Dark decay of the neutron

New decay channels for the neutron into dark matter plus other particles have been suggested for explaining a long-standing discrepancy between the neutron lifetime measured from trapped neutrons versus those decaying in flight. Many such scenarios are already ruled out by their effects on neutron stars, and the decay into dark matter plus photon has been experimentally excluded. Here we explore the decay into a dark Dirac fermion $\chi$ and a dark photon $A'$, which can be consistent with all constraints if $\chi$ is a subdominant component of the dark matter. Neutron star constraints are evaded if the dark photon mass to coupling ratio is $m_{A'}/g' \lesssim (45-60)\,$MeV, depending upon the nuclear equation of state. $g'$ and the kinetic mixing between U(1)$'$ and electromagnetism are tightly constrained by direct and indirect dark matter detection, supernova constraints, and cosmological limits.


I. INTRODUCTION
Recently a long-standing experimental discrepancy in the lifetime of the neutron has been highlighted: the decay rate of neutrons stored in a bottle [1][2][3][4][5] seems to be larger than that determined by detecting the decay products from neutrons in flight [6,7]. The difference in the associated lifetimes is ∼ 8 s with a 4σ tension. Ref. [8] pointed out that this could be explained by a hitherto undetected extra decay channel into dark particles, if the neutron mixes with fermionic dark matter (DM) χ whose mass is in a narrow range m p − m e < m χ < m n consistent with proton stability. The simplest decay channel, n → χγ, comes from the transition magnetic moment interactionnσ µν χF µν induced by the n-χ mixing. This has already been ruled out by a null search for the monochromatic photon [9]. (It was also argued that any such solution of the lifetime problem would exacerbate tensions in discrepant determinations of the neutron axial-vector coupling g A [10]).
Even if the new decay channel is completely invisible, for example n → χφ where φ is a dark scalar, neutron stars ostensibly rule out the scenario [11][12][13], since χ typically has a much softer equation of state than nuclear matter at high density. The conversion of n to χ in a neutron star (NS) then leads to the maximum possible mass of the star being well below the largest values observed (near 2M ). Ref. [11] pointed out that this constraint can be evaded if χ has additional pressure from the repulsive self-interactions that would come from dark photon (A ) exchange, if the mass-to-coupling ratio m A /g is sufficiently large, though the minimum required value was not determined. One of our goals is to find this value.
This suggests that a consistent picture can be made if the new dark decay channel is n → χA . Astrophysical and cosmological considerations substantially constrain DM in the mass range required to explain the neutron lifetime anomaly. However, χ could still be a subdomi-nant component of the total dark matter.
In the following, we start by deriving constraints on our scenario that depend only upon the low-energy particle content at the GeV scale or below, in Section II. This is followed by analysis of the effects upon neutron stars, Sec. III, where an upper bound on m A /g is derived.
We introduce the microscopic renormalizable model in Sec. IV and discuss the implications for cosmology in Sec. V. The constraints from direct and indirect detection are derived in Sections VI and VII. Various constraints on the kinetic mixing parameter between the U(1) and electromagnetic U (1) field strengths are compiled in Sec. VIII, followed by our conclusions, Sec. IX.

