# Novel neutron decay mode inside neutron stars

Wasif Husain1 & Anthony W. Thomas1 1ARC Centre of Excellence for Dark Matter
Particle Physics and CSSM, Department of Physics, University of Adelaide, SA
5005, Australia<EMAIL_ADDRESS>
###### Abstract

We explore the suggestion that the neutron lifetime puzzle might be resolved
by neutrons decaying into dark matter through the process, n
$\rightarrow\chi\chi\chi$, with $\chi$ having a mass one third of that of the
neutron. In particular, we examine the consequences of such a decay mode for
the properties of neutron stars. Unlike an earlier suggested decay mode, in
order to satisfy the constraints on neutron star mass and tidal deformability,
there is no need for a strong repulsive force between the dark matter
particles. This study suggests the possibility of having hot dark matter at
the core of the neutron star and examines the possible signal of neutrons
decaying in this way inside the neutron star right after its birth.

##  1 Introduction

The observation of two different values for the neutron lifetime has been a
mystery for some years. Neutrons in a beam show a longer lifetime than those
trapped in a bottle [1]. The lifetime of neutrons in a beam [2, 3, 4, 5] is
measured by counting the total number of protons produced, while the lifetime
of neutrons trapped in a bottle is measured by counting the number of neutrons
remaining after a known period of time [6, 7, 8, 9]. The lifetime of neutrons
in a beam appears to be approximately 8 seconds or 4$\sigma$ longer than that
of neutrons in a bottle. A recent precise measurement of the lifetime of
neutrons trapped in a bottle by the UCN$\tau$ Collaboration [10], found the
lifetime of the neutron was 877.75 $\pm$ 0.28 (stat.) + 0.22 - 0.16 (syst.)
seconds, which is similar to other recent measurements [11, 12].

A few years ago, Fornal and Grinstein proposed that the explanation for this
discrepancy might be that neutrons have a small branching ratio for decay
through an unknown dark channel (dark matter). This naturally explains why the
lifetime of neutrons in a beam, with the decay process observed, appears
longer than that for neutrons in a bottle experiment, where the decay mode is
unseen [13, 14, 15]. This new decay channel

$n\rightarrow\chi+\phi,$ (1)

must have a branching ratio around 1%. $\phi$ is an extremely light boson that
goes undetected and $\chi$ is a dark matter fermion with baryon number one and
mass in the narrow range 937.9 MeV $<$ $m_{\chi}$ $<$ 938.7 MeV [13, 16].

References [17, 18] ruled out the possibility of $\phi$ being a photon. It was
very quickly realized that this decay mode could be tested against the known
properties of neutron stars [19, 20, 21, 22]. Indeed, if the dark baryon is
non-interacting the maximum neutron star mass is reduced to 0.7 M⊙, which is
unacceptable. Thus, for this explanation to survive the dark matter particles
must be very strongly self-repulsive.

Recently, an interesting alternative was proposed by Strumia [23], in which it
was suggested that the dark decay mode might involve neutrons decaying into
three identical dark matter particles, each having baryon number 1/3. It was
claimed that this model does not require dark matter to be self-repulsive to
survive the neutron star maximum mass constraint. Here, we carry out a
detailed examination of the consequences for neutron star properties of the
model suggested in Ref. [23].

The layout of this article is as follows. Section two explains the model used
to describe the nuclear and dark matter and the corresponding equations of
state (EoS). Section three briefly explains how these EoS are used to
calculate the properties of the neutron stars (NS). This is followed by
section four, where the NS properties are calculated and the effects of
neutron decay on those properties are studied. Section five summarises the
main conclusions of this study.

##  2 Neutron star model and Equation of state (EoS)

Neutron stars are often considered to be laboratories for testing the physics
at ultra-high densities because they contain matter in extreme conditions,
which cannot be prepared in a laboratory. Neutron stars cover a wide range of
energy densities, possibly exceeding six times nuclear matter density at the
core. At such high densities there is no consensus concerning the state of
matter, for example whether it is still hadronic or has made a transition to
deconfined quark matter [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44]. Here we consider two EoSs which are based on
the quark meson coupling model, one involving nucleons only and the other
including hyperons at the higher energy densities in the core. The details of
the QMC model can be found in Refs. [45, 46, 47, 48, 49, 50, 51, 52]. For our
purposes it is sufficient that these EoS produce NS with maximum mass, tidal
deformability and radii consistent [53, 54] with current constraints from
gravitational wave [55, 56] and satellite observations [57].

