# Neutrino mean free path in neutron stars in the presence of hyperons

Jesper Leong<EMAIL_ADDRESS>CSSM and ARC Centre of Excellence
for Dark Matter Particle Physics, Department of Physics, University of
Adelaide, SA 5005 Australia  Parada T. P. Hutauruk ID<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Pukyong National University (PKNU),
Busan 48513, Korea  Anthony W. Thomas ID<EMAIL_ADDRESS>CSSM
and ARC Centre of Excellence for Dark Matter Particle Physics, Department of
Physics, University of Adelaide, SA 5005 Australia

###### Abstract

We investigate the neutrino elastic differential cross-section (NDCS) and
corresponding mean free path for neutral current scattering in the dense
matter of a neutron star. A wide range of observed neutron star (NS) masses is
considered, including the presence of $\Lambda$, $\Xi^{-}$, and $\Xi^{0}$
hyperons in the heaviest stars. Their presence significantly decreases the
total neutrino mean free path in the heavier stars.

neutrino mean free path, neutral current, dense nuclear matter, neutron star,
quark-meson coupling model, hyperons

††preprint: PKNU-NuHaTh-2024, ADP-24-03/T1242

##  I Introduction

During the evolution process of binary neutron star (NS) mergers
LIGOScientific:2017ync ; LIGOScientific:2017vwq ; Friman:1979ecl  and
supernova collapse Brown:1982cpw ; Burrows:1987zz  neutrino emission plays a
crucial role. Neutrinos are one of the group of multi-messengers, which along
with gravitational waves LIGOScientific:2017ync ; LIGOScientific:2017vwq ;
LIGOScientific:2016aoc , electromagnetic radiation LIGOScientific:2017ync  and
X-ray bursts LIGOScientific:2017ync  offer vital information on the physical
processes at play. Neutrino emission is expected to provide new insights into
NS properties, with the information obtained able to be used to constrain the
equation of state (EoS) of nuclear matter at extreme densities. The latter is
still poorly determined and raises many open questions. The neutrinos produced
in the process of NS mergers interact with the constituent matter of the NS
through the weak interaction. This may involve either neutral current (NC) or
charged current (CC) scattering. In this work, we will concentrate on neutrino
NC scattering, which is particularly important for muon neutrinos
Bollig:2017lki .

Thus far a number of studies of neutrino NC scattering and CC absorption in
compact stars have been made using either relativistic Reddy:1997yr ;
Hutauruk:2021cgi ; Hutauruk:2023nqc  or non-relativistic models Reddy:1997yr ;
Hutauruk:2023nqc ; Mornas:2004vt . However, most available calculations for
the neutrino scattering or absorption have only considered matter consisting
of protons and neutrons, with very few calculations considering the existence
of the $\Lambda$ hyperon in the neutrino mean free path (NMFP) calculation
Mornas:2004vt ; Mornas:2004vs ; Reddy:1996tw ; Rios:2006ic . On the other
hand, the vast differences in the time scales for weak interactions compared
with NS formation time, along with considerations of $\beta$-equilibrium, lead
one to expect that, even at zero temperature, hyperons must appear as stable
constituents of the dense matter as the central baryon density increases and
the NS become more massive. Of course, at the high temperatures experienced in
the few milliseconds after a merger, hyperons will be abundant Stone:2019blq ;
Oertel:2016xsn ; Sekiguchi:2011mc ; Panda:2010zz ; Sedrakian:2022ata .

Given the expected appearance of the $\Lambda$, $\Xi^{-}$ and even $\Xi^{0}$
hyperons at higher baryon number density, it is necessary to consider the
interactions of the neutrinos with these hyperons in the NS. Here we compute
the neutrino interactions with nucleons and include the $\Lambda$, $\Xi^{-}$,
and $\Xi^{0}$ hyperons, which naturally appear in the high density EoS within
the quark-meson coupling (QMC) model. It is both interesting and important to
understand their contribution to the total NMFP. In this context, it is worth
noting that the NMFP is required as input for supernova simulations
Fiorillo:2023frv ; Liebendoerfer:2003es .

In this paper, we compute the neutrino differential cross-section (NDCS) and
NMFP for neutrinos scattering through the NC from baryons within NS of various
masses. This necessarily includes neutrino-neutron, neutrino-proton,
neutrino-$\Lambda$, neutrino-$\Xi^{-}$, and neutrino-$\Xi^{0}$ elastic
scatterings. We do not consider the neutrino-electron and neutrino-muon
scattering in the present work, as their contributions to the NMFP are small
compared to those for the baryons Niembro:2001hd ; Hutauruk:2018cgu .

In order to calculate the NDCS and NMFP, we model the NS nuclear matter using
the QMC model Guichon:1987jp ; Guichon:1995ue ; Martinez:2018xep ;
Stone:2016qmi ; Saito:2005rv ; Saito:1994ki ; Leong:2023lmw ;
Whittenbury:2015ziz . This is a relativistic model which takes into account
the modification of the internal quark structure of the baryons in-medium in
response to the strong scalar mean field present. The $\sigma$, $\delta$,
$\omega$, $\rho$, and $\pi$ mesons carry the interactions between baryons by
coupling to the quarks confined in MIT bags. While pions are involved only
through Fock terms, the $\sigma$, $\delta$, and the time components of the
$\omega$ and $\rho$ fields constitute mean fields. Taking account of the
change in the internal structure of the baryons as a result of the scalar
mean-field is equivalent to the introduction of repulsive many-body forces
Guichon:2004xg ; Guichon:2006er  between all the baryons, nucleons and
hyperons. Because the meson interactions are defined at the quark level, no
additional parameters are associated with either the hyperon-meson couplings
or the many-body forces; they are all calculated in the model.