II. MODEL-INDEPENDENT ANALYSIS
Before presenting a more complete particle physics model, we start with the low-energy effective Lagrangian that contains almost everything needed for the phenomenology. It has mixing of the Dirac χ with the righthanded component of the neutron (this restriction does come from UV considerations, to be explained later) and the usual terms for χ interacting with a dark photon A : where D µ = ∂ µ − ig A µ and µ n is the neutron magnetic dipole moment. Kinetic mixing with the photon is included so that A will eventually decay to photons through a loop diagram, or electrons if m A > 2m e . We assume that χ is a Dirac particle, since UV models in which χ is Majorana generically lead to dinucleon decays such as 16 O(pp) → 14 Cπ + π − [14]. Diagonalization of the n-χ mass matrix requires rotations of (n R , χ R ) and (n L , χ L ) respectively by angles θ R and θ L with Since m χ is close to m n , these angles are roughly equal, θ = θ R ∼ = θ L ∼ = δm/(2(m n − m χ )). Going to the mass basis, theχ / A χ interaction then leads to an off-diagonal term, that leads to the dark decay n → χA assuming that m n > m χ + m A . Since A is unstable because of kinetic mixing with the photon, we assume that it is either sufficiently long-lived to escape detection, or its decay products e + e − or 3γ have not yet been excluded by observations of neutron decay. The decay rate for n → χA can be expressed in terms of the ratios x = (m n −m χ )/m n and r = m A /(m χ −m n ), where x ∈ [0, 1.78 × 10 −3 ] (the upper limit is imposed by the stability of 9 Be [8,15]) and r ∈ [0, 1]: where we have ignored negligible higher order corrections in x, and in the second line eliminated θ using (2). The limit m A → 0 is not singular, despite appearances, when A gets its mass from spontaneous symmetry breaking, as we explicitly show in section IV. g δm/m A remains finite in this limit.
To resolve the neutron lifetime discrepancy, the partial width must be 7 × 10 −30 GeV [8]. For definiteness, we adopt the reference value m χ = 937.9 MeV, which maximizes the dark decay rate with m n − m χ ∼ = 1.67 MeV. Moreover we consider two benchmark values of m A = 1.35 MeV and 0.5 MeV to illustrate the differences between being above or below the 2m e threshold for A decays. We refer to the two cases as scenarios A and B, respectively. We then find g δm m A ∼ = 7.2 (3.5) × 10 −13 , scenario A (B) (5) With the neutron star bound m A /g 60 MeV to be derived in section III, this implies δm < 4 (2) × 10 −11 MeV, which gives a limit on the mixing angle for the two scenarios. Although the decay χ → pe −ν e is kinematically allowed if m χ > m p + m e − = 938.8 MeV, such a small mixing endows χ with a lifetime well beyond the age of the universe. Therefore χ is always a component of the DM, even when it is not absolutely stable.
To satisfy the constraint on n → χγ from ref. [9], δm must be smaller by a factor of approximately 2.6 than the value needed for explaining the lifetime discrepancy in terms of this decay. 1 Using the value of δm determined 1 Ref. [9] does not quite exclude the model with minimum mχ = in [8], this leads to the limit δm 3.6 × 10 −11 MeV (7) which is similar to those coming from (5) combined with the bound on m A /g from neutron stars, that we discuss in the next section. This result can also be translated into to a limit on g using Eqn. 5:

III. NEUTRON STAR CONSTRAINTS
Although the mixing angle (2) between the neutron and χ is very small, it is sufficient to bring the two into chemical equilibrium in neutron stars [11]. If χ is noninteracting, then dense neutron matter converts to χ in order to lower the pressure. Integrating the Tolman-Oppenheimer-Volkoff equations [16,17] that determine the NS structure, one finds that the maximum NS mass attained as a function of its radius falls below the largest observed masses, close to 2M [18]. Ref. [11] notes that this can be avoided if m χ > 1.2 GeV, which would preclude the n → χ decays, or if the DM has repulsive selfinteractions, such as would arise from Coulomb repulsion if χ is charged under a U(1) gauge symmetry. If the U(1) is spontaneously broken as we assume, so the gauge boson is massive, this reduces the pressure since the range of the interaction becomes m −1 A . We would like to know 937.9 that we consider, which would lead to a photon of energy 1.66 MeV. The maximum photon energy excluded in [9] is 1.62 MeV. We assume that this narrow window will be closed by a wider search.
