Return of the X-rays: A New Hope for Fermionic Dark Matter at the keV Scale

A long time ago (in 2014), in galaxies and galaxy clusters far, far away, several groups have reported hints for a yet unidentified line in astrophysical X-ray signals at an energy of 3.5\,keV. While it is not unlikely that this line is simply a reflection of imperfectly modeled atomic transitions, it has renewed the community's interest in models of keV-scale dark matter, whose decay would lead to such a line. The alternative possibility of dark matter annihilation into monochromatic photons is far less explored, a lapse that we strive to amend in this paper. More precisely, we introduce a novel model of fermionic dark matter $\chi$ with $\mathcal{O}(\text{keV})$ mass, annihilating to a scalar state $\phi$ which in turn decays to photons, for instance via loops of heavy vector-like quarks. The resulting photon spectrum is box-shaped, but if $\chi$ and $\phi$ are nearly degenerate in mass, it can also resemble a narrow line. We discuss dark matter production via two different mechanisms -- misalignment and freeze-in -- which both turn out to be viable in vast regions of parameter space. We constrain the model using astrophysical X-ray data, and we demonstrate that, thanks to the velocity-dependence of the annihilation cross section, it has the potential to reconcile the various observations of the 3.5\,keV line. We finally address the $\phi$-mediated force between dark matter particles and its possible impact on structure formation.

Many recent papers in astroparticle physics start out by exposing the waning of traditional dark matter (DM) candidates, especially the Weakly Interacting Massive Particle (WIMP) [1][2][3]. The present article is no exception. And while it is certainly too early to give up on the elegance of the thermal freeze-out mechanism for DM production, a look beyond is now more motivated than ever.
In this paper, we will dwell on the possibility that DM is a fermion with a mass of only a few keV. Such DM candidates, which often go by the name of "sterile neutrinos" are well known for their potential to improve predictions for small scale structure (see for instance ref. [4] and references therein), their clean observational signatures in the form of X-ray lines [4][5][6][7][8][9], and their tendency to evoke animated discussions among physicists [10][11][12][13][14]. Many such discussions were incited by recent claims for a yet-unidentified line at ∼ 3.5 keV in the X-ray spectra from galaxies and galaxy clusters [8,[10][11][12][13][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Whether the origin of this lines is indeed related to DM physics [14,, or simply to imperfect modeling of atomic physics effects [10,13,33,85], the controversy surrounding it cannot belie the fact that precision observations of the X-ray sky are a prime tool to search for keV-scale DM. While the overwhelming majority of studies on this topic focus on line signals from DM decay, we will in the following explore the possibility that such signals arise from DM annihilation.
We will introduce a toy model in which keV-scale Dirac fermions χ annihilate to pairs of new real scalars φ, which in turn decay to photons, see fig. 1 (a). These photons will have a box-shaped spectrum, which can be indistinguishable from a monochromatic line within experimental resolutions if χ and φ are nearly degenerate in mass. We will see below that the model can also explain the observed DM relic abundance. At low energies, the phenomenology of our model is captured by an effective Lagrangian consisting of just the φ couplings to photons and χ: where y is a new dimensionless coupling constant, F µν is the electromagnetic field strength tensor, α is the electromagnetic fine structure constant, and Λ is the suppression scale of the dimension-5 interaction. We will see below that explaining the 3.5 keV line while satisfying all constraints requires y ∼ few × 10 −5 and Λ ∼ 10-100 PeV.
Equation (1) can be completed to a renormalizable model for instance by introducing heavy vector-like leptons L with charge assignment (0, 1, where D µ is the gauge covariant derivative, m L is the mass of the heavy leptons, and g is the dimensionless L-φ coupling constant. In this UV completion, where µ is the renormalization scale, which we take equal to m φ , and the loop function corresponding to the diagram in fig. 1 (b) is given by [86] f (τ ) = DM Annihilation. The cross section for the DM annihilation processχχ → φφ depends on the relative mass difference δ ≡ (m χ − m φ )/m χ of χ and φ and on the relative velocity v rel of the annihilating DM particles. In the Milky Way, v rel ∼ 200 km/sec, while in galaxy clusters, v rel ∼ 1 000 km/sec. If |δ| v 2 rel , the annihilation cross section is For δ v 2 rel , we find Note the different dependence on v rel in these two limiting cases. A small value of δ could be naturally explained in a supersymmetric extension of the model, with χ and φ residing in the same supermultiplet [87][88][89].
