Antineutrino Opacity in Neutron Stars in Models Constrained by Recent Terrestrial Experiments and Astrophysical Observations

Abstract

Inthe present paper, we investigate neutral current (NC) antineutrino scattering with the constituents of neutron star (NS) matter at zero temperature. The modeling of standard matter in NS is constructed within the framework of both extended relativistic mean-field (E-RMF) and nonrelativistic Korea-IBS-Daegu-SKKU energy density functional (KIDS-EDF) models. In the E-RMF model, we use a new parameter, G3(M), which was constrained by the recent PREX II experiment measurement of neutron distribution in ²⁰⁸Pb, while the KIDS-EDF models are constrained by terrestrial experiments, gravitational-wave signals, and astrophysical observations. Using both realistic and well-constrained matter models, we then calculate the antineutrino differential cross-section (ADCS) and antineutrino mean free path (AMFP) for the interaction between antineutrinos and neutron star (NS) matter constituents using linear response theory. It is found that the AMFP for the KIDS0 and KIDSA models are smaller compared to the SLy4 model and the E-RMF model with the G3(M) parameter. The AMFP result of the Skyrme model with the SLy4 parameter set is found to have a prediction almost similar to that of the E-RMF model with the G3(M) parameter. Contributions of each nucleon to the total AMFP are also presented for the G3(M) model.

1. Introduction

Neutrino emissions are also important to multi-messenger astronomy, providing fundamental information about the properties of stellar objects and compact stars. This complements gravitational waves, first detected by the Laser Interferometer Gravitational-Wave Observatory LIGO/Virgo Collaboration, which observed the waveform from the merger of binary neutron stars (NSs) [1], cosmic rays, and electromagnetic radiation. Even though neutrinos weakly interact with matter, the giant detector has successfully detected a few astrophysical neutrinos. For example, a burst of neutrinos from Supernova SN1987A was detected by Super-Kamiokande II, marking the first direct evidence of neutrino astronomy [2,3]. Since then, neutrino emissions have been detected from high-energy astrophysical flux objects. A notable observation of high-energy astrophysical neutrinos was recently reported by IceCube [4]. However, the origins of neutrino emissions are not yet fully understood, and further improvements in detectors and analyses are needed to address this question.

In supernova explosions, many neutrinos are produced, where the white dwarfs from a pre-supernova star collapse into a proto-neutron star (PNS). During this explosion process, the neutrinos are expected to diffuse from the neutron-rich matter (NM) of the PNS. During diffusion, a small percentage of neutrinos interact with the matter outside the PNS. These interactions can occur through neutral-current scattering or charged-current absorption. Instead of neutrinos, antineutrinos can also be produced during the supernova explosion. It was found that the DCS of the antineutrinos is considerably smaller than that of the neutrinos [5]. This implies that the mean free path of antineutrinos is considerably larger than that for neutrinos. This occurs if the weak magnetism of the nucleon is considered in the calculation. This finding was also supported by Hutauruk et al., in Ref. [6], where the weak magnetism and neutrino form factors are included in the calculation. Motivated by these studies, this work focuses on antineutrino’s neutral-current scattering using a well-constrained matter model based on terrestrial experiments and astrophysical observation data.

To realistically model NS matter, it is important to ensure the matter model’s stability. The model must reproduce a binding energy per nucleon $E_B/A=$ 15 MeV at a saturation density of $n_0=$ 0.16 fm⁻³. In the E-RMF model, coupling constants are dictated by fitting the binding energy per nucleon at the saturation density and reproducing the neutron skin thickness measured in the PREX II experiment [7]. Details of the fitting strategy used to extract the coupling constants in the E-RMF model are presented in Refs. [8,9]. The fitting strategy in the KIDS-EDF model is rather different from that in the E-RMF model. In the EDF model, the unknown parameters are matched with the conventional Skyrme force parameters, where the Skyrme force parameters are extracted by reproducing the skin thickness or charge radius measured in the terrestrial experiments, as well as reproducing the NS radius or mass taken from astrophysical observations, such as LIGO/VIRGO [1,10] and the Neutron Star Interior Composition Explorer (NICER) [11,12]. Details about the fitting strategy of the KIDS-EDF model can be found in Ref. [13]. It is important to note that a well-constrained EoS and antineutrino/neutrino mean free path ((A)NMFP) are essential for supernova simulations [14,15].

In this paper, we compute the DCSs of antineutrinos and AMFP in terms of neutral current scattering in the frameworks of the E-RMF and KIDS-EDF models [8,9,13]. Prior to calculating the DCS and AMFP in the scattering, we analyze the matter model used as input to calculate the DCS and AMFP. To do this, we evaluate the EoS for the E-RMF and KIDS-EDF models, where the coupling constants of the models are constrained by the terrestrial laboratory experiments and the astrophysical observations, to guarantee the stability of the matter. Also, the EoS must reproduce the binding energy per nucleon at saturation density. Thus, we performed calculations of the ADCS and AMFP for both E-RMF and KIDS-EDF models. The prediction for the AMFP in the KIDS0 model shows that it decreases as the $n_B/n_0$ increases. This result is also observed in the KIDSA and SLy4 models. However, starting at around $n_B/n_0=$ 3.5, the AMFP of the KIDS0 model is lower than that of the KIDSA and SLy4 models. This can be due to the KIDS0 model having a larger effective nucleon mass at that baryon number density. Similarly, it is also shown that the AMFP for the E-RMF model with the G3(M) parameter decreases as baryon density increases. In addition, we also found that the AMFP of the E-RMF model with the G3(M) parameter slowly decreases in comparison with those of the KIDS0, KIDSA, and SLy4 models.

The present paper is organized as follows. In Section 2, we briefly introduce the theoretical framework of the model of the nuclear matter constructed in the E-RMF and KIDS-EDF models. The results for the EoS, effective nucleon mass, and particle fraction for both models are also presented. In Section 3, we describe the neutrino-NS constituent scattering, constructed from the effective Lagrangian. The final expressions for the antineutrino and neutrino DCS and MFP are presented. Section 4 discusses the DCS of antineutrinos and the AMFP for both E-RMF and KIDS-EDF models. A summary and conclusion are presented in Section 5.

