# Nuclear Matter and Neutron Stars from Relativistic Brueckner-Hartree-Fock
Theory

Hui Tong (童辉)  College of Physics and Materials Science, Tianjin Normal
University, Tianjin 300387, China  Strangeness Nuclear Physics Laboratory,
RIKEN Nishina Center, Wako, 351-0198, Japan  Chencan Wang (王宸璨)  School of
Physics, Nankai University, Tianjin 300071, China  Sibo Wang (王锶博)
<EMAIL_ADDRESS>Department of Physics, Chongqing University, Chongqing
401331, China

###### Abstract

The momentum and isospin dependence of the single-particle potential for the
in-medium nucleon are the key quantities in the Relativistic Brueckner-
Hartree-Fock (RBHF) theory. It depends on how to extract the scalar and the
vector components of the single-particle potential inside nuclear matter. In
contrast to the RBHF calculations in the Dirac space with the positive-energy
states (PESs) only, the single-particle potential can be determined in a
unique way by the RBHF theory together with the negative-energy states (NESs),
i.e., the RBHF theory in the full Dirac space. The saturation properties of
symmetric and asymmetric nuclear matter in the full Dirac space are
systematically investigated based on the realistic Bonn nucleon-nucleon
potentials. In order to further specify the importance of the calculations in
the full Dirac space, the neutron star properties are investigated. The direct
URCA process in neutron star cooling will happen at density
$\rho_{\rm{DURCA}}=0.43,~{}0.48,~{}0.52$ fm-3 with the proton fractions
$Y_{p,\rm{DURCA}}=0.13$. The radii of a $1.4M_{\odot}$ neutron star are
predicated as $R_{1.4M_{\odot}}=11.97,~{}12.13,~{}12.27$ km, and their tidal
deformabilities are $\Lambda_{1.4M_{\odot}}=376,~{}405,~{}433$ for potential
Bonn A, B, C. Comparing with the results obtained in the Dirac space with PESs
only, full-Dirac-space RBHF calculation predicts the softest symmetry energy
which would be more favored by the gravitational waves (GW) detection from
GW170817. Furthermore, the results from full-Dirac-space RBHF theory are
consistent with the recent astronomical observations of massive neutron stars
and simultaneous mass-radius measurement.

##  1 Introduction

Compact matter in the universe, such as neutron stars, provides an intriguing
interplay between nuclear processes and astrophysical observables (Lattimer &
Prakash, 2004; Burgio et al., 2021). The central density in massive neutron
stars reaches several times the empirical nuclear matter saturation density
$(\rho_{0}\approx 0.16~{}\textrm{fm}^{-3})$, which is far from those
encountered in terrestrial experiments. Thus, astrophysical observations of
neutron star properties are crucial in constraining the equation of state
(EOS) for dense nuclear matter, since the latter constitutes the basic input
quantity for the theoretical reconstruction of a neutron star.

During the past decade, considerable progress has been achieved with precise
measurements for the massive neutron stars PSR J1614-2230 ($1.928\pm
0.017M_{\odot}$) (Demorest et al., 2010; Fonseca et al., 2016), PSR J0348+0432
($2.01\pm 0.04M_{\odot}$) (Antoniadis et al., 2013), and PSR J0740+6620
($2.14_{-0.09}^{+0.10}M_{\odot}$) (Cromartie et al., 2020). Recently, for the
first time the mass and radius of PSR J0030+0451 were simultaneously measured
by the Neutron star Interior Composition Explorer (NICER) collaboration
(Raaijmakers et al., 2019). From NICER’s data, two independent analysis groups
have reported a mass of $1.44_{-0.14}^{+0.15}M_{\odot}$ with a radius of
$13.02_{-1.06}^{+1.24}$km (Miller et al., 2019) and a mass of
$1.34_{-0.16}^{+0.15}M_{\odot}$ with a radius of $12.71_{-1.19}^{+1.14}$km
(Riley et al., 2019). In addition, the advanced LIGO and Virgo Collaborations
detected a gravitational wave (GW) signal from a binary neutron star (BNS)
merger, i.e., GW170817 (Abbott et al., 2017). This has greatly advanced
multimessenger astronomy and stimulated an intense research activity in the
field of nuclear physics. Besides the masses and radii of BNS, a significant
signature carried by GW is the tidal deformability which represents the mass
quadrupole moment response of a neutron star to the strong gravitational field
induced by its companion (Damour et al., 1992; Hinderer, 2008; Flanagan &
Hinderer, 2008; Damour & Nagar, 2009; Postnikov et al., 2010). In particular,
the limits on the tidal deformability inferred from the GW signal have been
widely used to constrain the neuron star radius (Fattoyev et al., 2018; Annala
et al., 2018; Most et al., 2018; Tews et al., 2018), the neutron skin
thickness of 208Pb (Fattoyev et al., 2018), and the nuclear matter EOS (Malik
et al., 2018; Zhang et al., 2018; Tong et al., 2020). Two years after
GW170817, the detection of a compact binary coalescence involving a
$22.2-24.3M_{\odot}$ black hole and a compact object with a mass of
$2.50-2.67M_{\odot}$ was reported by the LIGO and Virgo Collaborations, i.e.,
GW190814 (Abbott et al., 2020). The secondary object of GW190814 still
challenges the physics community nowadays, since it may be either the heaviest
neutron star or the lightest black hole ever discovered (Fattoyev et al.,
2020; Vattis et al., 2020; Safarzadeh & Loeb, 2020; Yang et al., 2020;
Tsokaros et al., 2020; Huang et al., 2020; Bombaci et al., 2021).

Theoretically, many attempts have also been made to obtain the EOS of neutron
star matter via different nuclear many-body theories. They can roughly be
divided into two categories: density functional theories (DFTs) with effective
nucleon-nucleon $(NN)$ interactions and ab-initio methods with realistic ones.

The effective $NN$ interactions in DFTs, either non-relativistic or
relativistic, are constructed to reproduce the properties of finite nuclei and
nuclear matter around saturation density, such as Skyrme (Skyrme, 1956;
Vautherin & Brink, 1972), Gogny (Dechargé & Gogny, 1980), or relativistic
mean-field (RMF) models (Boguta & Bodmer, 1977; Reinhard, 1989; Ring, 1996;
Meng et al., 2006; Meng, 2016). However, since the isovector channels of the
density functionals are loosely constrained in the fitting procedures, their
predictions for the nuclear matter properties such as symmetry energy at
higher densities still have large uncertainties (Li et al., 2008; Jiang et
al., 2015).

In ab-initio methods, the realistic $NN$ interactions are adopted, such as
Bonn (Machleidt, 1989) potentials determined by fitting the $NN$ scattering
data. The relativistic Bonn potential has been successfully applied in
relativistic Brueckner-Hartree-Fock (RBHF) theory (Shen et al., 2019), to
study both nuclear matter and neutron stars (e.g. Horowitz & Serot, 1984;
Brockmann & Machleidt, 1990; Engvik et al., 1994; de Jong & Lenske, 1998;
Alonso & Sammarruca, 2003; van Dalen et al., 2005; Krastev & Sammarruca, 2006;
Katayama & Saito, 2013; Tong et al., 2018, 2020). Comparing with non-
relativistic BHF theory, it is well known that the RBHF theory can reproduce
the empirical saturation properties of symmetric nuclear matter (SNM) without
an extra three-body force (Brockmann & Machleidt, 1990).

In the RBHF calculations, one of the most important procedure is the self-
consistent determination of the single-particle potential of the nucleons with
the effective $G$ matrix inside the nuclear medium. According to the
limitations of symmetries (Serot & Walecka, 1986), the single-particle
potential operator $\mathcal{U}$ is generally divided into scalar and vector
components. However, due to the complexity of the procedure, there exists
different approaches to extract the components of single-particle potential in
the RBHF theory, i.e., the momentum-independence approximation (Brockmann &
Machleidt, 1990), the projection method (Horowitz & Serot, 1987; Nuppenau et
al., 1989; Gross-Boelting et al., 1999) and the RBHF theory in the full Dirac
space (Wang et al., 2021).

In the momentum-independence approximation (Brockmann & Machleidt, 1990), the
momentum dependence of the scalar potential $U_{S}$ and the timelike component
of vector potential $U_{0}$ are neglected by regarding them as constants at a
given baryon density. When investigating the isospin asymmetric nuclear matter
(ANM), this method can not determine the correct behavior of the isospin
dependence of the single-particle potential (Ulrych & Müther, 1997; Schiller &
Müther, 2001). In the projection method (Horowitz & Serot, 1987; Nuppenau et
al., 1989; Gross-Boelting et al., 1999), the effective $G$ matrix is projected
to Lorentz invariant amplitudes which are used to determine the scalar and
vector components of the single-particle potential. This method is more
accurate than the momentum-independence approximation, since it keeps the
spacelike component of the vector potential and the momentum dependence of the
single-particle potential. However, this method is still restricted to the
positive-energy states (PESs) without considering the negative-energy states
(NESs). In paticular, contradictory results for the isospin dependence of the
single-particle potential are found between the two approximations (Ulrych &
Müther, 1997).

Recently, to avoid the approximations used in the momentum-independence
approximation and the projection method in the Dirac space with PESs only, a
new self-consistent RBHF calculation in the full Dirac space for both SNM and
ANM has been developed (Wang et al., 2021, 2022). The Lorentz structure and
the momentum dependence of the single-particle potential are determined
uniquely by decomposing the matrix elements of $\mathcal{U}$ in the full Dirac
space. In particular, the long-standing controversy of the isospin dependence
of the single-particle potential has been clarified for the opposite tendency
predicted by the momentum-independence approximation and projection method.

In this work, it is timely and necessary to study the nuclear matter
properties above the saturation density and neutron star properties with the
latest RBHF theory in the full Dirac space using the Bonn potentials. In order
to compare with the results obtained in the Dirac space with PESs only, we
also perform the calculations with the momentum-independence approximation and
the projection method.

This paper is arranged as follows. The theoretical framework of the RBHF
theory in the full Dirac space, the nuclear matter properties, and the neutron
star properties are reviewed in Section 2. In Section 3, the properties of
nuclear matter and neutron star are presented and discussed. The summary is
given in Section 4.

##  2 Theoretical framework

###  2.1 Relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac
space

In the RBHF theory, the essential point is to describe the single-particle
motion in nuclear matter by using the Dirac equation

$\left\\{\bm{\alpha}\cdot\bm{p}+\beta\left[M+\mathcal{U}_{\tau}(\bm{p})\right]\right\\}u_{\tau}(\bm{p},s)=E_{\bm{p},\tau}u_{\tau}(\bm{p},s),\quad\tau=n,p,$
(1)

where $\bm{\alpha}$ and $\beta$ are the Dirac matrices, $M$ is the nucleon
mass, $\bm{p}$ and $E_{\bm{p},\tau}$ are the momentum and single-particle
energy, $s$ and $\tau$ denote the spin and isospin. According to the
translational and rotational invariance, time-reversal invariance,
hermiticity, and parity conservation, the single-particle potential
$\mathcal{U}_{\tau}$ in the infinite, uniform nuclear matter can be expressed
as (Serot & Walecka, 1986)

$\mathcal{U}_{\tau}(\bm{p})=U_{S,\tau}(p)+\gamma^{0}U_{0,\tau}(p)+\bm{\gamma\cdot\hat{p}}U_{V,\tau}(p),$
(2)

where $U_{S,\tau}$ is the scalar part of single-particle potential,
$U_{0,\tau}$ and $U_{V,\tau}$ are the timelike and spacelike parts of the
vector potential. $\hat{\bm{p}}=\bm{p}/|\bm{p}|$ is the unit vector. It should
be emphasized that the single-particle potential in Eq. (2) is the function of
the nucleon momentum $\bm{p}$.

With the definitions of the following effective quantities:

$\displaystyle\bm{p}^{*}_{\tau}=$ $\displaystyle\
\bm{p}+\hat{\bm{p}}U_{V,\tau}(p),$ (3a) $\displaystyle M^{*}_{\bm{p},\tau}=$
$\displaystyle\ M+U_{S,\tau}(p),$ (3b) $\displaystyle E^{*}_{\bm{p},\tau}=$
$\displaystyle\ E_{\bm{p},\tau}-U_{0,\tau}(p),$ (3c)

the solutions of Eq. (1) in the full Dirac space are

$u_{\tau}(\bm{p},s)=\
\sqrt{\frac{E_{\bm{p},\tau}^{*}+M_{\bm{p},\tau}^{*}}{2M_{\bm{p},\tau}^{*}}}\begin{bmatrix}1\\\
\frac{\bm{\sigma}\cdot\bm{p}^{*}_{\tau}}{E_{\bm{p},\tau}^{*}+M_{\bm{p},\tau}^{*}}\end{bmatrix}\chi_{s}\chi_{\tau},~{}~{}~{}v_{\tau}(\bm{p},s)=\
\gamma^{5}u_{\tau}(\bm{p},s),$ (4)

where $u_{\tau}$ and $v_{\tau}$ are in-medium baryon spinors with positive and
negative energy, $\chi_{s}$ and $\chi_{\tau}$ are the spin and isospin wave
functions. The baryon spinor can be obtained once the single-particle
potentials are determined. The single-particle potentials in Eq. (2) can be
obtained uniquely from three matrix elements $\Sigma^{++}_{\tau}$,
$\Sigma^{-+}_{\tau}$, and $\Sigma^{--}_{\tau}$ of the operator
$\mathcal{U}_{\tau}$ in the full Dirac space (Anastasio et al., 1981;
Poschenrieder & Weigel, 1988),

$\displaystyle U_{S,\tau}(p)=$ $\displaystyle\
\frac{\Sigma^{++}_{\tau}(p)-\Sigma^{--}_{\tau}(p)}{2},$ (5a) $\displaystyle
U_{0,\tau}(p)=$ $\displaystyle\
\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}\frac{\Sigma^{++}_{\tau}(p)+\Sigma^{--}_{\tau}(p)}{2}-\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}\Sigma^{-+}_{\tau}(p),$
(5b) $\displaystyle U_{V,\tau}(p)=$ $\displaystyle\
-\frac{p^{*}_{\tau}}{M^{*}_{\bm{p},\tau}}\frac{\Sigma^{++}_{\tau}(p)+\Sigma^{--}_{\tau}(p)}{2}+\frac{E^{*}_{\bm{p},\tau}}{M^{*}_{\bm{p},\tau}}\Sigma^{-+}_{\tau}(p).$
(5c)

This indicates that one needs the matrix elements not only for the positive-
energy solutions given in Eq. (4), but also the elements coupling positive-
with negative-energy solutions given in Eq. (4) and those for the negative-
energy solutions, i.e., $\Sigma^{++}_{\tau}$, $\Sigma^{-+}_{\tau}$, and
$\Sigma^{--}_{\tau}$. These three matrix elements can be evaluated in the the
mean-field approximation through the effective $NN$ interactions $G$ matrix in
the full Dirac space

$\displaystyle\Sigma^{++}_{\tau}(p)=$ $\displaystyle\
\sum_{s^{\prime}\tau^{\prime}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\langle\bar{u}_{\tau}(\bm{p},1/2)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{++++}(W)|u_{\tau}(\bm{p},1/2)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle,$
(6a) $\displaystyle\Sigma^{-+}_{\tau}(p)=$ $\displaystyle\
\sum_{s^{\prime}\tau^{\prime}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\langle\bar{v}_{\tau}(\bm{p},1/2)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{-+++}(W)|u_{\tau}(\bm{p},1/2)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle,$
(6b) $\displaystyle\Sigma^{--}_{\tau}(p)=$ $\displaystyle\
\sum_{s^{\prime}\tau^{\prime}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\langle\bar{v}_{\tau}(\bm{p},1/2)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{-+-+}(W)|v_{\tau}(\bm{p},1/2)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle,$
(6c)

where $k_{F}^{\tau^{\prime}}$ specifies the Fermi momentum for neutrons and
protons, which has a relation to the total baryonic density
$\rho=\rho_{n}+\rho_{p}$. $\bar{G}$ is the antisymmetrized $G$ matrix, where
$\pm$ in the superscript denotes PESs or NESs.

