Detecting Boosted Dark Photons with Gaseous Detectors

We search for indirect signals of $\mathscr{O}$(keV) dark matter annihilating or decaying into $\mathscr{O}$(eV) dark photons. These dark photons will be highly boosted and have decay lengths larger than the Milky Way, and can be absorbed by neutrino or dark matter experiments at a rate dependent on the photon-dark photon kinetic mixing parameter and the optical properties of the experiment. We show that current experiments can not probe new parameter space, but future large-scale gaseous detectors with low backgrounds (i.e. CYGNUS, NEXT, PANDAX-III) may be sensitive to this signal when the annihilation cross section is especially large.


I. INTRODUCTION
Dark photons are a renormalizable extension of the Standard Model (SM), with a Lagrangian Here A ′ and F ′ are the dark field and dark field strength respectively, F is the Standard Model (SM) photon field strength, m A ′ is the mass of the dark photon, and ϵ is the kinetic mixing (first explored in [1], see [2] for a recent review).This kinetic mixing gives the dark photon a coupling to SM electromagnetic currents, leading to observable interactions.The dark photon mass is assumed throughout to be of a Stückelberg-type.
In this paper we focus on models in which cold dark matter (CDM) is partially comprised of fermionic χ particles that are charged particles of the dark sector with Lagrangian where m χ is the mass of the fermion, and g D is the dark electric charge of χ.
We consider a scenario where both χ and its antiparticle χ are present in the Milky Way.If m χ ≫ m A , then the annihilation of a χ and χ will produce ultrarelativistic dark photons, allowing them to be seen in terrestrial, low-background experiments (i.e.neutrino experiments).Similar analyses have been done using other decay/annihilation products of dark matter [3][4][5] or with other astrophysical sources [6].
This scenario allows us to explore the signal from lowmass dark matter (DM) that couple to dark photons.Until recent years, this low-mass DM was relatively unconstrained by direct detection experiments.The difficulty low-mass DM presents is that the recoil energy deposited is proportional to the DM mass, typically falling below the detector threshold for masses less than a few GeV.While low-threshold detector technologies have made advances in recent years, new strategies and materials have great promise to lead the field in constraining low-mass DM .
The layout of this paper is as follows: In Sec.II, we will discuss the distribution of χ within the Milky Way in light of annihilations and the corresponding dark photon flux.In Sec.III we describe the interaction of dark photons with matter, in particular how the optical properties of experiments can enhance or suppress dark photon absorption.In Sec.IV we show results from existing and projected experiments.Sec.V covers existing constraints on this model, while Sec.VI discusses a similar signal arising from decaying dark matter.

II. χ DISTRIBUTION AND DARK PHOTON FLUX
In this work, we consider the distributions of χ and χ to be identical, and will define ρ χ to be the combined mass density distribution of χ and χ (i.e. the density distribution of just χ is ρ χ /2).In this paper, we consider large χ annihilation cross sections, so this necessarily makes our distribution time-dependent.Annihilations deplete the energy distribution of χ according to [39,40] for s−wave annihilation.We take ρ χ to be proportional to the NFW profile [41] at every location in space, with a proportionality constant which depends on time (see Appendix A for more comments on this assumption), where ρ s = 0.184 GeV cm −3 and r s = 24.42kpc [42].
If we let f χ (t = 0) = f i , then the time evolution of this arXiv:2402.00941v2[hep-ph] 22 May 2024 Where ⟨σv⟩ is the present-day thermally averaged cross section and the volume integral is taken over a sphere 60 kpc in radius matching the upper limit from [42] 1 .We can use this to find the present-day (t = t 0 ) density of χ, and compute the mono-energetic flux of dark photons using a line-of-sight integral [42] Φ where x is the distance from the Sun, and θ is the angle between our line-of-sight and a line pointed towards the galactic center.We can relate this to r by , with r ⊙ being the galactic radius of the sun.Technically, the time used for the density t = t 0 − x v to account for the propagation time of the dark photons, but this effect is small compared to cosmological time scales necessary for appreciable annihilation.We can decompose this flux using Eq. 4, and then find the annihilation cross section which provides the largest flux, namely This result is seen clearly in Fig. 1.For much of the paper, we will assume this optimum annihilation cross section in order to make strong statements about which experiments are unable to probe this signal.However, if we were to take other values of ⟨σv⟩, the flux roughly scales as In this work we remain agnostic about the origin of χ in the early Universe.While thermal relics are already excluded for most dark matter coupled to dark photons [43] and for warm dark matter below 5 keV [44], a small window remains.We note that the scenario considered here lies outside of this region, as the values of ⟨σv⟩ max are simply too large.
In principle, the dark photon flux at our detector would also be shaped by dark photon decays en route to Earth and attenuation when passing through Earth.However, for the parameters considered in this paper, the decay length is much larger than galactic scales and the cross section is not large enough for significant attenuation to occur.