2. Theoretical Framework

In this section, we present the EoS of the nuclear matter model that describes the constituent matter of NS. The E-RMF model [16,17], inspired by the effective field theory, with the modified G3(M) parameter, is employed to describe NS matter. This model has successfully been applied to various phenomena in nuclear physics [18,19]. Apart from the E-RMF model, nonrelativistic EDF models have also been used to describe NS matter, such as the KIDS-EDF model, which was initially developed for homogeneous NM [13]. The KIDS-EDF model is structured around the expansion of Fermi momentum. In fact, the E-RMF and KIDS-EDF models are based on fundamentally different principles. Therefore, both models have been chosen for use in the present work.

2.1. Modeling of Nuclear Matter

Here, the description of NS matter is modeled using both E-RMF and KIDS-EDF models. Again, we emphasize that both models reproduce binding energy $E_B/A=$ 15 MeV at the saturation density $n_0=$ 0.16 fm⁻³ to ensure matter stability. Detailed descriptions of both models are presented in Section 2.1.1 and Section 2.1.2.

2.1.1. E-RMF Model

In the E-RMF model, commonly, the interactions among nucleons and mesons, self-interacting mesons, and meson–meson crossing interactions can be constructed within the framework of the effective Lagrangian. The expression for the effective Lagrangian in the E-RMF model is compactly presented as follows [16,17]:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{ERMF}}& =& {\mathcal{L}}_{\mathrm{NM}}+{\mathcal{L}}_{\sigma }+{\mathcal{L}}_{\omega }+{\mathcal{L}}_{\rho }+{\mathcal{L}}_{\delta }+{\mathcal{L}}_{\sigma \omega \rho },\hfill \end{array}$$

(1)

where the first term of the effective interaction Lagrangian for nucleons and mesons is given by [16,17], as follows:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{NM}}& =& \sum _{j=n,p}{\overline{\psi }}_j\left[i{\gamma }^{\mu }{\partial }_{\mu }-(M-g_{\sigma }\sigma -g_{\delta }{\tau }_j·\delta )\right.-\left(g_{\omega }{\gamma }^{\mu }{\omega }_{\mu }+{\displaystyle \frac{1}{2}}{g}_{\rho }{\gamma }^{\mu }{\tau }_j·{\rho }_{\mu }\right){\psi }_j,\hfill \end{array}$$

(2)

where M denotes the nucleon mass in free space (vacuum), and ${\tau }_j$ denotes isospin matrices. The total effective Lagrangian for interactions between protons and neutrons is expressed by a sum, symbolized mathematically. The coupling constants of $g_{\sigma }$, $g_{\omega }$, $g_{\rho }$, and $g_{\delta }$ denote the $\sigma$, $\omega$, $\rho$, and $\delta$ mesons, respectively. The second, third, fourth, and fifth terms of the Lagrangian for self-interactions for the $\sigma$, $\omega$, $\rho$, and $\delta$ mesons are, respectively, defined by [16,17], as follows:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\sigma }& =& {\displaystyle \frac{1}{2}}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -m_{\sigma }^2{\sigma }^2\right)-{\displaystyle \frac{{\kappa }_3}{6M}}{g}_{\sigma }{m}_{\sigma }^2{\sigma }^3-{\displaystyle \frac{{\kappa }_4}{24M^2}}{g}_{\sigma }^2{m}_{\sigma }^2{\sigma }^4,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\omega }& =& -{\displaystyle \frac{1}{4}}{\omega }_{\mu \nu }{\omega }^{\mu \nu }+{\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }+{\displaystyle \frac{1}{24}}{\xi }_0{g}_{\omega }^2{\left({\omega }_{\mu }{\omega }^{\mu }\right)}^2,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\rho }& =& -{\displaystyle \frac{1}{4}}{\rho }_{\mu \nu }·{\rho }^{\mu \nu }+{\displaystyle \frac{1}{2}}{m}_{\rho }^2{\rho }_{\mu }·{\rho }^{\mu },\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\delta }& =& {\displaystyle \frac{1}{2}}{\partial }_{\mu }\delta ·{\partial }^{\mu }\delta -{\displaystyle \frac{1}{2}}{m}_{\delta }^2{\delta }^2.\hfill \end{array}$$

(6)

The symbols of ${\kappa }_3$, ${\kappa }_4$, and ${\xi }_0$ are the self-interaction coupling constants. The symbols of $m_{\sigma }$, $m_{\omega }$, $m_{\rho }$, and $m_{\delta }$ are the masses of the corresponding mesons. The field tensors of ${\omega }^{\mu \nu }$ and ${\rho }^{\mu \nu }$ are, respectively, for the $\omega$ and $\rho$ mesons, which can be defined as ${\omega }^{\mu \nu }={\partial }^{\mu }{\omega }^{\nu }-{\partial }^{\nu }{\omega }^{\mu }$, and ${\rho }^{\mu \nu }={\partial }^{\mu }{\rho }^{\nu }-{\partial }^{\nu }{\rho }^{\mu }-g_{\rho }({\rho }^{\mu }\times {\rho }^{\nu })$. The sixth term of the nonlinear crossing interaction Lagrangian for $\sigma$, $\omega$, and $\rho$ mesons is expressed by [16,17], as follows:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\sigma \omega \rho }& =& {\displaystyle \frac{{\eta }_1}{2M}}{g}_{\sigma }{m}_{\omega }^2\sigma {\omega }_{\mu }{\omega }^{\mu }+{\displaystyle \frac{{\eta }_2}{4M^2}}{g}_{\sigma }^2{m}_{\omega }^2{\sigma }^2{\omega }_{\mu }{\omega }^{\mu }+{\displaystyle \frac{{\eta }_{\rho }}{2M}}{g}_{\sigma }{m}_{\rho }^2\sigma {\rho }_{\mu }·{\rho }^{\mu }\hfill \\ & +& {\displaystyle \frac{{\eta }_{1\rho }}{4M^2}}{g}_{\sigma }^2{m}_{\rho }^2{\sigma }^2{\rho }_{\mu }·{\rho }^{\mu }+{\displaystyle \frac{{\eta }_{2\rho }}{4M^2}}{g}_{\omega }^2{m}_{\rho }^2{\omega }_{\mu }{\omega }^{\mu }{\rho }_{\mu }·{\rho }^{\mu },\hfill \end{array}$$

(7)

where ${\eta }_1$, ${\eta }_2$, ${\eta }_{\rho }$, ${\eta }_{1\rho }$, and ${\eta }_{2\rho }$ are the coupling constants for the crossed interactions. The values of all coupling constants in the effective Lagrangian of the E-RMF model in Equations (2)–(7) are tabulated in

Table 1. The complete parameters of G3(M) [9] constrained by the PREX II data [7]. The nucleon mass in free space, M, is 939 MeV, and the coupling constants are dimensionless. The $k_3$ constant is in units of fm⁻¹, while $f_{\omega }/4=$ 0.220, $f_{\rho }/4=$ 1.239, and ${\beta }_{\sigma }=$ −0.0087.