To derive the effective interactive $G$ matrix in nuclear matter, one of the
most widely used equations in the RBHF theory is the in-medium covariant
Thompson equation (Brockmann & Machleidt, 1990),

$\begin{split}G_{\tau\tau^{\prime}}(\bm{q}^{\prime},\bm{q}|\bm{P},W)=&\
V_{\tau\tau^{\prime}}(\bm{q}^{\prime},\bm{q}|\bm{P})+\int\frac{d^{3}k}{(2\pi)^{3}}V_{\tau\tau^{\prime}}(\bm{q}^{\prime},\bm{k}|\bm{P})\\\
&\times\frac{M^{*}_{\bm{P}+\bm{k},\tau}M^{*}_{\bm{P}-\bm{k},\tau^{\prime}}}{E^{*}_{\bm{P}+\bm{k},\tau}E^{*}_{\bm{P}-\bm{k},\tau^{\prime}}}\frac{Q_{\tau\tau^{\prime}}(\bm{k},\bm{P})}{W-E_{\bm{P}+\bm{k},\tau}-E_{\bm{P}-\bm{k},\tau^{\prime}}}G_{\tau\tau^{\prime}}(\bm{k},\bm{q}|\bm{P},W),\end{split}$
(7)

where $\tau\tau^{\prime}=nn,~{}pp$ or $np$, $W$ is the starting energy,
$V_{\tau\tau^{\prime}}$ denotes a realistic bare $NN$ interaction and it is
constructed in terms of effective Dirac spinors in the full Dirac space as
explained in Eq. (4). Three parametrizations of Bonn potentials, namely, Bonn
A, B, and C (Machleidt, 1989) are chosen here.
$\bm{P}=\frac{1}{2}({\bm{k}}_{1}+{\bm{k}}_{2})$ is the center-of-mass momentum
and $\bm{k}=\frac{1}{2}({\bm{k}}_{1}-{\bm{k}}_{2})$ is the relative momentum
of the two interacting nucleons with momenta ${\bm{k}}_{1}$ and
${\bm{k}}_{2}$. $\bm{q},\bm{k}$ and $\bm{q}^{\prime}$ are the initial,
intermediate, and final relative momenta of the two nucleons scattering in
nuclear matter, respectively. $Q_{\tau\tau^{\prime}}(\bm{k},\bm{P})$ is the
Pauli operator, which prevents $NN$ scattering into occupied states in the
nuclear medium.

The relativistic $G$ matrix is self-consistently calculated with the single-
particle potentials in the standard RBHF iterative procedure through equations
(1), (5), (6), and (7). Once the iteration has converged, the binding energy
per nucleon in nuclear matter can be calculated by

$\begin{split}E/A=&\
\frac{1}{\rho}\sum_{s,\tau}\int^{k^{\tau}_{F}}_{0}\frac{d^{3}p}{(2\pi)^{3}}\frac{M^{*}_{\bm{p},\tau}}{E^{*}_{\bm{p},\tau}}\langle\bar{u}_{\tau}(\bm{p},s)|\bm{\gamma}\cdot\bm{p}+M|u_{\tau}(\bm{p},s)\rangle-M\\\
&\
+\frac{1}{2\rho}\sum_{s,s^{\prime},\tau,\tau^{\prime}}\int^{k^{\tau}_{F}}_{0}\frac{d^{3}p}{(2\pi)^{3}}\int^{k^{\tau^{\prime}}_{F}}_{0}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{M^{*}_{\bm{p},\tau}}{E^{*}_{\bm{p},\tau}}\frac{M^{*}_{\bm{p}^{\prime},\tau^{\prime}}}{E^{*}_{\bm{p}^{\prime},\tau^{\prime}}}\\\
&\
\times\langle\bar{u}_{\tau}(\bm{p},s)\bar{u}_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})|\bar{G}^{++++}(W)|u_{\tau}(\bm{p},s)u_{\tau^{\prime}}(\bm{p}^{\prime},s^{\prime})\rangle.\end{split}$
(8)

###  2.2 Nuclear matter properties

The binding energy per nucleon of ANM can be generally expressed as a power
series in the asymmetry parameter $\alpha=(\rho_{n}-\rho_{p})/\rho$,

$E/A(\rho,\alpha)=E/A(\rho,0)+E_{\mathrm{sym}}(\rho)\alpha^{2}+\mathcal{O}(4),$
(9)

where $E/A(\rho,0)$ is the binding energy per nucleon in SNM and
$E_{\mathrm{sym}}(\rho)$ is the nuclear symmetry energy,

$E_{\mathrm{sym}}(\rho)=\frac{1}{2}\frac{\partial^{2}E/A(\rho,\alpha)}{\partial\alpha^{2}}\bigg{|}_{\alpha=0}.$
(10)

The binding energy per nucleon in SNM can be expanded around the saturation
density $\rho_{0}$,

$E/A(\rho,0)=E/A(\rho_{0},0)+\frac{K_{\infty}}{2}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{2}+\mathcal{O}(3),$
(11)

where $E/A(\rho_{0},0)$ denotes the binding energy per nucleon at $\rho_{0}$.
The second derivative of $E/A(\rho,0)$ with respect to $\rho$ is given by the
incompressibility $K(\rho)$,

$K(\rho)=9\rho^{2}\frac{\partial^{2}E/A(\rho,0)}{\partial\rho^{2}},$ (12)

and $K_{\infty}$ is the value at $\rho_{0}$. Around the saturation density
$\rho_{0}$, the symmetry energy can be similarly expanded as

$E_{\mathrm{sym}}(\rho)=E_{\mathrm{sym}}(\rho_{0})+L\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)+\frac{K_{\mathrm{sym}}}{2}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{2}+\mathcal{O}(3),$
(13)

where $E_{\mathrm{sym}}(\rho_{0})$ is the value of the symmetry energy at
saturation density, $L$ and $K_{\mathrm{sym}}$ are the slope parameter and
curvature parameter of the nuclear symmetry energy at $\rho_{0}$:

$\displaystyle L$ $\displaystyle=$ $\displaystyle 3\rho\frac{\partial
E_{\mathrm{sym}}(\rho)}{\partial\rho}\bigg{|}_{\rho=\rho_{0}},$ (14)
$\displaystyle K_{\mathrm{sym}}$ $\displaystyle=$ $\displaystyle
9\rho^{2}\frac{\partial^{2}E_{\mathrm{sym}}(\rho)}{\partial\rho^{2}}\bigg{|}_{\rho=\rho_{0}}.$
(15)

Furthermore, $K_{\mathrm{asy}}=K_{\mathrm{sym}}-6L$ has been widely used to
describe the isospin dependence of the incompressibility of ANM in Refs.
(Baran et al., 2005; Chen et al., 2005; Li et al., 2008; Sun et al., 2008;
Centelles et al., 2009).

###  2.3 Neutron star properties

The neutron star is described as the $\beta$-stable nuclear matter system in
this work, which consists of nucleons and leptons (mainly $e^{-}$ and
$\mu^{-}$). The chemical potentials of nucleons and leptons satisfy the
equilibrium conditions

$\mu_{p}=\mu_{n}-\mu_{e},\quad\mu_{\mu}=\mu_{e},$ (16)

where $\mu_{e}$, $\mu_{\mu}$, $\mu_{p}$, and $\mu_{n}$ are the chemical
potentials of leptons, proton and neutron. In addition, charge neutrality is
required as

$\rho_{p}=\rho_{e}+\rho_{\mu}.$ (17)

With these constraints, the energy density of the $\beta$-stable nuclear
matter is then obtained as

$\varepsilon=\rho\left[E(\rho,1-2Y_{p})/A+Y_{p}M_{p}+(1-Y_{p})M_{n}\right]+\varepsilon_{e}+\varepsilon_{\mu},$
(18)

where the leptons are treated as non-interacting Fermi gas (Bombaci &
Lombardo, 1991; Maieron et al., 2004) and
$Y_{i}=\rho_{i}/\rho~{}(i=e,~{}\mu,~{}n,~{}p)$ are the equilibrium particle
fractions. The various chemical potential for each particle $i$ is

$\mu_{i}=\frac{\partial\varepsilon/\rho}{\partial Y_{i}}.$ (19)

Through solving Eqs. (16), (17), (18), and (19) simultaneously, one can obtain
the particle fractions $Y_{i}$ at a given density $\rho$. Then the energy
density $\varepsilon$ can be easily obtained from Eq. (18). The pressure $P$
for the $\beta$-stable nuclear matter is defined as

$P=-\frac{\partial(\varepsilon/\rho)}{\partial(1/\rho)}=\rho\frac{\partial\varepsilon}{\partial\rho}-\varepsilon.$
(20)

Once the EOS of $\beta$-stable nuclear matter in the form $P(\varepsilon)$ is
obtained, the mass and radius of a cold, spherically symmetric, static, and
relativistic star can be described by the Tolman-Oppenheimer-Volkov (TOV)
equations (Oppenheimer & Volkoff, 1939; Tolman, 1939),

$\displaystyle\frac{dP(r)}{dr}=$ $\displaystyle\
-\frac{[P(r)+\varepsilon(r)][M(r)+4\pi r^{3}P(r)]}{r[r-2M(r)]},$ (21a)
$\displaystyle\frac{dM(r)}{dr}=$ $\displaystyle\ 4\pi r^{2}\varepsilon(r),$
(21b)

where $P(r)$ is the pressure at neutron star radius $r$, $M(r)$ is the total
neutron star mass inside a sphere of radius $r$. For a given EOS, the TOV
equations have the unique solution that depends on the conditions of
$\beta$-stable matter at the center, such as the central density or the
central pressure. In addition, to solve the TOV equations, the EOS must cover
full regions of the neutron star from the crust to the core. In the present
work, we mainly concentrate on discussing the core region within the RBHF
theory. For the crust, the EOS from the Baym-Pethick-Sutherland (BPS) (Baym et
al., 1971b) and the Baym-Bethe-Pethick (BBP) (Baym et al., 1971a) model is
used.

Besides the masses and radii, another important property of neutron star, the
tidal deformability $\Lambda$ is defined as

$\Lambda=\frac{2}{3}k_{2}C^{-5},$ (22)

which represents the mass quadrupole moment response of a neutron star to the
strong gravitational field induced by its companion (Damour et al., 1992;
Hinderer, 2008; Flanagan & Hinderer, 2008; Damour & Nagar, 2009; Postnikov et
al., 2010). $C=M/R$ is the compactness parameter, $M$ and $R$ are the neutron
star mass and radius, and $k_{2}$ is the second love number

$\displaystyle k_{2}=$
$\displaystyle\frac{8C^{5}}{5}(1-2C)^{2}[2-y_{R}+2C(y_{R}-1)]\times\\{6C[2-y_{R}+C(5y_{R}-8)]$
(23) $\displaystyle+4C^{3}[13-11y_{R}+C(3y_{R}-2)+2C^{2}(1+y_{R})]$
$\displaystyle+3(1-2C)^{2}[2-y_{R}+2C(y_{R}-1)]\ln(1-2C)\\}^{-1},$

where $y_{R}=y(R)$ can be calculated by solving the following differential
equation:

$r\frac{dy(r)}{dr}+y^{2}(r)+y(r)F(r)+r^{2}Q(r)=0,$ (24)

with

$\displaystyle F(r)$
$\displaystyle=\left[1-\frac{2M(r)}{r}\right]^{-1}\left\\{1-4\pi
r^{2}[\varepsilon(r)-P(r)]\right\\},$ (25a) $\displaystyle Q(r)$
$\displaystyle=\left\\{4\pi\left[5\varepsilon(r)+9P(r)+\frac{\varepsilon(r)+P(r)}{\frac{\partial
P}{\partial\varepsilon}(r)}\right]-\frac{6}{r^{2}}\right\\}\times\left[1-\frac{2M(r)}{r}\right]^{-1}$
$\displaystyle~{}~{}-\left[\frac{2M(r)}{r^{2}}+2\times 4\pi
rP(r)\right]^{2}\times\left[1-\frac{2M(r)}{r}\right]^{-2}.$ (25b)

The differential equation (24) can be integrated together with the TOV
equations with the boundary condition $y(0)=2$.

