Dark matter (DM) accumulation sites can provide a valid strategy to potentially identify hints of DM proper existence as well as its nature and properties. In particular, one could think of the concentration of this type of matter inside astrophysical bodies as a consequence of gravitational interaction and, provided suitable ranges of masses and cross sections, thermalization with ordinary—baryonic—matter constituting these objects. Effects such as self-annihilation and coannihilation with a different species or decay have been exhaustively studied as possible multimessenger signatures taking place in the Sun, planets, main sequence stars, and more compact objects like white dwarfs or even neutron stars [1–11].

Regarding DM itself, beyond standard model (BSM) candidates have flourished in the literature over the past decades. For a review, see, e.g., [12]. Some of them are now well constrained from the coordinated effort of different communities focusing on direct, indirect, and collider searches [13]. One of the currently accepted possible realizations considers that DM could generate a relic density via annihilation into so-called dark mediators and subsequent SM particles. These types of models are also referred to as “secluded” in the sense that DM dramatically reduces its couplings to SM states by an intermediate state, a decaying mediator, Y. Examples in the literature include coupling strengths ranging from weakly interacting DM particles [14] to strongly interacting ones [15]. According to the duration of a lifetime for these metastable particles, there is a further division among short-lived or long-lived mediators, each giving rise to dramatic differences in the predicted indirect signal [16,17]. Annihilation of DM into two generic different mediators χχ¯→M1M2 can take place in the s- or p-wave scheme, depending on the Lorentz structure of the DM-mediator interactions [18]. There are some works that have focused on particular realizations of these secluded models [19].

Indirect signals from DM could also be expected from the possibility of annihilation through long-lived mediators into gamma rays in astrophysical environments, the Galactic center, dwarf spheroidals, the CMB [14,20–22], and also into neutrinos [23,24]. From the experimental side, current DM searches are actively constraining the available mass and DM-nucleon cross-section, mχ-σχ,N, phase space. Efforts include those from colliders like, e.g., Belle [25] or LHC [19]. Regarding indirect searches [26] in the neutrino channel, secluded DM models have been constrained by ANTARES [27], while in the gamma rays relevant searches are performed by Fermi, H.E.S.S., and AMS Collaborations [28].

On general grounds, considering dark metastable mediators enriches the picture by which DM interacts with ordinary SM matter. The main characteristic in this scenario comes from the fact that these particles have a finite lifetime. While short-lived mediators would be essentially indistinguishable in most of the searches from models where DM is not secluded, long-lived mediators allow an injection of SM states not directly related to places with enhanced DM density. Therefore, it could produce detectable signals far from the production site. Besides, this mechanism has been quoted [22] to introduce anisotropies of prompt species—positrons and photons—produced in the decays of long-lived mediators. Popular mediators such as the dark photon or the dark Higgs have been featured in a number of recently proposed dark sector models [29] although several model-independent DM scenarios also feature long-lived particles [30].

In this work, we consider the annihilation of light DM particles inside white dwarfs (WDs). We restrict our interest to dark candidates in the sub-GeV mass range as they can probe some particularly interesting astrophysical scenarios [31]. Recent constraints show a window for masses mχ≲1  GeV and χ-N cross sections σχ,N≲10-29  cm2 (see Fig. 6 in [32]). Electrons are also relevant species inside WDs. However, their scattering cross section with sub-GeV DM is typically much smaller than that of nucleons [33] (less than ∼10-38  cm2 for thermal DM when form factors do not play a role). Let us emphasize at this point that since the mass range beyond a few GeV is nowadays better probed, the less constrained light mass phase space seems the next interesting region to explore [34] with proposals allowing strongly interacting candidates (SIMPs) as recently described in [35].