$m_s/M$	$m_{\omega }/M$	$m_{\rho }/M$	$m_{\delta }/M$	${\beta }_{\omega }$	$g_s/4\pi$	$g_{\omega }/4\pi$	$g_{\rho }/4\pi$	$g_{\delta }/4\pi$
0.559	0.832	0.820	1.043	0.782	0.923	0.872	0.160	2.606
$k_3$	$k_4$	${\beta }_{\omega }$	${\xi }_0$	${\eta }_1$	${\eta }_2$	${\eta }_{\rho }$	${\eta }_{1\rho }$	${\eta }_{2\rho }$	${\alpha }_1$	${\alpha }_2$
1.694	−0.484	1.010	0.424	0.114	0.645	0.000	18.257	2.0000	−1.468	0.220

.

Although we do not explicitly include the electron and muon contributions in the ADCS and AMFP calculations in this work, for the sake of completeness of the Lagrangian model, we present the Lagrangian density for the electron and muon. The expression for the Lagrangian density of the electron and muon is given by the following:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_l& =& \sum _{l=e,\mu }{\overline{\psi }}_l(i{\gamma }_{\mu }{\partial }^{\mu }-m_l){\psi }_l,\hfill \end{array}$$

(8)

with $m_l$ denoting the lepton mass in free space. It is worth noting that the electron and muon masses do not change in a nuclear medium. Using the given Lagrangian, the particle populations or fractions of the NS standard matter with n, p, e, and $\mu$ can be calculated via the $\beta$-equilibrium constraint, which shows the relation of the chemical potentials as follows:

$$\begin{array}{ccc}\hfill {\mu }_n-{\mu }_p& =& {\mu }_e,{\mu }_e={\mu }_{\mu },\hfill \end{array}$$

(9)

and the charge neutrality must be satisfied, which is given by the following:

$$\begin{array}{ccc}\hfill n_p& =& n_e+n_{\mu },\hfill \end{array}$$

(10)

where, in this case, the leptons are assumed as relativistic ideal Fermi gases. The chemical potential for protons and neutrons can be obtained through the equation ${\mu }_{p,n}={\displaystyle \frac{\partial \mathcal{E}}{{\rho }_{p,n}}}$, and the $n_p$, $n_e$, and $n_{\mu }$ stand for the proton, electron, and muon densities, respectively. Because the lepton masses do not change with respect to the baryon number density. Then, the chemical potentials for lepton can be calculated by ${\mu }_{l=e,\mu }=\sqrt{k_{F_{l=e,\mu }}^2+m_{l=e,\mu }^2}$. Note that the total baryon number density is defined by $n_B=n_p+n_n$ where $n_n$ is the neutron number density.

As explained earlier, the G3(M) parameter was well-constrained using data from the terrestrial PREX II experiment. The difference between G3(M) and other parameter sets of the E-RMF model is that the G3(M) parameter has nonzero ${\eta }_1$, ${\eta }_2$, ${\eta }_{\rho }$, ${\alpha }_1$, ${\alpha }_2$, $f_{\omega }$, $f_{\rho }$, ${\beta }_{\sigma }$, and ${\beta }_{\omega }$ coupling constants, while other E-RMF parameter sets, such as the well-known TM1e and FSU Garnet parameters [9], set these coupling constants to zero. The complete G3(M) parameters are depicted in Table 1.

Apart from the differences in those coupling constants, the value of the $g_{\delta }$ coupling constant is also different. This coupling constant contributes to stiffening the equation of state (EoS) at high densities and affects the nuclear symmetry energy at sub-saturation densities. It is with noting that nonlinear crossing coupling constants play a crucial role in obtaining a better EoS for PNM. This will affect the particle populations/fractions, ADCS, and AMFP. Note that the nonlinear crossing coupling constant ${\eta }_{2\rho }$ and the coupling constant $g_{\rho }$ in the G3(M) parameter are determined by fine-tuning ${\eta }_{2\rho }$ and $g_{\rho }$ parameters to reproduce the neutron skin thickness of ²⁰⁸Pb from the PREX II data, marking a crucial difference between the G3(M) parameter and other E-RMF parameters.

Employing the effective Lagrangian in Equation (1), the energy density $\mathcal{E}$ and pressure P for NM can be evaluated using the energy-momentum tensor, as follows: $T_{\mu \nu }={\sum }_k{\partial }_{\nu }{\varphi }_k{\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }^{\mu }{\varphi }_k\right)}}-g_{\mu \nu }\mathcal{L}$ where ${\varphi }_k$ denotes each corresponding field in the effective Lagrangian of the E-RMF model in Equation (1). Straightforwardly, the energy density is obtained by calculating the zeroth component of the energy-momentum tensor, $\langle T_{00}\rangle$, which gives $\mathcal{E}=\langle T_{00}\rangle$, and the pressure is obtained from the third component of the energy-momentum tensor, $\langle T_{kk}\rangle$, which gives $P={\sum }_k{\displaystyle \frac{1}{3}}\langle T_{kk}\rangle$. The final expressions for the energy density and pressure expressions for PNM are, respectively, given by [16,17], as follows:

$$\begin{array}{ccc}\hfill \mathcal{E}& =& \sum _{i=n,p}{\displaystyle \frac{2}{{\left(2\pi \right)}^3}}{\int }_0^{k_F^i}{d}^{3}k\sqrt{{\overrightarrow{k}}_i^2+M_i^{*2}}+n_B{g}_{\omega }\omega -{\displaystyle \frac{1}{24}}{\xi }_0{g}_{\omega }^2{\omega }^4\hfill \\ & +& {\displaystyle \frac{1}{2}}{m}_{\sigma }^2{\sigma }^2\left(1+{\displaystyle \frac{{\kappa }_3}{3M}}{g}_{\sigma }\sigma +{\displaystyle \frac{{\kappa }_4}{12M^2}}{g}_{\sigma }^2{\sigma }^2\right)\hfill \\ & -& {\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }^2\left(1+{\displaystyle \frac{{\eta }_1}{M}}{g}_{\sigma }\sigma +{\displaystyle \frac{{\eta }_2}{2M^2}}{g}_{\sigma }^2{\sigma }^2\right)\hfill \\ & -& {\displaystyle \frac{1}{2}}{m}_{\rho }^2{\rho }^2\left(1+{\displaystyle \frac{{\eta }_{\rho }}{M}}{g}_{\sigma }\sigma +{\displaystyle \frac{{\eta }_{1\rho }}{2M^2}}{g}_{\sigma }^2{\sigma }^2\right)-{\displaystyle \frac{{\eta }_{2\rho }}{2}}{g}_{\rho }^2{g}_{\omega }^2{\rho }^2{\omega }^2\hfill \\ & +& {\displaystyle \frac{1}{2}}{\rho }_3{g}_{\rho }\rho +{\displaystyle \frac{1}{2}}{m}_{\delta }^2{\delta }^2,\hfill \end{array}$$

(11)

and

$$\begin{array}{ccc}\hfill P& =& \sum _{i=n,p}{\displaystyle \frac{1}{3}}{\displaystyle \frac{2}{{\left(2\pi \right)}^3}}{\int }_0^{k_F^i}{d}^{3}k{\displaystyle \frac{{\overrightarrow{k}}_i^2}{\sqrt{{\overrightarrow{k}}_i^2+M_i^2}}}+{\displaystyle \frac{1}{24}}{\xi }_0{g}_{\omega }^2{\omega }^4\hfill \\ & -& {\displaystyle \frac{1}{2}}{m}_{\sigma }^2{\sigma }^2\left(1+{\displaystyle \frac{{\kappa }_3}{3M}}{g}_{\sigma }\sigma +{\displaystyle \frac{{\kappa }_4}{12M^2}}{g}_{\sigma }^2{\sigma }^2\right)\hfill \\ & +& {\displaystyle \frac{1}{2}}{m}_{\omega }^2{\omega }^2\left(1+{\displaystyle \frac{{\eta }_1}{M}}{g}_{\sigma }\sigma +{\displaystyle \frac{{\eta }_2}{2M^2}}{g}_{\sigma }^2{\sigma }^2\right)\hfill \\ & +& {\displaystyle \frac{1}{2}}{m}_{\rho }^2{\rho }^2\left(1+{\displaystyle \frac{{\eta }_{\rho }}{M}}{g}_{\sigma }\sigma +{\displaystyle \frac{{\eta }_{1\rho }}{2M^2}}{g}_{\sigma }^2{\sigma }^2\right)+{\displaystyle \frac{{\eta }_{2\rho }}{2}}{g}_{\rho }^2{g}_{\omega }^2{\rho }^2{\omega }^2\hfill \\ & -& {\displaystyle \frac{1}{2}}{m}_{\delta }^2{\delta }^2,\hfill \end{array}$$

(12)

where the baryon Fermi momentum $k_F$ can be defined from the baryon number density $n_B={\displaystyle \frac{k_F^3}{3{\pi }^2}}$, and the isospin baryon density can be defined by $n_3=n_p-n_n$. Numerical results of the E-RMF model with the G3(M) parameter for pressure, the binding energy per nucleon, the effective nucleon mass, and the particle fractions are shown in Figure 1.

Figure 1a shows the pressure results for the G3(M) parameter compared to those from chiral perturbation theory (ChPT) [20] and heavy ion collision (HIC) [21]. The results for the nucleon’s pressure and binding energy are consistent with other calculations [20,21]. The nucleon effective mass and particle population fractions for the G3(M) parameter are depicted in Figure 1c,d, respectively.

2.1.2. KIDS-EDF Model

In this section, we describe the KIDS-EDF model [13,22]. The KIDS-EDF model has widely been applied to homogeneous NM [13] and finite nuclei. The EDF-KIDS model in NM was transformed into the Skyrme functional form in finite nuclei, where the specific values of the effective masses could be emulated by adjusting the functional parameters. The energy of a nucleon for the KIDS-EDF model was described in the homogeneous NM as an expansion of the power of the Fermi momentum $k_F$ at zero temperature. The expression of the KIDS-EDF model energy per nucleon at zero temperature is explicitly given by the following [13,22]:

$$\mathcal{E}(n_B,\delta )=\mathcal{T}(n_B,\delta )+\sum _{j=0}^3{c}_j\left(\delta \right)n_B^{[1+a_j]},$$

(13)

where $n_B=n_n+n_p$ is the baryon number density with $n_n$ and $n_p$ denoting the neutron and proton densities, respectively. The quantities $a_j=j/3$ and $\delta =(n_n-n_p)/n_B$ represent the isospin asymmetry parameter. $\delta =0$ denotes symmetric NM and $\delta =1$ denotes pure NM. The kinetic energy of the first term in Equation (13) can be defined by the following [13,22]:

$$\mathcal{T}(n_B,\delta )={\displaystyle \frac{3}{5}}{\left(3{\pi }^2{n}_B\right)}^{{\displaystyle \frac{2}{3}}}\left[{\displaystyle \frac{h}{4\pi M_p}}{\left({\displaystyle \frac{1-\delta }{2}}\right)}^{{\displaystyle \frac{5}{3}}}+{\displaystyle \frac{h}{4\pi M_n}}{\left({\displaystyle \frac{1+\delta }{2}}\right)}^{{\displaystyle \frac{5}{3}}}\right],$$

(14)

where $M_n$ and $M_p$ denote the neutron and proton masses in free space, respectively. It is worth noting that the variable $c_j\left(\delta \right)$ in the potential term of Equation (13) can be defined by $c_j\left(\delta \right)={\alpha }_j+{\delta }^2{\beta }_j$. The parameter ${\alpha }_j$ can be fixed using the symmetric NM properties and ${\beta }_j$ can be fixed using the EoS of the asymmetric NM.

