Constraining dark boson decay using neutron stars¹¹1Invited Review Article for a Special Edition on Neutron decay anomalies for UNIVERSE.

Wasif Husain, Dipan Sengupta and A W Thomas ARC Centre of Excellence for Dark Matter Particle Physics, Department of Physics, The University of Adelaide, SA 5005, Australia. [email protected],[email protected],[email protected]

Abstract

Inspired by the well known anomaly in the lifetime of the neutron, we investigate its consequences inside neutron stars. We first assess the viability of the neutron decay hypothesis suggested by Fornal and Grinstein within neutron stars, in terms of the equation of state and compatibility with observed properties. This is followed by an investigation of the constraint information on neutron star cooling can place on the decay rate of the dark boson into standard model particles, in the context of various BSM ideas.

I Introduction

Neutrons are a fundamental constituent of our universe. It has been over almost a century since they were discovered but their lifetime Wietfeldt and Greene (2011) still presents a challenging problem to solve. In particular, current experiments appear to show a difference in the neutron lifetime when measured with different methods. In the bottle method Serebrov et al. (2018); Pattie et al. (2018); Tan (2019); Steyerl et al. (2012) the neutrons are trapped and the number counted after a fixed time, with no specific determination of the decay mode. In contrast, using beam method Yue et al. (2013); Otono (2017); Olive (2016) one actually observes the protons produced in $\beta$ decay. Of course, the lifetime of the neutron should be the same, regardless of the method of measurement. However, the lifetime of the neutron shows a discrepancy. Using the bottle method, Ref. Gonzalez et al. (2021) found the lifetime of the neutron to be 877.75 $\pm 0.28_{\rm stat}+0.22-0.16_{\rm syst}$ s, which is very close to the lifetime measured in Ref. Pattie et al. (2018) and Ref. Serebrov et al. (2018). On the other hand, using the beam method the lifetime measured has been measured to be 887.7 $\pm 0.7\,(stat)\,+0.4/-0.2\,(sys)$ s Pattie et al. (2018).

A resolution of this discrepancy in the lifetime of the neutrons could potentially lead to new physics. A solution along those lines was recently proposed by Fornal and Grinstein Fornal and Grinstein (2018); Grinstein et al. (2019); Fornal and Grinstein (2020), who proposed an extra decay channel of the neutron into dark matter. Based on the difference in lifetimes, they suggested that roughly 1% of the time neutrons decay into dark matter, wile for the remaining 99% of the time they undergo $\beta$ decay. In the beam method the dark matter would go undetected and uncounted, while in the bottle method the effect of dark matter is automatically included.

According to the hypothesis of Fornal and Grinstein the dark decay mode of the neutron is

n\longrightarrow\chi+\phi\,,

(1)

where $\chi$ is the dark fermion and $\phi$ the dark boson. This hypothesis has attracted the interest of many physicists. For example, experimental studies showed very quickly that the $\phi$ particle could not be a photon Tang et al. (2018); Serebrov et al. (2008). The hypothesis was also very rapidly subject to tests using the properties the neutron stars, which indicated that the dark fermions, $\chi$ , has to experience a strong vector repulsion in order to be consistent with the observations Motta et al. (2018a, b); Baym et al. (2018); McKeen et al. (2018). A recent study Husain et al. (2022) suggested that there might be an observable signal of this decay if one could observe the neutron star right after its birth. An interesting discussion about the neutron decay can be found in Ivanov et al. (2018).

An alternative to the Fornal and Grinstein hypothesis was proposed by Strumia in Ref. Strumia (2021), where the author suggested that the neutrons might decay into three identical dark fermions, $\chi$ ,

n\longrightarrow\chi+\chi+\chi,

(2)

with each of them having baryon number 1/3 and mass $m_{\chi}$ = (mass of neutron)/3. An advantage of this proposal was that, to be consistent with the constraint on the maximum mass of neutron stars, the dark fermions, $\chi$ , are not required to be self-interacting. Our recent study in Ref. Husain and Thomas (2022) on the Strumia hypothesis agrees with this claim and indicated that one could find observable signals of neutron decay similar to those found within the Fornal and Grinstein hypothesis.

Within the Fornal and Grinstein hypothesis, shown in Eq. (1), the mass of the decay products must be in the range 937.9 MeV ¡ $m_{\chi}+m_{\phi}$ ¡ 938.7 MeV for the known stable nuclei to remain stable Fornal and Grinstein (2018); Grinstein et al. (2019). To date, most of the studies on the hypothesis using neutron stars, have considered the $\phi$ as an extremely light particle which escapes the neutron star immediately and treating the $\chi$ as almost degenerate with the neutron. There is a possibility that if $\phi$ boson has a mass close to the difference of the masses of the neutron and the $\chi$ , for example of order 1 MeV, then it may remain trapped inside the neutron star.