how large m A /g can be while still attaining a 2 solar mass neutron star. We have followed the procedure outlined in [11] to answer this question. The DM is treated as a degenerate Fermi gas with self-interactions, so that its energy density is given by where k F is the Fermi momentum and n χ = k 3 F /3π 2 . k F is determined by the chemical potential The DM pressure is given by p χ = −ρ χ +µ χ n χ . Chemical equilibrium implies that µ χ equals the baryon (neutron) chemical potential µ B , which is determined by the nuclear equation of state (EoS). χ will be produced at any radius within the NS where µ B > m χ . The DM contributions to pressure and energy density are added to those of the nuclear matter to find the modified EoS. The nuclear EoS can be specified in terms of the internal energy of a nucleon, E n (x), as a function of the dimensionless nuclear density x = n n /n 0 , where n 0 = 0.16 fm −3 is the saturation density. We adopt a polytropic EoS of the form [19] that has been fit to the results of quantum Monte Carlo calculations incorporating realistic 3-nucleon forces. The nuclear pressure is given by p n = n 2 n dE n /dn = n 0 (aαx α +bβx β ), and the energy density is ρ n = n n (m n + E n ). The chemical potential is then µ B = (p n + ρ n )/n n . This approximation is adequate at high densities near the center of the NS, but at some radius as the density increases, the effect of nuclear binding becomes important, allowing µ B to fall below m n (which otherwise would not happen) and also m χ . At this radius the χ density falls to zero and the dark matter can no longer soften the EoS. We model this effect by subtracting the binding energy E b = 8 MeV (which follows from the semi-empirical mass formula of nuclear physics [20]) from µ B .
Carrying out the above procedure we find the NS mass as a function of its radius for a range of central densities of the NS, to produce the mass-radius curves corresponding to different values of m A /g . The results are shown in fig. 1 for several equations of state from ref. [19]. The solid lines (labeled V 3π + V R ) correspond to a moderately stiff EoS with a (b) = 13.0 (3.21) MeV, α (β) = 0.49 (2.47), while the dashed ones (labeled UIX) pertain to a more stiff EoS with a (b) = 13.4 (5.62) MeV, α (β) = 0.514 (2.436). The dotted curves (labeled V P W 2π +V R µ=150 ) are for a soft EoS that cannot produce a 2M NS even in the absence of dark matter. To obtain a maximum mass compatible with 2M requires depending upon the nuclear EoS.

IV. UV MODEL
Direct and indirect signals of subdominant χ dark matter can depend upon details of the particle physics beyond the low-energy effective description (1). In particular a light scalar particle must be in the spectrum, to account for the mass of A through the Higgs mechanism.
Here we build a minimal model that could consistently give rise to the Lagrangian (1). It is strongly constrained by the gauge symmetries, giving us little freedom in its construction. We demand that the U(1) gauge symmetry be only spontaneously broken, requiring a scalar φ that carries U(1) charge, with a potential Moreover a renormalizable interaction of χ with a quark needs a scalar Φ 1 that is fundamental under SU(3) c and also charged under U(1) . To connect this to the other quarks in the neutron, a second triplet scalar Φ 2 , neutral under U(1) is needed, along with a cubic scalar interaction, where µ has dimensions of mass. To satisfy SU(2) L gauge symmetry we take the triplets to couple only to righthanded quarks, Φ * 1 , χ and φ all carry unit dark U (1) charge with coupling g . The vacuum expectation value (VEV) of φ breaks U (1) and gives the dark photon mass m A = g v .
We can consistently assign baryon number to all the new fields: Therefore the model is free from constraints on B-violating processes such as neutron-antineutron oscillations and dinucleon decay. The need to kinematically forbid decays of 9 Be (if m χ ≥ 937.9 MeV) also stabilizes the proton and forbids possibly problematic decays of Λ baryons to final states including a χ, such as Λ 0 → χK 0 , as well as decays of light mesons toχχ.
The virtual exchange of the heavy Φ 2 generates a contact interaction that can be Fierz-transformed to The coefficient is constrained by measurements of dijet angular distributions at the LHC [21], m Φ2 /λ 2 4 TeV. The scalar triplets, which could decay to two jets or a jet and missing energy, must be at the TeV scale to satisfy LHC constraints on their direct production. [22]. Integrating them out generates the effective interaction where β ∼ = 0.014 GeV 3 from lattice QCD [23], and the scalar masses saturate the ATLAS limit m Φi > 1.55 TeV. When φ gets its VEV, the off-diagonal mass δm = ηv between n R and χ L is generated. Since m A /g = v , we can combine (18) with (5) to determine If λ 1 ∼ λ 2 ∼ 0.4 (consistent with the previous LHC limits), then µ can be near the TeV scale, like the triplet scalar masses. But v must be at the much lower scale 60 MeV from (12).