The astrophysical scenario we consider is that of a WD where DM particles annihilate inside the stellar medium so that the metastable mediators produced in the reaction χχ¯→YY can lose energy while propagating (on-shell) outwards. Eventually, they will decay into photons, Y→γγ, either inside or outside the star, as this is governed by their energy-dependent lifetime and dissipation. White dwarfs are compact stars made mostly of carbon and oxygen and formed at the end of the lifetime of main sequence stars with masses up to ∼8M⊙. They represent around ∼95% of all the stars in our Galaxy. It is believed that a fraction up to 20% may harbor magnetic fields with a strength up to several hundred MG. Since, at their final stages, no fusion reactions can happen in their interior, they are supported by electron pressure. Typical masses range MWD∼(0.3–1)M⊙ and radii RWD∼(0.01–0.03)R⊙. Thus, these stars are essentially cold, with effective temperatures T∼103–104  K, and dense enough ρ∼104–107  g/cm3 so that density effects cannot be neglected. In order to avoid many of the limitations set by particular model settings, we will consider a generic scenario of annihilating fermionic DM into photons via dark mediators making a minimal set of assumptions. Depending on how long or short-lived these mediators are and the importance of dissipation, we will show how gamma ray emission and luminosities from WDs can end up being attenuated. For example, if the mediator lifetime is large enough, it could decay outside the stellar radius modifying the expected energy flux value with respect to that arising from decay in central regions. In this scenario, medium effects have to be dealt with as, generically, a mediator will loose energy when passing through the ordinary matter, provided the decay length, λD, is larger than the interaction length, λI. These boosted mediators will be produced with initial velocity, vY,0, that will be attenuated while propagating inside the star, thus affecting their survival probability and the energy deposited by them at the decaying site. However, let us emphasize that for less dense stellar bodies such as the sun or planets, average densities are less than ∼102  g/cm3, thus much lower than central densities in WDs and dissipation effects are less important.

The paper is structured as follows. In Sec. II, we describe the DM annihilation into dark mediators as well as particle energy losses inside the WD scenario. We calculate the survival probabilities for mediators with energy dependent lifetimes, their spectrum and the associated photon luminosities. In Sec. III, we show results concerning gamma-ray luminosities comparing to those for cold WDs in M4 globular cluster (GC) and further discuss possible constraints on dark mediator lifetimes. Finally, in Sec. IV, we conclude.

In this section, we describe the process where the photon emission arises from DM annihilation via dark mediators. We consider fermionic dark matter particles, χ, that annihilate into metastable mediators, Y, through reactions χχ→YY with subsequent decay Y→2γ. Additional reactions from radiative emission processes or three body decay of mediators [36] induce small corrections, including anisotropies [22] that we will not consider here. It is well known that considering DM candidates not restricted to weakly interacting particles introduces the possibility that they could undergo self-interactions with 3-to-2 or 4-to-2 annihilations [15,37]; however, we expect our results will not be qualitatively modified as stellar DM densities remain small.

The metastable mediator has a lifetime at rest that can be related to its decay width as τrest=ΓY-1 (we use ℏ=c=1). As initially created in the boosted 2→2 process, they possess a Lorentz factor γY,0=11-vY,02=mχ/mY. In the stellar environment, the mediator will, in principle, interact with the medium decreasing its energy and velocity, vY, from initial values. Attenuation of dark candidates has been considered as a source of DM depletion on terrestrial detectors [38,39]. This stopping power is a crucial aspect that could largely impact the energetic yields of annihilation processes in a medium.

From the accumulated DM present inside the WD, dark mediators are produced from the two-body annihilation reaction χχ¯→YY. Regarding the kinematics of the dark mediator propagation, we will use a radial treatment inside the star for the light DM mass range and (relatively) strong interacting cross sections motivated by previous findings of small-deflection angle approximation [38]. In the medium, they may suffer from interaction with ordinary matter composed of nuclei and a gas of electrons. Such a possibility is realized when they live long enough to experience the scattering processes we consider, e.g., if their decay length λD is larger than the interaction length, λI∼1σY,ini. ni is the number density of scattering centers of ith-type in the stellar volume (nuclei of baryonic number A end electrons) and σY,i is the scattering cross section describing mediator interactions with those matter constituents. Besides, the possible interaction with electrons σY,e is unknown but we take it to be bounded by that of DM, i.e., σY,e≲σχ,e according to experimental constraints [33]. In the situation where scattering of DM with electrons is less frequent than for nuclei λχ,e>λχ,A or, equivalently, σχ,e≲2Aσχ,N, the effect of electrons can be safely neglected in the sub-GeV mass range. In this work, we will constrain all decaying mediators to have rest lifetimes τrest<1s to evade nucleosynthesis constraints [40].