Here, the focus is on trapped $\phi$ bosons and their effects on neutron star heating. This leads to a strong constraint on the lifetime of the $\phi$ , which is then compared with limits from other studies of dark matter candidates. The manuscript is divided into sections as follows. Section II covers the necessary model for the equation of state of nuclear matter inside the neutron star and explains the change associated with neutron decay into $\chi$ and $\phi$ . This is followed by section III, where the consequence of trapping the $\phi$ boson are explored. In section IV the decay modes of the $\phi$ boson have been studied in detail. Finally, section V presents a summary of our findings.

II Neutron stars

Neutron stars are comparatively small objects that come into existence when an ordinary star of mass 8 M_⊙ - 15 M_⊙ dies. Neutrons are not surprisingly the dominant component of a neutron star and if neutrons decay into $\chi$ and $\phi$ then this decay must also take place inside the neutron star. Therefore, the neutron stars must contain $\chi$ fermions and in the circumstances explained earlier also $\phi$ bosons, and their presence inside neutron stars must change their properties Mukhopadhyay et al. (2017); Bertone and Fairbairn (2008); Kouvaris (2008); Ciarcelluti and Sandin (2011); Sandin and Ciarcelluti (2009); Leung et al. (2011); Ellis et al. (2018); Bell et al. (2020); Husain and Thomas (2021); Mielke and Schunck (2000); Blinnikov and Khlopov (1983); McKeen et al. (2018); Horowitz and Reddy (2019); Bertoni et al. (2013); Berryman et al. (2022); McKeen et al. (2021); de Lavallaz and Fairbairn (2010); Busoni (2021); Sen and Guha (2021); Guha and Sen (2021). There are some strong constraints on the properties of neutron stars imposed by observations that a realistic neutron star model must follow. For example, Ref. Abbott et al. (2018) showed that a neutron star of mass 1.4 M_⊙ should have a radius 10 - 14 km. PSR J1614-2230 Demorest et al. (2010) and PSR J0348+0432 Antoniadis et al. (2013) have masses 1.928 and 2.01 M_⊙, respectively, so the neutron star model must predict maximum mass of neutron stars at least 2 M_⊙ ²²2Note that recent observations of pulsars PSR J0030+0451Riley et al. (2019), and PSR J0740+6620 Riley et al. (2021) constrain their masses and radii to be 1.34 M_⊙ and 12.71 km and 2.072 M_⊙ and 12.39 km respectively. See also Miller et al. (2019, 2021). . The tidal deformability should be consistent with the discovery Abbott et al. (2017, 2019) of gravitational wave detection by LIGO and VIRGO observatories. Therefore neutrons stars can be very helpful in testing the Fornal and Grinstein hypothesis.

Neutron star interiors covers a wide range of densities right from the surface to the core Lawley et al. (2006); Whittenbury et al. (2014, 2016); Bodmer (1971); Witten (1984); Bombaci et al. (2004); Ren and Zhang (2020); Terazawa (1989); Bednarek et al. (2012); Vidaña (2015); Oertel et al. (2016); Akmal et al. (1998); Balberg and Gal (1997); Glendenning (1985); Kaplan and Nelson (1986); Glendenning et al. (1997); Glendenning and Moszkowski (1991); Glendenning (1997); Haensel and Zdunik (2017); Shuryak (1980); Weber et al. (2007); Spinella and Weber (2019); Weber (2016); Terazawa (2001); Husain and Thomas (2021); Lattimer and Prakash (2001); Zhao and Lattimer (2020); Drischler et al. (2021); Cierniak and Blaschke (2021); Shahrbaf et al. (2022); Nishizaki et al. (2002); Yamamoto et al. (2017, 2022); Motta and Thomas (2022), with the cores containing the most dense matter in the universe. To model the neutron star, one needs to adopt a suitable equation of state for the nuclear matter. At the core of a neutron star the density could be as high as 6 times the density of normal matter. Therefore, one needs to chose a model capable of describing the physics at such high densities. In this study quark meson coupling (QMC) model Guichon (1988); Guichon et al. (1996); Stone et al. (2016); Rikovska Stone et al. (2007) is adopted to model the neutron star matter. A brief description of the QMC model is presented below.