##  3 Results and discussions

###  3.1 Nuclear matter properties

Figure 1:  Single-particle potentials $U_{S}$, $U_{0}$, and $U_{V}$ at the
Fermi momentum $k_{F}$ as functions of the asymmetric parameter $\alpha$
calculated by the RBHF theory in the full Dirac space. The solid (red) and
dashed (blue) lines correspond to neutron and proton. The total baryon
densities are given at $\rho=0.16$ fm-3, 0.32 fm-3, and 0.48 fm-3.

In ANM, one of the most important questions is the isospin and density
dependence of the single-particle potentials (van Dalen et al., 2005), which
is related to a wildly-discussed topic concerning the proton-neutron mass
splitting. In Fig. 1, the isospin and density dependence of single-particle
potentials $U_{S,\tau},U_{0,\tau}$ and $U_{V,\tau}$ at the Fermi momentum
$k^{\tau}_{F}$ for densities $\rho=0.16~{}\text{fm}^{-3}$,
$0.32~{}\text{fm}^{-3}$ and $0.48~{}\text{fm}^{-3}$ are obtained from RBHF
calculation in the full Dirac space. At the density
$\rho=0.16~{}\text{fm}^{-3}$, the values of $U_{V,\tau}$ for both neutron and
proton are close to zero. For the other two components, the RBHF calculation
in the full Dirac space gives $U_{S,n}<U_{S,p}$ and $U_{0,n}>U_{0,p}$. It is
also found that with increasing neutron excess, $U_{S,n}$ and $U_{V,n}$
decrease, while $U_{0,n}$ increases. The case for proton shows an opposite
behavior. In the RMF theory (Ring, 1996), the isospin splittings of the scalar
and the timelike part of the vector potential can be calculated as (Ulrych &
Müther, 1997)

$\displaystyle U_{S,n}-U_{S,p}=$ $\displaystyle\
-2\frac{g^{2}_{\delta}}{m^{2}_{\delta}}(\rho_{s,n}-\rho_{s,p}),$ (26a)
$\displaystyle U_{0,n}-U_{0,p}=$ $\displaystyle\
+2\frac{g^{2}_{\rho}}{m^{2}_{\rho}}(\rho_{n}-\rho_{p}).$ (26b)

The parameters $g_{i}$ and $m_{i}\ (i=\delta,\rho)$ refer to the meson-nucleon
coupling constants and masses of the mesons. $\rho_{s,\tau}$ is the scalar
density. For ANM, Eqs. (26) lead to $U_{S,n}<U_{S,p}$ and $U_{0,n}>U_{0,p}$,
which is consistent with the results by the RBHF theory in the full Dirac
space. This consistence underlines the crucial importance for the
$\delta$-meson exchange in the RMF theory (Kubis & Kutschera, 1997; Hofmann et
al., 2001; Liu et al., 2002; Roca-Maza et al., 2011). Further, since neutron
star radii and tidal deformabilities are sensitive to the nuclear matter
properties at supra-saturation density (Fattoyev et al., 2018; Zhang, Nai-Bo &
Li, Bao-An, 2019; Tong et al., 2020). We also study the isospin dependence of
single-particle potentials at supra-saturation densities
$\rho=0.32~{}\text{fm}^{-3}$ and $0.48~{}\text{fm}^{-3}$. The differences
between neutron and proton becomes larger when increasing nuclear matter
density. In comparison to the density $\rho=0.16~{}\text{fm}^{-3}$, the
amplitudes of $U_{V,\tau}$ for both neutron and proton are much larger than
zero. However, $U_{V,\tau}$ is neglected in the momentum-independence
approximation even at supra-saturation densities. It means that determining
the correct behavior of the isospin dependence of the single-particle
potentials at supra-saturation densities is important to describe nuclear
matter and neutron star properties.

Figure 2:  Binding energies per nucleon $E/A$ as functions of the density
$\rho$ calculated by the RBHF theory in the full Dirac space with the
asymmetry parameter $\alpha$ ranging from $0$ to $1$. For comparison, the EOS
for SNM and pure neutron matter (PNM) obtained with the variational method
(APR) (Akmal et al., 1998) are also shown with black circles and green
squares, respectively.

From single-particle potentials one can calculate the binding energies per
nucleon $E/A$ for ANM. In Fig. 2, $E/A$ are depicted as functions of the
density $\rho$ with the asymmetry parameter $\alpha$ ranging from $0$ to $1$.
The binding energy and the saturation density become progressively smaller
when $\alpha$ increases. The EOS of ANM exhibits a minimum which disappears
for $\alpha\geqslant 0.8$. For comparison, Fig. 2 also displays the EOS for
SNM and PNM obtained with the variational calculations (Akmal et al., 1998)
using the Argonne $v_{18}$ interaction (Wiringa et al., 1995) with
relativistic boost corrections to the two-nucleon interaction, as well as
three-nucleon interactions modeled with the Urbana force (Pudliner et al.,
1995). It can be seen that the results obtained with two _ab initio_ methods
are comparable.

Table 1: Nuclear matter properties (as described in the text) at saturation density $\rho_{0}$ calculated by the RBHF theory in the full Dirac space, in comparison with the results obtained by the RBHF calculation with the projection method and the momentum-independence approximation (Mom.-ind.app). Results for Bonn potentials A, B, and C are shown. The empirical values are also listed in the last row. Model | Potential | $\rho_{0}$ | $E/A$ | $K_{\infty}$ | $E_{\mathrm{sym}}$ | $L$ | $K_{\mathrm{sym}}$ | $K_{\mathrm{asy}}$
---|---|---|---|---|---|---|---|---
(fm-3) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV)
| A | 0.188 | -15.40 | 258 | 33.1 | 65.2 | -67.3 | -463
Full Dirac Space | B | 0.164 | -13.36 | 206 | 28.9 | 48.6 | -64.1 | -356
| C | 0.144 | -12.09 | 150 | 25.8 | 38.0 | -58.1 | -286
| A | 0.179 | -16.15 | 254 | 34.7 | 68.8 | -70.1 | -483
Projection method | B | 0.161 | -14.60 | 206 | 31.3 | 54.3 | -68.7 | -395
| C | 0.149 | -13.67 | 157 | 29.1 | 45.5 | -65.3 | -339
| A | 0.178 | -15.36 | 273 | 33.2 | 67.3 | -65.7 | -469
Mom.-ind.app | B | 0.162 | -13.45 | 233 | 29.7 | 55.0 | -63.2 | -393
| C | 0.150 | -12.14 | 201 | 27.1 | 46.9 | -59.6 | -341
Empirical |  | 0.166 | -16 | 240 | 32 | 88 |  |
|  $\pm$0.018 |  $\pm$1 |  $\pm$20 |  $\pm$2 |  $\pm$25 |  |

In Table 1, we summarize our results for the properties of nuclear matter from
RBHF theory. In the second row, we show our results calculated by the RBHF
theory in the full Dirac space. The results obtained by the projection method
(Gross-Boelting et al., 1999) with the $ps$ representation for the subtracted
$T$ matrix (van Dalen et al., 2004) are shown in the third row. The results
obtained by the momentum-independence approximation are also shown in the
fourth row, the scalar potential and the time component of the vector
potential are extracted directly from the single-particle potential energy at
two casually selected momenta, $0.5k_{F}^{\tau}$ and $k_{F}^{\tau}$. The one-
boson-exchange potential (OBEP) is used for the realistic $NN$ interactions,
which is defined as a sum of one-particle-exchange amplitudes of six non-
strange bosons with given spin-parity properties (Machleidt, 1989). The
coupling strengths and form-factor parameters are determined by fitting to the
$NN$ scattering data and deuteron properties. Three different parametrizations
of OBEP were constructed for the application to relativistic nuclear structure
physics, which were denoted by Bonn A, B, and C. The three potentials differ
essentially in the $\pi NN$ form-factor parameter. Since the nuclear tensor
force is essentially provided by the pion, the main difference among the three
potentials is the strength of the tensor force which can be reflected in their
predictions for $D$-state probability of the deuteron
($P_{D}=4.5\%,~{}5.1\%,~{}5.5\%$ for Bonn A, B, C, respectively) (Machleidt et
al., 1987). As is well known, the strength of the tensor force determines the
location of the nuclear matter saturation point (Coester et al., 1970;
Brockmann & Machleidt, 1990). Compared to potential Bonn B and C, all three
methods by using potential Bonn A with the weakest tensor force can reproduce
the empirical value (Bethe, 1971; Sprung, 1972) for saturation density
$\rho_{0}$ and binding energy $E/A$. With the potential Bonn A, the
incompressibility of SNM at saturation density is 258 MeV for the RBHF
calculation in the full Dirac space and 254 MeV in the projection method,
which agree well with the commonly accepted value of 240 $\pm$ 20 MeV (Garg et
al., 2007; Garg & Colò, 2018). As for ANM, the symmetry energy
$E_{\mathrm{sym}}$ and the slope parameter $L$ in the full Dirac space with
potential Bonn A are $33.1$ MeV and $65.2$ MeV, which are smaller than the
values suggested in the projection method and the momentum-independence
approximation. This implies that the full Dirac space predicts the softest
symmetry energy at higher densities.

Figure 3:  $K_{\mathrm{sym}}$ (left panel) and $E_{\mathrm{sym}}$ (right
panel) as functions of $L$ calculated by RBHF theory using the potential Bonn
A in the full Dirac space (red star), in comparison with results obtained by
the projection method (blue down-pointing triangle), the momentum-independence
approximation (gray up-pointing triangle) and various density functionals
(circles and squares) (Sagawa et al., 2007; Zhao et al., 2010). The shaded
regions denote a recent constraint on $E_{\mathrm{sym}}$, $L$ and
$K_{\mathrm{sym}}$ by combining astrophysical data with PREX-II and chiral
effective field theory (Essick et al., 2021). The solid orange line is a
linear fit to the results of density functionals.

In several previous works, slope parameter $L$ and curvature parameter
$K_{\mathrm{sym}}$ of symmetry energy $E_{\mathrm{sym}}$, as well as their
correlations, were shown to have significant effects on the neutron star
properties, e.g., the composition of the core (Lattimer et al., 1991), crust-
core transition density and pressure (e.g., Vidaña et al., 2009; Zhang et al.,
2018; Li & Magno, 2020). At the saturation density, $E_{\mathrm{sym}}$ has
been well constrained by recent terrestrial nuclear experiments and
astrophysical observations. However, $L$ and $K_{\mathrm{sym}}$ still remain
relatively large uncertainties which characterize the $E_{\mathrm{sym}}$ at
supra-saturation densities. In Fig. 3, $K_{\mathrm{sym}}$, $L$ and
$E_{\mathrm{sym}}$ are calculated by a series of DFTs and the _ab initio_ RBHF
theory. The DFTs are adopted from nonrelativistic models (e.g., SLy4 and those
starting with S) (Dutra et al., 2012), to relativistic models (e.g., PK1, NL3,
DD-ME) (Long et al., 2006; Dutra et al., 2014). _Ab initio_ calculations have
been carried out with the RBHF theory using the potential Bonn A in the full
Dirac space, together with results obtained by the projection method and the
momentum-independence approximation. $E_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$
obtained from DFTs exhibit a positive correlation with $L$, and the results of
RBHF theory are all in good agreement with this correlation. We also perform a
linear fitting to this correlation,

$\displaystyle K_{\mathrm{sym}}=$ $\displaystyle\ -331.32+3.49L,$ (27a)
$\displaystyle E_{\mathrm{sym}}=$ $\displaystyle\ 27.03+0.09L,$ (27b)

with the correlation coefficients $r=0.93$ and $r=0.92$. The results of
fitting are depicted with the solid orange line. Our results are close to
$K_{\mathrm{sym}}=-307.862+3.292L$ with $r=0.972$ given in Ref. (Dong et al.,
2012) which used several different DFTs from the present work. It should be
mentioned that $K_{\mathrm{sym}}$ predicted by RBHF theory are negative
values, i.e., -67.3 MeV in full Dirac space, -70.1 MeV in the projection
method and -65.7 MeV in the momentum-independence approximation. In addition,
the shaded regions in Fig. 3 denote a recent constraint on $E_{\mathrm{sym}}$,
$L$, and $K_{\mathrm{sym}}$ by combining astrophysical data with PREX-II and
chiral effective field theory (Essick et al., 2021). It can be seen that the
results in the full Dirac space located inside the region labeled by $\chi$EFT
Astro+PREX-II posterior. Our results are also consistent with other recent
works, such as $K_{\mathrm{sym}}=-120_{-100}^{+80}$ MeV at $68\%$ confidence
level from a Bayesian analysis of neutron star properties (Xie & Li, 2020) and
$K_{\mathrm{sym}}=-209_{-182}^{+270}$ MeV at $95\%$ confidence level from
combining the neutron skin data and the pure neutron matter EOS from chiral
effective field theory calculations (Newton & Crocombe, 2021).

###  3.2 Neutron star properties

Figure 4: Particle fractions $Y_{i}~{}(i=n,p,e,\mu)$ of $\beta$-stable matter
as a function of the baryon number density. The results are calculated by RBHF
theory using the potential Bonn A in the full Dirac space (left panel),
together with the projection method (middle panel), and the momentum-
independence approximation (right panel).

On the basis of the Eqs. (16), (17), (18), and (19), the particle fractions
$Y_{i}~{}(i=n,p,e,\mu)$ for $\beta$-stable matter in Fig. 4 are calculated by
RBHF theory using the potential Bonn A in the full Dirac space, together with
the projection method, and the momentum-independence approximation. The
neutron fractions $Y_{n}$ decrease gradually with increasing $\rho$ in three
methods. In contrast, the values of $Y_{p}$, $Y_{e}$, and $Y_{\mu}$ grow
steadily. According to the energetic constraints (Wiringa et al., 1988), muons
only emerge at higher densities, and the onset densities for muons from three
methods are 0.144 fm-3, 0.132 fm-3 and 0.142 fm-3. The muons appear latest in
the full Dirac space, since it has the softest symmetry energy and generates
the smallest proton fraction. We also observe that the behavior of particle
fractions in the full Dirac space is qualitatively similar to the one seen in
other two methods.