III. DARK PHOTON INTERACTIONS WITH MATTER
The absorption rate of dark photons is affected by the optical properties of the detector.The in-medium photon propagator leads to an effective mixing parameter of where Π T (Π L ) is the transverse (longitudinal) polarization tensor of the medium [45][46][47].In our work, we are interested in transversely polarized dark photons, as in the annihilation of χ particles at the Galactic Center these are the dominant annihilation products in the limit E χ ≃ m χ ≫ m A ′ .This result is demonstrated in Appendix B. For an isotropic and non-magnetic (the relative permeability is 1) material, we can relate the polarization tensor to the (complex) index of refraction n ref via [47] (see also Appendix A of [48] or [49]) The index of refraction for a single element can be related to atomic scattering factors f 1 (ω) and f 2 (ω) (available from the Lawrence Berkeley Lab database [50]), and calculated via where n A is the number density of atoms, and r 0 = 2.82 × 10 −15 m is the classical radius of the electron.For molecular detectors, the scattering factors for each atom are added2 .Fig. 2 gives an example of the index of refraction over the range of energies for which we are interested.While in general the electric permittivity ε(ω, ⃗ k)/ε 0 = n 2 ref (ω, ⃗ k) (not to be confused with the kinetic mixing parameter ϵ) depends on the dark photon energy ω and three-momentum ⃗ k, for the dark photon energies and momentum transfer considered here the dipole approximation is a good approximation, and consequently the dependence of the electric permittivity on ⃗ k is suppressed.A further discussion on this point can be found in Appendix C. The absorption rate for a single dark photon within our material is Γ = −|ϵ eff | 2 Im(Π T (ω))/ω [51].Our rate of dark photon events within the detector is therefore where V det is the volume of our detector and we have used the fact that ω = m χ for our flux of dark photons. As We can see from Eq. 10 that for m 2 A ′ ≫ Π T , we have ϵ eff = ϵ, while for m 2 A ′ ≪ Π T , the effective kinetic mixing is suppressed by O(m 2 A ′ /Π T ).Avoiding this suppression leads us to look at low density detectors where n ref is close to unity.We also note that there is an enhancement when m 2 A ′ = Re(Π T ).

IV. RESULTS
Our model is dependent upon 5 parameters Oftentimes, we will mention ⟨σv⟩ instead of g D .We can relate the two up to an order 1 factor via [53] where α D = g 2 D /4π.In Fig. 3, we fix the value of f i = 0.1 and m χ = 1 keV, and let ⟨σv⟩ = ⟨σv⟩ max be given by Eq. 8 with our chosen m χ value.We vary m A ′ from 0.01 to 100 eV, and find the corresponding value of ϵ which gives the a priori event rate.
If we were to consider to consider arbitrary ⟨σv⟩, then because , our sensitivity would roughly scale as Alternatively, in Fig. 4, we instead fix the values of f i = 0.1 and m A ′ .We vary m χ from 0.3 to 30 keV, and at every value, we set ⟨σv⟩ to be the corresponding value of ⟨σv⟩ max .As before, the value of ϵ which provides the desired event rate is found and graphed.

A. Liquid Xenon/Argon
The first detector materials we consider are liquid xenon and argon, which are used in low background tonscale experiments such as dark matter direct detection [55][56][57][58] and neutrinoless double-beta decay (for the case of argon [59] and xenon [60]).For these noble liquids, the polarization tensor Π T ∼ O(100eV 2 ), which we can see from the kinks in ϵ − m A ′ plots in Figs.3a & 3b.Because of the relatively large (compared to gasses) inmedium effect, these liquid detectors are unable to probe any new parameter space.

B. Solid Germanium
Similar to liquid xenon and argon, germanium detectors have been used in dark matter direct detection [61] and neutrinoless double-beta decay experiments [62].While germanium crystals are not isotropic and so do not satisfy our assumption of isotropy, here we only consider incident dark photons with O(keV) energies and higher.For absorption the momentum-transfer is also of this scale, and thus we expect neglecting the band structure and the anisotropy of the crystals to be a good approximation.Again, the relatively high densities of these detectors make them unable to probe new parameter space, as can be seen in Fig. 3c.