II.1 Quark Meson Coupling model

The quark meson coupling model was initially proposed by Guichon Guichon (1988) and further developed by Guichon, Thomas and collaborators Guichon et al. (1996); Saito et al. (2007). In this model the nucleons are treated as a collection of 3 quarks confined in an MIT bag DeGrand et al. (1975). The internal structure of the nucleon is treated with great importance, unlike other models where nucleons are considered as a point like objects. In the QMC model the interaction between baryons is generated by the exchange of mesons, which couple self-consistently to the confined quarks. The strong scalar mean field in particular drives significant changes in the structure of the bound baryons.

The equation of state based on the QMC model has shown been shown to lead to an acceptable description of neutron star properties Rikovska Stone et al. (2007); Motta and Thomas (2022); Husain and Thomas (2022). The effective mass of the nucleon in-medium may be expressed in terms of the scalar polarisability, ’ $d$ ’, the mass of the free nucleon, $M_{N}$ , and the coupling constant of the $\sigma$ field to the nucleon in free space, $g_{\sigma}$ , as

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

(3)

The details of QMC model can be found in Refs. Guichon (1988); Guichon et al. (1996, 2018). For simplicity, in this study it is assumed that neutron stars do not contain hyperons or strange matter at the higher energy densities, a nucleon only equation of state is used Husain et al. (2022).

II.2 Formalism including neutron decay

According to the hypothesis given in Eq. (1), the neutron stars must contain $\chi$ and $\phi$ . As mentioned above there is no a priori constraint on the mass of the $\phi$ , within the small window allowed. We have chosen to study the case $m_{\phi}$ = 1 MeV, $m_{\chi}$ = 937.7 MeV in this work, where $m_{\phi}$ is the mass of $\phi$ boson and $m_{\chi}$ the mass of the $\chi$ fermion. In this case the velocity of the $\phi$ boson is sufficiently low that it will be trapped inside the neutron star.

Inside the neutron star the $\phi$ bosons must condense, according the Bose-Einstein condensation theory Husain et al. (2022); Motta et al. (2019). But the total contribution of the $\phi$ will be far too small to make any significant changes in the existing mass, radius, and tidal deformability constraints, even after condensation. In fact, we will see in later sections that the contribution of the $\phi$ bosons to the total mass is only about 1/10⁶ M_⊙. Although this is small, nevertheless amount of mass may contribute to the heating of the neutron star if the $\phi$ bosons decay into standard model particles. Therefore the focus of this study is on the $\phi$ boson decay into standard model particles inside neutron stars.

The equations are solved using Hartree-Fock approximation. The full Hartree-Fock terms can be found in Husain et al. (2022); Motta et al. (2019); Krein et al. (1999). Although the $\phi$ bosons will be in the lowest possible quantum state, the dark fermions will constitute a gas of fermions. The dark fermions are assumed to be self-interacting, in order to survive against the observational constraints on the neutron star properties. The self-interaction of the dark fermions is assumed to be similar to the neutron- $\omega$ interaction.

The presence of $\chi$ and $\phi$ inside the neutron stars changes their composition. Therefore, the chemical equilibrium equations are Motta et al. (2018b, a)

\mu_{n}=\mu_{\chi}+m_{\phi}\qquad\mu_{n}=\mu_{p}+\mu_{e}\qquad\mu_{\mu}=\mu_{e}\qquad n_{p}=n_{e}+n_{\mu}\,\qquad

(4)

where $\mu$ represents the chemical potential of the different associated particles, and $n_{p}$ , $n_{p}$ and $n_{e}$ stand for the number of neutrons, protons, electrons. The dark fermions, dark bosons and nuclear matter particles are assumed to not interact with each other. Therefore, the neutron star contains two non-interacting fluids. However, because the contribution of the dark matter compared to nuclear matter is very small, using the two-fluid TOV equation is not necessary. Therefore for ease of the calculation one fluid TOV is used.