The direct URCA (DURCA) process

$n\rightarrow p+e^{-}+\bar{\nu}_{e},\quad p\rightarrow n+e^{+}+\nu_{e},$ (28)

is one of the scenario of neutron star cooling (Pethick, 1992; Prakash et al.,
1992; Prakash, 1994). This process is allowed when the momenta for nucleons
and leptons can be conserved, which can predict the threshold density of the
DURCA process. It has been pointed out in Ref. (Lattimer & Prakash, 2004) that
the $\beta$-stable matter with a proton-to-neutron ratio in excess of $1/8$,
or the proton fraction $Y_{p}\geqslant 1/9$ can be cooled by the DURCA
process. As for the results from RBHF theory, the DURCA process happens after
muon occurs for three methods. The threshold densities are calculated as
$\rho=0.432$ fm-3 and $Y_{p}=0.127$ in the full Dirac space, $\rho=0.372$ fm-3
and $Y_{p}=0.126$ in the projection method, and $\rho=0.464$ fm-3 and
$Y_{p}=0.127$ in the momentum-independence approximation. These results from
three methods are consistent with the astronomical observations, which does
not allow the DURCA process to be too early.

Figure 5:  Neutron star mass-radius relations obtained by RBHF theory using
the potential Bonn A in the full Dirac space (red solid line), together with
the projection method (blue dashed line), and the momentum-independence
approximation (gray dotted line). The shaded areas represent the constraints
from astronomical observables for the massive neutron star (Demorest et al.,
2010; Fonseca et al., 2016; Antoniadis et al., 2013; Cromartie et al., 2020).
The inner and outer contours indicate the allowed area of mass and radius of
neutron stars by NICER’s analysis of PSR J0030+451 (Miller et al., 2019). The
excluded causality region is also shown (Lattimer & Prakash, 2007).

Based on the EOS from RBHF theory, the mass-radius relations of neutron stars
can be calculated from the TOV equation. In Fig. 5, the neutron star mass-
radius relations obtained are shown from the RBHF theory in the full Dirac
space with potential Bonn A, together with the projection method, and the
momentum-independence approximation. The radii $R_{1.4M_{\odot}}$ of a
$1.4M_{\odot}$ neutron star from three methods are 11.97 km, 12.38 km, and
12.35 km, respectively. The prediction from the full Dirac space leads to the
smallest radius for $R_{1.4M_{\odot}}$. The relatively small neutron star
radius suggested by full Dirac space implies that the symmetry energy at
higher densities is soft. It should be noted that soft EOSs, which produce
small values of radii for fixed $M$ are favored by the GW170817 (Fattoyev et
al., 2018). It can be found that our results for neutron star radii are also
consistent with other works, such as $R_{1.4M_{\odot}}\leqslant 13.76$ km
(Fattoyev et al., 2018), $R_{1.4M_{\odot}}\leqslant 13.6$ km (Annala et al.,
2018), 12.00 km $\leqslant R_{1.4M_{\odot}}\leqslant 13.45$ km (Most et al.,
2018) and 9.0 km $\leqslant R_{1.4M_{\odot}}\leqslant 13.6$ km (Tews et al.,
2018) from the tidal deformability (Abbott et al., 2017, 2019),
$R_{1.4M_{\odot}}=10.9_{-1.5}^{+1.9}$ km from the Bayesian approach (Kumar &
Landry, 2019) and the recent constraints by NICER (Miller et al., 2019) for
the mass and radius of PSR J0030+0451, i.e., mass
$1.44_{-0.14}^{+0.15}M_{\odot}$ with radius $13.02_{-1.06}^{+1.24}$ km. The
$68\%$ and $95\%$ contours of the joint probability density distribution of
mass and radius from the NICER analysis are also shown.

The neutron star matter is only considered to be composed of nucleons and
leptons here, the central densities of massive neutron stars in this work can
reach $5$ to $7$ times the nuclear saturation density $\rho_{0}=0.16$ fm-3 of
neutrons and protons found in laboratory nuclei, which is far away from the
region $\rho\leqslant 0.57$ fm-3 where RBHF theory in the full Dirac space is
applicable (Wang et al., 2022). Therefore, for RBHF theory in the full Dirac
space, we also follow the strategy in Refs. (Rhoades & Ruffini, 1974; Gandolfi
et al., 2012; Wang et al., 2022). The maximally stiff or causal EOS given by
$p(\varepsilon)=c^{2}\varepsilon-\varepsilon_{c}$ is used to replace the EOS
for neutron star matter above a critical density $\rho_{c}$, where $p$ is the
pressure, $\varepsilon$ is the energy density, $c$ is the speed of light, and
$\varepsilon_{c}$ is a constant. This EOS predicts the most rapid increase of
$p$ with $\varepsilon$ without violating causality limit and the upper bound
on the maximum mass of the neutron star. In our calculations, $\rho_{c}$ is
given as 0.57 fm-3 and $\varepsilon_{c}$ is determined by ensuring that the
energy density is continuous between the low- and high-density EOS. The upper
bound maximum masses of neutron stars are predicted as 2.43$M_{\odot}$ in the
full Dirac space, which is still smaller than the compact object with a mass
of 2.50-2.67$M_{\odot}$ by LIGO Scientific and Virgo collaborations in
GW190814 (Abbott et al., 2020). Moreover, the maximum masses are predicted as
2.31$M_{\odot}$ and 2.18$M_{\odot}$ from the projection method and the
momentum-independence approximation, which are consistent with the
measurements of massive neutron stars, such as PSR J1614-2230 (Demorest et
al., 2010; Fonseca et al., 2016), PSR J0348+0432 (Antoniadis et al., 2013),
and PSR J0740+6620 (Cromartie et al., 2020).

Figure 6: Tidal deformabilities $\Lambda$ as functions of neutron star mass
$M$ obtained by RBHF theory using the potential Bonn A in the full Dirac space
(red solid line), together with the projection method (blue dashed line), and
the momentum-independence approximation (gray dotted line). Constraints from
the BNS merger event GW170817 for the tidal deformability,
$\Lambda_{1.4M_{\odot}}=190_{-120}^{+390}$ (Abbott et al., 2018), is also
shown.

In the multimessenger era, another important constraint of the neutron star is
the tidal deformability $\Lambda$. In Fig. 6, the tidal deformabilities
$\Lambda$ as functions of neutron star mass $M$ from RBHF theory with
potential Bonn A in the full Dirac space, the projection method, and the
momentum-independence approximation are shown. These tidal deformabilities
decrease with increasing the neutron star mass. For a given neutron star mass,
the method which has a softer symmetry energy yield smaller neutron star radii
and correspondingly smaller tidal deformabilities. The tidal deformabilities
of a $1.4M_{\odot}$ neutron star from three methods are predicted as
$\Lambda_{1.4M_{\odot}}$ = 376, 473, and 459, respectively. It can be found
that the tidal deformabilities from the full Dirac space are significantly
lower than those from the projection method and the momentum-independence
approximation. The initial estimation for tidal deformability
$\Lambda_{1.4M_{\odot}}$ has an upper bound $\Lambda_{1.4M_{\odot}}<800$
(Abbott et al., 2017) from the observation of BNS merger event GW170817. Then
a revised analysis from LIGO and Virgo collaborations gives
$\Lambda_{1.4M_{\odot}}=190_{-120}^{+390}$ (Abbott et al., 2018). It is
important to underscore that the results from three methods are located in
these regions, especially that from the full Dirac space, which is the closest
one to the central values of the GW170817 shown as the circle symbol.

Table 2: Neutron star properties calculated by the RBHF theory in the full Dirac space, together with the results obtained by the RBHF calculation with the projection method and the momentum-independence approximation. Results for Bonn potentials A, B, and C are shown. Method | Potential | $\rho_{\rm{DURCA}}$ | $Y_{p,\rm{DURCA}}$ | $M_{\rm{DURCA}}$ | $R_{1.4M_{\odot}}$ | $\rho_{1.4M_{\odot}}$ | $\Lambda_{1.4M_{\odot}}$ | $M_{\rm{max}}$ | $R_{\rm{max}}$
---|---|---|---|---|---|---|---|---|---
(fm-3) | ($M_{\odot}$) | (km) | (fm-3) | ($M_{\odot}$) | (km)
| A | 0.43 | 0.13 | 1.28 | 11.97 | 0.46 | 376 | 2.43 | 10.93
Full Dirac Space | B | 0.48 | 0.13 | 1.55 | 12.13 | 0.45 | 405 | 2.43 | 10.98
| C | 0.52 | 0.13 | 1.78 | 12.27 | 0.43 | 433 | 2.44 | 11.04
| A | 0.37 | 0.13 | 1.16 | 12.38 | 0.42 | 473 | 2.31 | 11.27
Projection method | B | 0.41 | 0.13 | 1.41 | 12.53 | 0.41 | 505 | 2.31 | 11.34
| C | 0.45 | 0.13 | 1.64 | 12.64 | 0.40 | 532 | 2.32 | 11.42
| A | 0.46 | 0.13 | 1.52 | 12.35 | 0.43 | 459 | 2.18 | 10.99
Mom.-ind.app | B | 0.54 | 0.13 | 1.79 | 12.49 | 0.43 | 484 | 2.18 | 11.05
| C | 0.63 | 0.13 | 1.97 | 12.59 | 0.42 | 508 | 2.18 | 11.10

In Table 2, our results for neutron star properties, such as threshold of
DURCA process, the radii and tidal deformability for $1.4M_{\odot}$, and
central densities, are obtained by the RBHF theory in the full Dirac space,
the projection method and the momentum-independence approximation with the
potential Bonn A, B, C. It was pointed out that the strength of tensor force
plays an important role in the investigation of neutron star properties by the
RBHF theory in the momentum-independence approximation (Wang et al., 2020).
Therefore, it is interesting to investigate the effect of tensor force on the
neutron star properties in the RBHF theory, especially in the full Dirac
space. The threshold densities for DURCA process are listed in the third
column. As for the full Dirac space, $\rho_{\rm{DURCA}}$ from potential Bonn
A, B, C are $0.43$ fm-3, $0.48$ fm-3 and 0.52 fm-3, with the same DURCA proton
fractions $Y_{p}=0.13$. An inverse correlation can be found between the DURCA
threshold density and the symmetry energy, since Bonn A generates the largest
symmetry energy at a given baryon density with the weakest tensor force. The
critical mass for DURCA process are also listed in the fifth column,
$M_{\rm{DURCA}}$ from potential Bonn A, B, C are $1.28M_{\odot}$,
$1.55M_{\odot}$, and $1.78M_{\odot}$ in the full Dirac space. Recently, the
analysis of a neutron star in the transient system MXB 1659-29 has founded
that the critical mass is around $1.6M_{\odot}$ (Brown et al., 2018), and we
found that the results from Bonn B is the closest one to the current
astrophysical observations. The neutron star radii and tidal deformabilties at
1.4$M_{\odot}$ from the full Dirac space are shown in the sixth and the eighth
column, which are given as $R_{1.4M_{\odot}}$ = 11.97 km and
$\Lambda_{1.4M_{\odot}}=376$ for Bonn A, $R_{1.4M_{\odot}}$ = 12.13 km and
$\Lambda_{1.4M_{\odot}}=405$ for Bonn B, and $R_{1.4M_{\odot}}$ = 12.27 km and
$\Lambda_{1.4M_{\odot}}=433$ for Bonn C. We found that $R_{1.4M_{\odot}}$ and
$\Lambda_{1.4M_{\odot}}$ predicted by Bonn A are smaller than those from Bonn
B and C. This indicates that Bonn A with the lowest strength of tensor force
has the softest EOS and Bonn C has the stiffest EOS. In the ninth column, the
upper bound on the maximum masses of neutron stars from the full Dirac space
are shown as 2.43$M_{\odot}$, 2.43$M_{\odot}$, 2.44$M_{\odot}$ with Bonn A, B,
C. The difference among the three kinds of results is smaller than
0.01$M_{\odot}$. This is reasonable since neutrons dominate the nucleonic
component of the maximum masses of neutron stars while the main difference
between the three Bonn potentials is in their tensor force strength in the
$(T=0)$ ${}^{3}S_{1}$-${}^{3}D_{1}$ state that does not contribute to the
$(T=1)$ neutron-neutron state.

##  4 Summary

The single-particle potential can be used to describe the quasi-particle
properties of a nucleon inside a strongly interacting nuclear matter system.
However, because the incompleteness of the Dirac space, different
approximations have been introduced to extract the momentum and isospin
dependence of the single-particle potential in the RBHF theory. In contrast to
the RBHF calculations in the Dirac space with PESs only, the single-particle
potential can be determined in a unique way from RBHF theory in the full Dirac
space. The scalar component for neutron is found to be smaller than that for
proton at and above nuclear saturation density, i.e., $U_{S,n}<U_{S,p}$, which
leads to the neutron-proton Dirac mass splitting $M^{*}_{D,n}<M^{*}_{D,p}$. As
for the timelike component of the vector potential, the RBHF theory in the
full Dirac space gives $U_{0,n}>U_{0,p}$. At higher densities, the amplitudes
of $U_{V,\tau}$ for both neutron and proton are much larger than zero, it
means $U_{V,\tau}$ at higher densities can not be neglected. The saturation
properties of symmetric and asymmetric nuclear matter are systematically
investigated based on the Bonn potentials. The symmetry energy
$E_{\mathrm{sym}}$ and its slope parameter $L$ at the saturation density are
$33.1$ and $65.2$ MeV for potential Bonn A, which are consistent with the
empirical and experimental values. Especially, the slope parameter $L$ and the
curvature parameter $K_{\mathrm{sym}}$ predicted in the full Dirac space agree
well with the recent results from $\chi$EFT Astro+PREX-II posterior.

