Hugging Face's logo

Buckets:
huggingchat
/
papers-content

Files
xet
huggingchat/papers-content / 2501 /2501.01234.md
preview code
|
download
raw
105 kB

Title: Impact of QCD sum rules coupling constants on neutron stars structure

URL Source: https://arxiv.org/html/2501.01234

Published Time: Fri, 03 Jan 2025 02:19:27 GMT

Markdown Content: ††thanks: Corresponding author

H.R.Moshfegh a,b\XeTeXLinkBoxhmoshfegh@ut.ac.irK.Azizi a,c\XeTeXLinkBoxkazem.azizi@ut.ac.ira Department of Physics, University of Tehran, North Karegar Avenue, Tehran 14395-547, Iran

b Centro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud, 150, URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil

c Department of Physics, Doğuş University, Dudullu-Ümraniye, 34775 Istanbul, Türkiye

(January 2, 2025)

Abstract

We present a detailed investigation on the structure of neutron stars, incorporating the presence of hyperons within a relativistic model under the mean-field approximation. Employing coupling constants derived from QCD sum rules, we explore the particle fraction in beta equilibrium and establish the mass-radius relationship for neutron stars with hyperonic matter. Additionally, we compute the stellar Love number (𝒦 2 subscript 𝒦 2\mathcal{K}_{2}caligraphic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) and the tidal deformability parameter (Λ Λ\varLambda roman_Λ), providing valuable insights into the dynamical properties of these celestial objects. Through comparison with theoretical predictions and observational data, our results exhibit good agreement, affirming the validity of our approach. These findings contribute significantly to refining the understanding of neutron star physics, particularly in environments containing hyperons, and offer essential constraints on the equation of state governing such extreme astrophysical conditions.

I Introduction

The study of matter under extreme conditions, characterized by high densities and temperatures, has been a major focus in theoretical physics in recent decades. Models and concepts emerging from the study of nuclear matter have significantly contributed to unraveling some mysteries about the strong and weak interaction of matter. Compact stars serve as natural laboratories for investigating such exotic matter, allowing for the examination of processes involving all four known fundamental forces of nature. Existence of Neutron Stars (NSs) was proposed by Baade and Zwicky in 1934 Baade:1934wuu, sparking subsequent development of various models to describe their physics and properties Chin:1974sa; Huber:1997mg; Heiselberg:1999mq; Lattimer:2000nx; Weber:2004kj. Theoretical studies are currently being performed more than ever to elucidate neutron star physics through the equation of state (EoS) for dense matter. Despite the numerous assumptions and considerable efforts made to calculate the mass limits and explore the internal composition of neutron stars, several unresolved issues persist, partly due to the likely complex structures of neutron stars. The masses and radii of neutron stars are particular interest in observational experiments. The neutron star population’s members with the largest masses and their radii are particularly important for testing neutron star EoSs Ozel:2016oaf. Due to precise observations of neutron stars, such as the Shapiro delay measurement of a binary millisecond PSR J1614-2230 Demorest:2010bx, the radius measurement of PSR J0740+6620 Fonseca:2021wxt from the Neutron Star Interior Composition Explorer (NICER) and X-ray Multi-Mirror (XMMNewton) data, the discovery of PSR J0952-0607 Romani:2022jhd by the Low-Frequency Array (LOFAR) radio telescope, and observational data of PSR J0537-6910 Ho:2015vza; Ho:2020vxt with NICER timing, we have gained valuable insights into the properties of these celestial objects. Additionally, the direct detection of gravitational wave (GW) signals from a binary neutron star merger such as GW170817 LIGOScientific:2018hze, and GW190425 LIGOScientific:2020aai, observed by Advanced LIGO and Virgo detectors, have imposed a strong constraint on the mass-radius relation of neutron stars. Especially, the tidal deformability of a neutron star Flanagan:2007ix; Hinderer:2007mb plays an important role in constructing the EoS for neutron star matter. Assuming both NSs in the merger possess have the same EoS, this has led to stringent constraints on the tidal deformability, represented by Λ 1.4= 190−120+390 subscript Λ 1.4 superscript subscript 190 120 390\varLambda_{1.4}\leavevmode\nobreak\ =\leavevmode\nobreak\ 190_{-120}^{+390}roman_Λ start_POSTSUBSCRIPT 1.4 end_POSTSUBSCRIPT = 190 start_POSTSUBSCRIPT - 120 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + 390 end_POSTSUPERSCRIPT and on the radius of a 1.4 solar mass neutron star, bounded by 11.82<R 1.4<13.72 11.82 subscript 𝑅 1.4 13.72 11.82<R_{1.4}<13.72 11.82 < italic_R start_POSTSUBSCRIPT 1.4 end_POSTSUBSCRIPT < 13.72 K⁢m 𝐾 𝑚 Km italic_K italic_m NS for GW170817 event LIGOScientific:2018cki; Lim:2019som; Malik:2018zcf. Collectively, these groundbreaking detections have provided a substantial corpus of reliable observational data crucial for the investigation of the structure and internal composition of neutron stars Raaijmakers:2019qny; Miller:2021qha; Chatziioannou:2020pqz; Li:2021crp. However, a pressing question continues to challenge researchers: the uncertainty regarding whether neutron stars contain additional degrees of freedom, such as hyperons, alongside nucleons. Moreover, the necessary interactions for the emergence of these new degrees of freedom in the context of massive neutron stars remain elusive Xu:2023gmc. Consequently, the true internal composition of neutron stars remains an open and compelling question, driving ongoing research efforts in the field. Hyperons may appear in the inner core of neutron stars where densities are around 2 - 3 times the nuclear saturation density, ρ 0 subscript 𝜌 0\rho_{0}italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPTSchulze:1998jf; Djapo:2008au; Vidana:2015rsa; Lonardoni:2013gta. This leads to a softening of the equation of state (EoS) and consequently, the maximum mass of the neutron star could be lower than what current observations suggest, which is about 2 M⊙subscript 𝑀 direct-product M_{\odot}italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPTDemorest:2010bx; NANOGrav:2017wvv; NANOGrav:2019jur. The potential presence of hyperons in the interior of neutron stars poses a significant challenge for our current understanding because it contradicts our observations of high-mass neutron stars. This challenge, often termed the ”hyperon puzzle” in the literature, is currently a hot topic of research in nuclear astrophysics.

In this research, we will utilize by using the σ 𝜎\sigma italic_σ-ω 𝜔\omega italic_ω-ρ 𝜌\rho italic_ρ model Chin:1974sa; Mueller:1996pm; Walecka:1974qa; Uechi:2006pz to describe the EOS of the neutron star matter. The main goal is to explore the impact of coupling constants obtained from the QCD sum rules method on the equation of state of static and non-rotating NSs composed of baryons and leptons. Our aim is to identify a relativistic mean-field model that incorporates various baryon-meson coupling constants to accurately describe nuclear matter and address the hyperon puzzle challenge. The manuscript is organized as follows: In Section II, we introduce our theoretical model and provide a brief overview of QCD sum rules. Section III is dedicated to presenting our results and discussions. Within this section and its subsections, we will compare the predictions from our proposed models with observational data of neutron stars to identify the most suitable set of coupling constants. This comparative analysis will enable us to refine our understanding of the equation of state and gain insights into the internal composition of neutron stars. Finally, we conclude with remarks in Section IV.