The scalar φ cannot be arbitrarily heavy. Its mass is m φ = 2 √ λv hence m φ /m A = 2 √ λ/g . The upper limit from partial wave unitarity is λ ≤ 4π. For our benchmark models this would give a mass of order m φ ∼ 70 MeV. The decay channel n → χφ is closed, and only leads to a subdominant mode n → χA A through virtual φ exchange. However, when m φ m A , the scalar decays via φ → A A , and as χχ → φA is an allowed process, it can have non-negligible effects on both cosmology and indirect detection signals. For simplicity we assume that λ is large and m φ takes the benchmark value of 70 MeV. This is large enough so that we can ignore the effects of φ on neutron decay, but we will consider its effect on DM physics. We also note the cubic coupling (14) could lead to interesting cascade decays of the heavier of the two color triplet scalars Φ i produced at the LHC, but this is beyond the scope of the present work.

V. COSMOLOGY
In the limit of low relative velocity, the cross section for χχ elastic scattering by dark photon exchange takes the form with the lower bound derived using the limit on m A /g from neutron stars in Eqn. 12. It is interesting that the lowest allowed value of σ χχ is of the appropriate strength to explain observations of astrophysical structure on small scales that are in tension with the predictions of cold dark matter [24]. However constraints from the Cosmic Microwave Background (CMB) and astrophysics discussed in Section VII will ultimately limit χ to be a small fraction of the total DM, preventing our model from addressing these issues. 2 Larger values of σ χχ could run afoul of constraints on DM self-scattering from the Bullet Cluster [26], but this limit is avoided when χ is subdominant; up to around 10% of the total DM population might have such strong self-interactions [27,28]. The relic density of χ is determined by χχ → A A and χχ → φA annihilations, assuming there is no asymmetry between χ andχ. There are also a tree level process that give annihilation into quarks, χχ → dd, but this process is greatly suppressed by the TeV scale mass of the Φ 2 . The cross sections for the relevant processes are given by [29] σ ann v rel χχ→A A = g 4 16πm 2 where η A ,φ = m 2 A ,φ /m 2 χ . For χ to make up all of the DM, the total annihilation cross section should be close to 10 −25 cm 3 /s [30]. If m A m χ , m φ = 70 MeV, and χ constitutes less than 10% of the total DM, we need a 10 times larger cross section, leading to a lower bound of g > 0.041. If this bound were saturated, then the NS limit (12) would read m A < 2.5 MeV. For a strong coupling g = 1, the χ relic density would be 3 × 10 −7 of the total DM density.
Since χ is a Dirac particle, it is possible that it has an asymmetry from some unknown mechanism operating in the early universe. Then it would be possible to have large g and a more significant fraction of χ being the dark matter. Moreover, if the asymmetric component number density happens to be close to the symmetric contribution, 3 then one CP state of χ (either χ or its antiparticle) will have a suppressed density, which could weaken indirect detection constraints. We do not pursue this loophole in the present work, but it should be kept in mind.
χ and A chemically decouple at a dark sector temperature of m χ /30 ≈ 31 MeV. At this time the A are still relativistic with a large number density, which can lead to several problems: if they decay during the time of Big Bang Nucleosynthesis (BBN) their energetic decay products can disturb the production of light nuclei by diluting the baryon-photon ratio as well as causing photodissociation of the nuclei, and decays during and after recombination can distort the CMB temperature fluctuations. Moreover when the dark photons become non-relativistic, an early period of matter-domination can occur which would change the expansion rate of the universe and cause further problems for BBN.