To further study the importance of the calculations in the full Dirac space at
supra-saturation densities, the neutron star properties are studied with the
latest RBHF theory in the full Dirac space. The threshold densities of the
DURCA process are calculated as $\rho=0.43$ fm-3 and $Y_{p}=0.13$ in the full
Dirac space. These results are consistent with the astronomical observations,
which does not allow the DURCA process to be too early. The radii of a
$1.4M_{\odot}$ neutron star are calculated as
$R_{1.4M_{\odot}}=11.97,~{}12.13,~{}12.27$ km, and their tidal deformabilities
are $\Lambda_{1.4M_{\odot}}=376,~{}405,~{}433$ for potential Bonn A, B, C.
Comparing the results obtained with the projection method and the momentum-
independence approximation, the RBHF calculations in the full Dirac space
predict the softest EOS which would be more favored by GW170817. Especially,
the tidal deformability from the full Dirac space is the closest one to the
central values of the GW170817 from a revised analysis by LIGO and Virgo
collaborations. Among the potential Bonn A, B, C, it is also found that RBHF
calculations with the potential Bonn A has the softest EOS and potential Bonn
C has the stiffest EOS since the former one has the lowest strength of tensor
force. Furthermore, the results from RBHF theory are consistent with the
recent astronomical observations of massive neutron stars and simultaneous
mass-radius measurement.

S.W. thanks Qiang Zhao for helpful discussions. This work was partly supported
by the National Key R&D Program of China under Contracts No. 2017YFE0116700,
the National Natural Science Foundation of China (NSFC) under Grants 12147102,
the Fundamental Research Funds for the Central Universities under Grants No.
2020CDJQY-Z003 and No. 2021CDJZYJH-003, the MOST-RIKEN Joint Project “Ab
initio investigation in nuclear physics”. Part of this work was achieved by
using the supercomputer OCTOPUS at the Cybermedia Center, Osaka University
under the support of Research Center for Nuclear Physics of Osaka University.

## References

  * Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, PhRvL, 119, 161101, doi: 10.1103/PhysRevLett.119.161101
  * Abbott et al. (2018) —. 2018, PhRvL, 121, 161101, doi: 10.1103/PhysRevLett.121.161101
  * Abbott et al. (2019) —. 2019, PhRvX, 9, 011001, doi: 10.1103/PhysRevX.9.011001
  * Abbott et al. (2020) Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, ApJL, 896, L44, doi: 10.3847/2041-8213/ab960f
  * Akmal et al. (1998) Akmal, A., Pandharipande, V. R., & Ravenhall, D. G. 1998, PhRvC, 58, 1804, doi: 10.1103/PhysRevC.58.1804
  * Alonso & Sammarruca (2003) Alonso, D., & Sammarruca, F. 2003, PhRvC, 67, 054301, doi: 10.1103/PhysRevC.67.054301
  * Anastasio et al. (1981) Anastasio, M. R., Celenza, L. S., & Shakin, C. M. 1981, PhRvC, 23, 2273, doi: 10.1103/PhysRevC.23.2273
  * Annala et al. (2018) Annala, E., Gorda, T., Kurkela, A., & Vuorinen, A. 2018, PhRvL, 120, 172703, doi: 10.1103/PhysRevLett.120.172703
  * Antoniadis et al. (2013) Antoniadis, J., et al. 2013, Sci, 340, 1233232, doi: 10.1126/science.1233232
  * Baran et al. (2005) Baran, V., Colonna, M., Greco, V., & Di Toro, M. 2005, PhR, 410, 335, doi: https://doi.org/10.1016/j.physrep.2004.12.004
  * Baym et al. (1971a) Baym, G., Bethe, H. A., & Pethick, C. J. 1971a, NuPhA, 175, 225 , doi: https://doi.org/10.1016/0375-9474(71)90281-8
  * Baym et al. (1971b) Baym, G., Pethick, C., & Sutherland, P. 1971b, ApJ, 170, 299, doi: 10.1086/151216
  * Bethe (1971) Bethe, H. A. 1971, Annu. Rev. Nucl. Sci., 21, 93, doi: 10.1146/annurev.ns.21.120171.000521
  * Boguta & Bodmer (1977) Boguta, J., & Bodmer, A. 1977, NuPhA, 292, 413, doi: https://doi.org/10.1016/0375-9474(77)90626-1
  * Bombaci et al. (2021) Bombaci, I., Drago, A., Logoteta, D., Pagliara, G., & Vidaña, I. 2021, PhRvL, 126, 162702, doi: 10.1103/PhysRevLett.126.162702
  * Bombaci & Lombardo (1991) Bombaci, I., & Lombardo, U. 1991, PhRvC, 44, 1892, doi: 10.1103/PhysRevC.44.1892
  * Brockmann & Machleidt (1990) Brockmann, R., & Machleidt, R. 1990, PhRvC, 42, 1965, doi: 10.1103/PhysRevC.42.1965
  * Brown et al. (2018) Brown, E. F., Cumming, A., Fattoyev, F. J., et al. 2018, PhRvL, 120, 182701, doi: 10.1103/PhysRevLett.120.182701
  * Burgio et al. (2021) Burgio, G., Schulze, H.-J., Vidaña, I., & Wei, J.-B. 2021, PrPNP, 120, 103879, doi: https://doi.org/10.1016/j.ppnp.2021.103879
  * Centelles et al. (2009) Centelles, M., Roca-Maza, X., Viñas, X., & Warda, M. 2009, PhRvL, 102, 122502, doi: 10.1103/PhysRevLett.102.122502
  * Chen et al. (2005) Chen, L.-W., Ko, C. M., & Li, B.-A. 2005, PhRvC, 72, 064309, doi: 10.1103/PhysRevC.72.064309
  * Coester et al. (1970) Coester, F., Cohen, S., Day, B., & Vincent, C. M. 1970, PhRvC, 1, 769, doi: 10.1103/PhysRevC.1.769
  * Cromartie et al. (2020) Cromartie, H. T., et al. 2020, NatAs, 4, 72, doi: 10.1038/s41550-019-0880-2
  * Damour & Nagar (2009) Damour, T., & Nagar, A. 2009, PhRvD, 80, 084035, doi: 10.1103/PhysRevD.80.084035
  * Damour et al. (1992) Damour, T., Soffel, M., & Xu, C. 1992, PhRvD, 45, 1017, doi: 10.1103/PhysRevD.45.1017
  * de Jong & Lenske (1998) de Jong, F., & Lenske, H. 1998, PhRvC, 57, 3099, doi: 10.1103/PhysRevC.57.3099
  * Dechargé & Gogny (1980) Dechargé, J., & Gogny, D. 1980, PhRvC, 21, 1568, doi: 10.1103/PhysRevC.21.1568
  * Demorest et al. (2010) Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Natur, 467, 1081, doi: 10.1038/nature09466
  * Dong et al. (2012) Dong, J., Zuo, W., Gu, J., & Lombardo, U. 2012, PhRvC, 85, 034308, doi: 10.1103/PhysRevC.85.034308
  * Dutra et al. (2012) Dutra, M., Lourenço, O., Sá Martins, J. S., et al. 2012, PhRvC, 85, 035201, doi: 10.1103/PhysRevC.85.035201
  * Dutra et al. (2014) Dutra, M., Lourenço, O., Avancini, S. S., et al. 2014, PhRvC, 90, 055203, doi: 10.1103/PhysRevC.90.055203
  * Engvik et al. (1994) Engvik, L., Hjorth-Jensen, M., Osnes, E., Bao, G., & Østgaard, E. 1994, PhRvL, 73, 2650, doi: 10.1103/PhysRevLett.73.2650
  * Essick et al. (2021) Essick, R., Tews, I., Landry, P., & Schwenk, A. 2021, PhRvL, 127, 192701, doi: 10.1103/PhysRevLett.127.192701
  * Fattoyev et al. (2020) Fattoyev, F. J., Horowitz, C. J., Piekarewicz, J., & Reed, B. 2020, PhRvC, 102, 065805, doi: 10.1103/PhysRevC.102.065805
  * Fattoyev et al. (2018) Fattoyev, F. J., Piekarewicz, J., & Horowitz, C. J. 2018, PhRvL, 120, 172702, doi: 10.1103/PhysRevLett.120.172702
  * Flanagan & Hinderer (2008) Flanagan, E. E., & Hinderer, T. 2008, PhRvD, 77, 021502(R), doi: 10.1103/PhysRevD.77.021502
  * Fonseca et al. (2016) Fonseca, E., et al. 2016, ApJ, 832, 167, doi: 10.3847/0004-637X/832/2/167
  * Gandolfi et al. (2012) Gandolfi, S., Carlson, J., & Reddy, S. 2012, PhRvC, 85, 032801, doi: 10.1103/PhysRevC.85.032801
  * Garg & Colò (2018) Garg, U., & Colò, G. 2018, PrPNP, 101, 55, doi: https://doi.org/10.1016/j.ppnp.2018.03.001
  * Garg et al. (2007) Garg, U., Li, T., Okumura, S., et al. 2007, NuPhA, 788, 36, doi: https://doi.org/10.1016/j.nuclphysa.2007.01.046
  * Gross-Boelting et al. (1999) Gross-Boelting, T., Fuchs, C., & Faessler, A. 1999, NuPhA, 648, 105, doi: https://doi.org/10.1016/S0375-9474(99)00022-6
  * Hinderer (2008) Hinderer, T. 2008, ApJ, 677, 1216.  http://stacks.iop.org/0004-637X/677/i=2/a=1216
  * Hofmann et al. (2001) Hofmann, F., Keil, C. M., & Lenske, H. 2001, PhRvC, 64, 034314, doi: 10.1103/PhysRevC.64.034314
  * Horowitz & Serot (1984) Horowitz, C., & Serot, B. D. 1984, PhLB, 137, 287, doi: https://doi.org/10.1016/0370-2693(84)91717-9
  * Horowitz & Serot (1987) —. 1987, NuPhA, 464, 613, doi: https://doi.org/10.1016/0375-9474(87)90370-8
  * Huang et al. (2020) Huang, K., Hu, J., Zhang, Y., & Shen, H. 2020, ApJ, 904, 39, doi: 10.3847/1538-4357/abbb37
  * Jiang et al. (2015) Jiang, L. J., Yang, S., Dong, J. M., & Long, W. H. 2015, PhRvC, 91, 025802, doi: 10.1103/PhysRevC.91.025802
  * Katayama & Saito (2013) Katayama, T., & Saito, K. 2013, PhRvC, 88, 035805, doi: 10.1103/PhysRevC.88.035805
  * Krastev & Sammarruca (2006) Krastev, P. G., & Sammarruca, F. 2006, PhRvC, 74, 025808, doi: 10.1103/PhysRevC.74.025808
  * Kubis & Kutschera (1997) Kubis, S., & Kutschera, M. 1997, PhLB, 399, 191, doi: https://doi.org/10.1016/S0370-2693(97)00306-7
  * Kumar & Landry (2019) Kumar, B., & Landry, P. 2019, PhRvD, 99, 123026, doi: 10.1103/PhysRevD.99.123026
  * Lattimer et al. (1991) Lattimer, J. M., Pethick, C. J., Prakash, M., & Haensel, P. 1991, PhRvL, 66, 2701, doi: 10.1103/PhysRevLett.66.2701
  * Lattimer & Prakash (2004) Lattimer, J. M., & Prakash, M. 2004, Sci, 304, 536, doi: 10.1126/science.1090720
  * Lattimer & Prakash (2007) —. 2007, PhR, 442, 109, doi: https://doi.org/10.1016/j.physrep.2007.02.003
  * Li et al. (2008) Li, B.-A., Chen, L.-W., & Ko, C. M. 2008, PhR, 464, 113, doi: https://doi.org/10.1016/j.physrep.2008.04.005
  * Li & Magno (2020) Li, B.-A., & Magno, M. 2020, PhRvC, 102, 045807, doi: 10.1103/PhysRevC.102.045807
  * Liu et al. (2002) Liu, B., Greco, V., Baran, V., Colonna, M., & Di Toro, M. 2002, PhRvC, 65, 045201, doi: 10.1103/PhysRevC.65.045201
  * Long et al. (2006) Long, W.-H., Giai, N. V., & Meng, J. 2006, PhLB, 640, 150 , doi: https://doi.org/10.1016/j.physletb.2006.07.064
  * Machleidt (1989) Machleidt, R. 1989, Adv. Nucl. Phys., 19, 189, doi: 10.1007/978-1-4613-9907-0_2
  * Machleidt et al. (1987) Machleidt, R., Holinde, K., & Elster, C. 1987, PhR, 149, 1, doi: https://doi.org/10.1016/S0370-1573(87)80002-9
  * Maieron et al. (2004) Maieron, C., Baldo, M., Burgio, G. F., & Schulze, H.-J. 2004, PhRvD, 70, 043010, doi: 10.1103/PhysRevD.70.043010
  * Malik et al. (2018) Malik, T., Alam, N., Fortin, M., et al. 2018, PhRvC, 98, 035804, doi: 10.1103/PhysRevC.98.035804
  * Meng (2016) Meng, J. 2016, Relativistic Density Functional for Nuclear Structure (WORLD SCIENTIFIC), doi: 10.1142/9872
  * Meng et al. (2006) Meng, J., Toki, H., Zhou, S., et al. 2006, PrPNP, 57, 470, doi: https://doi.org/10.1016/j.ppnp.2005.06.001
  * Miller et al. (2019) Miller, M. C., Lamb, F. K., Dittmann, A. J., et al. 2019, ApJ, 887, L24, doi: 10.3847/2041-8213/ab50c5
  * Most et al. (2018) Most, E. R., Weih, L. R., Rezzolla, L., & Schaffner-Bielich, J. 2018, PhRvL, 120, 261103, doi: 10.1103/PhysRevLett.120.261103
  * Newton & Crocombe (2021) Newton, W. G., & Crocombe, G. 2021, PhRvC, 103, 064323, doi: 10.1103/PhysRevC.103.064323
  * Nuppenau et al. (1989) Nuppenau, C., Lee, Y., & MacKellar, A. 1989, NuPhA, 504, 839, doi: https://doi.org/10.1016/0375-9474(89)90011-0
  * Oppenheimer & Volkoff (1939) Oppenheimer, J. R., & Volkoff, G. M. 1939, PhRe, 55, 374, doi: 10.1103/PhysRev.55.374
  * Pethick (1992) Pethick, C. J. 1992, RvMP, 64, 1133, doi: 10.1103/RevModPhys.64.1133
  * Poschenrieder & Weigel (1988) Poschenrieder, P., & Weigel, M. K. 1988, PhRvC, 38, 471, doi: 10.1103/PhysRevC.38.471
  * Postnikov et al. (2010) Postnikov, S., Prakash, M., & Lattimer, J. M. 2010, PhRvD, 82, 024016, doi: 10.1103/PhysRevD.82.024016
  * Prakash (1994) Prakash, M. 1994, PhR, 242, 297, doi: https://doi.org/10.1016/0370-1573(94)90165-1
  * Prakash et al. (1992) Prakash, M., Prakash, M., Lattimer, J. M., & Pethick, C. J. 1992, ApJ, 390, L77, doi: 10.1086/186376
  * Pudliner et al. (1995) Pudliner, B. S., Pandharipande, V. R., Carlson, J., & Wiringa, R. B. 1995, PhRvL, 74, 4396, doi: 10.1103/PhysRevLett.74.4396
  * Raaijmakers et al. (2019) Raaijmakers, G., Riley, T. E., Watts, A. L., et al. 2019, ApJ, 887, L22, doi: 10.3847/2041-8213/ab451a
  * Reinhard (1989) Reinhard, P. G. 1989, Rep. Prog. Phys., 52, 439, doi: 10.1088/0034-4885/52/4/002
  * Rhoades & Ruffini (1974) Rhoades, C. E., & Ruffini, R. 1974, PhRvL, 32, 324, doi: 10.1103/PhysRevLett.32.324
  * Riley et al. (2019) Riley, T. E., Watts, A. L., Bogdanov, S., et al. 2019, ApJ, 887, L21, doi: 10.3847/2041-8213/ab481c
  * Ring (1996) Ring, P. 1996, PrPNP, 37, 193, doi: https://doi.org/10.1016/0146-6410(96)00054-3
  * Roca-Maza et al. (2011) Roca-Maza, X., Viñas, X., Centelles, M., Ring, P., & Schuck, P. 2011, PhRvC, 84, 054309, doi: 10.1103/PhysRevC.84.054309
  * Safarzadeh & Loeb (2020) Safarzadeh, M., & Loeb, A. 2020, ApJ, 899, L15, doi: 10.3847/2041-8213/aba9df
  * Sagawa et al. (2007) Sagawa, H., Yoshida, S., Zeng, G.-M., Gu, J.-Z., & Zhang, X.-Z. 2007, PhRvC, 76, 034327, doi: 10.1103/PhysRevC.76.034327
  * Schiller & Müther (2001) Schiller, E., & Müther, H. 2001, EPJA, 11, 15, doi: 10.1007/s100500170092
  * Serot & Walecka (1986) Serot, B. D., & Walecka, J. D. 1986, Adv. Nucl. Phys., 16, 1
  * Shen et al. (2019) Shen, S., Liang, H., Long, W. H., Meng, J., & Ring, P. 2019, PrPNP, 109, 103713, doi: https://doi.org/10.1016/j.ppnp.2019.103713
  * Skyrme (1956) Skyrme, T. H. R. 1956, Phil. Mag., 1, 1043, doi: 10.1080/14786435608238186
  * Sprung (1972) Sprung, D. W. L. 1972, Adv. Nucl. Phys., 5, 225, doi: 10.1007/978-1-4615-8231-1_2
  * Sun et al. (2008) Sun, B. Y., Long, W. H., Meng, J., & Lombardo, U. 2008, PhRvC, 78, 065805, doi: 10.1103/PhysRevC.78.065805
  * Tews et al. (2018) Tews, I., Margueron, J., & Reddy, S. 2018, PhRvC, 98, 045804, doi: 10.1103/PhysRevC.98.045804
  * Tolman (1939) Tolman, R. C. 1939, PhRe, 55, 364, doi: 10.1103/PhysRev.55.364
  * Tong et al. (2018) Tong, H., Ren, X.-L., Ring, P., et al. 2018, PhRvC, 98, 054302, doi: 10.1103/PhysRevC.98.054302
  * Tong et al. (2020) Tong, H., Zhao, P., & Meng, J. 2020, PhRvC, 101, 035802, doi: 10.1103/PhysRevC.101.035802
  * Tsokaros et al. (2020) Tsokaros, A., Ruiz, M., & Shapiro, S. L. 2020, ApJ, 905, 48, doi: 10.3847/1538-4357/abc421
  * Ulrych & Müther (1997) Ulrych, S., & Müther, H. 1997, PhRvC, 56, 1788, doi: 10.1103/PhysRevC.56.1788
  * van Dalen et al. (2004) van Dalen, E., Fuchs, C., & Faessler, A. 2004, NuPhA, 744, 227, doi: https://doi.org/10.1016/j.nuclphysa.2004.08.019
  * van Dalen et al. (2005) van Dalen, E. N. E., Fuchs, C., & Faessler, A. 2005, PhRvL, 95, 022302, doi: 10.1103/PhysRevLett.95.022302
  * Vattis et al. (2020) Vattis, K., Goldstein, I. S., & Koushiappas, S. M. 2020, PhRvD, 102, 061301, doi: 10.1103/PhysRevD.102.061301
  * Vautherin & Brink (1972) Vautherin, D., & Brink, D. M. 1972, PhRvC, 5, 626, doi: 10.1103/PhysRevC.5.626
  * Vidaña et al. (2009) Vidaña, I., Providência, C., Polls, A., & Rios, A. 2009, PhRvC, 80, 045806, doi: 10.1103/PhysRevC.80.045806
  * Wang et al. (2020) Wang, C., Hu, J., Zhang, Y., & Shen, H. 2020, ApJ, 897, 96, doi: 10.3847/1538-4357/ab994b
  * Wang et al. (2022) Wang, S., Tong, H., Zhao, Q., et al. 2022, arXiv:2203.05397.  https://arxiv.org/abs/2203.05397
  * Wang et al. (2021) Wang, S., Zhao, Q., Ring, P., & Meng, J. 2021, PhRvC, 103, 054319, doi: 10.1103/PhysRevC.103.054319
  * Wiringa et al. (1988) Wiringa, R. B., Fiks, V., & Fabrocini, A. 1988, PhRvC, 38, 1010, doi: 10.1103/PhysRevC.38.1010
  * Wiringa et al. (1995) Wiringa, R. B., Stoks, V. G. J., & Schiavilla, R. 1995, PhRvC, 51, 38, doi: 10.1103/PhysRevC.51.38
  * Xie & Li (2020) Xie, W.-J., & Li, B.-A. 2020, ApJ, 899, 4, doi: 10.3847/1538-4357/aba271
  * Yang et al. (2020) Yang, Y., Gayathri, V., Bartos, I., et al. 2020, ApJ, 901, L34, doi: 10.3847/2041-8213/abb940
  * Zhang et al. (2018) Zhang, N.-B., Li, B.-A., & Xu, J. 2018, ApJ, 859, 90, doi: 10.3847/1538-4357/aac027
  * Zhang, Nai-Bo & Li, Bao-An (2019) Zhang, Nai-Bo, & Li, Bao-An. 2019, EPJA, 55, 39, doi: 10.1140/epja/i2019-12700-0
  * Zhao et al. (2010) Zhao, P. W., Li, Z. P., Yao, J. M., & Meng, J. 2010, PhRvC, 82, 054319, doi: 10.1103/PhysRevC.82.054319