II Theoretical Framework

II.1 Lagrangian Density

This study will utilize a Lagrangian to describe the behavior of particles in the core. The Lagrangian density incorporates terms for leptons, baryons, and mesons, along the interaction terms. It is expressed as follows Glendenning:1984jr:

ℒ=∑B ℒ B+∑ℳ ℒ ℳ+∑λ ℒ λ+ℒ i⁢n⁢t.ℒ subscript 𝐵 subscript ℒ 𝐵 subscript ℳ subscript ℒ ℳ subscript 𝜆 subscript ℒ 𝜆 subscript ℒ 𝑖 𝑛 𝑡\mathscr{L}=\sum_{B}\mathscr{L}{B}+\sum{\mathcal{M}}\mathscr{L}{\mathcal{M}% }+\sum{\lambda}\mathscr{L}{\lambda}+\mathscr{L}{int}.script_L = ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT script_L start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT script_L start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT script_L start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT + script_L start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT .(1)

Baryons are represented by B, mesons by ℳ ℳ\mathcal{M}caligraphic_M, and leptons by λ 𝜆\lambda italic_λ. The Lagrangian for the core of a NS can be constructed using a theory that incorporates three types of mesons: the sigma (σ 𝜎\sigma italic_σ) , omega (ω 𝜔\omega italic_ω) , and rho (ρ 𝜌\rho italic_ρ). The generalized Lagrangian density, based on the σ,ω,ρ 𝜎 𝜔 𝜌\sigma,\omega,\rho italic_σ , italic_ω , italic_ρ theory, is:

ℒ=∑B ψ¯B(i γ μ∂μ−m B+g B⁢B⁢σ σ−g B⁢B⁢ω γ μ ω μ−1 2 g B⁢B⁢ρ γ μ 𝝉.𝝆 μ)ψ B+1 2⁢(∂μ σ⁢∂μ σ−m σ 2⁢σ 2)−1 4⁢ω μ⁢ν⁢ω μ⁢ν+1 2⁢m ω 2⁢ω μ⁢ω μ−1 4⁢𝝆 μ⁢ν.𝝆 μ⁢ν+1 2⁢m ρ 2⁢𝝆 μ.𝝆 μ−1 3⁢b⁢m n⁢(g N⁢N⁢σ⁢σ)3−1 4⁢c⁢(g N⁢N⁢σ⁢σ)4+∑λ ψ¯λ⁢(i⁢γ μ⁢∂μ−m λ)⁢ψ λ.\begin{split}\mathscr{L}=&\sum_{B}\bar{\psi}{B}(i\gamma{\mu}\partial^{\mu}-m% {B}+g{BB\sigma}\sigma-g_{BB\omega}\gamma_{\mu}\omega^{\mu}-\dfrac{1}{2}g_{BB% \rho}\gamma_{\mu}\bm{\tau}.\bm{\rho}^{\mu})\psi_{B}\ &+\dfrac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-m^{2}{\sigma}\sigma^{% 2})-\dfrac{1}{4}\omega{\mu\nu}\omega^{\mu\nu}+\dfrac{1}{2}m^{2}{\omega}% \omega{\mu}\omega^{\mu}\ &-\dfrac{1}{4}\bm{\rho}{\mu\nu}.\bm{\rho}^{\mu\nu}+\dfrac{1}{2}m^{2}{\rho}% \bm{\rho}{\mu}.\bm{\rho}^{\mu}-\dfrac{1}{3}bm{n}(g_{NN\sigma}\sigma)^{3}-% \dfrac{1}{4}c(g_{NN\sigma}\sigma)^{4}\ &+\sum_{\lambda}\bar{\psi}{\lambda}(i\gamma{\mu}\partial^{\mu}-m_{\lambda})% \psi_{\lambda}.\ \end{split}start_ROW start_CELL script_L = end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT over¯ start_ARG italic_ψ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_i italic_γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ∂ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ω end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ρ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT bold_italic_τ . bold_italic_ρ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ) italic_ψ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_σ ∂ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_σ - italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 4 end_ARG bold_italic_ρ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT . bold_italic_ρ start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT bold_italic_ρ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT . bold_italic_ρ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 3 end_ARG italic_b italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_c ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT over¯ start_ARG italic_ψ end_ARG start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_i italic_γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ∂ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - italic_m start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ) italic_ψ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT . end_CELL end_ROW(2)

Here, we assume that baryons consist of neutrons, protons, and Hyperons (Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT, and Λ Λ\Lambda roman_Λ), while mesons include σ,ω 𝜎 𝜔\sigma,\omega italic_σ , italic_ω, and ρ 𝜌\rho italic_ρ. Additionally, leptons are electron and muons. In the Lagrangian density ∂μ superscript 𝜇\partial^{\mu}∂ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT represents the four-derivatives and γ μ subscript 𝛾 𝜇\gamma_{\mu}italic_γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT are the (covariant) Dirac Matrices. 𝝆 μ⁢ν subscript 𝝆 𝜇 𝜈\bm{\rho}{\mu\nu}bold_italic_ρ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT and ω μ⁢ν subscript 𝜔 𝜇 𝜈\omega{\mu\nu}italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT are the ρ 𝜌\rho italic_ρ and ω 𝜔\omega italic_ω field strength tensors, respectively. The spinors are shown by ψ 𝜓\psi italic_ψ, while ψ¯≡ψ†⁢γ 0¯𝜓 superscript 𝜓†subscript 𝛾 0\bar{\psi}\equiv\psi^{\dagger}\gamma_{0}over¯ start_ARG italic_ψ end_ARG ≡ italic_ψ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT represent their adjoint. The coupling constants in the above Lagrangian density are shown with g B⁢B⁢σ subscript 𝑔 𝐵 𝐵 𝜎 g_{BB\sigma}italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT, g B⁢B⁢ω subscript 𝑔 𝐵 𝐵 𝜔 g_{BB\omega}italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ω end_POSTSUBSCRIPT, g B⁢B⁢ρ subscript 𝑔 𝐵 𝐵 𝜌 g_{BB\rho}italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ρ end_POSTSUBSCRIPT, b, and c, that indicate the interactions between baryons and mesons as well as scalar self-interactions (b and c). In addition, 𝝉 𝝉\bm{\tau}bold_italic_τ are the isospin matrices. A relativistic mean-field approximation (MFA) is employed to investigate high density nuclear and neutron matter in this study. To parameterise the density dependence of the energy functional, nonlinear interactions between the fields are introduced based on idea from effective field theory Mueller:1996pm. Specially, we focus on different types of nonlinearity involving scalar-isoscalar (σ 𝜎\sigma italic_σ), vector-isoscalar (ω 𝜔\omega italic_ω), and vector-isovector (ρ 𝜌\rho italic_ρ) fields. Within the MFA framework, it is assumed that the particles do not interact with each other but rather experience an average effect from surrounding of the system.

II.2 Chemical Potential and Meson Fields

As noted by Glendenning Glendenning:1984jr, the chemical equilibrium problem can be addressed without considering individual reactions. For frozen static matter, the chemical potential (μ 𝜇\mu italic_μ) represents the energy of a particle at the highest level within the Fermi sea. By solving the Euler-Lagrange equations and using mean-field approximation for the meson field, we can determine the fields and their eigenvalues. This allows us to express the baryon fields in momentum representation as:

[γ μ(k μ−g B⁢B⁢ω ω μ−1 2 g B⁢B⁢ρ 𝝉.𝝆 μ)−(m B−g B⁢B⁢σ σ)]ψ B(k)=0,[\gamma_{\mu}(k^{\mu}-g_{BB\omega}\omega^{\mu}-\dfrac{1}{2}g_{BB\rho}\bm{\tau}% .\bm{\rho}^{\mu})-(m_{B}-g_{BB\sigma}\sigma)]\psi_{B}(k)=0,[ italic_γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_k start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ω end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ρ end_POSTSUBSCRIPT bold_italic_τ . bold_italic_ρ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ) - ( italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ ) ] italic_ψ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_k ) = 0 ,(3)

and the eigenvalues for particles and antiparticles given by:

e B⁢(k)=g B⁢B⁢ω⁢ω 0+g B⁢B⁢ρ⁢ρ 03⁢I B⁢3+k 2+(m B−g B⁢B⁢σ⁢σ)2,subscript 𝑒 𝐵 𝑘 subscript 𝑔 𝐵 𝐵 𝜔 subscript 𝜔 0 subscript 𝑔 𝐵 𝐵 𝜌 subscript 𝜌 03 subscript 𝐼 𝐵 3 superscript 𝑘 2 superscript subscript 𝑚 𝐵 subscript 𝑔 𝐵 𝐵 𝜎 𝜎 2 e_{B}(k)=g_{BB\omega}\omega_{0}+g_{BB\rho}\rho_{03}I_{B3}+\sqrt{k^{2}+(m_{B}-g% _{BB\sigma}\sigma)^{2}},italic_e start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_k ) = italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ω end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ρ end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 03 end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_B 3 end_POSTSUBSCRIPT + square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,(4)

e¯B⁢(k)=−g B⁢B⁢ω⁢ω 0−g B⁢B⁢ρ⁢ρ 03⁢I¯B⁢3+k 2+(m B−g B⁢B⁢σ⁢σ)2.subscript¯𝑒 𝐵 𝑘 subscript 𝑔 𝐵 𝐵 𝜔 subscript 𝜔 0 subscript 𝑔 𝐵 𝐵 𝜌 subscript 𝜌 03 subscript¯𝐼 𝐵 3 superscript 𝑘 2 superscript subscript 𝑚 𝐵 subscript 𝑔 𝐵 𝐵 𝜎 𝜎 2\bar{e}{B}(k)=-g{BB\omega}\omega_{0}-g_{BB\rho}\rho_{03}\bar{I}{B3}+\sqrt{k% ^{2}+(m{B}-g_{BB\sigma}\sigma)^{2}}.over¯ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_k ) = - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ω end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ρ end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 03 end_POSTSUBSCRIPT over¯ start_ARG italic_I end_ARG start_POSTSUBSCRIPT italic_B 3 end_POSTSUBSCRIPT + square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .(5)