Assuming that the matter in the NS is hadronic, it is mostly made of neutrons
and if the neutrons decay into dark matter then there should be enough dark
matter particles inside the neutron star to make some changes in its
properties [58, 59, 60, 61, 62, 63, 64, 65, 66, 22, 67, 68, 69, 70, 71, 72].
Therefore, we may expect to test this model for neutron decay using the
properties of neutron stars.

###  2.1 Neutron decay model

Following Strumia [23], we consider the possible decay of the neutron as

$n\rightarrow\chi\chi\chi,$ (2)

where $\chi$ is a dark matter fermion with baryon number 1/3. The mass of dark
matter particle must be extremely close to $m_{n}$/3, in order to ensure that
nuclei remain stable. For this decay to be kinematically allowed the mass of
the dark matter particle must be $m_{\chi}<m_{n}/3\approx$ 313.19 MeV, while
for the nuclear decay to be forbidden one must have $m_{\chi}$ $>$
($m_{n}-E_{Be}$)/3 $\approx$ 312.63 MeV, because the strongest bound comes
from 8Be. Stability of the proton requires that $m_{\chi}>(m_{p}-m_{e})$/3 =
312.59 MeV, and for $m_{\chi}<(m_{p}+m_{e})$/3 $\approx$ 312.93 MeV the decay
of a Hydrogen atom, H $\rightarrow\chi\chi\chi\nu_{e}$, would be allowed.
While it is difficult to argue that such fine tuning yields a natural
explanation of the neutron lifetime puzzle, it is important to explore every
possible test of the idea.

###  2.2 Equation of state

The decay of neutrons into dark matter will change the chemical composition of
neutron star. The conditions of $\beta$-equilibrium inside the neutron star
require that the following equations hold for the nucleon-only EoS,

$n_{p}=n_{e}+n_{\mu}\,,\qquad\mu_{n}=\mu_{p}+\mu_{e},\qquad\mu_{\mu}=\mu_{e},\qquad\mu_{\chi}=\mu_{n}/3\qquad\,.$
(3)

For simplicity we choose $m_{\chi}$ = $m_{n}$/3 in all numerical calculations.
For our second EoS for nuclear matter, we allow hyperons to appear as the
chemical potential increases. With hyperons included in the EoS we must have

$\displaystyle
n_{p}=n_{e}+n_{\mu}+n_{\Xi^{-}}\,,\qquad\mu_{n}=\mu_{p}+\mu_{e},\qquad\mu_{\mu}=\mu_{e},\qquad\mu_{\chi}=\mu_{n}/3,\qquad$
(4) $\displaystyle\mu_{n}=\mu_{\Lambda}=\mu_{\Xi^{0}}\qquad\,,$ (5)

and $\mu_{\Xi^{-}}=\mu_{n}+\mu_{e}$. We note that the repulsion experienced by
the $\Sigma$ baryons in the QMC model [73] means that they do not appear at
any density of interest and so have not been shown explicitly. The hyperons
begin to appear when the density exceeds 3 times nuclear matter density, with
the $\Lambda$ appearing first, followed by the $\Xi^{-}$ and only at the
highest central densities the $\Xi^{0}$. The energy density [53, 74] is
calculated using the Hartree-Fock method. The Hartree term is as follows

$\displaystyle\epsilon_{H}=\frac{1}{2}m^{2}_{\sigma}\sigma^{2}+\frac{1}{2}m^{2}_{\omega}\omega^{2}+\frac{1}{2}m^{2}_{\rho}\rho^{2}+\frac{1}{\pi^{2}}\int^{k_{F}^{n}}_{0}k^{2}\sqrt{k^{2}+M_{N}^{*}(\sigma)^{2}}dk+$
(6)
$\displaystyle\frac{1}{\pi^{2}}\int^{k_{F}^{p}}_{0}k^{2}\sqrt{k^{2}+M_{N}^{*}(\sigma)^{2}}dk+\frac{1}{\pi^{2}}\int^{k_{F}^{e}}_{0}k^{2}\sqrt{k^{2}+m_{e}^{2}}dk$
(7)
$\displaystyle+\frac{1}{\pi^{2}}\int^{k_{F}^{\mu}}_{0}k^{2}\sqrt{k^{2}+m_{\mu}^{2}}dk+\frac{1}{\pi^{2}}\int^{k_{F}^{\chi}}_{0}k^{2}\sqrt{k^{2}+m_{\chi}^{2}}dk\,,$
(8)

where $M_{N}^{*}$ is the effective mass of the nucleon given by

$M_{N}^{*}(\sigma)=M_{N}-g_{\sigma}\sigma+\frac{d(g_{\sigma}\sigma)^{2}}{2}\,.$
(9)