C. Gaseous Xenon
Ton-scale gaseous xenon detectors have been proposed in the context of searching for double-beta decay such as the PANDAX-III [63] and NEXT [64] experiments.For gases, the size of Π T depends upon the target density.We can see in Figs.3d &  FIG. 3. Exclusion plots for various materials and assumed sensitivities for mχ = 1 keV, fi = 0.1, and the annihilation cross section taken to be ⟨σv⟩max.Included are existing dark photon bounds from [2] and χ bounds from [52].
(the density at standard temperature and pressure) can probe new parameter space if the experiment is sensitive to rates O(10 ton −1 yr −1 ) at keV scale energies.This may be overly optimistic for double-beta decay experiments, which are designed to be most sensitive near the Q-value of the isotope in question (2.46 MeV for xenon-136).However, a future gasesous xenon detector with a low energy threshold and strong background discrimination could look for this signal.
We would also like to point out Fig. 5, in which we consider the hypothetical sensitivities of gaseous xenon detectors with the same sensitivity but different xenon densities.We see that lowering the density allows the experiment to probe lower dark photon masses.

D. Helium / Sulfur Hexafluoride
Gaseous helium and sulfur hexafluoride detectors interest us because of the proposed CYGNUS experiment [65], a large volume dark matter detector which contains a mixture of the two gasses.This experiment has been considered to reach sizes of 1000 m 3 and beyond, with very low energy thresholds, making it an interesting detector FIG. 4. Exclusion plots for various materials and assumed sensitivities for fi = 0.1, with the annihilation section taken to be ⟨σv⟩max.Compared to Fig. 3, here the χ mass is varied with values for m A ′ taken to be near the resonance (which is material-dependent as seen in Fig. 3).The horizontal line shows the existing dark photon bound from [2] for our chosen m A ′ .Also included are bounds from χ scattering in the early universe, obtained from [54].We shade the helium results beyond 10 keV because our dipole assumption for absorption breaks down beyond this point (see Appendix C for more details).
Calculating the true sensitivity bounds in this region is beyond the scope of this paper, but using our method, the sensitivity is already subdominant to solar emitted A ′ by mχ = 10 keV.The cooling and CMB bounds should still be valid in this shaded region.
to search for these boosted dark photons.We would like to make a special note of helium gas in Fig. 3e.Although helium has a relatively small cross-section for x-rays, that same property leads to a strong resonance in Eq. 10.At a single density, the helium detector would be sensitive to a small range of dark photon masses.However, one could imagine a varying density (either in time or across modules) that would allow for an improved search over a larger range of masses.

A. Dark Photon Limits
One can consider an extension to the Standard Model given only by Eq. 1.This model has a rich phenomenology and many constraints on m A ′ and ϵ as shown in Figure 1 of [2] and regularly updated in corresponding website.Note that we disregard the bounds which arise when the dark photon is the dark matter candidate, as that is different than the scenario considered here.In our region of interest (m A ′ = O(10 −2 − 10 1 eV)) the most stringent constraints consider the Sun as a source of dark photons, and constrain either solar cooling [66] or direct detection of these emitted A ′ [67].

B. SMχ Scattering
Our model allows for χ to scatter with electrically charged SM particles by exchanging a virtual dark photon which mixes with the SM photon.We consider m χ ∼ O(1-100 keV) with virialized velocity, leading to kinetic energies of O(10 −3 − 10 −1 eV).These energies are too small to probe with modern-day direct detection ex- periments.However, in the early universe, scattering between χ and SM particles can leave an observable imprint on the Cosmic Microwave Background (CMB) [54,68].We can translate constraints on cross section into constraints on ϵ, g D , m χ , and f i by assuming that χ is the only component of dark matter to interact with SM particles.One could also consider the energy injection into the early universe from the annihilation of χ as in [69], but the resulting bounds are much weaker than those from scattering.
We do not consider constraints in which χ interacts with long-range electric or magnetic fields.Although there is mixing between the SM and dark photons, the dark photon masses considered here are large enough that the effects of long-range SM fields are exponentially suppressed.

C. χ Driven Stellar Cooling
Similar to dark photons, χ χ pairs can be produced in stars with temperature T ≳ m χ .The strongest of these constraints come from horizontal branch stars, which limit q D = g D ϵ/e < 2 × 10 −14 [52].

D. χχ Scattering
Gravitational measurements of galactic mergers and the ellipticity of galactic dark matter halos place constraints on the self-interactions of dark matter [70], which in our model can be translated into constraints on m χ , g D , f i , and m A ′ .The simplest way to avoid these constraints is to take f i ∼ O(0.1), so that any effects of χ will be smaller than the uncertainty on the measurements.