Where, I B⁢3 subscript 𝐼 𝐵 3 I_{B3}italic_I start_POSTSUBSCRIPT italic_B 3 end_POSTSUBSCRIPT represents the isospin projection for baryon B. In addition, σ,ω 0 𝜎 subscript 𝜔 0\sigma,\omega_{0}italic_σ , italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ρ 03 subscript 𝜌 03\rho_{03}italic_ρ start_POSTSUBSCRIPT 03 end_POSTSUBSCRIPT are meson fields in the uniform static matter, and we can write them as:

ω 0=∑B g B⁢B⁢ω m ω 2⁢ρ B,subscript 𝜔 0 subscript 𝐵 subscript 𝑔 𝐵 𝐵 𝜔 superscript subscript 𝑚 𝜔 2 subscript 𝜌 𝐵\omega_{0}=\sum_{B}\dfrac{g_{BB\omega}}{m_{\omega}^{2}}\rho_{B},italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT divide start_ARG italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ω end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ,(6)

ρ 03=∑B g B⁢B⁢ρ m ρ 2⁢I B⁢3⁢ρ B,subscript 𝜌 03 subscript 𝐵 subscript 𝑔 𝐵 𝐵 𝜌 superscript subscript 𝑚 𝜌 2 subscript 𝐼 𝐵 3 subscript 𝜌 𝐵\rho_{03}=\sum_{B}\dfrac{g_{BB\rho}}{m_{\rho}^{2}}I_{B3}\rho_{B},italic_ρ start_POSTSUBSCRIPT 03 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT divide start_ARG italic_g start_POSTSUBSCRIPT italic_B italic_B italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_I start_POSTSUBSCRIPT italic_B 3 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ,(7)

m σ 2⁢σ=−b⁢m N⁢g N⁢N⁢σ⁢(g N⁢N⁢σ⁢σ)2−c⁢g N⁢N⁢σ⁢(g N⁢N⁢σ⁢σ)3+∑B 2⁢J B+1 2⁢π 2⁢g B⁢B⁢σ⁢∫0 k B m B−g B⁢B⁢σ⁢σ k 2+(m B−g B⁢B⁢σ⁢σ)2⁢k 2⁢𝑑 k.superscript subscript 𝑚 𝜎 2 𝜎 𝑏 subscript 𝑚 𝑁 subscript 𝑔 𝑁 𝑁 𝜎 superscript subscript 𝑔 𝑁 𝑁 𝜎 𝜎 2 𝑐 subscript 𝑔 𝑁 𝑁 𝜎 superscript subscript 𝑔 𝑁 𝑁 𝜎 𝜎 3 subscript 𝐵 2 subscript 𝐽 𝐵 1 2 superscript 𝜋 2 subscript 𝑔 𝐵 𝐵 𝜎 superscript subscript 0 subscript 𝑘 𝐵 subscript 𝑚 𝐵 subscript 𝑔 𝐵 𝐵 𝜎 𝜎 superscript 𝑘 2 superscript subscript 𝑚 𝐵 subscript 𝑔 𝐵 𝐵 𝜎 𝜎 2 superscript 𝑘 2 differential-d 𝑘 m_{\sigma}^{2}\sigma=-bm_{N}g_{NN\sigma}(g_{NN\sigma}\sigma)^{2}-cg_{NN\sigma}% (g_{NN\sigma}\sigma)^{3}\ +\sum_{B}\dfrac{2J_{B}+1}{2\pi^{2}}g_{BB\sigma}\int_{0}^{k_{B}}\dfrac{m_{B}-g_% {BB\sigma}\sigma}{\sqrt{k^{2}+(m_{B}-g_{BB\sigma}\sigma)^{2}}}k^{2}dk.italic_m start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ = - italic_b italic_m start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_c italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT divide start_ARG 2 italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + 1 end_ARG start_ARG 2 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ end_ARG start_ARG square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_k .(8)

The spin of a baryon is denoted by J B subscript 𝐽 𝐵 J_{B}italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. The total baryon number density (ρ B subscript 𝜌 𝐵\rho_{B}italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT) is related to the Fermi momentum (k F subscript 𝑘 𝐹 k_{F}italic_k start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT) by the following relation,

ρ t⁢o⁢t⁢a⁢l=∑i ρ i=∑i k F i 3 3⁢π 2,subscript 𝜌 𝑡 𝑜 𝑡 𝑎 𝑙 subscript 𝑖 subscript 𝜌 𝑖 subscript 𝑖 superscript subscript 𝑘 subscript 𝐹 𝑖 3 3 superscript 𝜋 2\rho_{total}=\sum_{i}\rho_{i}=\sum_{i}\dfrac{k_{F_{i}}^{3}}{3\pi^{2}},italic_ρ start_POSTSUBSCRIPT italic_t italic_o italic_t italic_a italic_l end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG 3 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,(9)

where, i 𝑖 i italic_i is the set of baryons in the model.

II.3 Equation of State and Equilibrium Conditions

By applying the Energy-Momentum tensor to the Lagrangian density (Eq. 2), we obtain the following expressions for energy and pressure per particle:

ε=1 3⁢b⁢m N⁢(g N⁢N⁢σ⁢σ)3+1 4⁢c⁢(g N⁢N⁢σ⁢σ)4+1 2⁢m σ 2⁢σ 2+1 2⁢m ω 2⁢ω 0 2+1 2⁢m ρ 2⁢ρ 03 2+∑B 2⁢J B+1 2⁢π 2⁢∫0 k B k 2+(m B−g B⁢B⁢σ⁢σ)2⁢k 2⁢𝑑 k+∑λ 1 π 2⁢∫0 k λ k 2+m λ 2⁢k 2⁢𝑑 k,𝜀 1 3 𝑏 subscript 𝑚 𝑁 superscript subscript 𝑔 𝑁 𝑁 𝜎 𝜎 3 1 4 𝑐 superscript subscript 𝑔 𝑁 𝑁 𝜎 𝜎 4 1 2 superscript subscript 𝑚 𝜎 2 superscript 𝜎 2 1 2 superscript subscript 𝑚 𝜔 2 superscript subscript 𝜔 0 2 1 2 superscript subscript 𝑚 𝜌 2 superscript subscript 𝜌 03 2 subscript 𝐵 2 subscript 𝐽 𝐵 1 2 superscript 𝜋 2 superscript subscript 0 subscript 𝑘 𝐵 superscript 𝑘 2 superscript subscript 𝑚 𝐵 subscript 𝑔 𝐵 𝐵 𝜎 𝜎 2 superscript 𝑘 2 differential-d 𝑘 subscript 𝜆 1 superscript 𝜋 2 superscript subscript 0 subscript 𝑘 𝜆 superscript 𝑘 2 superscript subscript 𝑚 𝜆 2 superscript 𝑘 2 differential-d 𝑘\begin{split}\varepsilon=&\dfrac{1}{3}b\leavevmode\nobreak\ m_{N}(g_{NN\sigma}% \sigma)^{3}+\dfrac{1}{4}c(g_{NN\sigma}\sigma)^{4}+\dfrac{1}{2}m_{\sigma}^{2}% \sigma^{2}+\dfrac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\dfrac{1}{2}m_{\rho}^{2}% \rho_{03}^{2}\ &+\sum_{B}\dfrac{2J_{B}+1}{2\pi^{2}}\int_{0}^{k_{B}}\sqrt{k^{2}+(m_{B}-g_{BB% \sigma}\sigma)^{2}}k^{2}dk\ &+\sum_{\lambda}\dfrac{1}{\pi^{2}}\int_{0}^{k_{\lambda}}\sqrt{k^{2}+m_{\lambda% }^{2}}k^{2}dk,\ \end{split}start_ROW start_CELL italic_ε = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 3 end_ARG italic_b italic_m start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_c ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT 03 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT divide start_ARG 2 italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + 1 end_ARG start_ARG 2 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_k end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_k , end_CELL end_ROW(10)