#  STATISTICAL ANALYSIS OF THERMAL CONDUCTIVITY
EXPERIMENTALLY MEASURED IN ETHYLENE GLYCOL-BASED NANOFLUIDS

Tielke J.* and Avila  M.
*Author for correspondence
Center of Applied Space Technology and Microgravity
University of Bremen
Bremen  28359
Germany
E-mail<EMAIL_ADDRESS>
## ABSTRACT

We collected literature data of thermal conductivity experimentally measured
in ethylene glycol-based nanofluids and investigated the influence of
concentration, temperature and nanoparticle size. We implemented statistical
linear regression analysis of all data points and examined four separate
nanoparticle materials – alumina, titania, copper oxide and carbon-nanotubes.
We found that the statistical correlations are in good agreement with
Maxwell’s effective medium theory, despite large scatter in the data. The
thermal conductivity increases linearly with concentration, and in the case of
carbon-nanotubes with temperature, whereas the nanoparticle size shows
significant influence for alumina and titania. The large scatter in the
experimental data is one of the main problems. We suggest that there is a need
for careful, detailed characterizations and measurements to quantify the
potential of nanofluids.

NOMENCLATURE

$k_{eff}$ | [W/mK] | Effective thermal conductivity of suspension
---|---|---
$k_{f}$ | [W/mK] | Thermal conductivity base fluid
$k_{p}$ | [W/mK] | Thermal conductivity nanoparticle material
$T$ | [K] | Temperature
$S$ | [1/nm] | Nanoparticle surface
$C$ | [-] | Regression coefficient

Special characters

$\varphi$ | [-] | Particle volume fraction ratio
---|---|---
$\beta$ | [-] | Standardized regression coefficient

## INTRODUCTION

The term Nanofluids was introduced by Lee et al. [1], who dispersed metallic
nanoparticles in heat transfer fluids and measured the strongly increased
thermal conductivity. A first description of the increase in thermal
conductivity of a fluid by the dispersion of particles is given by Maxwell’s
[2] effective medium theory

$\frac{k_{\text{eff}}}{k_{\text{f}}}=1+\frac{3\varphi(k_{\text{p}}-k_{\text{f}})}{3k_{\text{f}}+(1-\varphi)(k_{\text{p}}-k_{\text{f}})}.$
(1)

Maxwell’s effective medium theory assumes homogeneously dispersed particles
and gives the lower limit in the bounds of thermal conductivity derived by
Hashin and Shtrikman [3]. Buongiorno et al. [4] conducted a benchmark study,
which showed large scatter in the experimental data due to differences in the
characterization and measurements. First attempts were made to increase the
quality of measurements by critically examining the different measuring
techniques [5, 6, 7, 8], with the conclusion that insufficient
characterizations were mainly responsible for scatter. However, general
correlations to calculate the thermal conductivity of nanofluids were still
developed resulting in various versions [9, 10, 11]. A first attempt using
statistical linear regressions was used by Khanafer et al. [12], again with
the aim of having a general equation describing the increase in thermal
conductivity.

In this study, we use linear regressions to quantify the influence of the
parameters concentration, temperature and nanoparticle size on the
experimental measured thermal conductivity. We further compare the results to
Maxwell’s effective medium theory for separate materials to estimate their
potential.

## MATERIAL AND METHODS

We collected literature data and set up a database with ${N=1167}$ data points
of experimentally measured thermal conductivity in ethylene glycol nanofluids.
We only considered studies in which the concentration, temperature and
nanoparticle size are specified. The preparation of the nanofluids and method
of thermal conductivity measurement are explained in each study. The database
contains data for 18 different materials out of $59$ publications. First we
analysed all data points together. Additionally, four separate materials are
analysed, in which at least $4$ different publications with $50$ data points
are necessary to ensure statistical significance. The separate analyzed
materials are:

Alumina ($Al_{2}O_{3}$, $N=292$): [13, 14, 15, 16, 17, 18, 19, 1, 20, 21, 22,
23, 24, 25, 26, 27, 28]

Titania ($TiO_{2}$, $N=105$) [29, 30, 20, 31, 23, 32, 33]

Carbon-Nanotubes (CNT, $N=76$) [34, 35, 36, 37, 38]

Copper oxide (CuO, $N=61$) [39, 15, 18, 1, 36, 26, 40, 41]

The remaining materials do not have $50$ data points out of $4$ publications
and are thus not analyzed separately [42, 43, 44, 45, 15, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 23, 57, 58, 26, 40, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70].

### Linear Statistical Model

We employed a linear statistical model to analyze the effect of the
concentration $\varphi$, the temperature $T$ and the particle size through the
specific surface $S$ on the normalized thermal conductivity

$k^{*}(\varphi,T,S)=\frac{k_{\text{{eff}}}(\varphi,T,S)}{k_{\text{f}}(T)}$ (2)

where $k_{\text{f}}(T)$ is the thermal conductivity of the base fluid
(ethylene glycol) as a function of the measurement of temperature. With the
use of linear regressions, the predictors were implemented in the following
equation with the coefficients ${C_{i}~{}(i={0,\varphi,\text{T},\text{S}})}$
given from each regression:

$k^{*}(\varphi,T,S)=C_{0}+C_{\varphi}\,\varphi+C_{\text{T}}\,T^{*}+C_{\text{S}}\,S^{*},$
(3)

The determination of the linear model and the statistical method is fully
described in a previous paper [71], where it is applied to water-based
nanofluids.

## RESULTS

The results of the statistical analysis of thermal conductivity modeled with
the linear regression eq. (3) for all data points and the separate materials
are shown in table 1 and fig. 1, respectively. The scatter of the data,
expressed by the corrected correlation coefficients $R^{2}\ \text{in}\
[0.30;0.80]$ suggest that there are large uncertainties in the measurements.
However, the regression for all data points is in good approximation with
Maxwell’s effective medium theory. The influence of each parameter, given by
the standardized regression coefficients
$\beta_{j}~{}(j={\varphi,\text{T},\text{S}})$, show only a concentration-
dependence with $\beta_{\varphi}=0.55$ and insignificant coefficients towards
the temperature or surface with $\beta_{\text{T}}=0.08$ and
$\beta_{\text{S}}=0.03$, respectively. Most of the data, especially of the
separately analyzed materials, lie within the $\pm 10\%$ interval, as
displayed in fig. 1. But there are materials and measurements that
significantly differ from the results with modeled thermal conductivities
above $k^{*}>1.5$.

Table 1: Results of the linear regressions fitted to the entire database and for separate materials. Given are the number of data points ($N$), corrected correlation coefficient ($R^{2}$), model coefficients ($C_{i}$), standardized regression coefficients ($\beta_{j}$). Material | $N$ | $R^{2}$ | $C_{0}$ | $C_{\varphi}$ | $\beta_{\varphi}$ | $C_{T}$ | $\beta_{T}$ | $C_{S}$ | $\beta_{S}$
---|---|---|---|---|---|---|---|---|---
all | $1167$ | $0.30$ | $1.03$ | $3.46$ | $0.55$ | $0.23$ | $0.08$ | $0.05$ | $0.03$
$Al_{2}0_{3}$ | $292$ | $0.54$ | $0.99$ | $3.10$ | $0.70$ | $0.09$ | $0.04$ | $0.45$ | $0.36$
$TiO_{2}$ | $105$ | $0.69$ | $1.00$ | $2.24$ | $0.71$ | $0.01$ | $0.01$ | $0.37$ | $0.35$
$CNT$ | $76$ | $0.50$ | $1.06$ | $2.85$ | $0.71$ | $0.70$ | $0.28$ | $0.01$ | $0.03$
$CuO$ | $61$ | $0.80$ | $0.99$ | $4.57$ | $0.94$ | $0.16$ | $0.15$ | $0.58$ | $0.03$

Figure 1: Experimentally measured versus modeled normalized thermal
conductivity ($k^{*}$) for ethylene glycol-based nanofluids. The ideal values
of $k^{*}$ and a $\pm 10\%$ interval are displayed as a solid line and a grey
area, respectively. Colors are used to distinguish single materials (analyzed
separately) from other materials (see the legend).