At the high densities of the nuclear matter associated with the heaviest NS,
Pauli effects arising from the overlap of the finite-size baryons may lead to
additional repulsion, beyond that experienced at normal nuclear matter
density. We follow Leong et al. Leong:2023yma  in treating this
phenomenologically, in a manner that ensures that the properties of symmetric
nuclear matter are unchanged at saturation density. With this additional
repulsion, the model generates NS including hyperons with masses as large as
2.2 M⊙.

Our computations reveal interesting predictions for the NDCS and NMFP arising
from neutrino-$\Lambda$ and neutrino-$\Xi$ scattering at higher baryon number
densities. That $\Sigma$ hyperons do not occur in dense matter in
$\beta$-equilibrium in the QMC model has been explained in earlier work
Guichon:2018uew . The reason is that the gluonic hyperfine interaction, which
splits the masses of the $\Lambda$ and $\Sigma$ hyperons in free space, is
enhanced in-medium as the mean scalar field increases Guichon:2008zz , raising
their chemical potential so they cannot appear. The NDCS for
neutrino-$\Xi^{-}$ hyperon scattering begins to dominate beyond $n_{B}\simeq$
4.0 $n_{0}$. This increase in NDCS leads to a decrease in the NMFP at higher
densities and therefore in heavier NS.

This paper is organized as follows. Section II briefly describes the modeling
of nuclear matter and the resultant EoS in the QMC model, including the
phenomenological overlap correction. The effective nucleon, hyperon masses,
and particle fractions for different NS masses, serve as inputs to the
neutrino-nucleon and neutrino-hyperon scattering calculations for the
corresponding NS masses. In Sec. III, we present the expressions for the NDCS
and NMFP for the neutrino-nucleon and neutrino-hyperon interactions using a
linear response theory approach at zero temperature. In Sec. IV, we present
and discuss our numerical results for the NDCS and NMFP for different NS
masses. Section V is devoted to a summary and concluding remarks.

##  II Equation of State of Nuclear Matter

Here we use the QMC model to calculate the properties of nuclear matter in
$\beta$-equilibrium. The model originated with Guichon Guichon:1987jp , who
considered the effect of the strong Lorentz scalar field in a dense medium on
the internal quark structure of the nucleons. Using the MIT bag model to
describe nucleon structure he showed that the self-consistent treatment of the
change in quark structure led to a novel saturation mechanism. Later
refinements of the model involved a number of technical improvements
Guichon:1995ue  as well as a generalization to include hyperons
Tsushima:1997cu ; Guichon:2008zz . Applications to the EMC effect
Thomas:1989vt ; Cloet:2005pp ; Cloet:2005rt  and the Coulomb sum rule
Cloet:2015tha  required replacing the MIT bag model with the covariant model
of Nambu and Jona-Lasinio Bentz:2001vc .

In the QMC model, the coupling constants of the meson fields to the valence
quarks are chosen such that a self-consistent calculation reproduces the
properties of nuclear matter at normal saturation density. The model has been
widely and successfully applied to many problems in nuclear physics, such as
finite nuclei Guichon:1995ue ; Martinez:2018xep ; Stone:2016qmi ;
Guichon:2006er  and hadron structure in a nuclear medium Saito:2005rv  as well
as neutron star properties Leong:2023yma ; Saito:1994ki ; Leong:2023lmw  and
possible hybrid stars Whittenbury:2015ziz . Here, we briefly review the EoS of
nuclear matter within the QMC model, including the additional phenomenological
repulsion which may arise at higher densities because of baryon overlap
Leong:2023yma . Within this model, the baryon energy density takes the form

$\displaystyle\epsilon_{B}$ $\displaystyle=$
$\displaystyle\frac{\langle\mathcal{H}_{B}\rangle+\langle\mathcal{V}_{\sigma}\rangle+\langle\mathcal{V}_{\omega}\rangle+\langle\mathcal{V}_{\rho}\rangle+\langle\mathcal{V}_{\delta}\rangle+\langle\mathcal{V}_{\pi}\rangle+\langle\mathcal{V}_{O}\rangle}{V}\,,$
(1)

where the baryon contribution in the first term of Eq. (1) is expressed as

$\displaystyle\frac{\langle\mathcal{H}_{B}\rangle}{V}$ $\displaystyle=$
$\displaystyle
2\sum_{f}\int^{k_{f}}\frac{d^{3}k}{(2\pi)^{3}}\sqrt{\vec{k}^{2}+M_{f}^{*2}}$
(2)

and the meson contributions, including Fock terms, are

$\displaystyle\frac{\langle\mathcal{V}_{\sigma}\rangle}{V}$ $\displaystyle=$
$\displaystyle\frac{m_{\sigma}^{2}\bar{\sigma}^{2}}{2}+\frac{\lambda_{3}g_{\sigma}^{3}\bar{\sigma}^{3}}{6}+\sum_{f}\left(\frac{\partial
M_{f}^{*}}{\partial\bar{\sigma}}\right)^{2}\int^{k_{f}}\int^{k_{f}}\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\frac{1}{(\vec{k}_{1}-\vec{k}_{2})^{2}+m_{\sigma}^{2}}\frac{M_{f}^{*2}}{\sqrt{(\vec{k}_{1}^{2}+M_{f}^{*2})(\vec{k}_{2}^{2}+M_{f}^{*2})}},$
(3) $\displaystyle\frac{\langle\mathcal{V}_{\omega}\rangle}{V}$
$\displaystyle=$
$\displaystyle\frac{m_{\omega}^{2}\bar{\omega}^{2}}{2}-\sum_{f}g_{\omega}^{f2}\int^{k_{f}}\int^{k_{f}}\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\frac{1}{(\vec{k}_{1}-\vec{k}_{2})^{2}+m_{\omega}^{2}},$
(4) $\displaystyle\frac{\langle\mathcal{V}_{\rho}\rangle}{V}$ $\displaystyle=$
$\displaystyle\frac{m_{\rho}^{2}\bar{\rho}^{2}}{2}-\sum_{f,f^{\prime}}g_{\rho}^{2}\int^{k_{f}}\int^{k^{\prime}_{f}}\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\frac{C_{m,m^{\prime}}\delta_{S,S^{\prime}}}{(\vec{k}_{1}-\vec{k}_{2})^{2}+m_{\rho}^{2}},$
(5) $\displaystyle\frac{\langle\mathcal{V}_{\delta}\rangle}{V}$
$\displaystyle=$
$\displaystyle\frac{m_{\delta}^{2}\bar{\delta}^{2}}{2}+\sum_{f,f^{\prime}}\left(g_{\delta}^{f}(\bar{\sigma})g_{\delta}^{f^{\prime}}(\bar{\sigma})\right)\int^{k_{f}}\int^{k^{\prime}_{f}}\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\frac{C_{m,m^{\prime}}\delta_{S,S^{\prime}}}{(\vec{k}_{1}-\vec{k}_{2})^{2}+m_{\delta}^{2}}\frac{M_{f}^{*}M_{f^{\prime}}^{*}}{\sqrt{(\vec{k}_{1}^{2}+M_{f}^{*2})(\vec{k}_{2}^{2}+M_{f^{\prime}}^{*2})}}\,.$
(6)