To avoid these issues, we follow [31] and require the dark photons to decay before they exceed half the energy density of the universe. The temperature of the SM photon bath at the time this occurs was shown to be where Y A = n A /s SM at χ freeze-out and f = 1/2 is the fraction of the universe's energy density made up of dark photons. Y A is given by whereg D /g SM is the ratio of relativistic degrees of freedom in the dark sector to that of the SM sector when the two sectors thermally decouple. As the regions of parameter space we are interested in lead to small scattering cross sections between dark and SM particles, we assume that the two sectors thermally decouple when all species are relativistic, sog D = 8,g SM = 106.75, and Y A ≈ 0.02. We then require the lifetime τ A to be less than H −1 (T dom ), the inverse Hubble rate at this temperature. This leads to constraints on the kinetic mixing parameter that we present in section VIII. We emphasize that the method we have used here to estimate the cosmological constraints on the dark photon lifetime is rough, and more detailed studies similar to those undertaken in [32][33][34][35] are needed to determine this limit with greater precision. Finally, DM annihilation close to recombination can also lead to distortions of the CMB anisotropies. Based on searches for this effect in temperature and polarization data, the Planck collaboration has determined the following 95% CL constraint [36]: (25) where f eff is an efficiency parameter that depends on the spectrum of injected electrons and photons, while Ω DM and Ω χ denote the ratio of the energy densities of the total dark matter and χ, respectively, to the critical density. For m χ = 937.9 MeV, f eff is approximately 0.5 for DM annihilations to 2e + e− or 3e + e− final states, and 0.4 for annihilation to 6γ or 9γ final states [37]. Therefore (Ω χ /Ω DM ) 2 σ ann v rel 1.5 × 10 −27 cm 3 /s for scenario A (A decays to e + e − ), while (Ω χ /Ω DM ) 2 σ ann v rel 1.9 × 10 −27 cm 3 /s for scenario B (A decays to 3γ).

VI. DIRECT DETECTION
Because of kinetic mixing, A couples to standard model (SM) particles of charge e with strength e. The cross section for χ to scatter on protons is where µ pχ = 469 MeV is the reduced mass of χ and the proton. The CRESST-III experiment [38] sets a limit of 10 −38 cm 2 for scattering on nucleons at m χ ∼ = 1 GeV, which is relaxed by a factor of ∼ 2 for our model where only protons scatter. This leads to the constraint A similar, but weaker bound, arises from χ-electron scattering, which has been constrained using XENON100 data in ref. [39]. Mixing of χ with the neutron leads to scattering mediated by the strong interactions of the neutron with nucleons, of order θ 4 σ np , where σ np ∼ = 20 b is the low-energy cross section for neutron-proton scattering. In view of (6), θ 4 σ np < 10 −67 cm 2 , far from observability. In models with m χ > m n , as suggested in ref. [11] for evading the NS constraint, an inelastic χN → nN cross section of order θ 2 σ np ∼ 10 −45 cm 2 would arise, with a highly distinctive signature of large energy deposited in the target material. This is still 7 orders of magnitude below current sensitivity of ordinary elastic scattering, but the large release of energy might make it more observable.
In addition to these contributions that only depend upon the low-energy effective description (1), exchange of the heavy colored scalar Φ 1 in the UV model leads to scattering of χ on down quarks, which after Fierztransforming comes from the operator The vector part of the quark current is simply related to that of nucleons (counting the number of down quarks in the nucleon). The corresponding cross section is 10 −40 cm 2 , several orders of magnitude below the current limit.