p=−1 3⁢b⁢m N⁢(g N⁢N⁢σ⁢σ)3−1 4⁢c⁢(g N⁢N⁢σ⁢σ)4−1 2⁢m σ 2⁢σ 2+1 2⁢m ω 2⁢ω 0 2+1 2⁢m ρ 2⁢ρ 03 2+1 3⁢∑B 2⁢J B+1 2⁢π 2⁢∫0 k B k 4⁢d⁢k k 2+(m B−g B⁢B⁢σ⁢σ)2+1 3⁢∑λ 1 π 2⁢∫0 k λ k 4⁢d⁢k k 2+m λ 2.𝑝 1 3 𝑏 subscript 𝑚 𝑁 superscript subscript 𝑔 𝑁 𝑁 𝜎 𝜎 3 1 4 𝑐 superscript subscript 𝑔 𝑁 𝑁 𝜎 𝜎 4 1 2 superscript subscript 𝑚 𝜎 2 superscript 𝜎 2 1 2 superscript subscript 𝑚 𝜔 2 superscript subscript 𝜔 0 2 1 2 superscript subscript 𝑚 𝜌 2 superscript subscript 𝜌 03 2 1 3 subscript 𝐵 2 subscript 𝐽 𝐵 1 2 superscript 𝜋 2 superscript subscript 0 subscript 𝑘 𝐵 superscript 𝑘 4 𝑑 𝑘 superscript 𝑘 2 superscript subscript 𝑚 𝐵 subscript 𝑔 𝐵 𝐵 𝜎 𝜎 2 1 3 subscript 𝜆 1 superscript 𝜋 2 superscript subscript 0 subscript 𝑘 𝜆 superscript 𝑘 4 𝑑 𝑘 superscript 𝑘 2 superscript subscript 𝑚 𝜆 2\begin{split}p=&-\dfrac{1}{3}b\leavevmode\nobreak\ m_{N}(g_{NN\sigma}\sigma)^{% 3}-\dfrac{1}{4}c(g_{NN\sigma}\sigma)^{4}-\dfrac{1}{2}m_{\sigma}^{2}\sigma^{2}+% \dfrac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\dfrac{1}{2}m_{\rho}^{2}\rho_{03}^{2}% \ &+\dfrac{1}{3}\sum_{B}\dfrac{2J_{B}+1}{2\pi^{2}}\int_{0}^{k_{B}}\dfrac{k^{4}dk% }{\sqrt{k^{2}+(m_{B}-g_{BB\sigma}\sigma)^{2}}}\ &+\dfrac{1}{3}\sum_{\lambda}\dfrac{1}{\pi^{2}}\int_{0}^{k_{\lambda}}\dfrac{k^{% 4}dk}{\sqrt{k^{2}+m_{\lambda}^{2}}}.\ \end{split}start_ROW start_CELL italic_p = end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 3 end_ARG italic_b italic_m start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_c ( italic_g start_POSTSUBSCRIPT italic_N italic_N italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_m start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT 03 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 3 end_ARG ∑ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT divide start_ARG 2 italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + 1 end_ARG start_ARG 2 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_k start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_d italic_k end_ARG start_ARG square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_B italic_B italic_σ end_POSTSUBSCRIPT italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 3 end_ARG ∑ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_k start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_d italic_k end_ARG start_ARG square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG . end_CELL end_ROW(11)

where, m N subscript 𝑚 𝑁 m_{N}italic_m start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and m B subscript 𝑚 𝐵 m_{B}italic_m start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT are mass of nucleon and baryons, respectively. The masses of the mesons are denoted by m σ subscript 𝑚 𝜎 m_{\sigma}italic_m start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT, m ω subscript 𝑚 𝜔 m_{\omega}italic_m start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT, and m ρ subscript 𝑚 𝜌 m_{\rho}italic_m start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT. This study employs a model of a cold NS comprising a combination of neutrons, protons, electrons, muons, and hyperons (Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ). Due to the high rest mass of the τ 𝜏\tau italic_τ lepton, its contribution is considered negligible. The possible processes to establish the beta equilibrium condition in the system are as follows:

n⟶p+e+ν¯e,⟶𝑛 𝑝 𝑒 subscript¯𝜈 𝑒 n\longrightarrow p+e+\bar{\nu}_{e},italic_n ⟶ italic_p + italic_e + over¯ start_ARG italic_ν end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ,(12)

and

p+e⟶n+ν e.⟶𝑝 𝑒 𝑛 subscript 𝜈 𝑒 p+e\longrightarrow n+\nu_{e}.italic_p + italic_e ⟶ italic_n + italic_ν start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT .(13)

We know these processes as ”Urca processes” Haensel1995. When the energy of electrons increases enough, muons can be produced as,

e⟶μ+ν¯μ+ν e.⟶𝑒 𝜇 subscript¯𝜈 𝜇 subscript 𝜈 𝑒 e\longrightarrow\mu+\bar{\nu}{\mu}+\nu{e}.italic_e ⟶ italic_μ + over¯ start_ARG italic_ν end_ARG start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT + italic_ν start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT .(14)

The appearance of each hyperon species in matter depends on the electric charge and isospin. Since nuclear matter has an excess of positive charge and negative isospin, the formation of hyperons is favoured by negative charge and positive isospin, as well as lower mass. The appearance of each hyperon species at a specific baryon number density is typically the outcome of various factors interacting with each other. However, a detailed quantitative analysis necessitates modeling the interactions at high densities Balberg:1998ug. At higher densities, Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ could be produced via the following processes,

n+e⟶Σ−+ν e,⟶𝑛 𝑒 superscript Σ subscript 𝜈 𝑒 n+e\longrightarrow\Sigma^{-}+\nu_{e},italic_n + italic_e ⟶ roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT + italic_ν start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ,(15)

and

n+n⟶n+Λ.⟶𝑛 𝑛 𝑛 Λ n+n\longrightarrow n+\Lambda.italic_n + italic_n ⟶ italic_n + roman_Λ .(16)

The thermodynamic equilibrium conditions for chemical potentials that govern the above equations are as follows:

μ i=B i⁢μ n−Q i⁢μ e.subscript 𝜇 𝑖 subscript 𝐵 𝑖 subscript 𝜇 𝑛 subscript 𝑄 𝑖 subscript 𝜇 𝑒\mu_{i}=B_{i}\mu_{n}-Q_{i}\mu_{e}.italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT .(17)

Where, B i subscript 𝐵 𝑖 B_{i}italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, and Q i subscript 𝑄 𝑖 Q_{i}italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT denote the baryon charge and electric charge of each species. For Eqs. (12) and (13), the chemical potential of neutrinos is zero since they escape the star as it evolves. Additionally, the system maintains charge neutrality, which is established internally, as:

ρ p=ρ e+ρ μ+ρ Σ−,subscript 𝜌 𝑝 subscript 𝜌 𝑒 subscript 𝜌 𝜇 subscript 𝜌 superscript Σ\rho_{p}=\rho_{e}+\rho_{\mu}+\rho_{\Sigma^{-}},italic_ρ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ,(18)

where ρ i subscript 𝜌 𝑖\rho_{i}italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the number density of each particle. Finally, number density conservation is expressed as:

ρ B=ρ n+ρ p+ρ Σ−+ρ Λ.subscript 𝜌 𝐵 subscript 𝜌 𝑛 subscript 𝜌 𝑝 subscript 𝜌 superscript Σ subscript 𝜌 Λ\rho_{B}=\rho_{n}+\rho_{p}+\rho_{\Sigma^{-}}+\rho_{\Lambda}.italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT .(19)

II.4 Coupling Constants

We have incorporated various baryon-meson coupling constants in the equation of state to describe stellar matter. Two types of coupling constants have been utilized. One type of parameters was extracted from Refs. Glendenning:1991es; Karimi:2022vxw, which is based on the σ 𝜎\sigma italic_σ-ω 𝜔\omega italic_ω-ρ 𝜌\rho italic_ρ model. The other method is based on the QCD sum rules, providing a more fundamental based approach. Quantum Chromodynamics Sum Rules (QCDSR), is a method that connects the hadronic and the QCD descriptions of matter. It was introduced in Ref. Shifman:1978bx for mesons and in Ref. Ioffe:1981kw for baryons, and has since become a powerful tool for the phenomenology of hadronic physics. Various hadronic properties can be determined via QCDSR, such as masses, decay rates, coupling constants, magnetic moments and weak form factors. These properties are expressed in terms of QCD parameters, such as quark masses, and quark and gluon condensates Aliev:2009jt; Agaev:2016dev; Azizi:2016dhy. The main idea of QCDSR is to construct a correlation function of two/three hadronic currents, which can be calculated in two different ways: on one hand, by using the operator product expansion (OPE) and the QCD perturbation theory at short distances; and on the other hand, by using the hadronic spectral representation and the dispersion relation at long distances Doi:2003cd. The idea behind the OPE method is that the properties of a hadron, such as its mass or decay constant, can be related to the behavior of quarks and gluons inside the hadron through a series expansion. This expansion expresses a physical quantity, such as the mass of a hadron, as a sum of terms involving quark and gluon field operators of increasing dimension, with coefficients known as Wilson coefficients. As mentioned, one of the applications of QCDSR method is to calculate the meson-baryon coupling constants, which are essential for understanding the hadronic interactions. For instance, to determine the coupling constant of a vector meson to the nucleon, one can consider the following three-point correlation function:

Π μ⁢(q)=i 2⁢∫d 4⁢x⁢∫d 4⁢y⁢e−i⁢p⋅x⁢e i⁢p′⋅y⁢⟨0|𝒯⁢(J N⁢(y)⁢J ℳ μ⁢(0)⁢J¯N⁢(x))|0⟩,superscript Π 𝜇 𝑞 superscript 𝑖 2 superscript 𝑑 4 𝑥 superscript 𝑑 4 𝑦 superscript 𝑒⋅𝑖 𝑝 𝑥 superscript 𝑒⋅𝑖 superscript 𝑝′𝑦 quantum-operator-product 0 𝒯 subscript 𝐽 𝑁 𝑦 subscript superscript 𝐽 𝜇 ℳ 0 subscript¯𝐽 𝑁 𝑥 0\Pi^{\mu}(q)=i^{2}\int d^{4}x\int d^{4}y\leavevmode\nobreak\ e^{-ip\cdot x}% \leavevmode\nobreak\ e^{ip^{\prime}\cdot y}\leavevmode\nobreak\ \langle 0|% \mathcal{T}\big{(}J_{N}(y)\leavevmode\nobreak\ J^{\mu}{\mathcal{M}}(0)% \leavevmode\nobreak\ \bar{J}{N}(x)\big{)}|0\rangle,roman_Π start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_q ) = italic_i start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∫ italic_d start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_x ∫ italic_d start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_y italic_e start_POSTSUPERSCRIPT - italic_i italic_p ⋅ italic_x end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⋅ italic_y end_POSTSUPERSCRIPT ⟨ 0 | caligraphic_T ( italic_J start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_y ) italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT ( 0 ) over¯ start_ARG italic_J end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x ) ) | 0 ⟩ ,(20)

where, 𝒯 𝒯\mathcal{T}caligraphic_T is the time ordering operator and q=p−p′𝑞 𝑝 superscript 𝑝′q=p-p^{\prime}italic_q = italic_p - italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is the transferred momentum. J N subscript 𝐽 𝑁 J_{N}italic_J start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and J ℳ μ subscript superscript 𝐽 𝜇 ℳ J^{\mu}_{\mathcal{M}}italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT denote the interpolating fields of the nucleon and meson, respectively. Their interpolating currents in terms of the quark fields, are:

J ℳ μ⁢(0)=q¯1⁢(0)⁢γ μ⁢q 2⁢(0),subscript superscript 𝐽 𝜇 ℳ 0 subscript¯𝑞 1 0 superscript 𝛾 𝜇 subscript 𝑞 2 0 J^{\mu}{\mathcal{M}}(0)=\bar{q}{1}(0)\leavevmode\nobreak\ \gamma^{\mu}% \leavevmode\nobreak\ q_{2}(0),italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT ( 0 ) = over¯ start_ARG italic_q end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) italic_γ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 0 ) ,(21)

and

J N⁢(y)=ϵ α⁢β⁢z⁢(u α T⁢(y)⁢C⁢γ ν⁢u β⁢(y))⁢γ 5⁢γ ν⁢d z⁢(y).subscript 𝐽 𝑁 𝑦 subscript italic-ϵ 𝛼 𝛽 𝑧 superscript 𝑢 superscript 𝛼 𝑇 𝑦 𝐶 subscript 𝛾 𝜈 superscript 𝑢 𝛽 𝑦 subscript 𝛾 5 superscript 𝛾 𝜈 superscript 𝑑 𝑧 𝑦 J_{N}(y)=\epsilon_{\alpha\beta z}\big{(}u^{\alpha^{T}}(y)\leavevmode\nobreak\ % C\leavevmode\nobreak\ \gamma_{\nu}\leavevmode\nobreak\ u^{\beta}(y)\big{)}% \gamma_{5}\leavevmode\nobreak\ \gamma^{\nu}\leavevmode\nobreak\ d^{z}(y).italic_J start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_y ) = italic_ϵ start_POSTSUBSCRIPT italic_α italic_β italic_z end_POSTSUBSCRIPT ( italic_u start_POSTSUPERSCRIPT italic_α start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_y ) italic_C italic_γ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( italic_y ) ) italic_γ start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT italic_γ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT italic_d start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ( italic_y ) .(22)

Here, q i subscript 𝑞 𝑖 q_{i}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are the light quarks, and α 𝛼\alpha italic_α, β 𝛽\beta italic_β, z 𝑧 z italic_z are the color indices, C 𝐶 C italic_C is the charge conjugation operator, and T 𝑇 T italic_T denotes the transpose with respect to the Dirac indices. By applying the OPE and the spectral representation on this correlation function, one can obtain a sum rule for the nucleon-nucleon-meson coupling constant. The Borel transformation is applied to both sides to suppress the contributions of the higher states and continuum Azizi:2015bxa. The above procedure can be applied to calculate the baryon-meson couplings, such as the N⁢N⁢ω 𝑁 𝑁 𝜔 NN\omega italic_N italic_N italic_ω, N⁢N⁢ρ 𝑁 𝑁 𝜌 NN\rho italic_N italic_N italic_ρ, Λ⁢Λ⁢ρ Λ Λ 𝜌\Lambda\Lambda\rho roman_Λ roman_Λ italic_ρ, Λ⁢Λ⁢ω Λ Λ 𝜔\Lambda\Lambda\omega roman_Λ roman_Λ italic_ω, etc. The results can be compared with the empirical values obtained from the One-Boson Exchange (OBE) models of the two-baryon interactions, as well as the SU(3)-flavor relations and the chiral symmetry constraints Aliev:2009ei; Wang:2007yt; Erkol:2006sa; Erkol:2006eq; Zamiralov:2013gva. The accuracy of the coupling constant calculation using QCDSR depends on a number of factors, such as the quality of the assumptions made in the OPE method and working intervals of some auxiliary parameters entering the sum rules. However, QCDSR have been successfully employed to calculate coupling constants for a broad range of processes involving hadrons, and have offered important insights into the behavior of quarks and gluons within QCD. In this study, we use the coupling constants obtained by two different methods ( σ−ω−ρ 𝜎 𝜔 𝜌\sigma-\omega-\rho italic_σ - italic_ω - italic_ρ Model and QCDSR method) as described in Refs. Karimi:2022vxw; Aliev:2009ei; Wang:2007yt; Erkol:2006sa; Erkol:2006eq; Zamiralov:2013gva, which are listed in Table 1, and divided into four sets according to Table 2.

Table 1: Meson-baryon coupling constants (dimensionless).

Table 2: Coupling constant sets.

III Results and discussion

III.1 Particle Fractions and EOS

To explore the structure and composition of a neutron star, it’s essential to determine the number density of each potential particle within the star. Thus, we assess the number density of each particle relative to the baryonic number density, representing it as a fraction of particles. To obtain the particle fractions in the NS matter, we solve the sets of equations of chemical potentials (17), charge neutrality (18), baryon number density (19) and meson fields (6)−(8)68(\ref{eqn:m-f-one})-(\ref{eqn:m-f-tree})( ) - ( ) at varying densities. The relative particle fractions as a function of the baryon number density of each coupling constant sets are presented in Fig. 1. This figure illustrates that when the electron chemical potential reaches a threshold equal to or surpassing the mass of muon, muon can emerge within the system. As depicted in the figure, the threshold for the onset of muon within the set of coupling constants derived from the QCDSR method (sets 2-4) is approximately around a baryon number density of 0.23 f⁢m−3 𝑓 superscript 𝑚 3 fm^{-3}italic_f italic_m start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT, exceeding what was obtained in Karimi:2022vxw using a simple parameterization of coupling constants. At low densities, the system is chiefly composed of neutron matter, but at high densities, there is a significant contribution from other baryons.