Here the isoscalar and isovector-scalar Motta:2019tjc  mean fields are denoted
$\bar{\sigma}$ and $\bar{\delta}$, respectively. The $\bar{\omega}$ and
$\bar{\rho}$ stand for the time components of the isoscalar and isovector-
vector mean fields and $m_{\sigma}$, $m_{\omega}$, $m_{\rho}$, and
$m_{\delta}$ are respectively $\sigma$, $\omega$, $\rho$, and $\delta$ meson
masses. The $\sigma$ meson-nucleon, $\omega$ meson-nucleon, $\rho$ meson-
nucleon, and $\delta$ meson-nucleon coupling constants in free space are
respectively given by $g_{\sigma}$, $g_{\omega}$, $g_{\rho}$, and
$g_{\delta}$. We follow Ref. Leong:2023yma  in choosing
$G_{\delta}\,(\equiv\,g_{\delta}^{2}/m_{\delta}^{2})\,=3$ fm2, with
$m_{\delta}=983$ MeV, $m_{\sigma}=700$ MeV, the $m_{\omega}$ and $m_{\rho}$
take their physical values and the other parameters are chosen to reproduce
the binding energy per nucleon $E_{B}/A=-$ 15.8 MeV and symmetry energy $S=30$
MeV at the saturation density $n_{0}=$ 0.16 fm-3, with $\lambda_{3}=0.02$
fm-1. The quantity
$C_{m,m^{\prime}}=\delta_{m,m^{\prime}}I_{m}^{f2}+(\delta_{m,m^{\prime}+1}+\delta_{m+1,m^{\prime}})I_{t}^{f}$,
where the subscripts $m$ and $t$ stand for the projection, and isospin,
respectively.

The effective mass $M_{f}^{*}$ for each flavor baryon, $f$, which is
particularly important in the present context, is given by

$\displaystyle M_{f}^{*}\left(\bar{\sigma},\bar{\delta}\right)$
$\displaystyle=$ $\displaystyle
M_{f}-g_{\sigma}^{f}(\bar{\sigma})\bar{\sigma}-g_{\delta}^{f}(\bar{\sigma})I_{m}^{f}\bar{\delta},$
(7) $\displaystyle=$ $\displaystyle
M_{f}-\omega_{\sigma}^{f}g_{\sigma}\bar{\sigma}+\tilde{\omega}_{\sigma}^{f}\frac{d}{2}(g_{\sigma}\bar{\sigma})^{2}-t_{\delta}^{f}g_{\delta}I_{m}^{f}\bar{\delta}+\tilde{d}g_{\sigma}g_{\delta}\bar{\sigma}I_{m}^{f}\bar{\delta}\,.$

Here the scalar polarizabilities, $d$ and $\tilde{d}$ (with numerical values
given in Ref. Motta:2019tjc ), are the origin of the repulsive many-body
forces, which appear naturally and are determined, within the quark model
used, with no new parameters. The sixth term on the right hand side of Eq. (1)
is the Fock term for pion exchange Rikovska-Stone:2006gml ; Krein:1998vc .

As explained earlier, following Ref. Leong:2023yma , we also include a purely
phenomenological repulsive contribution to the energy density which increases
as the density, and hence the degree of potential overlap of the baryons,
increases. This short-distance repulsive contribution was shown as the last
term in Eq. (1) and has the form

$\displaystyle\frac{\langle\mathcal{H}_{O}\rangle}{V}$ $\displaystyle=$
$\displaystyle
E_{0}~{}n_{B}\exp[-\left(\frac{n_{B}^{-\frac{1}{3}}}{b}\right)^{2}\Bigg{]}.$
(8)

Here $n_{B}^{-1/3}$ is a measure of the average distance between baryons in
the nuclear matter. The quantities $E_{0}$ and $b$ are free parameters, chosen
such that they do not change the properties of symmetric NM at nuclear
saturation density. The $b$ parameter can be interpreted as the quark core
size of the baryons, which is expected to be of order 0.5 fm.

In the NS, we impose the conditions of $\beta$ equilibrium and charge
neutrality. Then, the total energy density can be written as

$\displaystyle\epsilon_{\rm{total}}$ $\displaystyle=$
$\displaystyle\epsilon_{B}+\epsilon_{e}+\epsilon_{\mu}\,,$ (9)

where $\epsilon_{B}$ is defined in Eq. (1), while for each species of lepton,
the energy density is defined as
$\epsilon_{l}=2\int^{k_{F}^{l}}d^{3}k/(2\pi)^{3}\sqrt{\vec{k}_{l}^{2}+m_{l}^{2}}$,
with $m_{l}$ the lepton mass in free space.