VII. INDIRECT DETECTION
Indirect DM searches can give strong limits the on annihilation rate of χ particles. In particular scenario B, where A → 3γ, could lead to significant fluxes of gamma-rays from nearby dwarf spheroidal galaxies, and the Fermi -LAT collaboration has searched for this excess emission [40]. The spectrum of photons from a single decay of an A (in the rest frame of the A ) is given by [41] The spectrum of photons from the annihilation χχ → A A (in the center of mass frame of the two annihilating particles) is then given by the boost equation where x ± = m χ /m A ± (m χ /m A ) 2 − 1. In principle, annihilations of χχ to φA will also have some effect on the shape of the average photon spectrum, but for the regions of parameter space of interest the branching fraction to the φA final state is less than 1/5 (see Eqns. 21 and 22). Therefore we make the approximation that dN/dE is determined solely by annihilations of χχ to A A . With this spectrum we can determine the expected flux from a particular galaxy. Using the gamLike code [42], we calculate a likelihood function based on 6 years of Fermi -LAT observations of 15 dwarf spheroidal galaxies. With this likelihood we find that the quantity (Ω χ /Ω DM ) 2 σ ann v rel < 2.8 × 10 −28 cm 3 /s at 95% CL for 6-photon final states. Scenario A, in which the dark photons decay to electron positron pairs, could in theory give a large flux of positrons which could be detected by Voyager [43] and the AMS-02 [44,45] experiment. However, it was shown in [46] that significant uncertainties in the model of cosmic-ray propagation make it difficult to reliably constrain dark matter candidates with masses less than a GeV via these observations, so we do not consider them further. Limits can also be placed on the χχ annihilation rate in this scenario with Fermi observations of dwarf spheroidal galaxies, but these limits are subdominant to those from observations of the CMB, which we discussed in Sec. V.
As m A m χ , the DM annihilation cross section is subject to large Sommerfeld enhancement at small relative velocities. This leads to an enhanced annihilation rate in dwarf spheroidal galaxies and at the time of recombination. The enhanced cross section is given by σ ann v rel = S σ ann v rel 0 , where σ ann v rel 0 is the sum of Eqns. 21 and 22 and the Sommerfeld factor is approximately [50] with a ± = 1 + iu 1 ± 1 − x/u , x = g 2 /4πβ, u = 6βm χ /π 2 m A , and β = v rel /c. For all of the dwarf spheroidal galaxies, we make the approximation β ∼ 10 −4 (for a compilation of the velocity dispersions in each dwarf spheroidal of interest, see [51]) and we take β ∼ 10 −8 at times around recombination. Increasing g ultimately reduces the expected signal in searches for DM annihilation, as the increase in the annihilation rate from the increased cross section is more than offset by the reduced χ relic density. For scenario A (A decays to e + e − ), the CMB constraints described in Sec. V limit g 0.07, so χ can be no more than 1.2% of the total DM, while for scenario B (A decays to 3γ), g > 0.11. For scenario B, the gamma-ray limits are even stronger, with g limited to values greater than 0.14. This corresponds to χ making up only 0.08% of the total DM. These limits are displayed in Fig. 2. Lowering the mass of the dark photon increases the Sommerfeld enhancement and shifts the limits on g to higher values, as more suppression of the χ relic density is needed to avoid these constraints.

VIII. KINETIC MIXING CONSTRAINTS
The BaBar experiment has searched for production of A -photon pairs in e+ e− collisions, with the A taken to decay to invisible particles [49]. This leads to the constraint < 10 −3 . Since the dark photon in our model will decay outside the detector when = 10 −3 , this limit applies here. Further direct search limits come from beam dump experiments, where dark photons with mass 2 MeV can be produced (and detected if m A > 2m e ). Ref. [48] finds a limit of < 2.5 × 10 −8 from the E137 experiment. In addition, if A is emitted copiously from supernovae, the observed emission of neutrinos from supernova 1987A would have been attenuated. This robustly excludes the overlapping range ∈ [2 × 10 −9 , 1 × 10 −6 ] [47] (see also [52,53]). If m A is below the 2m e threshold, the beam dump limits do not apply, and the supernova limits shift to ∈ [2 × 10 −9 , 8 × 10 −6 ].