When determining the KIDS-EDF parameters based on the Skyrme force parameters, it is necessary to establish a correspondence between them. Before explaining the parameter matching procedure, we briefly describe the conventional Skyrme force interaction, as follows [23]:

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{i,j}(\mathbf{k},{\mathbf{k}}^{\prime })& =& t_0(1+x_0{P}_{\sigma })\delta ({\mathbf{r}}_i-{\mathbf{r}}_j)+{\displaystyle \frac{1}{2}}{t}_1(1+x_1{P}_{\sigma })\left[\delta ({\mathbf{r}}_i-{\mathbf{r}}_j){\mathbf{k}}^2+{{\mathbf{k}}^{\prime }}^2\delta ({\mathbf{r}}_i-{\mathbf{r}}_j)\right]\hfill \\ & +& t_2(1+x_2{P}_{\sigma }){\mathbf{k}}^{\prime }·\delta ({\mathbf{r}}_i-{\mathbf{r}}_j)\mathbf{k}+{\displaystyle \frac{1}{6}}{t}_3(1+x_3{P}_{\sigma }){\rho }^{\alpha }\delta ({\mathbf{r}}_i-{\mathbf{r}}_j)\hfill \\ & +& i{W}_0{\mathbf{k}}^{\prime }\times \delta ({\mathbf{r}}_i-{\mathbf{r}}_j)\mathbf{k}·({\sigma }_i-{\sigma }_j),\hfill \end{array}$$

(15)

where the spin exchange operator is denoted by $P_{\sigma }=\left(1+{\sigma }_1·{\sigma }_2\right)/2$ with $\sigma$ denoting the Pauli spin matrices. The relative momenta operating in the initial and final states are given by the following: $\mathbf{k}=({\nabla }_i-{\nabla }_j)/2i$ and ${\mathbf{k}}^{\prime }=({\nabla }_i^{\prime }-{\nabla }_j^{\prime })/2i$, respectively. It is worth noting that the strength of the spin–orbit coupling $W_0$ is not considered in the EDF for homogeneous NM in the Skyrme force. The energy density for infinite NM in Equation (13) can be decomposed in terms of the Skyrme parameters as follows [23]:

$$\begin{array}{ccc}\hfill \mathcal{E}(n_B,\delta )& =& \mathcal{T}(n_B,\delta )+{\displaystyle \frac{3}{8}}{t}_0{n}_B-{\displaystyle \frac{1}{8}}(2y_0+t_0)n_B{\delta }^2+{\displaystyle \frac{1}{16}}{t}_3{n}_B^{(\alpha +1)}\hfill \\ & -& {\displaystyle \frac{1}{48}}(2y_3+t_3)n_B^{(\alpha +1)}{\delta }^2+{\displaystyle \frac{1}{16}}(3t_1+5t_2+4y_2)\tau \hfill \\ & -& {\displaystyle \frac{1}{16}}\left[(2y_1+t_1)-(2y_2+t_2)\right]\tau {\delta }^2,\hfill \end{array}$$

(16)

where the definition of variable $y_i\equiv t_i{x}_i$ with $i=1,2,3$. It can be easily seen that matching Equations (13) and (16) can be used to determine the relations between KIDS parameters $c_j\left(\delta \right)$ in terms of the Skyrme coefficients ($t_i,y_i)$, and one obtains the following [23]:

$$\begin{array}{ccc}\hfill c_0\left(\delta \right)& =& {\displaystyle \frac{3}{8}}{t}_0-{\displaystyle \frac{1}{8}}(2y_0+t_0){\delta }^2,\hfill \\ \hfill c_1\left(\delta \right)& =& {\displaystyle \frac{1}{16}}{t}_{31}-{\displaystyle \frac{1}{48}}(2y_{31}+t_{31}){\delta }^2,\hfill \\ \hfill c_2\left(\delta \right)& =& {\displaystyle \frac{1}{16}}{t}_{32}-{\displaystyle \frac{1}{48}}(2y_{32}+t_{32}){\delta }^2\hfill \\ & +& {\displaystyle \frac{3}{5}}{\left({\displaystyle \frac{6{\pi }^2}{\nu }}\right)}^{{\displaystyle \frac{2}{3}}}{\displaystyle \frac{1}{16}}\left\{(3t_1+5t_2+4y_2)-\left[(2y_1+t_1)-(2y_2+t_2)\right]{\delta }^2\right\},\hfill \\ \hfill c_3\left(\delta \right)& =& {\displaystyle \frac{1}{16}}{t}_{33}-{\displaystyle \frac{1}{48}}(2y_{33}+t_{33}){\delta }^2,\hfill \end{array}$$

(17)

where the variable $\alpha =1/3$ is assigned to $t_{31}$ and $y_{31}$; $\alpha =2/3$ is assigned to $t_{32}$ and $y_{32}$; and $\alpha =1$ is assigned to $t_{33}$ and $y_{33}$. The spin and isospin degeneracy factor with $\nu =4$ is used for symmetric NM and $\nu =2$ is the spin and isospin degeneracy for the asymmetric NM. It is worth noting that the KIDS-EDF model commonly has 13 parameters in total to be determined in the Skyrme force. In the present work, we consider the various KIDS-EDF models: KIDS0, KIDSA, and SLy4 models.

Here, we specifically describe how to determine the parameters in the KIDS0 model. The parameters of ${\alpha }_0$, ${\alpha }_1$, and ${\alpha }_2$ have been determined to adjust to three symmetric NM data points (at saturation density ${\rho }_0=0.16$ fm⁻³, binding energy $E_{\mathrm{B}}=16$ MeV, and compressibility $K_0=240$ MeV). Four parameters of ${\beta }_i$’s are fitted to a PNM EoS. We then obtain the isoscalar effective mass $m_s^{*}\simeq 1.0$ M and the isovector one $m_v^{*}\simeq 0.8$ M as a result of the parameter fitting for the KIDS0 model.