Figure 1:  (a) QMC EoS is displayed as a function of $n_{B}$ with the energy
density (blue solid line) corresponding to the left vertical axis, and
pressure (red solid line) corresponding to the right vertical axis, (b)
Effective mass of each of the baryons falls in response to the increase in the
mean scalar field strength (see Eq. (7)).

The pressure and energy density used here are shown as a function of the
baryon number density in Fig. 1(a), while the nucleon and hyperon effective
masses are shown in Fig. 1(b). It is worth noting that the EoS in Fig. 1(a)
was used to describe the NS properties starting from the most common masses
(of order 1.5 $M_{\odot}$) up to the most massive stars LIGOScientific:2018cki
; Xie:2019sqb ; Antoniadis:2013pzd ; Ozel:2016oaf ; NANOGrav:2017wvv ;
Riley:2021pdl ; Fonseca:2021wxt ; LIGOScientific:2020aai ; Bassa:2017zpe . In
Fig. 1(b) we see that the effective masses for different flavor baryons
decrease as the baryon number density increases. The decrease is not as great
as in many other models because of the self-consistent nature of the QMC
model, with the internal structure of the baryons adjusting to oppose the
applied scalar field. It is worth noting that in Fig. 1, we only show the EoS
and nucleon and hyperon effective masses over the range of baryon densities
relevant to stable NS.

Figure 2:  Particle fractions for matter in $\beta$-equilibrium within the NS
are calculated using the QMC model over the range $0<n_{B}<1.2$ fm-3. Vertical
dashed lines from left to right indicate a NS with a mass of 1.4, 1.6, 1.8,
2.0, and 2.14 $M_{\odot}$ respectively, with the latter the maximum predicted
by the QMC EoS. All particle species falling to the left of the respective
mass star are present within that star.

Besides the EoS and nucleon and hyperon effective masses, we also compute the
particle fractions for different NS masses. Results for these particle
fractions for the $M_{\rm{NS}}=$ 1.4 $M_{\odot}$, which is the canonical NS
mass LIGOScientific:2018cki  are shown in Fig. 2. Of course, protons,
neutrons, electrons, and muons are well-known as the standard matter occurring
in NS. None of the hyperons appear in the low mass NS. This can be understood
in terms of the maximum baryon number density for $M_{\rm{NS}}=$ 1.4
$M_{\odot}$, which is only $n_{B}\sim$ 0.43 fm-3. A similar indication is
found for $M_{\rm{NS}}=$ 1.6 $M_{\odot}$, with the same particles appearing.
For $M_{\rm{NS}}=$ 1.8 $M_{\odot}$ the $\Lambda$ hyperon starts to appear at
threshold baryon number density of around $n_{B}=$ 0.55 fm-3, as shown in Fig.
2. As the baryon number density increases further the $\Xi^{-}$ hyperons begin
to appear – beyond the relevant baryon number density threshold, which lies
around $n_{B}=$ 0.58 fm-3. The threshold density for the $\Xi^{0}$ is greater
still at $n_{B}=0.87$ fm-3. The QMC model predicts a maximum mass neutron star
($M_{\rm{max}}=2.14\,M_{\odot}$) to have a central core density in excess of 6
$n_{0}$, which at its centre, is the highest density achievable. In
consequence, the composition of heavy neutron stars, such as PSR J0740+6620
with $M_{\rm{NS}}=2.072^{+0.067}_{-0.066}M_{\odot}$, is predicted by the QMC
model to have the $\Lambda$, $\Xi^{-}$, and $\Xi^{0}$ hyperons present within
the inner core. Finally, once each of the hyperons appears, they continue to
increase in abundance as the baryon number density grows (or the mass of the
NS increases), which is likely to affect the NDCS and NMFP.

As a further illustration of the physics of NS, in Fig. 3, we show the baryon
number density at various radii within the different chosen fixed mass NSs. In
Fig. 3(a), the baryon number density, $n_{B}$, is given at the mass contained
within a given radius, $m(r)$ in units of $M_{\odot}$. Figure 3(b) shows how
the baryon number density changes from the centre as we move radially outwards
towards the surface. In comparison to a canonical NS mass, $M_{\rm{NS}}=1.4$
$M_{\odot}$, the central density of our maximum mass NS is around 2.3 times
larger. This corresponds to the change of the energy density and pressure of
2.9 and 8.3 times larger, respectively. Even when going from $M_{\rm{NS}}=2.0$
$M_{\odot}$ to the maximum $M_{\rm{NS}}=2.14$ $M_{\odot}$, the central density
and consequently the energy density and pressure (see Fig. 1(a)), need to
increase by a large amount for relatively small gains in mass. This indicates
that the highest mass NSs have extremely high pressure to counteract the
gravitational force.

At this point, one can summarize that the underlying cold NS EoS used here is
derived from the QMC model which produces NS masses up to and beyond 2.0
$M_{\odot}$. The central densities of NS can be in excess of 6 times the
saturation density of symmetric nuclear matter. This is a density region where
additional interactions may be present Harvey:1980rva ; HALQCD:2018gyl  but
difficult to probe in laboratory experiments Liu:1993sc . At zero temperature
the EoS used here predicts that only the $\Lambda$, $\Xi^{0}$, and $\Xi^{-}$
hyperons will appear in even the heaviest stars, while $\Sigma$ hyperon will
not appear as previously explained. Next, we use the nucleon and hyperon
effective masses and particle fractions computed for different NS masses to
calculate the NDCS and NMPF.