II.3 Tolman Oppenheimer Volkoff (TOV) equations

To calculate the properties of the neutron star the equation of state is combined with the structural equations derived using the Einstein’s equations of general relativity. Therefore, the TOV equations Tolman (1934); Oppenheimer and Volkoff (1939); Ciarcelluti and Sandin (2011); Sandin and Ciarcelluti (2009) given as

\frac{dP}{dr}=-\frac{[\epsilon(r)+P(r)][4\pi r^{3}(P(r)+P(r))+m(r)]}{r^{2}(1-\frac{2m(r)}{r})},

(5)

m(r)=4\pi\int_{0}^{r}dr.r^{2}\epsilon(r),

(6)

are integrated from the centre of the neutron star towards the surface, using the boundary conditions that at the surface the pressure and energy density should be zero. Here, $P$ is the total pressure, $P=P_{nucl}+P_{DM}$ and $\epsilon=\epsilon_{nucl}+\epsilon_{DM}$ is the total energy density, including the energy density of nuclear matter and dark matter. The tidal deformability is calculated by using the method explained in Ref. Hinderer (2008); Hinderer et al. (2010).

III Results

In this section the consequences of the neutron decay on the properties of the neutron stars are given. The vector interaction of dark fermions is increased until it follows the constraint Motta et al. (2018b); Grinstein et al. (2019); Baym et al. (2018); Cline and Cornell (2018); Berryman et al. (2022); Strumia (2021); Rajendran and Ramani (2021); Tang et al. (2018); Berezhiani et al. (2021) on the properties of the neutron stars.

As shown in Fig. (1), the mass of the neutron star is reduced after the neutron decay. In fact, the maximum mass of the neutron star falls below 2 M_⊙ after the neutron decay Motta et al. (2018a); Baym et al. (2018); McKeen et al. (2018) if the dark fermions are considered to be non-self-interacting. However, neutron stars of mass above 2 M_⊙ Özel and Freire (2016); Demorest et al. (2010); Antoniadis et al. (2013) have been observed. Therefore, in order to survive, a neutron star model must predict neutron stars of maximum mass of at least 2 M_⊙. Fig. (1) indicates that the dark fermions must have self-repulsion with a strength parameter of order 26 fm² to be consistent with the observations. Moreover, there is a significant reduction in the radius of the neutron stars after the decay, which suggest that neutron stars should spin up during the decay.

Figure (2) shows the tidal deformability against the radius of the neutron star. The analysis of the gravitational waves Abbott et al. (2017, 2019); Bramante et al. (2018) indicated that a neutron star of mass 1.4 M_⊙ must have tidal deformability in the range 70 - 580, with 90% confidence level. Fig. (2) shows that dark fermions with a vector self-repulsion of strength 26 fm² satisfy the constraint on mass and tidal deformability.

Refer to caption — Figure 1: The total mass (given in solar masses) vs radius of the neutron star. Here $G$ is the $\chi$ - $\chi$ vector repulsion strength which is increased to satisfy the observational constraints.

Figure 3 shows the moment of inertia against the mass of the neutron star. The moment of inertia is reduced after the neutron decay. When the dark fermion vector-interaction is lowered the difference in moment of inertia increases. Thus we are only interested in the case when the dark fermions, $\chi$ , have a vector interaction strength $\geq$ 26 fm². Figure 3 indicates that the moment of inertia of heavier neutron stars is significantly reduced even when $G\geq$ 26fm², which should result in spinning up of the neutron star. That in turn may provide a signal of the neutron’s exotic decay.

Most studies indicate that neutron stars cool down very quickly by the standard Urca process. After approximately a million years the neutron stars have a luminosity of order 10^31.5 erg/s. Therefore, if the $\phi$ boson decays into photons it will contribute to the heating of the neutron star and after a million years it must not contribute a luminosity $\geq$ 10^31.5 erg/s. Based on the luminosity after 1 million years, we find that the lifetime ( $\tau$ ) of the $\phi$ bosons must be greater than 1.85 $\times$ 10¹¹ years. With such a long lifetime the luminosity stays essentially constant.

In the next section the consequences of $\phi$ boson decay into standard model particles are explored.

IV Decay modes of $\phi$ bosons

The decay products of the neutron, the $\chi$ and $\phi$ are BSM particles (particles beyond the Standard Model) that can originate from some UV complete theory or be considered within some low energy effective theory. While remaining agnostic about their origin we can comment on the constraints on them from a variety of sources. The massive fermion, $\chi$ , is an ideal candidate for dark matter and can form the bulk or all of the observed relic density today. We leave a detailed discussion on the details of this mechanism for a later expanded work. The other product of the decay, the boson, can originate from a BSM source. On general grounds and experimental considerations the possibility that the boson is a photon has been ruled out. Here we consider some simple possibilites for the bosons to couple SM particles, and constraints on the basis of findings in the previous sections.

IV.1 Scalars and Pseudoscalars

In the last few years light scalar and pseudo-scalar particles have emerged as leading new physics candidates that can be constrained from a variety of sources. While the primary motivation is derived from axions, simplified models with light scalars or pseudo-scalars have triggered a lot of attention. Here we assess their viability given our findings above.

