Absorption of Sub-MeV Fermionic Dark Matter by Electron Targets

We study a new class of signals where fermionic dark matter is absorbed by bound electron targets. Fermionic absorption signals in direct detection and neutrino experiments are sensitive to dark matter with sub-MeV mass, probing a region of parameter space in which dark matter is otherwise challenging to detect. We calculate the rate and energy deposition spectrum in xenon-based detectors, making projections for current and future experiments. We present two possible models that display fermionic absorption by electrons and study the detection prospects in light of other constraints.


I. INTRODUCTION
The observation of dark matter (DM) through its gravitational interactions is indisputable evidence of physics beyond the Standard Model. This has motivated experimental efforts to learn about dark matter by searching for its decays, annihilations, self-interactions, and scattering off Standard Model particles. If DM is sufficiently heavy, the scattering off a target material can deposit a detectable amount of kinetic energy in large-volume detectors such as time-projection chambers (TPCs). The energy deposited by the scattering of non-relativistic DM (χ) off a target (T ) via χT → χT is, at most, O(100) keV, demonstrating the need for sensitive, low-threshold direct detection experiments.
Recently, several of us proposed a class of novel and * jdror1@ucsc.edu † gelor@uw.edu ‡ rmcgehee@umich.edu § tientien@uoregon.edu distinct signatures arising from the absorption of sub-GeV fermionic DM [56,57]. The energy deposited in a fermionic absorption signal is largely independent of the dark matter velocity and parametrically larger than that of DM scattering. Thus, any large-exposure detector can be used to search for this class of signals. 1 In [57], we considered the specific signals arising from models in which the DM is absorbed by nuclear targets. Such signals can probe DM masses down to an MeV with searches in existing data and significantly below with the proposed lower-threshold experiments. If an atom-bound electron absorbs enough energy from incoming DM to be ionized, the ionized electron may be searched for in photoelectron signatures, known as S2, in TPCs [61][62][63] -see Fig. 1 for a schematic of the signal. Current xenon-based direct detection experiments such as XENON1T [61] and LZ [64], as well as future ones such as XENONnT [65], PandaX-4T [66], and DARWIN [67], are sensitive to fermionic absorption of DM with masses in the sub-MeV range by electrons. We explore searches for absorption by electron targets, with a focus on xenon detectors, although our discussion is applicable to other target materials (e.g., liquid argon).
Fermionic absorption by electrons can be induced by vector-and scalar-type operators, given respectively by, where Λ may be a ratio of a mediator mass to its coupling. We will consider the case where the mediator is always 1 e ! ⌫e (5) 3 ⇥ 10 10 sec ✓ 2 GeV 10 TeV GeV -10 TeV Baryon Asymmetry ⇠ 10 10 Absorption of fermionic DM by electrons in xenon (A = 54). A DM particle χ is absorbed by an electron in a target xenon atom of a direct detection experiment, emitting a neutrino. With enough energy, the electron is ionized and produces a distinctive signal. Fermionic absorption, unlike DM scattering, has a significant contribution from electrons bound to both inner and outer atomic shells.
heavier than the energy transferred during direct detection and hence the interaction is adequately described by an effective operator. DM capable of inducing fermionic absorption is inevitably unstable. Meta-stable DM changes the equation of state of the universe and can imprint signals in the Cosmic Microwave Background. If the decay products include (or subsequently emit) photons, decaying DM can also be detected with a range of telescopes, strengthening the discovery potential. These searches limit the types of operators as well as the DM mass range detectable by direct detection. In addition, any such operator will have constraints from overproduction near Big Bang Nucleosynthesis (BBN) and searches in colliders which further constrain the viable parameter space. The interplay between the direct detection rates, decay and directproduction bounds is model-dependent. To demonstrate the feasibility of fermionic absorption DM models, we discuss possible UV completions for both operator types. 2 The paper is organized as follows. In Sec. II, we present the details of the signal arising from fermionic absorption by electrons. Here, we discuss the rate and present the projected constraints at current and upcoming xenonbased experiments. We also briefly discuss and perform a fit to the recent XENON1T excess.  Sec. III, we present two UV completions that give rise to fermionic absorption and compute the decay and direct constraint bounds in each case. In addition, we include a model-independent discussion of the irreducible decay constraints. We conclude in Sec. IV. App. A contains a detailed derivation of the differential ionization cross section for fermionic absorption by bound electrons.