##  III Neutrino Interaction

In this section, we briefly introduce the general formalism for the neutrino
interaction with NS matter through NC scattering using linear response theory.
Earlier work involved the calculation of NDCS and NMFP, including weak
Hutauruk:2022bii  and electromagnetic interactions. That work took into
account the neutrino form factors such as the neutrino magnetic moment and
charge radius Hutauruk:2022bso ; Hutauruk:2020mhl ; Hutauruk:2018cgu ;
Hutauruk:2006re ; Kalempouw-Williams:2005zbp ; Sulaksono:2006eu . However, to
avoid complications, in the present work we limit our investigation to the
Standard Model weak interaction. The relevant interaction Lagrangian for the
neutrino-baryon and neutrino-hyperon NC scatterings in terms of the current-
current interaction is written as

$\displaystyle\mathcal{L}_{\mathrm{INT}}^{NC}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\Big{[}\bar{\nu}_{e}\gamma^{\mu}\left(1-\gamma_{5}\right)\nu_{e}\Big{]}\Big{[}\bar{\psi}_{B,Y}\Gamma_{\mu}^{[B,Y],NC}\psi_{B,Y}\Big{]}\,,$
(10)

where $G_{F}=1.023/M\times 10^{-5}$ is the weak coupling constant and for
standard nucleon vertex (including free space form factor of a nucleon) is
defined
$\Gamma_{\mu}^{[B,Y],NC}=\gamma_{\mu}\Big{[}C_{V}^{[B,Y]}-C_{A}^{[B,Y]}\gamma_{5}\Big{]}$,
where $B=p,n$ and $Y=\Lambda,\Xi^{-}$, and $\Xi^{0}$ hyperons. The values of
$C_{V}$ and $C_{A}$ for neutrons, protons, and $\Lambda$, $\Xi^{-}$, and
$\Xi^{0}$ hyperons can be found in Refs. Hutauruk:2022bii ; Savage:1996zd ;
Carrillo-Serrano:2014zta . They are summarized in Table. 1.

Table 1: Axial and vector couplings for proton, neutron, and $\Lambda$, $\Xi^{-}$, and $\Xi^{0}$ hyperons at $q^{2}=0$ for neutral-current reactions. In the numerical calculation, we use $\sin^{2}\theta_{w}=0.231$, $g_{A}=1.27$, $\kappa_{p}=1.793$ and $\kappa_{n}=-1.913$. The values of $D=0.774$ and $F=0.496$ are considered in this work.  Target |  $C_{A}$ |  $C_{V}$
---|---|---
$n$ |  $-(D+F)/2=-\frac{g_{A}}{2}=-0.635$ |  $-0.5$
$p$ |  $(D+F)/2=\frac{g_{A}}{2}=0.635$ |  $0.5-2\sin^{2}\theta_{w}=0.038$
$\Lambda$ |  $-F/2-D/6=-0.377$ |  $-0.5$
$\Xi^{-}$ |  $(D-3F)/2=-0.357$ |  $-\frac{3}{2}+2\sin^{2}\theta_{w}=-1.038$
$\Xi^{0}$ |  $-(D+F)=\frac{-g_{A}}{2}=-0.635$ |  $-0.5$

An expression for the double differential cross-section per volume for the
neutrino scattering can be computed from the interaction Lagrangian given in
Eq. (10) and it gives

$\displaystyle\frac{1}{V}\frac{d^{3}\sigma}{d^{2}\Omega^{\prime}dE_{\nu}^{\prime}}=-\frac{G_{F}^{2}}{32\pi^{2}}\frac{E_{\nu}^{\prime}}{E_{\nu}}\mathrm{Im}\Big{[}L_{\mu\nu}\Pi^{\mu\nu}_{[B,Y]}\Big{]}\,,$
(11)

where $E_{\nu}$ and $E^{\prime}_{\nu}=E_{\nu}-q_{0}$ are the initial and final
neutrino energies. The leptonic and hadronic tensors are respectively defined
by

$\displaystyle L_{\mu\nu}=8\Big{[}2k_{\mu}k_{\nu}+(k\cdot
q)g_{\mu\nu}-(k_{\mu}q_{\nu}+q_{\mu}k_{\nu})\mp i\epsilon_{\mu\nu\alpha\beta
k^{\alpha}q^{\beta}}\Big{]}\,,$ (12)

where the sign in the last term of the leptonic tensor is minus ($-$) for
neutrinos and plus ($+$) for antineutrinos. In addition, we need

$\displaystyle\Pi_{\mu\nu}^{[B,Y]}(q^{2})=-i\int\frac{d^{4}p}{(2\pi)^{4}}\mathrm{Tr}\Big{[}G^{[B,Y]}(p)\Gamma_{\mu}^{[B,Y]}G^{[B,Y]}(p+q)\Gamma_{\nu}^{[B,Y]}\Big{]}\,,$
(13)

where the nucleon and hyperon propagators in the nuclear medium are defined by

$\displaystyle
G^{[B,Y]}(p)=\Bigg{[}\frac{p\\!\\!\\!/^{*}+M_{f}^{*}}{p^{*2}-M_{f}^{*2}+i\epsilon}+i\pi\frac{p\\!\\!\\!/^{*}+M_{f}^{*}}{E^{*}}\delta\left(p_{0}^{*}-E^{*}\right)\Theta\left(p_{F}^{[B,Y]}-|\mathbf{p}|\right)\Bigg{]}\,,$
(14)

where $M_{f}^{*}$ is the nucleon or hyperon effective mass as defined in
Eq.(7) and $p_{F}$ being the appropriate Fermi momentum.