The decay rate of φ to two photons is Γ φ→γγ = . For DM annihilation in the Milky Way, this implies that the morphology of the photon signal traces the distribution of DM only if Λ 2.5 PeV × (m φ /keV) 3 . Then, the intermediate φ particles travel for 1 kpc before decaying, so the wash-out of the annihilation signal due to their long lifetime is degenerate with the large uncertainties in the Milky Way's DM halo profile at radius 1 kpc [90].
For δ 10 −5 -10 −4 so that the photon signal from DM annihilation appears monochromatic within experimental energy resolutions, we compare the signals predicted by our model to data in fig. 2. The shaded exclusion regions in this plot are based on reinterpreting existing limits on DM decay to monochromatic photons by equating the photon flux in the two cases: Here, Γ χ is the DM decay rate, ρ DM is the DM density, and f (l, Ω, v rel ) is the DM velocity distribution at  1) and (2). We show the relevant constraints as a function of the DM mass mχ and the Yukawa coupling between χ and the mediator φ, assuming a degeneracy parameter of δ ≡ (mχ − m φ )/mχ = 10 −4 . Shaded regions are excluded by X-ray observations of the Andromeda Galaxy (M31) [5], the Galactic Center ("NuSTAR Galactic Center") [91], the galactic ("Galactic diffuse") [7] and extragalactic ("EG. gal" and "EG. dwarfs") [6] diffuse X-ray background, and by Pauli blocking arguments ("Tremaine-Gunn") [92,93]. The red star [94] corresponds to the excess of monochromatic X-rays at 3.5 keV observed in ref. [8,15]. Horizontal blue lines indicate the suppression scale Λ of the φγγ coupling required to obtain the correct DM relic abundance via UV freeze-in, assuming TRH = 1 TeV. To the right of the dashed line, part of the initially produced abundance of χ gets transferred to φ via φφ ↔χχ at T ∼ keV. The region to the right of the dashed line can also be reached via freeze-in through misalignment (see text). The vertical green band indicates where low temperature freeze-in via φφ →χχ yields the correct relic abundance.
distance l along the line of sight in solid angle direction Ω. We obtain f (l, Ω, v rel ) numerically by evaluating Eddington's formula [95] (see [96] for possible shortcomings of this approach due to baryonic effects). The factor 4 on the right hand side of eq. (7) accounts for the number of photons produced in each annihilation χχ → φφ → 4γ. For the limits based on diffuse extragalactic X-rays from galaxies and dwarf galaxies ("EG gal." and "EG dwarfs"), eq. (7) receives redshiftdependent corrections [97,98]. Among these is a factor ∆ 2 (z) ≡ ∆ 2 (0)/(1 + z) 3 [97,99] that accounts for the stronger clumping of DM at late times in cosmological history. We conservatively choose ∆ 2 (0) = 10 6 [100]. We moreover assume that the DM velocity distributions f (l, Ω, v rel ) of distant galaxies and dwarf galaxies follow [101], we adopt observations from stacked galaxy clusters using the MOS and PN instruments on the XMM Newton satellite [8], from the Perseus cluster alone [8], from the Andromeda Galaxy (M31) [15], and from the Chandra Deep Field [102]. Limits are based on non-observations in M31 [5], dwarf spheroidal galaxies [18], and the Perseus cluster [21,85], and the regions above the solid lines are excluded.
those of the Milky Way and of Milky Way dwarfs, respectively.
Focusing specifically on the signal at 3.5 keV, fig. 3 demonstrates that our scenario can align the different observations and non-observations of the line. We have taken into account an uncertainty of roughly a factor of two in the DM velocity distribution for each astrophysical target, based on varying the parameters of the underly-ing NFW profiles within the ranges found in the literature [14]. For cumulative data sets from several sources, we sum eq. (7) over all of them, weighted by the individual exposure times. For quasidegenerate DM and φ masses with δ = 10 −4 , we find that the 1σ confidence regions overlap for all observations except those from the Chandra Deep Field [102]. The alignment between different observations is thus marginally better than for decaying DM [101]. For δ = 10 −4 , eqs. (5) and (6) contribute roughly equally for galaxy clusters, while eq. (6) dominates for galaxies and dwarf galaxies. When compared to exclusion limits, our scenario fares better than decaying DM because the limit from dwarf galaxies is less relevant thanks to the strong dependence of the annihilation cross section on the DM velocity. We remark that future high-resolution observations of the 3.5 keV line [103,104] could potentially distinguish our model from models of decaying DM and from astrophysical explanations of the line by studying the line shape.