As mentioned, inside the WD, we assume that a fraction of DM is present. The population number of DM particles inside the star, Nχ, will be the result of several processes. On one hand, the capture rate, Γcapt, collecting particles by gravitational capture from an existing galactic DM distribution. It is indeed expected at early ages for the star from the hydrogen dominated era, during the main sequence, and later, in the more compact configuration [5,41]. On the other hand, there are other processes yielding the opposite effect, such as annihilation and evaporation, each with a definite rate Γann, Γevap, respectively. See, e.g., [42–45]. Let us emphasize that the strength of the possible (indirect) gamma ray signal is to be directly related to the amount of DM that the star is able to capture and retain.

During the WD stage the DM capture rate can be written [46,47] as Γcapt=24πGρχMWDRWDmχv¯fχ,A[1-1-e-B2B2],(1)where G is the gravitational constant, MWD and RWD are the WD mass and radius, ρχ is the local DM density, mχ is the DM particle mass and v¯ is the velocity dispersion between the DM particle and the WD. fχ,A is the fraction of DM particles that undergo one or more scatterings with a nucleus of mass mA and baryonic number A while inside the star. In our work, we set A∼14 to account for a mixed C-O composition. fχ,A saturates to unity when it is larger than a typical scale σsat=πRWD2mAMWD, known as geometrical cross section. Thus fχ,A∼1, if σχ,A≳σsat, where σχ,A≃A2σχ,N is the DM-nucleus cross section [48–51], expressed in terms of the DM-nucleon cross section, σχ,N. Otherwise, fχ,A∼σχ,Aσsat. Finally, the bracketed term accounts for DM that scatters but it is not captured in the WD, being B2=6mχvesc2mAv¯2(mχmA-1)2,(2)and vesc=2GMWDRWD,(3)the escape velocity.

Additional decay or even coannihilation with a different species could be possible but, for the sake of brevity, we will not consider those channels here. We assume no DM self-interaction exists. In order to check the consistency of our argument neglecting evaporation for light DM, we can estimate the limiting evaporation mass by demanding that the typical DM particle velocity, v∼Tcmχ, at the WD central temperature, Tc, be less than the local escape velocity of the star, vesc. Thus, the evaporation mass value will somewhat depend on the thermodynamical properties of the star considered. For the WD masses MWD∈[0.2M⊙,0.95M⊙] and radii RWD∈[1.2810-2R⊙,2.1510-2R⊙], evaporation mass limits attain values mχ,evap∼Tcvesc2∼2  MeV when we set Tc∼106  K. The corresponding set of WD masses and radii have been obtained using the Lane-Emden solution for the nonrelativistic polytropic equation of state with n=3/2 [52]. Inside WDs, mass density can be reasonably taken as a slowly variating radial function, approximately equal to the central density ρ(r)∼ρc. In our case, the specific central density value range we use to obtain the selected MWD-RWD parameter space is ρc∈[1.66×105,3.78×106]  g/cm3.

As explained, in the range of χ masses, we will consider, well above evaporation limits, the number of DM particles in the WD is obtained by solving the differential equation, dNχdt=Γcapt-2Γann,(4)where Γann=12∫d3r→nχ2(r→)⟨σav⟩=12CaNχ2,(5)and ⟨σav⟩ is the annihilation cross section averaged over the initial DM states and nχ(r→) is the DM number density inside the star, which verifies ∫d3r→nχ(r→)=Nχ at a given time. For simplicity, the ⟨σav⟩ will be taken in what follows as that of a wimp-like candidate ⟨σav⟩∼10-26  cm3/s but this parameter will indeed depend on the interacting nature of the DM candidate and will discuss later on its impact in our calculation. In particular, this value will assure that during the typical WD lifetime of ∼Gyr, DM can effectively thermalize in the interior of the star. In such a case, the DM particle number density can be cast under a Gaussian form, nχ(r)=n0,χe-(rrth)2,(6)where n0,χ is the central number particle density value and the thermal radius is given [47,53] by rth=9T8πGρcmχ.(7)Using this expression, we can find an explicit form for Ca∼⟨σav⟩/rth3 in Eq. (5). Finally, solving Eq. (4), it is found that Γann≃Γcapt2 for times much larger than the typical equilibrium time scale τeq=1/ΓcaptCa.