After contracting the leptonic tensor and the polarization insertions
(hadronic and hyperonic tensors) for the neutrons, protons, and hyperons, the
final expression for the NDCS is given by

$\displaystyle\frac{1}{V}\frac{d^{3}\sigma}{dE_{\nu}^{{}^{\prime}}d^{2}\Omega^{\prime}}=\frac{G_{F}^{2}}{4\pi^{3}}\frac{E_{\nu}^{{}^{\prime}}}{E_{\nu}}q^{2}\Bigg{[}A\mathcal{R}_{1}+\mathcal{R}_{2}+B\mathcal{R}_{3}\Bigg{]}\,.$
(15)

Here $A=\left[2E_{\nu}(E_{\nu}-q_{0})+0.5q^{2}\right]/|\vec{\mathbf{q}}|^{2}$,
$B=2E_{\nu}-q_{0}$, and $\mathcal{R}_{1}$, $\mathcal{R}_{2}$, and
$\mathcal{R}_{3}$ are, respectively, given by

$\displaystyle\mathcal{R}_{1}$ $\displaystyle=$
$\displaystyle\left(C_{V}^{2}+C_{A}^{2}\right)\Big{[}\mathrm{Im}\Pi_{L}^{[B,Y]}+\mathrm{Im}\Pi_{T}^{[B,Y]}\Big{]},$
$\displaystyle\mathcal{R}_{2}$ $\displaystyle=$ $\displaystyle
C_{V}^{2}\Pi_{T}^{[B,Y]}+C_{A}^{2}\Big{[}\mathrm{Im}\Pi_{T}^{[B,Y]}-\mathrm{Im}\Pi_{A}^{[B,Y]}\Big{]},$
$\displaystyle\mathcal{R}_{3}$ $\displaystyle=$ $\displaystyle\pm
2C_{V}C_{A}\mathrm{Im}\Pi_{VA}^{[B,Y]}\,.$ (16)

The plus ($+$) sign in $\mathcal{R}_{3}$ is for the neutrinos and the minus
($-$) sign is for 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, protons,
and hyperons. These are, respectively, given by

$\displaystyle\mathrm{Im}\Pi_{L}^{[B,Y]}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{2\pi|\vec{\mathbf{q}}|^{3}}\left[\frac{q^{2}}{4}\left(E_{F}-E^{*}\right)+\frac{q_{0}}{2}\left(E_{F}^{2}-E^{*2}\right)+\frac{1}{3}\left(E_{F}^{3}-E^{*3}\right)\right],$
(17) $\displaystyle\mathrm{Im}\Pi_{T}^{[B,Y]}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi|\vec{\mathbf{q}}|}\Bigg{[}\left(M_{f}^{*2}+\frac{q^{4}}{4|\vec{\mathbf{q}}|^{2}}+\frac{q^{2}}{2}\right)\left(E_{F}-E^{*}\right)+\frac{q_{0}q^{2}}{2|\vec{\mathbf{q}}|^{2}}\left(E_{F}^{2}-E^{*2}\right)+\frac{q^{2}}{3|\vec{\mathbf{q}}|^{2}}\left(E_{F}^{3}-E^{*3}\right)\Bigg{]},$
(18) $\displaystyle\mathrm{Im}\Pi_{A}^{[B,Y]}$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi|\vec{\mathbf{q}}|}M_{f}^{*2}\left(E_{F}-E^{*}\right),$
(19) $\displaystyle\mathrm{Im}\Pi_{VA}^{[B,Y]}$ $\displaystyle=$
$\displaystyle\frac{iq^{2}}{8\pi|\vec{\mathbf{q}}|^{3}}\left[\left(E_{F}^{2}-E^{*2}\right)+q_{0}\left(E_{F}-E^{*}\right)\right]\,.$
(20)

The final inverse NMFP expression for the neutrino-nucleon and neutrino-
hyperon NC scatterings as a function of the initial energy of neutrino at zero
temperature is given by

$\displaystyle\lambda^{-1}(E_{\nu})=2\pi\int_{q_{0}}^{(2E_{\nu}-q_{0})}d|\vec{\mathbf{q}}|\int_{0}^{2E_{\nu}}dq_{0}\frac{|\vec{\mathbf{q}}|}{E_{\nu}E_{\nu}^{\prime}}\left[\frac{1}{V}\frac{d^{3}\sigma}{dE_{\nu}^{\prime}d^{2}\Omega^{\prime}}\right]\,.$
(21)

##  IV Numerical results and Discussions

The numerical results for the NDCS and NMFP of neutrino-nucleon and neutrino-
hyperon scattering for various NS masses, computed in the QMC model including
the repulsive baryon overlap term, are presented in Figs. 4-5 and in Figs.
6-7, respectively. The NDCS and NMFP are first computed at three-component
momentum transfer $|\vec{\textbf{q}}|=$ 2.5 MeV and initial neutrino energy
$E_{\nu}=$ 5 MeV (Figs. 4 and 6), which is typical of the neutrino energies in
NS during the cooling phase Horowitz:1990it . In Figs. 5 and 7 we also show
results of relevance to supernovae and NS mergers for the commonly used
neutrino energy around $E_{\nu}=$ 30 MeV Janka:2006fh ; Kresse:2020nto ;
Bruenn:1985en ; Kamiokande-II:1987idp .

As depicted in Fig. 2, the central densities of stars with mass in the range
1.4$-$1.6 $M_{\odot}$ are around $2.5-3.0$ $n_{0}$. Thus we compute the NDCS
starting at very low baryon number density $n_{B}\simeq$ 1.0 $n_{0}$ (the
crust-core boundary of the NS), then at $n_{B}\simeq$ 2.5 $n_{0}$, which is
the central core density of a $M_{\mathrm{NS}}=$ 1.4 $M_{\odot}$, and finally
at $n_{B}\simeq$ 3.0 $n_{0}$, which represents the central density of a $1.6$
$M_{\odot}$ NS.

The NDCS results for $M_{\rm{NS}}=$ 1.4 $M_{\odot}$ are shown in Figs.
4(a)-(b). For the canonical mass NS case, we find that only the proton and
neutron appear (see Fig. 2) and, as a consequence, only the nucleons
contribute to the NDCS. As shown in Fig. 2, the electrons and muons also
appear together with protons and neutrons but their NDCS are relatively small
compared to those for protons and neutrons and we ignore them in the present
calculation. As expected, the NDCS for the neutrons is greater than that of
the protons for all baryon number densities. This can be understood because of
the differences in the abundances of the particles present within the star,
where the neutron particle fractions are larger in comparison to those for the
protons.