Here $M_{N}$ is the mass of the nucleon, $g_{\sigma}$ is the scalar ($\sigma$)
coupling to the nucleon in free space and $d$ is the scalar polarisability,
which arises because of the change in nucleon structure caused by the strong
scalar mean field in dense nuclear matter [75, 47]. The expressions for the
Fock terms (including those for the pion), which are more complicated, can be
found in Ref. [53]. The pressure is calculated using

$P=\int\mu_{f}n_{f}-\epsilon.$ (10)

and the baryon number is given by

$B_{i}=4\pi\int_{0}^{R}\frac{r^{2}n_{i}(r)}{[1-\frac{2M(r)}{r}]^{1/2}}dr\,,$
(11)

where $i$ identifies the different baryons, such as neutron, proton and
hyperons, while the dark matter baryon number involves the number density
divided by a factor of three.

##  3 Structural equations for the neutron star

In order to avoid additional unknown parameters, we make the working
hypothesis that the nuclear matter does not interact with the dark matter.
Hence, the two fluid TOV [76, 77] equations must be used to calculate the
distribution of matter in the star. The following two fluid coupled TOV
equations are integrated from the centre of the neutron star to the surface.

$\frac{dP_{nucl}}{dr}=-\frac{[\epsilon_{nucl}(r)+P_{nucl}(r)][4\pi
r^{3}(P_{nucl}(r)+P_{DM}(r))+m(r)]}{r^{2}(1-\frac{2m(r)}{r})},$ (12)
$m_{nucl}(r)=4\pi\int_{0}^{r}dr.r^{2}\epsilon_{nucl}(r)\,,$ (13)
$\frac{dP_{DM}}{dr}=-\frac{[\epsilon_{DM}(r)+P_{DM}(r)][4\pi
r^{3}(P_{nucl}(r)+P_{DM}(r))+m(r)]}{r^{2}(1-\frac{2m(r)}{r})}\,,$ (14)
$m_{DM}(r)=4\pi\int_{0}^{r}dr.r^{2}\epsilon_{DM}(r)\,,$ (15)
$m(r)=m_{nucl}(r)+m_{DM}(r)\,.$ (16)

For the purpose of calculating the tidal deformability, the method provided in
Refs. [78, 79] is considered.

##  4 Effects of neutron decay

The effects of neutron decay on the neutron star properties are presented in
this section. Figure 1 shows that as the neutrons decay into dark matter the
maximum mass of the neutron star stays close to 2M⊙, whether we use the
nucleon-only EoS or hyperons-included EoS. Therefore, in agreement with
Strumia [23], we find that without any need for additional self repulsion
between the dark matter particles, the NS can maintain a maximum mass in the
region of 2M⊙, which is our maximum mass constraint.

Figure 2 tests the hypothesis of neutron decay against the tidal deformability
constraint [80, 81, 82], namely that a neutron star of mass 1.4 M⊙ should have
a tidal deformability in the range $70\leq\Lambda\leq 580$ . For both EoS the
value of tidal deformability of neutron star after the decay decreases
slightly, clearly satisfying this tidal deformability constraint.

Figure 1: Total mass versus radius of a NS, at T = 0 ${}^{\circ}K$, before and
after the neutrons decay. Figure 2: Total mass versus tidal deformability of
neutron star before and after the decay of neutrons.

Figure 3 shows that in the lighter neutron stars the fraction of the matter
converted to dark matter is quite small, although it increases significantly
in heavier neutron stars. In maximum mass neutron stars produced by both
hadronic EoS, we find around 4% of the total mass contributed by the dark
matter.

Figure 3: Total number of baryons versus total number of dark matter
particles. Each dark matter particle has baryon number $\frac{1}{3}$.

Figure 4 illustrates the moment of inertia for both hadronic EoS. Clearly the
neutron decay leads to a decrease in the moment of inertia of the NS and
therefore the spin of neutron star must increase after the decay.

Figure 4: Total mass versus moment of inertia of neutron star before and after
the neutrons decay. Figure 5: Distribution of nuclear matter and dark matter
inside the neutron star at T = 0 ${}^{\circ}K$ after the neutrons decay for
nucleons only EoS.

Figure 5 displays the distribution of energy density of the nuclear matter and
dark matter from the centre of the neutron star to the surface. Although we
only show the nucleon-only result, the hyperons-included case is essentially
identical. Both selected EoS, be it nucleon-only or hyperons-included, show
that the degenerate dark matter remains inside and does not envelop the the
neutron star. Most of the dark matter is trapped inside the inner core of the
NS.