The nuclear symmetry energy for the KIDSA model is determined to reproduce the recent constraints from NS observations. The nuclear symmetry energy parameters are consistent with the NS data within the uncertainty of the error bars. Note that the values of the effective masses in the KIDSA model are obtained in a manner similar to the isoscalar and isovector effective masses for the KIDS0 model. The difference between the KIDS0 and KIDSA models is clearly evident in their nuclear symmetry energy. However, the KIDS0 and SLy4 models yield similar results for the nuclear symmetry energy. Their difference is notably indicated by the effective nucleon masses, as shown in Figure 2c,d.

Based on the standard Skyrme force matching with the KIDS-EDF model parameters [23], the effective nucleon masses in the asymmetric NM can be defined in terms of the Skyrme parameters, as follows [13,22]:

$$\begin{array}{c}\hfill M_i^{*}=M_i{\left[1+{\displaystyle \frac{M_i}{8{\hslash }^2}}\rho {\mathsf{\Theta }}_s-{\displaystyle \frac{M_i}{8{\hslash }^2}}{\tau }_3^i\left(2{\mathsf{\Theta }}_v-{\mathsf{\Theta }}_s\right)\rho \delta \right]}^{-1},\end{array}$$

(18)

where parameter $M_i$ denotes the proton and neutron masses in free space ($i=n,p$). The parameters of the Skyrme force can be, respectively, defined by ${\mathsf{\Theta }}_s=3t_1+(5t_2+4y_2)$ and ${\mathsf{\Theta }}_v=(2t_1+y_1)+(2t_2+y_2)$. The values of $t_1$, $t_2$, $y_1$, and $y_2$ are presented in

Table 2. The complete parameters of the Skyrme force for the KIDS0 and KIDSA models [24]. The units of $t_0\mathrm{and}{y}_0$ are in MeV fm³, the units of $t_{31}\mathrm{and}{y}_{31}$ are in MeV fm⁴, and $W_0$ are in MeV fm⁵, and the unit of $y_{33}$ is in MeV fm⁶. $K_0$, J, and L are in units of MeV.

Parameters	KIDS0	KIDSA
$t_0$	$-1772.044$	$-1855.377$
$y_0$	$-127.524$	2182.404
$t_1$	275.724	276.058
$y_1$	0.000	0.000
$t_2$	$-161.507$	$-167.415$
$y_2$	0.000	0.000
$t_{31}$	12,216.730	14,058.746
$y_{31}$	−11,969.990	−73,482.960
$t_{32}$	571.074	$-1022.193$
$y_{32}$	29,485.421	122,670.831
$t_{33}$	0.000	0.000
$y_{33}$	−22,955.280	−73,105.329
$W_0$	108.359	92.023
${\mu }_s^{*}$	0.991	1.004
${\mu }_v^{*}$	0.819	0.827
$K_0$	240	230
J	32.8	33
L	49.1	66

. The third component of the isospin of a nucleon, with ${\tau }_3=+1,-1$ for the neutron and proton, respectively, defines their isospin states. Having the parameters of the Skyrme force of the nucleon isoscalar ${\mathsf{\Theta }}_s$ and isovector mass ${\mathsf{\Theta }}_v$, their ratios can be, respectively, given by the following [13,22]:

$$\begin{array}{c}\hfill {\mu }_s^{*}=m_s^{*}/M={\left(1+{\displaystyle \frac{M}{8{\hslash }^2}}\rho {\mathsf{\Theta }}_s\right)}^{-1},{\mu }_v^{*}=m_v^{*}/M={\left(1+{\displaystyle \frac{M}{4{\hslash }^2}}\rho {\mathsf{\Theta }}_v\right)}^{-1}.\end{array}$$

(19)

Through the equation above, it can be easily seen that the effective nucleon masses can be written in terms of the isoscalar and isovector masses; see via Equations (18) and (19). The values of the Skyrme force parameters, including the isoscalar and isovector effective masses, $K_0$, J, and L for the KIDS0, and KIDSA models, are summarized in Table 2.

Figure 2 shows the numerical results for the pressure, energy density, effective neutron mass, and particle fractions for the KIDS0, KIDSA, and SLy4 models. A comparison between the pressure results for the KIDS0, KIDSA, and SLy4 models, and ChPT and HIC predictions is given in Figure 2a.

3. Neutrino-NS Matter Constituent Scattering

In this section, we present a generic formalism for neutrino and antineutrino interactions with NS matter via neutral current (NC) scattering using free-space nucleon form factors at zero temperature. In the literature, researchers have calculated the DCS for neutrinos, considering both weak and electromagnetic interactions including neutrino form factors [25]; however, to avoid complicating the calculation, in this work, we limit the current study to the standard weak interaction. The relevant interaction Lagrangian for neutrino and antineutrino NC scattering, expressed in terms of current–current interactions, is given by [9,26], as follows:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{INT}}^{NC}& =& {\displaystyle \frac{G_F}{\sqrt{2}}}\left[{\overline{\nu }}_e{\gamma }^{\mu }\left(1-{\gamma }_5\right){\nu }_e\right]\left[\overline{\psi }{\mathsf{\Gamma }}_{\mu }^{[n,p],NC}\psi \right],\hfill \end{array}$$

(20)

where $G_F=1.023/M\times {10}^{-5}$ is the weak coupling constant, and the standard nucleon vertex (including the free-space form factor of a nucleon) is defined as ${\mathsf{\Gamma }}_{\mu }^{[n,p],NC}$$={\gamma }_{\mu }\left(C_V^{[n,p]}-C_A^{[n,p]}{\gamma }_5\right)$. The values of $C_V$ and $C_A$ for neutrons and protons can be found in Refs. [24,26,27,28,29,30,31,32]. It is worth noting that $C_V\equiv F_{1W}(Q^2=0)$ and $C_A\equiv G_A$, where $F_{1W}(Q^2=0)$ and $G_A$ denote, respectively, the weak vector nucleon form factor and axial vector coupling constant. The values of $G_A$ and $F_{1,2}^W$ at $q^2=0$ in free space are provided in

Table 3. Free space nucleon axial and vector form factor values at $q^2=0$. In this work, we use ${sin}^2{\theta }_w=0.231$, $g_A=1.260$, ${\kappa }_p=1.793$, and ${\kappa }_n=-1.913$.