In this section, we analyze the results obtained for the photon luminosity and flux to compare with existing experimental measurements of the coldest WDs. By comparing the expected internal warming due to the dark mediator in-medium decay, one could obtain some constraints to the lifetime of the mediator in the scenario considered. In order to maximize the possible DM effects in the stellar warming, we consider those WDs present in the M4 GC [58] where, in line with [6], we assume a DM density ρχ=2660ρχ,0, being ρχ,0=0.3  GeV cm-3 the solar-circle value, usually taken as reference in our Galactic DM distribution. The quoted value in M4 GC refers to the density at the largest radius where the WD data are observed [2,6], having a velocity dispersion of v¯=20  km/s. It is important to mention that in this GC the age of the WDs is set to t=12.7±0.7  Gyr [59], so that the assumption that the DM particles which have some impact on the luminosity through annihilation are those which are captured in the WD stage seems a reasonable hypothesis. Along the same lines, it is clear that at this age t>τeq and they are indeed thermalized.

In Fig. 2, we show the energy flux due to annihilation of DM with mχ=800  MeV and mY=100  MeV as a function of the photon energy at distance d=2R for a WD with mass MWD=0.28  M⊙ and radius RWD=0.04R⊙ and normalized to the product 2Γannmχ. We consider two different values of τrest=0.8  s (solid lines) and τrest=0.1  s (dashed lines) and σχ,N=10-39  cm2 (green) and σχ,N=5×10-39  cm2 (magenta). We assume σχ,N=σY,N and q=0.5. The case where more attenuation is obtained is that of the larger cross section where the available energy window for photoproduction is small and centered about Eγ=mY/2. For the considered cases it is fulfilled that λD>2R and as τrest grows the flux obtained is smaller since less mediators have been able to decay into photons at the radial distance 2R. Let us note at this point that, in principle, some modulating geometrical factors would be needed in order to precisely account for decays from mediators beyond stellar radius. As we will see later, since most lifetime values allowed are in the region τ>0.1  s, around one order of magnitude larger than the typical WD radius, we expect this approximate treatment not to substantially change our conclusions.

In Fig. 3, we show the WD internal luminosity, i.e., energy per unit time deposited inside the stellar volume, due to the annihilation of DM as a function of the WD mass for mχ=500  MeV and mY=375  MeV (solid lines) and mY=10  MeV (dash-dotted lines). We set different lifetimes τrest=0.001, 0.01, 0.1, 1 s. We fix σχ,N=σY,N=10-39  cm2 and q=0.5. We can see that the more similar to the DM particle mass, mχ, the mediator mass, mY, is, the higher the luminosity is. This happens because the mediator is produced almost at rest and it is equivalent to a prompt photoproduction (there is no energy attenuation for the mediator). Moreover, the luminosity decreases when τrest increases due to the fact that the higher τrest is, the smaller the probability of decaying inside the object. It is clear from the figure that we can exclude sets of parameters that yield an internal luminosity Lχ higher than that experimentally extracted for WDs in M4 GC (red points as given by [6]). In other words, for fixed values of mχ, mY and σχ,N (that is fixed to be equal to σY,N) there will be a limiting lower value of τrest below which the luminosity would be higher than that deduced from experimental data.