The sharp peak structure in NDCS, shown in Figs. 4 and 5 depends on the
maximum value of $q_{0}$,
$q_{0}^{\rm{max}}\approx|\vec{\mathbf{q}}|/\sqrt{(M_{f}^{*}/p_{F})^{2}+1}$
Hutauruk:2022bii , which increases as the effective mass decreases. Here we
note that the appearance of protons and neutrons for 1.4 $M_{\odot}$ NS is
consistent with the result found in Ref. Zhao:2015tra , except for the
appearance of the $\Lambda$ hyperon in that work. That depends, of course, on
the depth of the hyperon potential and coupling constant in the model. The QMC
model provides a relatively good description of the binding energies of known
hypernuclei, which are consistent with the available experimental results
Guichon:2008zz .

Results for the NDCS at densities relevant to a star with $M_{\rm{NS}}=$ 1.8
$M_{\odot}$ are shown in Figs. 4(a)-(c), where the baryon number density
varies from $n_{B}\simeq$ 1.0 and 2.0 to 3.0 $n_{0}$. In Fig. 4 we do not show
explicitly the NDCS result for $n_{B}\simeq$ 3.4 $n_{0}$, where the $\Lambda$
hyperons start to appear. At this density, which is just reached in a 1.8
$M_{\odot}$ star, the NDCS for neutrino-$\Lambda$ hyperon scattering does
begin to contribute to the total NDCS. However, as the number density is still
very low, the NDCS for the $\Lambda$ hyperon for $M_{\rm{NS}}=$ 1.8
$M_{\odot}$ is very small. The range of $q_{0}$ is also small compared to that
for the neutrons at $n_{B}\simeq$ 3.4 $n_{0}$.

For $M_{\rm{NS}}=$ 2.0 $M_{\odot}$, not only does the neutrino-$\Lambda$
hyperon NDCS contribute but the neutrino-$\Xi^{-}$ hyperon also starts to
contribute to the total. The contributions at $n_{B}\simeq$ 4.0 $n_{0}$ can be
seen in Fig. 4(d). This is consistent with the threshold baryon number density
for the appearance of the $\Xi^{-}$ hyperon as shown in Fig. 2. Figures 4(e)
and (f) show the NDCS relevant to a star with mass $M_{\rm{NS}}=M_{\rm{max}}$,
where the maximum baryon number density rises just beyond $n_{B}\simeq$ 6.0
$n_{0}$. Here the $\Lambda$ and $\Xi^{-}$ NDCS still contribute to the total
NDCS, while the NDCS of the neutrino-$\Xi^{0}$ NC scattering also contributes
to the total.

Figure 3:  (a) Baryon number density as a function of the $m(r)/M_{\odot}$
where $m(r)$ is the total energy within a certain radius in a star, (b) Baryon
number density as a function of radius. Figure 4:  Neutrino differential cross
section was calculated using $E_{\nu}=5$ MeV and $|\vec{\mathbf{q}}|=2.5$ MeV,
with the corresponding baryon number density given from (a) through to (f).
Figure 5:  Same as in Fig. 4, but for $E_{\nu}=$ 30 MeV.

As explained earlier, we also compute the NDCS for higher neutrino energy,
$E_{\nu}=$ 30 MeV, which is relevant to supernova and NS mergers, as given in
Fig. 5. We find that the NDCS for $E_{\nu}=$ 30 MeV has the same tendency and
interpretation as $E_{\nu}=$ 5 MeV for appropriate baryon number densities,
except for the order of magnitude. The NDCS increases remarkably, by a factor
of 50, at higher nuclear matter density, leading to a much lower NMFP, as
shown in Fig. 7(a).

Figure 6: (a) Neutrino mean free path for $E_{\nu}=5$ MeV as a function of the
baryon number density. The color scheme reflects the appearance of new baryon
species as the number density increases. In (b) Neutrino mean free path as a
function of NS radius $r$ for stars of various masses, from 1.4 to 2.14
$M_{\odot}$.  Figure 7: Same as in Fig. 6, but for $E_{\nu}=$ 30 MeV.

Next we turn to the neutrino mean free path. Results for the total NMFP for
$E_{\nu}=$ 5 MeV as a function of $n_{B}$ for $M_{\rm{NS}}=M_{\rm{max}}$ are
shown in Fig. 6(a), covering a large range of baryon number density. Here we
consider only the total NMFP at $M_{\rm{NS}}=M_{\rm{max}}$, treating it as
representative of other NS, since the same EoS is used to describe them. At
low density the only contribution to the NMFP comes from the nucleons (as
indicated by the blue line). This occurs up to around $n_{B}\simeq 0.54$ fm-3
where a small plateau precedes a slight but rapid decrease in the NMFP caused
by the appearance of the $\Lambda$ hyperon (green). There is a second more
significant decrease to the NMFP which closely follows the $\Lambda$ and is
associated with the $\Xi^{-}$ appearance at around $n_{B}\simeq 0.67$ fm-3
(mustard). Finally for the heaviest NS the central core density is in excess
of 0.83 fm-3 and the $\Xi^{0}$ hyperon contributions further decreases the
NMFP.

In Fig. 6 (b), we show the NMFP for $E_{\nu}=$ 5 MeV as a function of radius
for different NS masses. We see that the NMFP results for $M_{\rm{NS}}=$ 1.4
M⊙ and 1.6 M⊙ are almost the same because the contributions to the total NMFP
in both cases come from the protons and neutrons. For $M_{\rm{NS}}=$ 1.8 M⊙,
the $\Lambda$ hyperon begins to contribute to the total NMFP, in addition to
the protons and neutrons, leading to a decrease in the total NMFP around the
core of the NS, $r\simeq$ 2 km ($n_{B}\simeq$ 0.54 fm-3, see Fig. 2). This
indicates that the neutrino emission could be delayed around the NS core.

