L 1984 Phys. Rev. D 30 2379 Google Scholar
[28]	Holdom B, Ren J, Zhang C 2018 Phys. Rev. Lett. 120 222001 Google Scholar
[29]	Wang J T, Peng G X 2023 Int. J. Mod. Phys. E 32 2350033 Google Scholar
[30]	Fowler G, Raha S, Weiner R 1981 Z. Phys. C 9 271 Google Scholar
[31]	Peng G X, Chiang H C, Yang J J, Li L, Liu B 1999 Phys. Rev. C 61 015201 Google Scholar
[32]	Wen X J, Zhong X H, Peng G X, Shen P N, Ning P Z 2005 Phys. Rev. C 72 015204 Google Scholar
[33]	Chen H M, Xia C J, Peng G X 2022 Chin. Phys. C 46 055102 Google Scholar
[34]	Damour T, Nagar A 2009 Phys. Rev. D 80 084035 Google Scholar
[35]	Fonseca E, Pennucci T T, Ellis J A, Stairs I H, Nice D J, Ransom S M, Demorest P B, Arzoumanian Z, Crowter K, Dolch T, Ferdman R D, Gonzalez M E, Jones G, Jones M L, Lam M T, Levin L, McLaughlin M A, Stovall K, Swiggum J K, Zhu W 2016 Astrophys. J. 832 167 Google Scholar
[36]	Chu P C, Chen L W 2014 Astrophys. J 780 135 Google Scholar
[37]	Xu J F, Xia C J, Lu Z Y, Peng G X, Zhao Y P 2022 Nucl. Sci. Tech. 33 143 Google Scholar
[38]	Xu J F, Cui L, Xia C J, Lu Z Y 2024 Nucl. Phys. Rev. 41 325 Google Scholar
[39]	Cui S S, Peng G X, Lu Z Y, Peng C, Xu J F 2015 Nucl. Sci. Tech. 26 040503 Google Scholar
[40]	Yang L, Wen X J 2017 Phys. Rev. D 96 056023 Google Scholar
[41]	Chu P C, Li X H, Ma H Y, Wang B, Dong Y M, Zhang X M 2018 Phys. Lett. B 2502 447 Google Scholar
[42]	Felipe R G, Martinez A P, Rojas H P, Orsaria M 2008 Phys. Rev. C 77 015807 Google Scholar
[43]	Chu P C, Liu H, Du X B 2024 Acta Phys. Sin. 73 052101 Google Scholar
[44]	Pal S, Chaudhuri G 2024 Phys. Rev. D 110 123021 Google Scholar
[45]	Zheng X P, Yang S H, Li J R 2003 Astrophys. J. Lett. 585 L135 Google Scholar
[46]	Gourgoulhon E, Haensel P, Livine R, Paluch E, Bonazzola S, Marck J A 1999 Astron. Astrophys. 349 851 Google Scholar
[47]	Yuan W L, Li A 2024 Astrophys. J. 996 3 Google Scholar
[48]	Bai Y, Chen T K 2025 arXiv: 2502.20241 [hep-ph]

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