II. FERMIONIC ABSORPTION BY ELECTRONS
In this section, we present projected limits at current and upcoming xenon experiments for the two operators in Eqs. (1)-(2) -the details of possible renormalizable models that generate such operators, along with the associated model-dependent constraints, follow in Sec. III. We begin by calculating the rate and spectrum of fermionic absorption by (atom-bound) electrons. The results presented here are general and apply to bound electrons of any target material, up to specifying the material-dependent form factors.
Fermionic DM is absorbed by electron targets via the process χ+A → ν +A + +e − , where A denotes the atomic number of the target as shown in Fig. 1. The underlying interaction is given by which can be mediated by, for instance, the operators of Eqs. (1)- (2). In particular, we are interested in the outgoing electron which may be detected via scintillation signals or secondary ionizations. Since we are considering the regime where m χ < 2m e and DM has typical halo velocities v χ ∼ 10 −3 , the atom is sufficiently heavy so that the target electron is in a static background potential to good approximation.

3
The momentum of the outgoing neutrino is given by where v χ is the incoming DM velocity, q is the momentum transfer to the target electron, and θ qv is the angle between v χ and q, all evaluated in the lab frame. The absorbing electron in the shell (n, l) with binding energy E nl B < 0 is ionized with recoil energy E R (evaluated far from the atom potential). Energy conservation gives Typically, the initial momentum of the DM is negligible relative to the momentum of the neutrino, m χ v χ q. This holds when Therefore, we can drop O (v χ ) terms and energy conservation simplifies to Unless otherwise explicitly stated, all future instances of q will be implicitly dictated by the energy conservation condition in Eq. (6). Following the procedure discussed in the appendices of [4] and expanded on in App. A, we arrive at the average differential ionization cross section of an electron in the (n, l) shell, where |M| 2 is the matrix element squared averaged over initial spins and summed over final ones, and Θ is the Heaviside step function. f nl ion (k , q) 2 is the ionization form factor of an electron in the (n, l) shell with final momentum k = √ 2m e E R , and is given by [3,12] |f nl ion (k , q) where [· · · ] is the Wigner 3-j symbol and j L are the spherical Bessel functions. R nl (r) are the bound electron radial wavefunctions [68] and the radial wavefunctions R k l (r) of the outgoing unbound electrons are obtained by solving the radial Schrödinger equation with a central potential Z eff (r)/r. Z eff is determined from the initial electron wavefunction under the assumption that it is a bound state of the same potential. Note that this procedure assumes non-relativistic electron wavefunctions; once q m e , one must take into account relativistic corrections to the electron wavefunctions, which will increase the total cross-section. This enhancement to the crosssection ranges from a factor of a few to several orders of magnitude for q ∼ O(MeV) [69,70]. In our results, we denote the region m χ 100 keV in which relativistic corrections become important by dashed lines. The specific form of the differential ionization cross section depends on the underlying operator in the matrix element. For example, for the vector operator in Eq. (1) we find while the scalar operator in Eq. (2) results in where we have kept the terms of O(q 3 ). Note that the operators have a different parametric dependence on q.
In both cases, σ e ≡ m 2 χ / 4πΛ 4 is a useful parameterization of the cross section.
The differential cross section in Eq. (7) and the corresponding ionization rate are independent of the DM velocity distribution to leading order. As a result, this class of signals is free from the usual astrophysical uncertainties on the local DM velocity distribution that affect direct detection bounds (see e.g. [71][72][73][74][75][76]). The total differential ionization rate is found by summing over all possible (n, l) shells of the absorbing target electrons Here N T is the number of targets (each of which has all (n, l) shells of bound electrons) per unit mass and ρ χ ∼ 0.4 GeV/cm 3 [77] is the local DM energy density. Fermionic absorption by electrons can be searched for in a host of experiments and target materials. Specifically, if the incoming DM transfers enough energy to ionize an electron, the resulting ionization signal can be searched for in the S2 data set of TPC experiments [61][62][63]. For concreteness, we focus on the absorption of fermionic DM by electrons in liquid xenon, although our discussion is applicable to other noble liquid targets such as argon.
We consider absorption by electrons in the shells with binding energies E nl B shown in Table I. Note that the kinematics of fermionic absorption necessitate the inclusion of all the electron shells, in contrast to DM-electron scattering which is driven primarily by the outer shellelectrons. In Fig. 2, we plot the ionization form factor of each xenon shell for two benchmark DM masses. For larger DM masses, the incoming electron kinetic energy is small relative to the mass of χ resulting in form factors becoming localized around E R = m 2 χ /2m e , the recoil energy of a free electron absorbing the DM. In Fig. 3, we show a few representative examples of the differential ionization rate in liquid xenon. We project the sensitivities of XENON1T [61], LZ [64], PandaX-4T [66], XENONnT [65], and DARWIN [67] to the ionization rates in Eqs. (9) and (10), corresponding to the operators in Eqs. (1) and (2), in Fig. 4. 3 For both operators, the shaded regions are excluded by a combination of indirect detection constraints on DM decay, constraints from the overproduction of the DM, and direct constraints on the mediator. To calculate the differential ionization rate in XENON1T, we convolve Eq. (11) with their reported total (detector and selection) efficiency using a hard cutoff at E R = 1 keV [78]. We calculate the projected constraints on σ e for various m χ by requiring 10 or more events to have occurred over XENON1T's full exposure. We calculate the rates and projections for the other experiments similarly, assuming they have the same efficiency as XENON1T. As discussed above, relativistic corrections to the form factor and absorption rate start becoming relevant at higher m χ , which we denote by dashed lines. Additionally, the starting assumption that the initial DM kinetic energy and momentum are negligible is only valid when E R − E nl B m χ . Since we impose a hard recoil cutoff at 1 keV, we can only reliably calculate the rates down to m χ 3 keV and only show this range in Fig. 4. Existing searches for DM scattering can be recast as limits for fermionic absorption by electrons with the caveat that the kinematics begin to differ for m χ 100 keV due to relativistic corrections of the electron wavefunctions. We find that XENON1T has the potential to probe 20 keV m χ 1 MeV for the dark photon model, but is less effective for the scalar mediator model, as shown in Fig. 4. However, LZ, as well as future proposed experiments such as PandaX-4T, XENONnT and DARWIN, have the potential to probe both models. We show bounds from a combination of decays as well as direct and cosmological constraints. For both the vector and scalar cases, the constraints are driven by the overproduction of DM and decays. In addition, there are also direct bounds on the mediator which have been taken into account. The decay and direct constraints are sensitive to the models introduced in section III and may shift depending on the specific UV completion. In contrast, the direct detection signal only relies on the underlying effective operator and is robust given an operator type.
We also explore the possibility that fermionic absorption on electrons can explain the reported XENON1T excess [78]. 5 We preform a χ 2 fit to the XENON1T light as the phase-space packing bound, mχ 190 eV [79,80]. 5 It was claimed that a similar model could explain the XENON1T signal. As a background model, we adopt the "B 0 " model as presented by the XENON1T collaboration [78], for which χ 2 /dof = 47.2/29, and fit the fermionic absorption signal (floating DM mass and χ − e cross section) plus B 0 background model. The best fit point is (m χ , σ e ) = (56.5 keV, 1 × 10 −49 cm 2 ) which corresponds to χ 2 /dof = 35.4/27. The best fit to the XENON1T data is shown Fig. 4 as a red-star with 1σ and 2σ contours in mass and cross section. The excess region is easily accommodated by the dark photon UV completion, while for the scalar UV completion, it is firmly excluded by searches for DM decays into νγγ.