The decreases in NMFP begin to move to larger radii in the heavier stars. For
example, for $M_{\rm{NS}}=$ 2.0 M⊙ the NMFP starts to decrease rapidly inside
about 6 km, where the $\Lambda$ hyperons appear. A little further inside, at
around $r\simeq$ 5.5 km, the $\Xi^{-}$ begin to contribute. With the
contribution of the $\Xi^{0}$ to the total NMFP for $M_{\rm{NS}}=$ 2.14 M⊙,
one sees a greater decrease in NMFP at around $r\simeq$ 7.8 km. From these
results, one can conclude that mostly $\lambda>R_{\rm{NS}}(=r)$, indicating
the neutrinos can escape from the NS relatively easily, leading to fast NS
cooling. This also indicates that our results suggest very little chance of
neutrino trapping for any cool NS.

Next we compute the NMFP for $E_{\nu}=$ 30 MeV, showing the results in Figs.
7(a) and 7(b). The qualitative trend of the NMFP in Figs.7(a) and 7(b) was
found to be the same as that found in Figs. 6(a) and 6(b), respectively.
However, they are very different in magnitude, as small as some tens of
metres. Indeed, we found that the $\lambda<<R_{\rm{NS}}$ for all NS masses,
which is in contrast with the results in Fig. 6(b). This indicates that
neutrinos have a large probability of trapping in a hot NS environment.

##  V Summary and future perspectives

To summarize, we have investigated the neutrino differential cross section
(NDCS) and mean free path (NMFP) for neutrino-nucleon and neutrino-hyperon
neutral current scattering at densities relevant to a wide range of neutron
star masses. The QMC model was used to generate the equation of state for
matter in $\beta$-equilibrium. This was then used in the TOV equations to
calculate the structure of the NS. The relativistic effective masses of the
nucleons and hyperons were then used in the NDCS and NMFP calculations,
leading to predictions for the NMFP as a function of density and equivalently
radius in the NS.

For NDCS results for different NS masses, we found interesting results for the
NDCS of the neutrino-$\Lambda$, neutrino-$\Xi^{-}$ and neutrino-$\Xi^{0}$
scattering in the matter with higher baryon density ($n_{B}\gtrsim$ 3.4
$n_{0}$), corresponding to NS masses beyond $M_{\rm{NS}}=$ 1.8 $M_{\odot}$.
Within the QMC model with additional repulsion depending on the degree of
overlap of the baryons, we found that the threshold baryon number densities
for $\Lambda$, $\Xi^{-}$, and $\Xi^{0}$ hyperons are around 0.53 fm-3, 0.58
fm-3, and 0.87 fm-3, respectively. The NDCS for the $\Xi^{-}$ begins to
dominate beyond $n_{B}\simeq$ 4.0 $n_{0}$.

An increase in the NDCS of neutrino-$\Xi^{-}$ hyperons at higher baryon number
density implies that the NMFP decreases. As the $\Xi^{0}$ hyperon appears at a
very high threshold baryon number density, around $n_{B}\simeq$ 6.0 $n_{0}$,
the NDCS of the neutrino-$\Xi^{0}$ hyperon scattering starts to contribute to
the total NDCS at that density.

Next, we found that the NMFP for a star at the canonical NS mass,
$M_{\rm{NS}}=$ 1.4 $M_{\odot}$, decreases as density increases up to
$n_{B}/n_{0}\simeq$ 2.5. A similar behavior is shown by the total NMFP for
$M_{\rm{NS}}=$ 1.6 $M_{\odot}$. Note that, in this range of NS masses, only
the NMFP of protons and neutrons contribute to the total NMFP. This is because
the hyperons do not yet appear at the baryon number densities corresponding to
either NS mass. As the threshold baryon number density of the $\Lambda$
hyperon is reached, the NMFP of the neutrino-hyperon scattering for
$M_{\rm{NS}}=$ 1.8 $M_{\odot}$ starts to contribute to the total NMFP, in
addition to the NMFP of the neutrino-nucleon scattering. The NMFP contribution
of the neutrino scattering by a particular hyperon is signaled by a further
drop in the total NMFP just beyond the relevant threshold baryon number
density. There are several such thresholds shown in Figs. 6 and 7 for
$M_{\rm{NS}}=$ 2.0 $M_{\odot}$ and $M_{\rm{NS}}=M_{\rm{max}}$.

An important feature of the results presented here (in Fig. 6 versus Fig. 7)
is that the NMFP shows a dramatic decrease as the neutrino energy goes up. In
particular, a neutrino of energy 30 MeV has a mean free path of just 20 m in
the core of the most heavy NS. As hyperons exhibit no threshold at finite
temperatures and are more abundant at even 10 MeV than at zero temperature
Stone:2019blq ; Guichon:2023iev , the pressure associated with neutrino
trapping by hyperons will be especially important in proto-NS.

## Acknowledgment

The authors wish to thank the Asia Pacific Center for Theoretical Physics
(APCTP) for the support and warm hospitality and for organizing the 2023
workshop on Origin of Matter and Masses in the Universe: Hadrons in Free
Space, Dense Nuclear Medium, and Compact Stars, where the discussions on the
topic of this paper were initiated. P.T.P.H was supported by the National
Research Foundation of Korea (NRF) grants funded by the Korean government
(MSIT) (2018R1A5A1025563, 2022R1A2C1003964, and 2022K2A9A1A0609176). We also
acknowledge the support of the University of Adelaide and the Australian
Research Council through Discovery Project (AWT) DP230101791.

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