The model defined by eq. (1) and its UV-completion in eq. (2) (or minimal extensions thereof, see below) enable several new DM production mechanisms, all of them based on freeze-in of dark sector particles. For a numerical analysis, it is convenient to parameterize the DM abundance by the DM yield Y ≡ n DM /S(T ). Here, S(T ) ≡ 2π 2 g S * (T ) T 3 /45 is the entropy density of the Universe at temperature T , and the number of effective relativistic degrees of freedom is g S * [121]. The Boltzmann equation governing freeze-in for a general process AB → CD can be written as [122][123][124] Here, H(T ) 1.66 g * (T ) T 2 /M Pl is the Hubble rate, g * is the corresponding effective number of relativistic degrees of freedom (which for our purposes equals g S * ), and M Pl is the Planck mass. The squared matrix element |M| 2 AB→CD (summed over initial and final state spins) describes the relevant particle physics, s is the center of mass energy, the modified Bessel function of the second kind K 1 ( √ s/T ) describes Boltzmann suppression at high √ s, and is a kinematic factor. If more than one 2 → 2 process con-tributes to DM production, the right hand sides of eq. (8) should be summed over all relevant processes.
Ultraviolet (UV) Freeze-In Through the φ-Photon Couplings. The dominant DM production processes in the effective theory from eq. (1) are γf → φf andf f → γφ (where f denotes a SM fermion), followed by the decay φ →χχ. This decay can occur even if m φ < 2m χ because quantum corrections can raise the effective mass of φ at high temperature. If φ has a selfcoupling of the form λ 4! φ 4 , these corrections are of order (m eff φ ) 2 ∼ (λ/4!)(n φ /n eq φ ) T 2 [125], where, n φ is the number density of φ, n eq φ is its value in thermal equilibrium, and λ is a dimensionless coupling constant. As the φ selfcoupling is not forbidden by any symmetry, it should be included in eqs. (1) and (2) anyway.
By integrating eq. (8) with the upper integration limit set to the reheating temperature after inflation, T RH , the DM yield is found to be Here, the infrared divergence in γf → φf has been regularized by the effective photon mass in the plasma. The DM abundance today is then The subscript "UV" in the above expressions indicates that the DM abundance is set at T RH [124]. The blue lines in fig. 2 indicate the value of Λ needed to obtain the correct DM relic abundance today, Ωh 2 = 0.12 [126]. We see that a UV freeze-in scenario with Λ ∼ 65 PeV could explain the 3.5 keV line. To the right of the dashed diagonal blue line, φ and χ come into mutual equilibrium via φφ ↔χχ after freeze-in is completed. This reduces the resulting DM abundance by a factor 0.8, and slightly smaller Λ is required to compensate. In this regime, it is imperative that φ decays only after φ and χ have decoupled again at T keV. Otherwise, the DM abundance would be depleted too strongly. We have checked that this condition is always satisfied in the parameter regions not yet ruled out by X-ray constraints. A potential problem for parameter points to the right of the dashed line arises because the photons from φ → γγ decays at sub-keV temperatures could leave an observable imprint in the CMB or in the spectrum of extragalactic background light [127][128][129]. One way of avoiding these constraints is to postulate an additional decay mode for φ that is several order of magnitude faster than φ → γγ, for instance a decay to pairs of light ( keV) fermions χ χ . In this case, the coupling y in figs. 2 and 3 needs to be rescaled by a factor [BR(φ → γγ)] −1/4 . To avoid φ decaying while still in equilibrium with χ, BR(φ → γγ) 2.1 × 10 −3 (keV/m) 1/3 (y/10 −4 ) 16/9 (20 PeV/Λ) 10/9 is required, where y is the yet unrescaled coupling plotted in figs. 2 and 3.
Note also that parameter points with m φ few hundred keV and Λ 20 PeV (and correspondingly lower T RH according to eq. (10)) are disfavored because φ production in stars could violate bounds on anomalous stellar energy loss [128,130,131].