The first bosonic candidate is a scalar coupled to the electromagnetic field strength,

\mathcal{L}_{int}=\frac{C_{s}}{\Lambda}\phi F_{\mu\nu}F^{\mu\nu}+\frac{m_{f}}{\Lambda}\phi\bar{f}f+\cdots

(7)

where $\phi$ is the scalar field, $F_{\mu\nu}$ the electromagnetic field strength, and $f$ the Dirac spinor for the leptons. The overall normalization $\frac{C_{s}}{\Lambda}$ is model dependent, while $m_{l}$ is the mass of the lepton. The linear couplings can be generated by from the scalar coupling to Higgs, as $\phi H^{\dagger}H$ . A quadratic coupling can also be generated if $\phi$ carries a $Z_{2}$ symmtery,

\mathcal{L}=\frac{C_{q}}{\Lambda_{q}^{2}}\phi^{2}F_{\mu\nu}F^{\mu\nu}+\sum_{f}\frac{m_{f}}{\Lambda_{q}^{2}}\phi^{2}\bar{f}f+\cdots

(8)

In both cases the dots indicate any other couplings that may be induced.

Couplings to the neutron can be obtained by integrating out, for example, heavy fermions yielding dimension 6 operators, such that the effective neutron coupling can be written as,

\mathcal{L}\in L_{kin}+\lambda_{eff}n\chi\phi\,.

(9)

The linear (and quadratic) couplings induce a shift in the electromagnetic couplings that can be constrained from a variety of sources. A summary of these can be found in Ref. Antypas et al. (2022).

The next possibility is that of a pseudoscalar that couples like an axion (like particle) to photons, and derivatively to electrons

\mathcal{L}_{int}=\frac{C_{s\gamma}}{\Lambda}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{C_{f}}{2\Lambda}(\partial_{\mu}\phi)\bar{f}\gamma^{\mu}\gamma^{5}f+\cdots

(10)

For axion-like particles in Eq. 10, the effective ALP coupling to leptons generates a coupling,

\frac{C_{f}}{2\Lambda}(\partial_{\mu}\phi)\bar{f}\gamma^{\mu}\gamma^{5}f=-\frac{C_{f}m_{f}}{\Lambda}i\bar{f}\gamma^{\mu}\gamma^{5}f+\cdots

(11)

where the dots indicate terms proportional to $F\tilde{F}$ . The decay widths to charged fermions are given by,

\Gamma(a\to f\bar{f})=\frac{m_{f}^{2}m_{a}|C_{f}^{2}|}{4\pi\Lambda^{2}}\sqrt{1-\frac{4m_{f}^{2}}{m_{a}^{2}}}\,.

(12)

Analogous to scalars the effective ALP coupling to neutrons can be written as,

\mathcal{L}\in L_{kin}+\lambda_{eff}n\chi\gamma^{5}\phi\,.

(13)

A comprehensive account of UV complete models and their phenomenological consequences is left for future work. In principle, since the bosons in our case are heavy, the most general Lagrangian will contain interaction terms involving not only photons and leptons, but hadrons as well.

The decay widths for pseudoscalars to diphotons are given by,

\Gamma(a\to\gamma\gamma)=\frac{|C_{\gamma}^{2}|}{4\pi\Lambda^{2}}m_{a}^{3}\,.

(14)

The lifetime is

\tau(a\to\gamma\gamma)=1/\Gamma(a\to\gamma\gamma\times f)

(15)

where $f$ is the conversion factor from $\rm GeV^{-1}$ to seconds. From the estimates derived above, for a boson mass of 1 MeV, a lifetime of $\tau\geq\simeq 10^{11}$ years, and if this is only decay channel relevant, the effective coupling, $g_{eff}=C_{\gamma}/\Lambda\leq 10^{-17}$ . Note that a lifetime of $10^{11}$ years is about $10^{18}$ seconds. The lifetime of the universe is about $10^{18}$ seconds and therefore this boson is cosmologically stable and should add to the total relic density of the universe. The exact amount of dark matter density depends on the co-efficient $C_{f}$ , as well as the decay constant $\Lambda$ . Typically in axion like models, like the ones considered here, we can obtain a significant fraction of the dark matter with a $\mathcal{O}$ (1) misallignement angle.