VI. DECAYING DARK MATTER
The analysis considered in this paper could similarly be used to constrain a dark matter candidate ϕ which would decay into dark photons (ϕ → A ′ A ′ ).If we assume that ϕ is the entirety of dark matter, this model can be characterized by four parameters where Γ ϕ is the decay rate of ϕ.This will produce a galactic flux of dark photons which is given by Taking a value for Γ ϕ = 0.03t −1 0 , which is consistent with [71], we obtain a flux of dark photons comparable with the maximum allowed flux from χ χ annihilations.(In general, the dark photon flux could also have a contribution from extragalactic ϕ, but for now, only the galactic contributions are compared).In Fig. 6 we compare the two fluxes, along with the direct flux of dark matter at Earth, calculated via Φ χ = ρχ mχ v ⊙ where v ⊙ is the velocity of the Sun around the Milky Way.
Comparing the galactic dark photon flux from annihilating dark matter with annihilation cross section given by Eq. 8 and initial fraction fi = 0.1, and decaying dark matter with Γ ϕ = 0.03 t −1 0 .For annihilation and decay, "Energy" refers to the dark photon energy, while for the direct χ flux, it refers to mχ.

VII. CONCLUSIONS
In this paper, we have shown that for χ being a subcomponent of dark matter with large annihilation cross sections, there exists currently unconstrained parameter space that can lead to visible signals in future gaseous detectors.We do not provide exclusions or projections, as that would require essential features of future experiments, such as energy thresholds and estimates of backgrounds, that are beyond the scope of this work.However, we do provide code to determine the rate of dark photon events at a particular experiment (see the GitHub link in the abstract).We also wish to point out some detector properties that would be most useful in looking for this signal • Low Densities and Large Volumes: Clearly, a larger target mass allows for more dark photons to be absorbed, and we see in Fig. 5 that lower densities can be sensitive to lower m A ′ since the dark photons are predominantly transversely polarized.
A variable density (with significant exposure) could scan parameter space for an even larger range of dark photon masses.
• Low Energy thresholds and Good Energy Resolution: The flux of A ′ is largest for small m χ , but this flux would only be visible if the energy threshold is below m χ .Moreover, the flux of dark photons would be mono-energetic.Good energy resolution and reconstruction would have these dark photon events appear as a peak, reducing the number of events necessary to be significant.
• Good Spatial Resolution: Typical x-ray backgrounds, which may come from outside the detector or detector components, will mainly occur on the detector edges and be unable to penetrate to the center of the fiducial volume.On the other hand, the small interaction probability of dark photons would make them equally likely to appear anywhere within the detector, so topological discrimination could be used to reduce the background.the final state is found to be dominated by transversely polarized dark photons.Production of longitudinally podark photons is highly suppressed.In order to demonstrate the dominance of transverse polarizations, we subtract off the transversely polarized contribution from the total rate and compare the two.The process of χ(k 1 ) χ(k 2 ) → A ′,ν (p1)A ′,µ (p 2 ) at treelevel consists of the matrix element where t and u are the Mandelstam variables and ϵ µ (p) is a polarization vector, not to be confused with the kinetic mixing parameter.To compute the cross section of this process, we find |M | 2 and sum over the outgoing polarizations.We know that for a massive vector boson, the sum over polarizations (denoted by λ) returns (B4) When we apply the on-shell condition, the first term will be O(m 2 A ′ /m 2 χ ), which is small enough that we will ignore it.To find final value of the polarization sum, it is useful to define a new four-vector η µ (ω, p) (this is easily implemented using FeynCalc [74][75][76]).If we work in the low-velocity limit, expanding in both m A ′ and K defined as s = 4(m χ + K) 2 , we find, dark photon wavelength is larger than the radius of that inner electron) then retardation effects can be neglected and consequently e ikx = 1 to a good approximation (Section 69, Eq. 69.9, [77]).
For argon, Zαm e ≈ 65 keV, for fluorine Zαm e ≈ 32 keV, for germaniun Zαm e ≈ 116 keV, for sulphur Zαm e ≈ 60 keV, and for xenon Zαm e ≈ 200 keV.For these elements, our analysis is accordingly restricted to ω ≲ 30 keV to maintain the dipole approximation.For helium Zαm e ≈ 7 keV, however, such that the dipole approximation becomes invalid at energies ≳ 10 keV.In Fig. 4c, the region above 10 keV is highlighted to indicate the breakdown of the dipole approximation there.

FIG. 1 .
FIG.1.Flux of dark photons at Earth vs. ⟨σv⟩ obtained by computing Eq. 7 for different χ masses, all with fi = 0.1.We can clearly see the optimum ⟨σv⟩ for each mass.

FIG. 2 .
FIG. 2. The real and imaginary part of 1−n *ref for a xenon gas at ρXe = 5 kg m −3 .Note that for this plot, we are using the complex conjugate of the index of refraction, as Im(n ref ) > 0.

FIG. 7 .
FIG.7.Comparing the two equations in Eq.A3 as a function of radius r from the galactic center, with fi = 0.1 and mχ = 1 keV.