Let us finally remark that in deriving eqs. (9) and (10) we have assumed that DM production starts only when reheating is completed and the Universe follows standard cosmology from T RH onwards. It is, however, possible that reheating proceeds relatively slowly and the Universe maintains a temperature close to T RH for a relatively long time at the end of inflation, while Hubble expansion and inflaton decay balance each other. In such a situation, more time is available for DM production and the resulting Ωh 2 is increased unless Λ is increased appropriately.
Freeze-In Through the Misalignment Mechanism. [132][133][134] At the end of inflation, the field φ may be in a coherent state, with the same expectation value φ throughout the visible Universe. When the Hubble expansion rate H(T ) drops below m φ /3, the field begins to oscillate about its potential minimum at φ = 0. In the language of quantum field theory, such oscillations correspond to an abundance of φ particles. At later times, φ and χ come into thermal contact, and a fraction of the φ energy density is transferred to χ. (χ production via φ →χχ is not possible here since the dark sector will never become hot enough for thermal corrections to raise m eff φ above 2m χ .) The resulting relic abundance of χ is [55,64] where φ 0 is the initial value of the field. If the main production channel for dark sector particles is misalignment, Λ can be larger than for freeze-in through the φ-photon coupling.
Low Temperature Freeze-In. If the reheating temperature T RH is larger than the cutoff scale of the effective theory, the computation of the DM abundance must be based on a UV completion of eq. (1), such as eq. (2). In this case, for not too small Yukawa coupling g, the scalar φ will be in thermal equilibrium with L and with SM particles. χ can then freeze in via φ →χχ and φφ →χχ. The DM abundance is approximately The second line of eq. (12) corresponds to the narrow green band in fig. 2. In principle DM could also be produced directly in annihilations of heavy vector-like leptons,LL →χχ. However, for the large g and small m L required to generate the correct relic abundance this way, φ particles would be efficiently produced in stars (unless m φ few × 100 keV), which is excluded [128,130,131].
As for UV freeze-in, φ needs to be depleted after χ production is over, for instance by introducing an invisible decay mode like φ →χ χ . Note that φ particles abundant during Big Bang Nucleosynthesis (BBN) at T ∼ MeV do not violate the constraint on the effective number of neutrino species, N eff = 2.85 ± 0.28 [135]. The reason is that, by the time of BBN, the dark sector has been sufficiently diluted by entropy production in the SM sector, which occurs when heavy SM particles disappear from the primordial plasma.
Structure Formation. Some of the strongest constraints on keV-scale DM candidates are based on their potential impact on structure formation as observed using Lyman-α data [83,112,114,[117][118][119][136][137][138] and counts of galaxies [115] and dwarf galaxies [114,119]. Albeit suffering from systematic uncertainties that are difficult to control [116,139], these observations are sensitive to the suppression of structure at small ( Mpc) scales caused by DM particles whose kinetic energy is not completely negligible yet when structure formation begins. For DM with an initially thermal momentum spectrum, a lower mass bound of 4.65 keV at 95% CL has been quoted [118], while for sterile neutrinos produced through oscillations, the bound is significantly stronger [114,118] and appears to be in conflict with attempts to explain the 3.5 keV line in such scenarios.
In our model of annihilating DM, the initial momentum spectrum of χ depends on the DM production mechanism. For freeze-in via misalignment, it is very cold and therefore consistent with constraints. For UV freeze-in through the φ-photon coupling and for low temperature freeze-in, it is slightly harder than thermal because the freeze-in rate is biased towards higher energies. However, since the dark and visible sectors decouple at T 100 GeV, the effective temperature of the dark sector is reduced by an entropy dilution factor [g * (100 GeV)/g * (1 eV)] 1/3 2.9 compared to the photon temperature, improving the situation. For χ production via φ →χχ, each DM particles receives only half the energy of the parent particle, reducing the DM temperature by another factor of two. For UV freeze-in there is moreover the possibility that the dark sector cools via self-interactions, φφ → 4φ, mediated by the same quartic coupling λ 4! φ 4 that enabled thermal corrections to m φ . The corresponding increase in φ number density can be compensated by an appropriate adjustment of Λ, which controls the initial freeze-in. For m φ ∼ 7 keV, choosing λ ∼ 10 −3 boosts the rate of φφ → 4φ above the Hubble rate at m φ T 100 keV. We have checked that the inverse process 4φ → φφ never dominates over φφ → 4φ, even when the dark sector turns non-relativistic.