There are however significant constraints of models of this class. For scalars and pseudoscalars, one of the strongest constraints at this mass originates from the consideration that photons produced during ALP decays when the Universe is transparent should not exceed the total extragalactic background light (EBL) Cadamuro and Redondo (2012). For pseudoscalar ALPs, this limits lifetimes to $\tau\geq 10^{23}$ seconds, such that the effective ALP coupling is restricted $g_{eff}\leq 10^{-19}$ . Furthermore X-rays produced from ALP decays in galaxies must not exceed the known backgrounds. This limits $\tau\geq 10^{25}$ seconds leading to an effective coupling $g_{eff}\leq 10^{-20}$ Cadamuro and Redondo (2012).

IV.2 Spin-1

While the decay to a photon has been ruled out, a possible solution is that the spin-1 boson can be a dark (kinetically) mixed photon. The massless part of the most general theory of two $U(1)_{a,b}$ Abelian gauge bosons can be written as,

\mathcal{L}=-\frac{1}{4}F_{a\mu\nu}F^{\mu\nu}_{a}-\frac{1}{4}F_{b\mu\nu}F^{\mu\nu}_{b}-\frac{\varepsilon}{2}F_{a\mu\nu}F^{\mu\nu}_{b}

(16)

The masses of these can be obtained via a Stuckelberg mechanism, or via a spontaneously broken gauge symmetry

\mathcal{L}_{m}=\frac{1}{2}M_{a}^{2}A_{\mu}^{a}A^{a\mu}+\frac{1}{2}M_{b}^{2}A_{\mu}^{b}A^{b\mu}+M_{a}M_{b}A_{\mu}^{a}A^{\mu}_{b}

(17)

Consider a hypercharge mixing with the usual photon,

\mathcal{L}=\frac{\epsilon}{2\cos{\theta}_{W}}\tilde{F}^{{}^{\prime}}_{\mu\nu}B^{\mu\nu}\,.

(18)

Then, the effective Lagrangian becomes,

\mathcal{L}\in e\epsilon J_{\mu}A_{\mu}^{{}^{\prime}}+e^{\prime}\epsilon\tan{\theta}_{W}J_{\mu}^{{}^{\prime}}Z_{\mu}+e^{\prime}J_{\mu}^{{}^{\prime}}A_{\mu}^{{}^{\prime}}\,,

(19)

where $J_{\mu}^{{}^{\prime}}$ and $e^{\prime}$ are the dark sector current and the dark photon coupling to the dark sector. Once the $Z$ boson is integrated out we can see that the coupling of the dark photon to SM fermions is proportional to $e\epsilon$ , i.e., millicharged dark photons which are constrained from various sources. The effective coupling to neutrons can be written as,

\mathcal{L}\in e\epsilon(n\sigma^{\mu\nu}\chi F^{{}^{\prime}}_{\mu\nu})\,.

(20)

Below the two electron threshold, the constraints on dark photons originate from stellar cooling bounds and from the Xenon-1T experiments Caputo et al. (2021); Aprile et al. (2022). The constraints on the kinetic mixing parameter is $\epsilon\leq 10^{-13}$ for $m_{A^{\prime}}\simeq 1MeV$ .

If the dark photon is extremely light, if produced non-thermally like a condensate, like the axion with a misallignement mechanism. In this case the mass is generated by the Stuckelberg mechanism, and the generation of the relic follows like the usual axion.

Additionally, in most relevant models, the dark photon is accompanied to dark fermions. Here dark photon can account for the relic density through a freeze-in mechanism within the dark sector or through feeble couplings to SM.

V Conclusion

In this work, we explored the consequences of neutron decays into a dark sector inside neutron stars. Working on the hypothesis that $n\to\chi+\phi$ , we analyzed the feasibility of this decay by studying the effect of the corresponding equation of state on the properties of the neutron star, including its mass, radius and tidal deformability. We then focused on the possibility that the $\phi$ remains trapped inside the star leading to heating. We concluded that if the $\phi$ has a mass of around 1 MeV, based upon observations of luminosity of stars as a function of age, the $\phi$ must have a lifetime greater than $10^{11}$ years. Finally we studied the consequences and of a $\phi$ coupled to standard model particles within simplified ALP like models. An expanded work with cosmological consequences, as well as a study of UV completions, is left for future work.

Acknowledgements.

This study has been supported by a University of Adelaide International Scholarship (WH) and by the Australian Research Council through the ARC Centre for Dark Matter Particle Physics (CE200100008).

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Constraining dark boson decay using neutron stars111Invited Review Article for a Special Edition on Neutron decay anomalies for UNIVERSE.