Image 1: Refer to caption

Image 2: Refer to caption

(a)

Image 3: Refer to caption

(b)

Image 4: Refer to caption

(c)

Figure 1: Particle fraction Y i subscript 𝑌 𝑖 Y_{i}italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for npYe μ 𝜇\mu italic_μ system using coupling constants of (a): set 1, (b): set 2, (c): set 3, and (d): set 4.

The emergence of hyperons in NS cores transpires as the nucleon chemical potentials surpass the mass differences between nucleons and hyperons, while the threshold for hyperon appearance is determined by their interactions. As the Hyperon fraction increase, the nucleon fractions decrease due to the baryon number density conservation (Eq. 19). The particle fractions become particularly intriguing with the presence of hyperons (Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ) in the system. With the appearance of the Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT hyperon, the density of negatively charged leptons begins to decrease rapidly. This occurs because the charge neutrality condition (Eq. 18) is now fulfilled primarily by the Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT hyperon. Similarly, the emergence of the Λ Λ\Lambda roman_Λ hyperon will accelerate the disappearance of neutrons, as both are neutral particles. The Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT fraction reaches a saturation approximately 0.1. However, in the case of set 1 in Fig. 1, this saturation occurs around 0.2. Meanwhile, the Λ Λ\Lambda roman_Λ particles, unaffected by isospin-dependent forces, continue to increase until they are eventually saturated by short-range repulsion forces. The onset of hyperons is met when the sum of the chemical potentials of neutrons and electrons is greater than or equal to the mass of Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT (μ n subscript 𝜇 𝑛\mu_{n}italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + μ e subscript 𝜇 𝑒\mu_{e}italic_μ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT≥\geq≥m Σ−subscript 𝑚 superscript Σ m_{\Sigma^{-}}italic_m start_POSTSUBSCRIPT roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT), and when the chemical potential of neutrons is greater than or equal to the mass of Λ Λ\Lambda roman_Λ, respectively. With the exception of the configuration in set 4 (Fig. 1(c)), the muons are entirely annihilated, and the electron fraction drops below 1 %percent%%. In contrast, the electron fraction exceeds 10 %percent%% in nucleonic matter. It is widely acknowledged that hyperons start to emerge at a density of approximately 2-3 times the nuclear saturation density (ρ 0 subscript 𝜌 0\rho_{0}italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT) Schulze:1998jf; Djapo:2008au; Vidana:2015rsa; Lonardoni:2013gta, with the threshold values being influenced by the nature of their interactions. The threshold values for the presence of muons and hyperons for each set of coupling constants studied in this research are provided in Table 3. These values consistent with other studies’ reports, such as Refs. Schulze:1998jf; Djapo:2008au; Vidana:2015rsa; Lonardoni:2013gta. However, they exceed the usual reported values for set 4 of coupling constants. As seen in Table 2, the coupling constants of set 4 for ω 𝜔\omega italic_ω and ρ 𝜌\rho italic_ρ meson exchanges are lower than those for the other sets. Consequently, set 4 exhibits weaker interactions for hyperons compared to the other sets. This could explain the onset of hyperons at higher baryon number densities in set 4. However, such high baryon densities only occur within a small radius around the center of a neutron star. Therefore, the amount of hyperons does not play a significant role in determining the overall structure of the NS. Results clearly demonstrate the impact of coupling constants on the system, because, according to Eqs. (4) to (8), the values of the coupling constants affect the values of the meson fields which is turn affect the chemical potential. This can lead to different EOS and physics for each system.

Table 3: The threshold values (in f⁢m−3 𝑓 superscript 𝑚 3 fm^{-3}italic_f italic_m start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT) for the presence of muons and hyperons in each set.

The equation of state plays a crucial role in investigating the properties of NSs. Therefore, we present our calculated results with the presentation of our EOS findings. Fig. 2 shows the EOS for NSs when Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ hyperons are present for different sets of coupling constants, as well as for NS matter without hyperons. The line labeled ”P=ε 𝑃 𝜀 P=\varepsilon italic_P = italic_ε” corresponds to the causal limit, where the speed of sound is equal to the speed of light. At low densities, the curves coincide, indicating the existence of only nucleonic matter in β 𝛽\beta italic_β-equilibrium within the systems. The pressure-density diagram for all configurations is shown in Fig. 2. The bold points on the curves indicate the central densities in the stars with the maximum mass. We have checked the properties of nuclear matter at saturation point too. We observed that all investigated EOS can reproduce the saturation properties like saturation density and energy as well as incompatibility at saturation point in acceptable range. So in this study we focus on EOS predictions at high densities. The pressure-density relation contains all the essential information necessary for determining the macroscopic properties such as mass, radius, stability and gravitational wave emission of a NS. The pressure-density also determines the internal structure of NSs, such as the presence of different phases of matter including hyperons, quarks, or super fluids, in the core or the crust of the star. The stiffness of the equation of state is crucial in determining the maximum mass of a neutron star. A stiffer equation of state typically leads to a higher maximum mass for the star. In the figure, despite the presence of hyperons in the star, the equation of state is softer compared to the scenario where hyperons are absent. However, as depicted in the figure, the equations of state resulting from set 2 of coupling constants is sufficiently stiff to satisfy observational constraints related to maximum mass, surpassing 2 times the mass of the Sun. This observation is significant in addressing the hyperon puzzle, and we will explore this further in the following sections.

Image 5: Refer to caption

Image 6: Refer to caption

Figure 2: (a): Pressure vs. energy density, (b): Pressure vs. density for npYe μ 𝜇\mu italic_μ systems using the various sets of coupling constants.

III.2 Mass-Radius relation of Neutron Stars

The Tolman-Oppenheimer-Volkoff (TOV) Equation Tolman:1939jz; Oppenheimer:1939ne, are used to determine the stability of an EOS against gravitational collapse. These equations relate the change in pressure with radius to the state variables of the EOS. To solve the equations, the boundary condition that the pressure is zero at the surface of the star is applied. The TOV equations are:

d⁢P⁢(r)d⁢r=−G⁢M⁢(r)⁢ε⁢(r)c 2⁢r 2⁢(1+P⁢(r)ε⁢(r))⁢(1+4⁢π⁢r 3⁢P⁢(r)M⁢(r)⁢c 2)⁢(1−2⁢G⁢M⁢(r)r⁢c 2)−1,𝑑 𝑃 𝑟 𝑑 𝑟 𝐺 𝑀 𝑟 𝜀 𝑟 superscript 𝑐 2 superscript 𝑟 2 1 𝑃 𝑟 𝜀 𝑟 1 4 𝜋 superscript 𝑟 3 𝑃 𝑟 𝑀 𝑟 superscript 𝑐 2 superscript 1 2 𝐺 𝑀 𝑟 𝑟 superscript 𝑐 2 1\dfrac{dP(r)}{dr}=-\dfrac{GM(r)\varepsilon(r)}{c^{2}r^{2}}(1+\dfrac{P(r)}{% \varepsilon(r)})(1+\dfrac{4\pi r^{3}P(r)}{M(r)c^{2}})(1-\dfrac{2GM(r)}{rc^{2}}% )^{-1},divide start_ARG italic_d italic_P ( italic_r ) end_ARG start_ARG italic_d italic_r end_ARG = - divide start_ARG italic_G italic_M ( italic_r ) italic_ε ( italic_r ) end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( 1 + divide start_ARG italic_P ( italic_r ) end_ARG start_ARG italic_ε ( italic_r ) end_ARG ) ( 1 + divide start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_P ( italic_r ) end_ARG start_ARG italic_M ( italic_r ) italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ( 1 - divide start_ARG 2 italic_G italic_M ( italic_r ) end_ARG start_ARG italic_r italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ,(23)

and

d⁢M⁢(r)d⁢r=4⁢π⁢ε⁢(r)⁢r 2 c 2.𝑑 𝑀 𝑟 𝑑 𝑟 4 𝜋 𝜀 𝑟 superscript 𝑟 2 superscript 𝑐 2\dfrac{dM(r)}{dr}=\dfrac{4\pi\varepsilon(r)r^{2}}{c^{2}}.divide start_ARG italic_d italic_M ( italic_r ) end_ARG start_ARG italic_d italic_r end_ARG = divide start_ARG 4 italic_π italic_ε ( italic_r ) italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .(24)