A lower bound on comes from the requirement that A decays sufficiently fast (before recombination) so as MeV and the dark photon decays nearly entirely to electron pairs while the right plot shows limits for when m A = 0.5 MeV and the predominant decay is to 3 photons. Constraints shown include those from direct detection with the CRESST-III experiment [38] (red; see Sec. VI), the maximum allowed neutron star mass (yellow; see Sec. III), distortions of the CMB anisotropy power spectrum (green; see Sec. VII.1) from χ annihilations [36], observations of the neutrinos from supernova 1987A [47] (blue; see Sec. VIII), BBN constraints on the decay rate of the dark photons (orange; see Sec. V), results from the E137 beam dump experiment [48] (purple, left plot; see Sec. VIII), BaBar searches for dark photon production [49] (black, right plot; see Sec. VIII), and observations of gamma rays from dwarf spheroidal galaxies with the Fermi-LAT [40] (purple, right plot; see Sec. VII.1). The black shaded region in the left plot corresponds to values of g that cannot explain the neutron lifetime discrepancy due to constraints from searches for neutron decays to a photon and invisible particle [9] (see Sec. II).
not to disturb the CMB. For scenario A, the width of the dominant A → e + e − decay and corresponding lifetime are given by Using the rate for A → 3γ from [41], the lifetime for the A when m A < 2m e is τ 3γ ∼ = 2 × 10 12 s 10 −10 2 1 MeV m A
The exact upper limit on τ from the CMB has not been worked out in detail in the literature; published constraints on the allowed abundance of A versus lifetime only go down to τ = 10 12 s, where the constraints are rapidly weakening, but not yet gone; see for example fig. 7 of ref. [54] or fig. 4 of ref. [55]. The true limit is probably somewhat lower than 10 12 s; for example if τ 3γ < 10 11 s and m A = 0.5 MeV, we require > 4.5×10 −10 , but since this overlaps with the BBN limits shown in Fig. 2 (right), this uncertainty is not important for our constraints.
Concerning limits from BBN, following the argument of section V, the requirement for the lifetime of A is τ e + e − < 540 s for m A = 1.35 MeV and τ 3γ < 3920 s for m A = 0.5 MeV. This translates into lower bounds on of 2 × 10 −11 and 5 × 10 −5 for scenarios A and B respectively, as displayed in Fig. 2 along with the other constraints derived in previous sections. These plots show that the combination of direct detection and cosmological constraints strongly disfavor m A < 1.0 MeV-only when A is heavy enough to decay at tree level are the BBN constraints on sufficiently weak for the model to be viable. Lower values of m A require much larger to ensure that A decays before disturbing BBN.

IX. CONCLUSIONS
The possibility that the neutron decays into a dark fermion χ and a dark photon A is highly constrained by an array of laboratory and astrophysical observations. The mass of χ must lie in a narrow window between 937.9 and 939.6 MeV. The mass of the A must be greater than 1.022 MeV (2m e ) and less than 1.67 MeV. 4 The U(1) gauge coupling cannot be small (above 0.07), implying that χ comprises no more than ∼ 1% of the total DM. The kinetic mixing parameter must lie in a rather narrow range of 10 −11 -10 −9 . It is intriguing that despite the many restrictions, there is still viable parameter space.
The decay mode n → χγ is still present in our model, but its rate is suppressed relative to the invisible mode by Γ(n → χγ)/Γ(n → χA ) ∼ (µ n v ) 2 where µ n is the neutron magnetic moment and v is the VEV of the light scalar φ that breaks the U(1) symmetry. Since smaller values of v seem less likely in view of the hierarchy problem for scalar masses, the eventual detection of the visible decay mode could be expected. Dark neutron decay could also lead to effects in nuclear processes, with 11 Be decays discussed as a particularly promising candidate for study in [56]. The dark matter could be discovered by direct detection, despite its small relic density. The UV completion of our model also requires heavy scalar quarks that could be accessible at LHC and detectable through their decays to first generation quarks or a jet plus missing energy.
Note added. After completion of this work, ref. [57] reported new limits on the decay channel n → χe + e − . Our model predicts such events from off-shell dark pho-ton exchange, but at a very low rate. We computed the partial width for our benchmark models, finding 10 −37 − 10 −38 MeV, which is negligible. In addition, ref. [58] presented a new precise determination of the neutron axial vector coupling g A using lattice QCD. This value of the coupling is lower than the experimental average determined in [10], suggesting consistency between a dark decay contribution to the neutron lifetime and g A .