Target	${\mathit{G}}_{\mathit{A}}$	${\mathit{C}}_{\mathit{V}}={\mathit{F}}_1^{\mathbf{W}}$
n	$-{\displaystyle \frac{g_A}{2}}$	$-0.5$
p	${\displaystyle \frac{g_A}{2}}$	$0.5-2{sin}^2{\theta }_w$

. Note that, for antineutrinos, the sign of $G_A$ is replaced by $-G_A$, and $C_V\equiv F_{1W}$ is kept the same.

Neutral Current Interaction

As mentioned above, in this work, we consider the NC weak interaction where neutrinos interact with NS matter constituents through neutral Z gauge bosons, assuming that the momentum transfer is less than the masses of the weak gauge bosons. A generic expression for the double differential cross-section per volume for neutrino and antineutrino scattering at zero temperature is given by [9,26], as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{V}}{\displaystyle \frac{d^3\sigma }{d^2{\mathsf{\Omega }}^{\prime }d{E}_{\nu }^{\prime }}}=-{\displaystyle \frac{G_F^2}{32{\pi }^2}}{\displaystyle \frac{E_{\nu }^{\prime }}{E_{\nu }}}\mathrm{Im}\left[L_{\mu \nu }{\mathsf{\Pi }}^{\mu \nu }\right],\end{array}$$

(21)

where $E_{\nu }$ and $E_{\nu }^{\prime }=E_{\nu }-q_0$ are the initial and final neutrino energies. The leptonic and hadronic tensors are, respectively, given by the following:

$$\begin{array}{c}\hfill L_{\mu \nu }=8\left[2k_{\mu }{k}_{\nu }+(k.q)g)\mu \nu -(k_{\mu }{q}_{\nu }+q_{\mu }{k}_{\nu })\mp i{ϵ}_{\mu \nu \alpha \beta k^{\alpha }{q}^{\beta }}\right],\end{array}$$

(22)

where the sign in the last term of the leptonic tensor is a minus sign (−) for neutrinos and a plus sign (+) for antineutrinos. and

$$\begin{array}{c}\hfill {\mathsf{\Pi }}_{\mu \nu }^{[n,p]}\left(q^2\right)=-i\int {\displaystyle \frac{d^{4}p}{{\left(2\pi \right)}^4}}\mathrm{Tr}\left[G^{[n,p]}\left(p\right){\mathsf{\Gamma }}_{\mu }^{[n,p]}{G}^{[n,p]}(p+q){\mathsf{\Gamma }}_{\nu }^{[n,p]}\right],\end{array}$$

(23)

where the neutron and proton propagators in the nuclear medium can be defined by the following:

$$\begin{array}{c}\hfill G^{[n,p]}\left(p\right)=\left[{\displaystyle \frac{p{/}^{*}+M^{*}}{p^{*2}-M^{*2}+iϵ}}+i\pi {\displaystyle \frac{p{/}^{*}+M^{*}}{E*}}\delta \left(p_0^{*}-E^{*}\right)\mathsf{\Theta }\left(P_F^{[n,p]}-\left|\mathbf{p}\right|\right)\right].\end{array}$$

(24)

After contracting the leptonic tensor with the polarization insertions (hadronic tensor) for neutrons and protons, the final formula for the neutrino and antineutrino DCS is given by the following [9,26]:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{V}}{\displaystyle \frac{d^3\sigma }{d{E}_{\nu }^{\prime }{d}^2\mathsf{\Omega }}}={\displaystyle \frac{G_F^2}{4{\pi }^3}}{\displaystyle \frac{E_{\nu }^{\prime }}{E_{\nu }}}{q}^2\left[A{\mathcal{R}}_1+{\mathcal{R}}_2+B{\mathcal{R}}_3\right],\end{array}$$

(25)

where $A=\left[2E_{\nu }(E_{\nu }-q_0)+0.5q^2\right]/{\left|\mathbf{q}\right|}^2$, $B=2E_{\nu }-q_0$, and ${\mathcal{R}}_1$, ${\mathcal{R}}_2$, and ${\mathcal{R}}_3$ are, respectively, given by the following:

$$\begin{array}{ccc}\hfill {\mathcal{R}}_1& =& \left(C_V^2+C_A^2\right)\left[\mathrm{Im}{\mathsf{\Pi }}_L^{[n,p]}+\mathrm{Im}{\mathsf{\Pi }}_T^{[n,p]}\right],\hfill \\ \hfill {\mathcal{R}}_2& =& C_V^2{\mathsf{\Pi }}_T^{[n,p]}+C_A^2\left[\mathrm{Im}{\mathsf{\Pi }}_T^{[n,p]}-\mathrm{Im}{\mathsf{\Pi }}_A^{[n,p]}\right],\hfill \\ \hfill {\mathcal{R}}_3& =& \pm 2C_V{C}_A\mathrm{Im}{\mathsf{\Pi }}_{VA}^{[n,p]},\hfill \end{array}$$

(26)

where the plus sign (+) in ${\mathcal{R}}_3$ denotes the neutrinos and the minus sign (−) in ${\mathcal{R}}_3$ denotes the antineutrinos. In a nuclear medium, the polarization insertion can be decomposed into the polarization for the longitudinal, transversal, axial, and mixed vector-axial channels for the neutrons and protons, respectively, given by the following:

$$\begin{array}{ccc}\hfill \mathrm{Im}{\mathsf{\Pi }}_L^{[n,p]}& =& {\displaystyle \frac{q^2}{{2\pi \left|\mathbf{q}\right|}^3}}\left[{\displaystyle \frac{q^2}{4}}\left(E_F-E*\right)+{\displaystyle \frac{q_0}{2}}\left(E_F^2-E^{*2}\right)+{\displaystyle \frac{1}{3}}\left(E_F^3-E^{*3}\right)\right],\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \mathrm{Im}{\mathsf{\Pi }}_T^{[n,p]}& =& {\displaystyle \frac{1}{4\pi \left|\mathbf{q}\right|}}[\left(M^{*2}+{\displaystyle \frac{q^4}{{4\left|\mathbf{q}\right|}^2}}+{\displaystyle \frac{q^2}{2}}\right)\left(E_F-E^{*}\right)+{\displaystyle \frac{q_0{q}^2}{{2\left|\mathbf{q}\right|}^2}}\left(E_F^2-E^{*2}\right)\hfill \\ & +& {\displaystyle \frac{q^2}{{3\left|\mathbf{q}\right|}^2}}\left(E_F^3-E^{*3}\right)],\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \mathrm{Im}{\mathsf{\Pi }}_A^{[n,p]}& =& {\displaystyle \frac{i}{2\pi \left|\mathbf{q}\right|}}{M}^{*2}\left(E_F-E^{*}\right),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill \mathrm{Im}{\mathsf{\Pi }}_{VA}^{[n,p]}& =& {\displaystyle \frac{i{q}^2}{{8\pi \left|\mathbf{q}\right|}^3}}\left[\left(E_F^2-E^{*2}\right)+q_0\left(E_F-E^{*}\right)\right].\hfill \end{array}$$