Where P⁢(r)𝑃 𝑟 P(r)italic_P ( italic_r ), ε⁢(r)𝜀 𝑟\varepsilon(r)italic_ε ( italic_r ), and M⁢(r)𝑀 𝑟 M(r)italic_M ( italic_r ) represent the pressure, energy density and mass of the star, respectively. To solve the TOV equations, we require the relation between pressure and energy density. In this work, we use different equations of state (EOS) as discussed in the previous section for the core of the star, and BPS equation of state PPS for the crust of the neutron star (NS), to solve the TOV equations. We assume spherical symmetry and zero pressure gradient d⁢P d⁢r 𝑑 𝑃 𝑑 𝑟\dfrac{dP}{dr}divide start_ARG italic_d italic_P end_ARG start_ARG italic_d italic_r end_ARG at the center of the star (r=0). By using the Runge-Kutta method to solve the TOV differential equations with the boundary condition (P=0) at the surface of the star, we can determine the pressure at an arbitrary radius. Subsequently, by utilizing the given equation of state, we can calculate the energy density. Using calculated ε⁢(r)𝜀 𝑟\varepsilon(r)italic_ε ( italic_r ), we can then determine the mass within a given radius R′superscript 𝑅′R^{{}^{\prime}}italic_R start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT as

M⁢(R′)=∫0 R′4⁢π⁢r 2⁢ε⁢(r)⁢𝑑 r.𝑀 superscript 𝑅′superscript subscript 0 superscript 𝑅′4 𝜋 superscript 𝑟 2 𝜀 𝑟 differential-d 𝑟 M(R^{{}^{\prime}})=\int_{0}^{R^{{}^{\prime}}}4\pi r^{2}\varepsilon(r)dr.italic_M ( italic_R start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_R start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT 4 italic_π italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ε ( italic_r ) italic_d italic_r .(25)

The total mass of the star is given by M⁢(R)𝑀 𝑅 M(R)italic_M ( italic_R ), where R 𝑅 R italic_R is the radius of the star. By varying the central density, ρ C subscript 𝜌 𝐶\rho_{C}italic_ρ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, as an initial input, and repeating the integration for different values the mass-radius (M-R) relation is determined. Consequently, the maximum mass and corresponding radius of the neutron star can be found. In Fig. 3, we display the M-R relation and gravitational mass as a function of the central baryon density of NS. This include NSs consisting of nep, nep μ 𝜇\mu italic_μ (using coupling constants of set 2) and, nep μ⁢Σ−⁢Λ 𝜇 superscript Σ Λ\mu\Sigma^{-}\Lambda italic_μ roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT roman_Λ matter of each set. The results are compared with the masses of the heaviest known pulsars, including; PSR J0740+6620 Fonseca:2021wxt, PSR J0952-0607 Romani:2022jhd, PSR J1614-2230 Demorest:2010bx, PSR J0537-6910 Ho:2015vza; Ho:2020vxt, and binary neutron star observed in the GW190425, and GW170817 gravitational events LIGOScientific:2020aai; LIGOScientific:2018cki; Lim:2019som; Malik:2018zcf. The observational constraints are represented by color coded regions in this figure. The bold points in the figure indicate the maximum mass, and the numerical values of the masses and radii are given in Table 4. The nep and nep μ 𝜇\mu italic_μ diagrams shows that due to the low muon abundance in NSs, their effect on the M-R relation is negligible. The fraction of muons in NSs is typically less than 10 %percent%%, and their contribution to the pressure is small in comparison to nucleons and hyperons. We can observe that QCDSR coupling constant can create a stiff enough EOS compare to set 1, thereby enabling the existence of massive and larger NSs. The outcomes of our sets are in good agreement with the most massive observed objects.

Image 7: Refer to caption

Image 8: Refer to caption

(a)

Figure 3: (a): The mass-radius relation (left panel), and (b): gravitational mass as a function of the central baryon density (right panel) in the case of nucleon stars (upper curves) and hyperon stars (lower curves) in compared with observational data (color lines).

Table 4: The maximum masses (M m⁢a⁢x/M⊙subscript 𝑀 𝑚 𝑎 𝑥 subscript 𝑀 direct-product M_{max}/M_{\odot}italic_M start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT / italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT) and radii (R 𝑅 R italic_R) values of stellar matter for each set and observed object.

We have illustrated the distribution of baryons within neutron stars, as depicted in Fig. 4. Approximately 1.5 to 2.5 km in the surface of stars are predominantly composed of nucleons. Moving further in the presence of muon particles becomes significant. Closer to the core, the distribution changes. From the center of the stars extending up to about 7 km in stars sigma and lambda baryons are prevalent. Nucleons and hyperons are almost equally distributed within the inner core of these stars. Additionally, small amounts of leptons are observed in these inner core. This detailed mapping of baryon density provides valuable insights into the complex structure and composition of neutron stars, enhancing our understanding of these fascinating astronomical objects.

Image 9: Refer to caption

Image 10: Refer to caption

(a)

Image 11: Refer to caption

(b)

Figure 4: Baryon density as a function of stellar radius for npYe μ 𝜇\mu italic_μ system using coupling constants of (a): set 1, (b): set 2, and (c): set 3.

III.3 Calculation of Tidal deformability and Love number

General relativity predicts that moving bodies distort the surrounding space-time. Oscillations in their mass or motion can produce gravitational waves, which are ripples in space-time. The merging of massive binary systems, such as black holes or NSs, is one of the most powerful sources of gravitational waves. The discovery of gravitational waves, which was achieved by the advanced Laser Interferometer Gravitational wave Observatory (LIGO) detector LIGOScientific:2016aoc, is a significant milestone in astrophysics/cosmology. This was the first direct observation of these ripples in spacetime, which are predicted by the general theory of relativity. It was caused by the inspiral and coalescence of two black holes. It is expected that LIGO LIGOScientific:2016aoc, VIRGO Virgo:11, and KAGRA Kagra:11 will also detect gravitational waves from binary NSs (BNSs). This will provide valuable insights into the properties of highly compressed baryonic matter. Many studies have suggested that the tidal effects of BNSs can be measured by the current generation of gravitational wave (GW) detectors. A dimensionless parameter was introduced by the mathematician A. E. H. Love in Newtonian theory Love:1912 to describe the tidal deformation of the Earth caused by the gravitational attraction of the Moon and the Sun. This theory was later extended to general relativity Damour:2012yf; Binnington:2009bb, where it was found that there are electric and magnetic types of dimensionless gravitational Love numbers that characterize the tidal fields associated with the gravito-electric and gravito-magnetic interactions. The tidal deformability parameter λ 𝜆\lambda italic_λ is dependent on the EOS through the NS’s radius and Love number 𝒦 2 subscript 𝒦 2\mathcal{K}_{2}caligraphic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Flanagan and Hinderer Flanagan:2007ix provided the following expression for λ 𝜆\lambda italic_λ:

λ=2 3⁢𝒦 2⁢R 5,𝜆 2 3 subscript 𝒦 2 superscript 𝑅 5\lambda=\frac{2}{3}\mathcal{K}_{2}R^{5},italic_λ = divide start_ARG 2 end_ARG start_ARG 3 end_ARG caligraphic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_R start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT ,(26)

the dimensionless tidal deformability Λ Λ\varLambda roman_Λ is related to the compactness parameter C=M R 𝐶 𝑀 𝑅 C=\dfrac{M}{R}italic_C = divide start_ARG italic_M end_ARG start_ARG italic_R end_ARG as,

Λ=2⁢𝒦 2 3⁢C 5,Λ 2 subscript 𝒦 2 3 superscript 𝐶 5\varLambda=\dfrac{2\leavevmode\nobreak\ \mathcal{K}_{2}}{3\leavevmode\nobreak% \ C^{5}},roman_Λ = divide start_ARG 2 caligraphic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 3 italic_C start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG ,(27)

where M and R represent the mass and radius, respectively, of the isolated spherical star. We use units in which c = G = 1. To calculate the deformability parameter λ 𝜆\lambda italic_λ, it is necessary to obtain the Love number 𝒦 2 subscript 𝒦 2\mathcal{K}_{2}caligraphic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, which is the key quantity of deformation due to the gravitational attraction between the binary stars. To calculate the Love number, the TOV equations must be solved iteratively along with the following first-order differential equation Hinderer:2007mb. The initial boundary condition is y⁢(0)=2 𝑦 0 2 y(0)=2 italic_y ( 0 ) = 2.