As the neutron decay takes place the total energy and the total baryon number
inside the NS must remain conserved. The values given in Table 1 correspond to
a 1.8 M⊙ NS and show that for both EoS, if the total baryon number is kept
fixed, then after the decay the mass of the neutron star is lighter than
before, by approximately 0.007M⊙. Since neutron stars are very compact objects
and have an escape velocity comparable to the speed of light, there is no
mechanism that can give particles enough velocity to escape the star following
the decay.

The solution to this problem is that the dark matter will not be completely
degenerate but have effectively a finite temperature. The hot dark matter
inside the neutron star then contains the energy equivalent to the difference
of masses of neutron star before and after the decay, that is 0.007M⊙. We
calculated the NS properties with hot dark matter having heat energy
equivalent to 0.007M⊙ and they are summarised in Table 2. The dark matter
inside the neutron must have a temperature of approximately 6.5 MeV and 6 MeV
for nucleon-only and hyperons-included EoS, respectively. As there is no
significant change in the radius of the NS associated with dark matter
heating, the moment of inertia of the star at 6 MeV or 6.5 MeV remains almost
the same as the moment of inertia at 0${}^{\circ}K$. However, because of the
decay the change in radius of the neutron star is significant and the total
change in moment of inertia is approximately 5.75%. As a result the neutron
star must spin up during the decay.

| NS Mass | T (MeV) | R (Km) | I (kg.m2)
---|---|---|---|---
1 | 1.8 M⊙ (Nucleon only EoS before decay) | 0 | 12.25 | 2.25 $\times$ 1038
2 | 1.793644 M⊙ (Nucleon only EoS + DM after decay) | 0 | 11.96 | 2.12 $\times$ 1038
3 | 1.8 M⊙ (Hyperons included EoS before decay) | 0 | 12.20 | 2.24 $\times$ 1038
4 | 1.793577 M⊙ (Hyperons included EoS + DM after decay) | 0 | 11.96 | 2.11 $\times$ 1038
Table 1: Properties of a 1.8M⊙ neutron star with total baryon number 2.4445
$\times 10^{57}$ at temperature 0 MeV, before and after the neutrons decay.
Here, T stands for the temperature and M represents the NS mass, while R and I
denote the radius and the moment of inertia of the neutron star.

M⊙ | T (MeV) of DM | R (Km) | I (kg.m2)
---|---|---|---
1.8 M⊙ (Nucleon only EoS before decay) | 0 | 12.25 | 2.25 $\times$ 1038
1.793644 M⊙ (Nucleon only EoS + DM after decay) | 6.5 | 11.96 | 2.124 $\times$ 1038
1.8 M⊙ (Hyperons included EoS before decay) | 0 | 12.20 | 2.24 $\times$ 1038
1.793577 M⊙ (Hyperons included EoS + DM after decay) | 6 | 11.96 | 2.115 $\times$ 1038
Table 2: Properties of a 1.8M⊙ neutron star with total baryon number 2.4445
$\times 10^{57}$, before and after the neutrons decay. Here, T stands for the
temperature of the dark matter and M represents the NS mass, while R and I
denotes the radius and the moment of inertia of the neutron star.

##  5 Conclusion

From Fig. 1 it is evident that the decay of a neutron into three dark matter
particles is consistent with the neutron star maximum mass constraint, without
any need for dark matter self-repulsion. Whether it is hyperons or nucleons in
the core, the dark matter can sustain the neutron star against gravitational
collapse and give a NS with a maximum mass of approximately 2 $M_{\odot}$.
Figure 2 shows that the neutron decay into dark matter is also consistent with
the tidal deformability constraint.

As illustrated by Fig. 3 the presence of dark matter increases rapidly in
heavier neutron stars, suggesting that we should investigate heavier neutron
stars to study this mode of neutron decay. After the decay the moment of
inertia of the neutron star is reduced, as illustrated in Fig. 4, which
requires that after the decay the NS should spin up. From Fig. 5 we see that
the dark matter remains inside the NS, with most of the dark matter inside the
inner core.

From the conservation of total baryon number and total energy, Tables 1 and 2
indicate that for the decay to take place the dark matter inside the neutron
must heat up by around 6 MeV. Although the heating of dark matter does not
cause a significant change in radius or moment of inertia compared to the
neutron star properties at 0${}^{\circ}K$, the total change in radius and
moment of inertia before and after the decay is significant. Therefore, if
neutrons decay into dark matter in this way, the NS must spin-up. Whether or
not such a spin-up can be observed is a challenge for future work.

## Acknowledgements

This work was supported by the University of Adelaide International
Scholarship and by the Australian Research Council through grants CE200100008
and DP180100497.

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