The concentration is the main parameter in the statistical correlation. In all
cases, the standardized regression coefficient $\beta_{\varphi}$ is the
highest. The modeled coefficients $C_{\varphi}$ are in general agreement with
the the bounds derived from the linearized Maxwell equation
($C_{\varphi,~{}Maxwell}\in[-1.5,3]$, see [71]). We plotted the values of
$C_{\varphi}$ against the thermal conductivity of the particles $k_{p}$ in
fig. 2. We also show the bounds according to Hashin-Shtrikman [3] in their
linearized form as solid black lines. The lower HS-bound is Maxwell’s
effective medium theory with the maximum at $C_{\varphi}=3$. The upper HS-
bound includes all possible dispersion states. The modeled values for alumina
and carbon-nanotubes are in excellent agreement with the linearized Maxwell
equation, while the value for titania lies below it. If the data for higher
concentrations $(\varphi\geq 2.5vol\%)$ are excluded, the value for titania
also fits to Maxwell’s prediction. The value for copper oxide exceeds
Maxwell’s prediction, but is inside the bounds derived by Hashin-Shtrikman.

Figure 2: The colored symbols show the values of the model coefficient
$C_{\varphi}$ (with corresponding 95% confidence-interval, shown as error-
bars) as a function of the thermal conductivity $k_{\text{p}}$ of each
material.The linearized HS-bounds [2] are displayed as solid black lines,
enclosing the shaded area.

Similar to the concentration, we plotted the temperature- and surface-
dependent coefficients $C_{\text{T}}$ and $C_{\text{S}}$ in figures 3 and 4,
respectively. If the standardized regression coefficient $\beta<0.1$, the
corresponding parameter rendered insignificant. In the case of the temperature
coefficient, results gave insignificant coefficients for alumina with
$\beta_{\text{T}}=0.04$ and titania with $\beta_{\text{T}}=0.01$. We found a
marginally significant temperature-dependence for copper oxide with
$C_{\text{T}}=0.16\ (\beta_{\text{T}}=0.15)$. The results for carbon-nanotubes
have a significant temperature coefficient with $C_{\text{T}}=0.70\
(\beta_{\text{T}}=0.28)$. As seen in fig. 3, the results for carbon-nanotubes
go along with large error-bars. In contrast to the temperature, the size
dependence showed insignificant results for copper oxide and carbon-nanotubes,
both with $\beta_{\text{S}}=0.03$. The large error-bars for copper oxide as
seen in fig. 4 correspond to the small range of measured nanoparticle
diameters. Alumina and titania exhibit significant regression coefficients at
the same level with $C_{\text{S}}=0.45\ (\beta_{\text{S}}=0.36)$ for alumina
and $C_{\text{S}}=0.37\ (\beta_{\text{S}}=0.35)$ for titania.

Figure 3: The colored symbols show the values of the model coefficient
$C_{\text{T}}$ (with corresponding 95% confidence-interval, shown as error-
bars) as a function of the thermal conductivity $k_{\text{p}}$ of each
material.

Figure 4: The colored symbols show the values of the model coefficient
$C_{\text{S}}$ (with corresponding 95% confidence-interval, shown as error-
bars) as a function of the thermal conductivity $k_{\text{p}}$ of each
material.

## DISCUSSION

The main problem in the field of nanofluids seems to be the large scatter in
the measurements. Nanofluids with the apparently same characterizations
(concentration, temperature, size, material) exhibit different measurement
results depending on the publication. The use of different measuring
techniques is not the only reason [8]. There is a lack of understanding on how
to measure the thermal conductivity of dispersions; even in the measurement of
pure fluids, partially large scattering occurs [72]. In the case of
concentration, the use of volume-based units by dispersing colloidal
nanoparticles is less common than the use of weight-based units[72, 8]. The
determination of the nanoparticle size and homogeneous distribution should be
measured in all cases. A simple reference to the manufacturers information is
not sufficient, because the particles act differently in dispersion than when
stored dry. Comparing measurements of the sizes (e.g. TEM and DLS
measurements) should not only lead to more significant results, they would
also give a hint to the dispersion state and the agglomeration of the
particles [8]. Therefore it is important to examine and develop comparable
measuring techniques. A careful characterization of the nanoparticles
properties could lead to verifiable measurements.

The results shown in table 1 and fig. 2 suggest that the thermal conductivity
of nanofluids is in agreement with Maxwell’s effective medium theory. The
concentration mainly determines the thermal conductivity, the temperature and
nanoparticle size have lower significant influence. The temperature effect is
only significant for carbon-nanotubes and marginal for copper oxide (see fig.
3). As already stated by Prasher et. al., the influence of temperature on the
thermal conductivity coincides with agglomeration in the dispersion [73]. As
seen in fig. 2, the concentration coefficient for copper oxides exceed
Maxwell’s prediction. This suggests a non-homogeneous dispersion state in the
theory of Hashin-Shtrikman [3, 6] and agrees with the significant temperature
coefficient; copper oxides are known to build large agglomerates [74].
Percolation networks likely occur without the use of surfactants or other
surface treatments. Indeed, the data for copper oxide based on the statistical
regression used no surfactants at all. In the case of carbon-nanotubes, the
temperature effect is significant while the concentration coefficient fits to
Maxwell’s theory. In many cases surfactants or surface functionalizations are
used to ensure the homogeneous dispersion of the carbon-nanotubes, but
surfactants can also lead to a decrease in thermal conductivity [75]. Fig. 3
therefore shows that while the data for alumina, titania and copper oxide are
similar, the temperature coefficient and the corresponding error-bars for
carbon-nanotubes significantly differ from the other materials. Significant
size effects occur only for alumina and titania in our statistical results,
while the results for carbon-nanotubes and copper oxides are insignificant
(see fig. 4). These insignificant size effects mainly result for the limited
variation of nanoparticle sizes in the data sets. For carbon-nanotubes the
variation of the size is limited due to the shape. It is also unclear whether
the size of carbon-nanotubes is to be defined by the diameter or the length of
the tube [76]. The size coefficient and the standardized regression
coefficients for alumina and titania are in comparable ranges. This supports
that the thermal conductivity of the nanofluid increases with decreasing
particle sizes as already published [12]. However, the uncertainties of the
measurements and the difficulties in the stabilization of the well-dispersed
state, are not taken into account in the statistical analysis.

## CONCLUSION

With our statistical analysis we show that the thermal conductivity of
nanofluids can be well described by Maxwell’s effective medium theory. Only
copper oxides exceeds this prediction with $C_{\varphi}=4.57$, possibly due to
agglomeration effects. We found significant influence for the temperature in
carbon nanotubes ($C_{\text{T}}=0.70$) and for the particle size in alumina
($C_{\text{S}}=0.45$) and titania ($C_{\text{S}}=0.37$). The main limiting
factor is in general the lack of precise measurements and characterizations.
Furthermore, the formation of agglomerates and percolation networks increases
the viscosity of the dispersion, which makes it unfavourable for application.
With careful and comparable measurements, further statistical analyses would
allow to asses the full potential of nanofluids [71].