r⁢d⁢y⁢(r)d⁢r+y⁢(r)2+y⁢(r)⁢F⁢(r)+r 2⁢Q⁢(r)=0,𝑟 𝑑 𝑦 𝑟 𝑑 𝑟 𝑦 superscript 𝑟 2 𝑦 𝑟 𝐹 𝑟 superscript 𝑟 2 𝑄 𝑟 0 r\dfrac{dy(r)}{dr}+y(r)^{2}+y(r)F(r)+r^{2}Q(r)=0,italic_r divide start_ARG italic_d italic_y ( italic_r ) end_ARG start_ARG italic_d italic_r end_ARG + italic_y ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y ( italic_r ) italic_F ( italic_r ) + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_Q ( italic_r ) = 0 ,(28)

where,

F⁢(r)=r−4⁢π⁢r 3⁢[ε⁢(r)−P⁢(r)]r−2⁢M⁢(r),𝐹 𝑟 𝑟 4 𝜋 superscript 𝑟 3 delimited-[]𝜀 𝑟 𝑃 𝑟 𝑟 2 𝑀 𝑟 F(r)=\dfrac{r-4\pi r^{3}[\varepsilon(r)-P(r)]}{r-2M(r)},italic_F ( italic_r ) = divide start_ARG italic_r - 4 italic_π italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT [ italic_ε ( italic_r ) - italic_P ( italic_r ) ] end_ARG start_ARG italic_r - 2 italic_M ( italic_r ) end_ARG ,(29)

and,

Q⁢(r)=4⁢π⁢r⁢(5⁢ε⁢(r)+9⁢P⁢(r)+ε⁢(r)+P⁢(r)∂P⁢(r)∂ε⁢(r)−6 4⁢π⁢r 2)r−2⁢M⁢(r)−4⁢[M⁢(r)+4⁢π⁢r 3⁢P⁢(r)r 2⁢(1−2⁢M⁢(r)r)]2.𝑄 𝑟 4 𝜋 𝑟 5 𝜀 𝑟 9 𝑃 𝑟 𝜀 𝑟 𝑃 𝑟 𝑃 𝑟 𝜀 𝑟 6 4 𝜋 superscript 𝑟 2 𝑟 2 𝑀 𝑟 4 superscript delimited-[]𝑀 𝑟 4 𝜋 superscript 𝑟 3 𝑃 𝑟 superscript 𝑟 2 1 2 𝑀 𝑟 𝑟 2 Q(r)=\dfrac{4\pi r\Big{(}5\varepsilon(r)+9P(r)+\dfrac{\varepsilon(r)+P(r)}{% \frac{\partial P(r)}{\partial\varepsilon(r)}}-\frac{6}{4\pi r^{2}}\Big{)}}{r-2% M(r)}-4\Big{[}\dfrac{M(r)+4\pi r^{3}P(r)}{r^{2}(1-\frac{2M(r)}{r})}\Big{]}^{2}.italic_Q ( italic_r ) = divide start_ARG 4 italic_π italic_r ( 5 italic_ε ( italic_r ) + 9 italic_P ( italic_r ) + divide start_ARG italic_ε ( italic_r ) + italic_P ( italic_r ) end_ARG start_ARG divide start_ARG ∂ italic_P ( italic_r ) end_ARG start_ARG ∂ italic_ε ( italic_r ) end_ARG end_ARG - divide start_ARG 6 end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) end_ARG start_ARG italic_r - 2 italic_M ( italic_r ) end_ARG - 4 [ divide start_ARG italic_M ( italic_r ) + 4 italic_π italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_P ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - divide start_ARG 2 italic_M ( italic_r ) end_ARG start_ARG italic_r end_ARG ) end_ARG ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .(30)

In y=y 2⁢(R)𝑦 subscript 𝑦 2 𝑅 y=y_{2}(R)italic_y = italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_R ), the electric tidal Love numbers are given as below Damour:2009wj:

𝒦 2=8 5(1−2 C)2 C 5[2 C(y 2−1)−y 2+2]{2 C(4(y 2+1)C 4+(6 y 2−4)C 3+(26−22 y 2)C 2+3(5 y 2−8)C−3 y 2+6)−3(1−2 C)2(2 C(y 2−1)−y 2+2)l o g(1 1−2⁢C)}−1.subscript 𝒦 2 8 5 superscript 1 2 𝐶 2 superscript 𝐶 5 delimited-[]2 𝐶 subscript 𝑦 2 1 subscript 𝑦 2 2 superscript 2 𝐶 4 subscript 𝑦 2 1 superscript 𝐶 4 6 subscript 𝑦 2 4 superscript 𝐶 3 26 22 subscript 𝑦 2 superscript 𝐶 2 3 5 subscript 𝑦 2 8 𝐶 3 subscript 𝑦 2 6 3 superscript 1 2 𝐶 2 2 𝐶 subscript 𝑦 2 1 subscript 𝑦 2 2 𝑙 𝑜 𝑔 1 1 2 𝐶 1\begin{split}\mathcal{K}{2}=&\leavevmode\nobreak\ \frac{8}{5}(1-2C)^{2}C^{5}% \Big{[}2C(y{2}-1)-y_{2}+2\Big{]}\Biggl{{}2C\Big{(}4(y_{2}+1)C^{4}+(6y_{2}-4)% C^{3}+(26-22y_{2})C^{2}+3(5y_{2}-8)C\ &-3y_{2}+6\Big{)}-3(1-2C)^{2}\Big{(}2C(y_{2}-1)-y_{2}+2\Big{)}log(\dfrac{1}{1-% 2C})\Biggr{}}^{-1}.\ \end{split}start_ROW start_CELL caligraphic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = end_CELL start_CELL divide start_ARG 8 end_ARG start_ARG 5 end_ARG ( 1 - 2 italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_C start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT [ 2 italic_C ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 ) - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 ] { 2 italic_C ( 4 ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 1 ) italic_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + ( 6 italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 4 ) italic_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + ( 26 - 22 italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 3 ( 5 italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 8 ) italic_C end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - 3 italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 6 ) - 3 ( 1 - 2 italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_C ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 ) - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 ) italic_l italic_o italic_g ( divide start_ARG 1 end_ARG start_ARG 1 - 2 italic_C end_ARG ) } start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT . end_CELL end_ROW(31)

We used equations (26) to (31) to calculate the tidal deformability and Love number of compact stars. Fig. 5 shows dimensionless tidal deformability as a function of gravitational mass for NSs at the presence of Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ hyperons for set 1 to set 3. Also, The radius, tidal deformability, Love number and, the compactness values foe the NSs with 1.4 M⊙subscript 𝑀 direct-product M_{\odot}italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT mass are reported in Table 5. As can be seen from table 5, our results are in good agreement with the GW170817, and GW190425 gravitational events data and other papers.

Image 12: Refer to caption

Figure 5: Dimensionless tidal deformability as a function of gravitational mass for NSs at the presence of Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ hyperons for set 1 to set 3.

Table 5: The properties of the 1.4 M⊙subscript 𝑀 direct-product M_{\odot}italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT NS at the presence of Σ−superscript Σ\Sigma^{-}roman_Σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and Λ Λ\Lambda roman_Λ hyperons for different classes of the coupling constants, with the results compared to those of other references.

IV CONCLUSION

This paper examines the impact of baryon-meson coupling constants on the structure of neutron stars, utilising the relativistic mean-field (RMF) theory as the theoretical framework. Our findings indicate that the energy density and pressure of neutron star matter increase with the QCDSR coupling constants, resulting in a stiffer equation of state and a higher mass comparable to the most massive neutron stars observed in recent years. The utilisation of QCDSR coupling constants modifies the equation of state, thus resolving the hyperon puzzle problem. This enables the observation of a mass exceeding 2 solar masses, even in the presence of hyperons within a neutron star. Furthermore, the numerical results of tidal deformability and Love number values are in accordance with observational data.

Finally, it should be noted that the coupling constants employed in this study were derived in a vacuum, yet the results obtained were reasonably satisfactory. It is recommended that future research focus on the acquisition of these coupling constants within matter. It is hypothesized that this will result in slight improvements to the results obtained, although the observational data and the alignment of calculations with the QSDSR coupling constants suggest that even these coupling constants in matter, up to high densities, will not deviate significantly from the reported range. It is anticipated that these coupling constants will be capable of determining the lower limit in vacuum with greater certainty, and at high densities, they will reach the upper values of this range.

ACKNOWLEDGEMENTS

K. Azizi thanks Iran national science foundation (INSF) for the partial financial support provided under the elites Grant No. 4037888. He is also grateful to the CERN-TH department for their support and warm hospitality.

References

Xet Storage Details

Size:
105 kB
·
Xet hash:
e74b8889f390643ba5c81550f17e953d234b73232b33901c989719d6db249193

Xet efficiently stores files, intelligently splitting them into unique chunks and accelerating uploads and downloads. More info.