(30)

The formula for the inverse neutrino and antineutrino MFP for the NC scattering at zero temperature is given by the following [9,26]:

$$\begin{array}{c}\hfill {\lambda }^{-1}\left(E_{\nu }\right)=2\pi {\int }_{q_0}^{(2E_{\nu }-q_0)}d\left|\mathbf{q}\right|{\int }_0^{2E_{\nu }}d{q}_0{\displaystyle \frac{\left|\mathbf{q}\right|}{E_{\nu }{E}_{\nu }^{\prime }}}\left[{\displaystyle \frac{1}{V}}{\displaystyle \frac{d^3\sigma }{d{E}_{\nu }^{\prime }d\mathsf{\Omega }}}\right].\end{array}$$

(31)

4. Numerical Results and Discussions

Here, the numerical results for the antineutrino DCS and AMFP for the E-RMF, EDF-KIDS: KIDS0, KIDSA, and SLy4 models are presented in Figure 3, Figure 4, Figure 5 and Figure 6. The antineutrino DCS for neutrons and protons for the KIDS0, KIDSA, and SLy4 models at $n_b=$ 1.0 $n_0$ is shown in Figure 2a. It shows the dominant contributions of neutron and proton scattering to the total antineutrino DCS, justifying the reason not to include the electron and muon scatterings in the present work, as they contribute less to the antineutrino DCS. One can conclude that the shape and magnitude of the antineutrino DCS strongly depend on the proton and neutron effective masses, as shown in Figure 2c. Antineutrino DCS for the neutrons and protons of the KIDS0, KIDSA, and SLy4 models at different baryon densities $n_B=$ 2.0 $n_0$, $n_B=$ 3.0 $n_0$, and $n_B=$ 4.0 $n_0$ are, respectively, shown in Figure 3a–d. They are calculated at an initial neutrino energy $E_{\nu }=$ 5 MeV and a three-momentum transfer of $\left|\mathbf{q}\right|=$ 2.5 MeV. Neutron contributions to the total antineutrino DCS are more pronounced at higher baryon number densities for the KIDS0 and KIDSA models, as expected.

Similarly, the antineutrino DCS of neutrons and protons for the E-RMF model with the G3(M) parameter is computed. The results of ADCS are shown in Figure 4 with similar baryon number densities, as depicted in Figure 3. Figure 3a shows the antineutrino DCS of neutrons and protons for the E-RMF model with the G3(M) parameter at $n_B=$ 1.0 $n_0$. This result of the E-RMF model with the G3(M) parameter is almost similar to that for the SLy4 model. Again, it is obvious because of the effective masses of neutrons and protons, as their differences can be seen in Figure 1c and Figure 2c. As the baryon number density increases, the range coverage of $q_0$ increases, and the shape of the antineutrino DCS changes, as shown in Figure 4. However, the magnitude of the antineutrino DCS does not change significantly.

Using the antineutrino DCS results for the E-RMF and EDF-KIDS models, the AMFP can be straightforwardly calculated for the corresponding models. The results of AMFP for the KIDS0, KIDSA, and SLy4 models are shown in Figure 5. One can see that the larger AMFP is found for the SLy4 model. For the KIDS0 and KIDSA models, it shows that the AMFP for both models is the same up to around $n_B=$ 3.0 $n_0$, and it begins to diverge at $n_B>$ 3.5 $n_0$.

The AMFP results of neutrons and protons, as well as the total (neutrons + protons) for the E-RMF model with the G3(M) parameter, are depicted in Figure 6. Again, as antineutrino DCS for the SLy4 and E-RMF with the G3(M) parameter models, the antineutrino AMFP results for the SLy4 and E-RMF with the G3(M) parameter models are almost the same in magnitude, as expected. One conclusion is that the antineutrino DCS and AMFP strongly depend on the effective nucleon masses. The coverage range of $q_0$ is longer if the $q_0^{\max }>M_N^{*}$, which is related to the kinematic constraint of the nucleon.

5. Summary and Conclusions

In summary, we investigated the antineutrino DCS and MFP in the relativistic E-RMF model with the G3(M) parameter, inspired by an effective field theory (EFT), as well as in the nonrelativistic EDF-KIDS models: KIDS0, KIDSA, and the SLy4 model. To ensure the stability of neutron star matter, we also investigated the EoS for the G3(M), KIDS0, KIDSA, and SLy4 models to verify their reproduction of the binding energy per nucleon at saturation baryon number density.

From the present work, it was found that ADCS and AMFP strongly depend on the nucleon effective masses, and the range of $q_0$ increases as the baryon number density increases when the kinematic constraint $q_0^{\max }>M_N^{*}$ is fulfilled. This finding is similar to those found in the neutrino DCS and MFP.

In qualitative comparison with the neutrino MFP, it was found that the antineutrino MFP is larger than the neutrino MFP, which is consistent with the result found in Ref. [5]. Even though the neutrino DCS and NMFP are not explicitly presented in this paper, the results for neutrino DCS and NMFP for the E-RMF and KIDS-EDF models are confirmed in Refs. [9,24].

This work also shows that the AMFP predictions of the E-RMF and KIDS-EDF models have rather different predictions on ADCS and AMFP, meaning the AMFP strongly depends on the nuclear models. This indicates the need for more data from terrestrial experiments and astrophysical observations to constrain the nuclear models better. Another approach would be for astrophysical experiments or observations (i.e., involving high-energy neutrino flux from the IceCube neutrino observatory) to directly measure the neutrino mean free path. This would be a great challenge and opportunity to compare empirical data with theory calculations directly.