## References

  * [1] S Lee et al. “Measuring Thermal Conductivity of Fluids Containing Oxide Nanoparticles” In _Journal of Heat Transfer_ 121.2 ASME, 1999, pp. 280–289 URL: http://dx.doi.org/10.1115/1.2825978
  * [2] J.. Maxwell “A Treatise on Electricity and Magnetism” Clarendon Press, 1881, pp. 435
  * [3] Z Hashin et al. “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials” In _Journal of Applied Physics_ 33.10 American Institute of Physics, 1962, pp. 3125–3131 DOI: 10.1063/1.1728579
  * [4] Jacopo Buongiorno et al. “A benchmark study on the thermal conductivity of nanofluids” In _Journal of Applied Physics_ 106.9 American Institute of Physics, 2009, pp. 94312 DOI: 10.1063/1.3245330
  * [5] Pawel Keblinski et al. “Thermal conductance of nanofluids: is the controversy over?” In _Journal of Nanoparticle Research_ 10.7, 2008, pp. 1089–1097 DOI: 10.1007/s11051-007-9352-1
  * [6] Jacob Eapen et al. “The Classical Nature of Thermal Conduction in Nanofluids” In _Journal of Heat Transfer_ 132.10 ASME, 2010, pp. 102402–102414 URL: http://dx.doi.org/10.1115/1.4001304
  * [7] Carlos A Nieto de Castro et al. “Towards the Correct Measurement of Thermal Conductivity of Ionic Melts and Nanofluids” In _Energies_ 13.1, 2020 DOI: 10.3390/en13010099
  * [8] Sergio Bobbo et al. “Analysis of the Parameters Required to Properly Define Nanofluids for Heat Transfer Applications” In _Fluids_ 6.2, 2021 DOI: 10.3390/fluids6020065
  * [9] J A Eastman et al. “Thermal Transport in Nanofluids” In _Annual Review of Materials Research_ 34.1 Annual Reviews, 2004, pp. 219–246 DOI: 10.1146/annurev.matsci.34.052803.090621
  * [10] J Buongiorno “Convective Transport in Nanofluids” In _Journal of Heat Transfer_ 128.3 ASME, 2005, pp. 240–250 URL: http://dx.doi.org/10.1115/1.2150834
  * [11] Ibai Mugica et al. “A critical review of the most popular mathematical models for nanofluid thermal conductivity” In _Journal of Nanoparticle Research_ 22.5, 2020, pp. 113 DOI: 10.1007/s11051-020-4776-y
  * [12] Khalil Khanafer et al. “A critical synthesis of thermophysical characteristics of nanofluids” In _International Journal of Heat and Mass Transfer_ 54.19, 2011, pp. 4410–4428 DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2011.04.048
  * [13] Benigno Barbés et al. “Thermal conductivity and specific heat capacity measurements of Al2O3 nanofluids” In _Journal of Thermal Analysis and Calorimetry_ 111.2, 2013, pp. 1615–1625 DOI: 10.1007/s10973-012-2534-9
  * [14] Michael P Beck et al. “The effect of particle size on the thermal conductivity of alumina nanofluids” In _Journal of Nanoparticle Research_ 11.5, 2009, pp. 1129–1136 DOI: 10.1007/s11051-008-9500-2
  * [15] J A Eastman et al. “Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles” In _Applied Physics Letters_ 78.6 American Institute of Physics, 2001, pp. 718–720 DOI: 10.1063/1.1341218
  * [16] Mohammad Hemmat Esfe et al. “Experimental study on thermal conductivity of ethylene glycol based nanofluids containing Al2O3 nanoparticles” In _International Journal of Heat and Mass Transfer_ 88, 2015, pp. 728–734 DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.010
  * [17] Mohammad Hemmat Esfe et al. “Using artificial neural network to predict thermal conductivity of ethylene glycol with alumina nanoparticle” In _Journal of Thermal Analysis and Calorimetry_ 126.2, 2016, pp. 643–648 DOI: 10.1007/s10973-016-5506-7
  * [18] Raghu Gowda et al. “Effects of Particle Surface Charge, Species, Concentration, and Dispersion Method on the Thermal Conductivity of Nanofluids” In _Advances in Mechanical Engineering_ , 2010 DOI: 10.1155/2010/807610
  * [19] Nishant Kumar et al. “Experimental study of thermal conductivity, heat transfer and friction factor of Al2O3 based nanofluid” In _International Communications in Heat and Mass Transfer_ 90, 2018, pp. 1–10 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2017.10.001
  * [20] Giovanni A Longo et al. “Experimental Measurements of Thermophysical Properties of Al2O3– and TiO2–Ethylene Glycol Nanofluids” In _International Journal of Thermophysics_ 34.7, 2013, pp. 1288–1307 DOI: 10.1007/s10765-013-1478-z
  * [21] A V Minakov et al. “Measurement of the Thermal-Conductivity Coefficient of Nanofluids by the Hot-Wire Method” In _Journal of Engineering Physics and Thermophysics_ 88.1, 2015, pp. 149–162 DOI: 10.1007/s10891-015-1177-7
  * [22] Hamid Mohammadiun et al. “Experimental study of ethylene glycol-based Al2O3 nanofluid turbulent heat transfer enhancement in the corrugated tube with twisted tapes” In _Heat and Mass Transfer_ 52.1, 2016, pp. 141–151 DOI: 10.1007/s00231-015-1550-2
  * [23] S M Sohel Murshed “Simultaneous Measurement of Thermal Conductivity, Thermal Diffusivity, and Specific Heat of Nanofluids” In _Heat Transfer Engineering_ 33.8 Taylor & Francis, 2012, pp. 722–731 DOI: 10.1080/01457632.2011.635986
  * [24] Dong-Wook Oh et al. “Thermal conductivity measurement and sedimentation detection of aluminum oxide nanofluids by using the 3$\omega$ method” In _International Journal of Heat and Fluid Flow_ 29.5, 2008, pp. 1456–1461 DOI: https://doi.org/10.1016/j.ijheatfluidflow.2008.04.007
  * [25] María José Pastoriza-Gallego et al. “Thermal conductivity and viscosity measurements of ethylene glycol-based Al2O3 nanofluids” In _Nanoscale Research Letters_ 6.1, 2011, pp. 221 DOI: 10.1186/1556-276X-6-221
  * [26] Hrishikesh E Patel et al. “An experimental investigation into the thermal conductivity enhancement in oxide and metallic nanofluids” In _Journal of Nanoparticle Research_ 12.3, 2010, pp. 1015–1031 DOI: 10.1007/s11051-009-9658-2
  * [27] Elena V Timofeeva et al. “Thermal conductivity and particle agglomeration in alumina nanofluids: Experiment and theory” In _Physical Review E_ 76.6 American Physical Society, 2007, pp. 61203 DOI: 10.1103/PhysRevE.76.061203
  * [28] Huaqing Xie et al. “Thermal conductivity enhancement of suspensions containing nanosized alumina particles” In _Journal of Applied Physics_ 91.7 American Institute of Physics, 2002, pp. 4568–4572 DOI: 10.1063/1.1454184
  * [29] D Cabaleiro et al. “Thermal conductivity of dry anatase and rutile nano-powders and ethylene and propylene glycol-based TiO2 nanofluids” In _The Journal of Chemical Thermodynamics_ 83, 2015, pp. 67–76 DOI: https://doi.org/10.1016/j.jct.2014.12.001
  * [30] Rohit S Khedkar et al. “Experimental investigations and theoretical determination of thermal conductivity and viscosity of TiO2–ethylene glycol nanofluid” In _International Communications in Heat and Mass Transfer_ 73, 2016, pp. 54–61 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2016.02.004
  * [31] S M S Murshed et al. “Morphology and thermophysical properties of non-aqueous titania nanofluids” In _Heat and Mass Transfer_ 54.9, 2018, pp. 2645–2650 DOI: 10.1007/s00231-018-2308-4
  * [32] Shriram S Sonawane et al. “Effect of sonication time on enhancement of effective thermal conductivity of nano TiO2–water, ethylene glycol, and paraffin oil nanofluids and models comparisons” In _Journal of Experimental Nanoscience_ 10.4 Taylor & Francis, 2015, pp. 310–322 DOI: 10.1080/17458080.2013.832421
  * [33] N R Karthikeyan et al. “Effect of clustering on the thermal conductivity of nanofluids” In _Materials Chemistry and Physics_ 109.1, 2008, pp. 50–55 DOI: https://doi.org/10.1016/j.matchemphys.2007.10.029
  * [34] Sivasankaran Harish et al. “Enhanced thermal conductivity of ethylene glycol with single-walled carbon nanotube inclusions” In _International Journal of Heat and Mass Transfer_ 55.13, 2012, pp. 3885–3890 DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2012.03.001
  * [35] Xiaoke Li et al. “Experimental investigation of $\beta$-cyclodextrin modified carbon nanotubes nanofluids for solar energy systems: Stability, optical properties and thermal conductivity” In _Solar Energy Materials and Solar Cells_ 157, 2016, pp. 572–579 DOI: https://doi.org/10.1016/j.solmat.2016.07.030
  * [36] MinSheng Liu et al. “Enhancements of thermal conductivities with Cu, CuO, and carbon nanotube nanofluids and application of MWNT/water nanofluid on a water chiller system” In _Nanoscale Research Letters_ 6.1, 2011, pp. 297 DOI: 10.1186/1556-276X-6-297
  * [37] Meisam Shamaeil et al. “The effects of temperature and volume fraction on the thermal conductivity of functionalized DWCNTs/ethylene glycol nanofluid” In _Journal of Thermal Analysis and Calorimetry_ 126.3, 2016, pp. 1455–1462 DOI: 10.1007/s10973-016-5548-x
  * [38] Huaqing Xie et al. “Nanofluids containing multiwalled carbon nanotubes and their enhanced thermal conductivities” In _Journal of Applied Physics_ 94.8 American Institute of Physics, 2003, pp. 4967–4971 DOI: 10.1063/1.1613374
  * [39] Benigno Barbés et al. “Thermal conductivity and specific heat capacity measurements of CuO nanofluids” In _Journal of Thermal Analysis and Calorimetry_ 115.2, 2014, pp. 1883–1891 DOI: 10.1007/s10973-013-3518-0
  * [40] José R Vázquez Peñas et al. “Measurement of the thermal conductivity of nanofluids by the multicurrent hot-wire method” In _Journal of Applied Physics_ 104.4 American Institute of Physics, 2008, pp. 44314 DOI: 10.1063/1.2970086
  * [41] Mohan Leena et al. “Evaluation of acoustical parameters and thermal conductivity of TiO2-ethylene glycol nanofluid using ultrasonic velocity measurements” In _Nanotechnology Reviews_ 4.5 De Gruyter, 2015, pp. 449–456 DOI: https://doi.org/10.1515/ntrev-2015-0016
  * [42] Ravi Agarwal et al. “Comparison of Experimental Measurements of Thermal Conductivity of Fe2O3 Nanofluids Against Standard Theoretical Models and Artificial Neural Network Approach” In _Journal of Materials Engineering and Performance_ 28.8, 2019, pp. 4602–4609 DOI: 10.1007/s11665-019-04202-z
  * [43] Suleiman Akilu et al. “Rheology and thermal conductivity of non-porous silica (SiO2) in viscous glycerol and ethylene glycol based nanofluids” In _International Communications in Heat and Mass Transfer_ 88, 2017, pp. 245–253 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2017.08.001
  * [44] Faris Mohammed Ali et al. “The effect of volume fraction concentration on the thermal conductivity and thermal diffusivity of nanofluids: Numerical and experimental” In _Review of Scientific Instruments_ 81.7 American Institute of Physics, 2010, pp. 74901 DOI: 10.1063/1.3458011
  * [45] Ahmad Amiri et al. “The Specific Heat Capacity, Effective Thermal Conductivity, Density, and Viscosity of Coolants Containing Carboxylic Acid Functionalized Multi-Walled Carbon Nanotubes” In _Journal of Dispersion Science and Technology_ 37.7 Taylor & Francis, 2016, pp. 949–955 DOI: 10.1080/01932691.2015.1074588
  * [46] Mohammad Hemmat Esfe et al. “Thermal conductivity modeling of MgO/EG nanofluids using experimental data and artificial neural network” In _Journal of Thermal Analysis and Calorimetry_ 118.1, 2014, pp. 287–294 DOI: 10.1007/s10973-014-4002-1
  * [47] Xin Fang et al. “Effects of inclusion size on thermal conductivity and rheological behavior of ethylene glycol-based suspensions containing silver nanowires with various specific surface areas” In _International Journal of Heat and Mass Transfer_ 81, 2015, pp. 554–562 DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2014.10.043
  * [48] Wenwen Guo et al. “Measurement of the thermal conductivity of SiO2 nanofluids with an optimized transient hot wire method” In _Thermochimica Acta_ 661, 2018, pp. 84–97 DOI: https://doi.org/10.1016/j.tca.2018.01.008
  * [49] Hyun Uk Kang et al. “Estimation of Thermal Conductivity of Nanofluid Using Experimental Effective Particle Volume” In _Experimental Heat Transfer_ 19.3 Taylor & Francis, 2006, pp. 181–191 DOI: 10.1080/08916150600619281
  * [50] A Kazemi-Beydokhti et al. “Experimental Investigation of Parameters Affecting Nanofluid Effective Thermal Conductivity” In _Chemical Engineering Communications_ 201.5 Taylor & Francis, 2014, pp. 593–611 DOI: 10.1080/00986445.2013.782291
  * [51] Nishant Kumar et al. “Experimental study of Fe2O3/water and Fe2O3/ethylene glycol nanofluid heat transfer enhancement in a shell and tube heat exchanger” In _International Communications in Heat and Mass Transfer_ 78, 2016, pp. 277–284 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2016.09.009
  * [52] D Madhesh et al. “Experimental study on the heat transfer and flow properties of Ag–ethylene glycol nanofluid as a coolant” In _Heat and Mass Transfer_ 50.11, 2014, pp. 1597–1607 DOI: 10.1007/s00231-014-1370-9
  * [53] Alejandra Mariano et al. “Thermal conductivity, rheological behaviour and density of non-Newtonian ethylene glycol-based SnO2 nanofluids” In _Fluid Phase Equilibria_ 337, 2013, pp. 119–124 DOI: https://doi.org/10.1016/j.fluid.2012.09.029
  * [54] Alejandra Mariano et al. “Co3O4 ethylene glycol-based nanofluids: Thermal conductivity, viscosity and high pressure density” In _International Journal of Heat and Mass Transfer_ 85, 2015, pp. 54–60 DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2015.01.061
  * [55] Monisha Michael et al. “Thermo-physical properties of pure ethylene glycol and water–ethylene glycol mixture-based boron nitride nanofluids” In _Journal of Thermal Analysis and Calorimetry_ 137.2, 2019, pp. 369–380 DOI: 10.1007/s10973-018-7965-5
  * [56] Majid Moosavi et al. “Fabrication, characterization, and measurement of some physicochemical properties of ZnO nanofluids” In _International Journal of Heat and Fluid Flow_ 31.4, 2010, pp. 599–605 DOI: https://doi.org/10.1016/j.ijheatfluidflow.2010.01.011
  * [57] Nader Nikkam et al. “The effect of particle size and base liquid on thermo-physical properties of ethylene and diethylene glycol based copper micro- and nanofluids” In _International Communications in Heat and Mass Transfer_ 86, 2017, pp. 143–149 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2017.05.026
  * [58] M J Pastoriza-Gallego et al. “Thermophysical profile of ethylene glycol-based ZnO nanofluids” In _The Journal of Chemical Thermodynamics_ 73, 2014, pp. 23–30 DOI: https://doi.org/10.1016/j.jct.2013.07.002
  * [59] B Jacob Rubasingh et al. “Predicting thermal conductivity behaviour of ZnO, TiO2 and ball milled TiO2/ZnO based nanofluids with ethylene glycol as base fluid” In _Materials Research Express_ 6.9 IOP Publishing, 2019, pp. 95702 DOI: 10.1088/2053-1591/ab2bc5
  * [60] C Selvam et al. “Thermal conductivity enhancement of ethylene glycol and water with graphene nanoplatelets” In _Thermochimica Acta_ 642, 2016, pp. 32–38 DOI: https://doi.org/10.1016/j.tca.2016.09.002
  * [61] Merve Seyhan et al. “The effect of functionalized silver nanoparticles over the thermal conductivity of base fluids” In _AIP Advances_ 7.4 American Institute of Physics, 2017, pp. 45101 DOI: 10.1063/1.4979554
  * [62] Huaqing Xie et al. “Thermal Conductivity of Suspensions Containing Nanosized SiC Particles” In _International Journal of Thermophysics_ 23.2, 2002, pp. 571–580 DOI: 10.1023/A:1015121805842
  * [63] Huaqing Xie et al. “MgO nanofluids: higher thermal conductivity and lower viscosity among ethylene glycol-based nanofluids containing oxide nanoparticles” In _Journal of Experimental Nanoscience_ 5.5 Taylor & Francis, 2010, pp. 463–472 DOI: 10.1080/17458081003628949
  * [64] K.. Yashawantha et al. “Experimental investigation on physical and thermal properties of graphite nanofluids” In _AIP Conference Proceedings_ 2039.1 American Institute of Physics, 2018, pp. 20057 DOI: 10.1063/1.5079016
  * [65] Wei Yu et al. “Experimental investigation on the thermal transport properties of ethylene glycol based nanofluids containing low volume concentration diamond nanoparticles” In _Colloids and Surfaces A: Physicochemical and Engineering Aspects_ 380.1, 2011, pp. 1–5 DOI: https://doi.org/10.1016/j.colsurfa.2010.11.020
  * [66] Wei Yu et al. “Experimental investigation on thermal conductivity and viscosity of aluminum nitride nanofluid” In _Particuology_ 9.2, 2011, pp. 187–191 DOI: https://doi.org/10.1016/j.partic.2010.05.014
  * [67] Gaweł Żyła et al. “Experimental studies on viscosity, thermal and electrical conductivity of aluminum nitride–ethylene glycol (AlN–EG) nanofluids” In _Thermochimica Acta_ 637, 2016, pp. 11–16 DOI: https://doi.org/10.1016/j.tca.2016.05.006
  * [68] Gaweł Żyła et al. “Huge thermal conductivity enhancement in boron nitride – ethylene glycol nanofluids” In _Materials Chemistry and Physics_ 180, 2016, pp. 250–255 DOI: https://doi.org/10.1016/j.matchemphys.2016.06.003
  * [69] Gaweł Żyła “Viscosity and thermal conductivity of MgO–EG nanofluids” In _Journal of Thermal Analysis and Calorimetry_ 129.1, 2017, pp. 171–180 DOI: 10.1007/s10973-017-6130-x
  * [70] Gaweł Żyła et al. “Viscosity, thermal and electrical conductivity of silicon dioxide–ethylene glycol transparent nanofluids: An experimental studies” In _Thermochimica Acta_ 650, 2017, pp. 106–113 DOI: https://doi.org/10.1016/j.tca.2017.02.001
  * [71] Julia Tielke et al. “Statistical analysis of thermal conductivity experimentally measured in water-based nanofluids” In _Proceedings of the Royal Society A_ 477, 2021, pp. 20210222
  * [72] Robert D Chirico et al. “Improvement of Quality in Publication of Experimental Thermophysical Property Data: Challenges, Assessment Tools, Global Implementation, and Online Support” In _Journal of Chemical & Engineering Data_ 58.10 American Chemical Society, 2013, pp. 2699–2716 DOI: 10.1021/je400569s
  * [73] Ravi Prasher et al. “Effect of aggregation on thermal conduction in colloidal nanofluids” In _Applied Physics Letters_ 89.14, 2006, pp. 143119 DOI: 10.1063/1.2360229
  * [74] Sunita Jadhav et al. “Copper Oxide Nanoparticles: Synthesis, Characterization and Their Antibacterial Activity” In _Journal of Cluster Science_ 22.2, 2011, pp. 121–129 DOI: 10.1007/s10876-011-0349-7
  * [75] Sedong Kim et al. “Experimental investigation of dispersion characteristics and thermal conductivity of various surfactants on carbon based nanomaterial” In _International Communications in Heat and Mass Transfer_ 91, 2018, pp. 95–102 DOI: https://doi.org/10.1016/j.icheatmasstransfer.2017.12.011
  * [76] Ali Eatemadi et al. “Carbon nanotubes: properties, synthesis, purification, and medical applications” In _Nanoscale Research Letters_ 9.1, 2014, pp. 393 DOI: 10.1186/1556-276X-9-393