In order to analyze these specific constraints, we plot in Fig. 4, the excluded values of τrest (colored region) as a function of the logarithm (base 10) of σχ,N for mY/mχ→1. Being conservative, we exclude values of τrest which provide luminosities beyond a 50% tolerance for the complete set of all experimental data, i.e., Lχ>1.5Lexp. We fix this error band since experimentally deduced luminosities for WDs are accurate only to the first or second significant figures. Incidentally, this happens for luminosities above 50% of the value for the first experimental data point considered in the series (MWD=0.28  M⊙, RWD=2.7×109  cm for ρ∼3.3×105  g/cm3). On this plot we show the boundary of the colored region, whose physical meaning is that of the minimum value of τrest below which lifetimes for a decaying mediator produced at rest are not allowed for a given σχ,N. In this case, since the mediator decays at rest, it does not suffer attenuation so that there is no dependence on σY,N. As it is obtained, it is indeed a lower limit of allowed τrest independent on mχ, mY, and σY,N. Besides, the green line indicates the corresponding Y-lifetime where τrest=λY,A, i.e., the value for which at least one scattering between the mediator Y and a nucleus A inside the WD will take place, assuming σY,N=σχ,N. Below this limiting value situation is equivalent to that with no energy losses. In the figure, it is shown that for saturated values of the capture rate in the WD, i.e., σχ,N>σsat=1.1×10-39  cm2 there is no further change in the limiting value of τrest. We refer to this as τlimit. For smaller values σχ,N<σsat there is a quenching of τlimit as smaller values of τrest are not excluded from luminosity constraints. We expect that our results could be, in principle, extended up to σχ,N∼10-29  cm2 covering the targeted region in the phase space for sub-GeV DM as we comment in the Introduction. In other more general cases, for a fixed value of σχ,N and given mχ, mY the maximum excluded value of τrest, τlimit, will be lower than that obtained in the extreme mY∼mχ case. In other words, the most restrictive τrest limit would be that in which is independent on σY,N and mχ, mY.

In Fig. 5, τlimit (boundary of the excluded region for τrest) is shown as a function of the ratio mY/mχ for different values of σχ,N, σY,N. In these cases, when mY/mχ≠1, there is effective contribution from energy attenuation and decay effects. We fixed σχ,N=σY,N=10-39  cm2 (solid lines), σχ,N=σY,N=10-38  cm2 (short-dashed lines), σχ,N=σY,N=10-34  cm2 (dash-dotted lines) and σχ,N=10σY,N=10-38  cm2 (long-dashed lines). We can see that the smaller the ratio mY/mχ is, the smaller the lifetime limit is, since the photons would yield luminosities compatible with experimental bounds, thus being a weak constraint. In the case of σχ,N≲1.1×10-39  cm2 (solid line), there is a further effect not present in the other cases considered due to the reduction of the WD capture rate of DM as the saturation factor fχ,A<1. It can be seen that the effect of reducing σY,N fixing σχ,N is quenching τlimit. Although all decay probabilities are naturally allowed in our setting, let us remind that results shown in Figs. 3–5 dealing with the internal luminosity will effectively be sensitive to decays only inside the stellar radius, by construction, while those in Fig. 2 consider decays both inside and outside the stellar volume.

We have studied the gamma ray emission in WDs from annihilation of DM in their interior through metastable mediators. We have considered the combined effect of energy attenuation and finite decaying lifetime. Using an approximation where the energy loss can be described in a continuous way through mediator scattering with nuclei inside the stellar volume we have derived the internal luminosities and fluxes. We have compared these luminosities to those from cold WDs in M4 GC. We find that in the case where σχ,N≲10-40  cm2 the attenuation is negligible and the only effect comes determined from the lifetime. However, for larger cross sections, up to σχ,N∼10-29  cm2, there are nontrivial effects that further constrain the lifetime bounds with a monotonic increase in the mY/mχ ratio. The effect of the expected seclusion of DM from nucleons can be seen by imposing the nondegeneracy of σχ,N and σY,N values. We find that the smaller the σY,N is the less restrictive effect on the allowed Y-lifetimes. Note that the annihilation rate of DM or, in other words, the value of the thermally averaged annihilation cross section could somewhat modify our results concerning obtained luminosities and flux (and lifetimes) as an increase factor would yield larger values of τlimit. As for energy flux, we find the more attenuation there is, the sharper the photoproduction results. Let us keep in mind that the considerations in the discussion about the restrictions on the decaying mediator lifetimes in cold WDs must also include, apart from other astrophysical scenarios, current efforts from colliders. As a new strategy to follow, they have shifted towards an alternative simplified model paradigm that includes these additional mediators and the search for a displaced secondary vertex, characterised by the mass of the particle and its lifetime, see [60]. This has led to an extensive effort amongst both theorists and experimentalists at the LHC to establish a systematic program to characterize DM searches using simplified models. A multidirectional and complementary approach from different search contexts will most surely provide valuable information on this type of model.