III. MODELS
We now discuss how the direct detection signal arises in explicit models, effectively resolving the scale Λ that gives rise to the higher dimensional operators as in Eqs. (1) and (2). Importantly, we compute the constraints in each case emphasizing some general features for both the scalarand vector-mediated models. Constraints will arise from direct searches for the mediators, overproduction of χ near BBN, and various decays modes.
Regardless of the details of the UV completion, operators giving rise to fermionic absorption will always lead excess [81]. This work omitted form factors in the matrix elements, which are critical to calculating the absorption rate. In addition, [81] took DM decay bounds to be of order the age of the universe, while current bounds from indirect detection are significantly more stringent. to some decays computed by inserting these operators into loops of Standard Model states. The list of leading diagrams resulting in DM decay is given in Table II. Loop-induced decays contain several different scales -an electroweak boson mass, the mass of the mediator making up the direct detection operator, and an electron or χ mass. If the mediator mass is above the weak scale, then the loop momentum can often be of order the weak scale, greatly enhancing the induced decay rate. If, on the other hand, the mediator is well below the weak scale (but has small couplings to Standard Model fields to avoid other bounds), then the induced decays are smaller as the loopmomentum can at most be the mediator mass and we will work in this limit.
DM models with a detectable fermionic absorption rate typically have interactions that induce absorption at leading order and decays at higher coupling and/or loop order. We consider two example UV completions treating the left handed neutrino and electron as independent components. Therefore, above the electroweak scale new states must appear to absorb divergences until renormalization group flow. It is straightforward to extend the models presented here to be fully renormalizable by introducing couplings with the Standard Model Higgs and integrating out additional fields above the weak scale. For the scalar operator, we consider a scalar mediator that couples off-diagonally between DM and a neutrino as a consequence of a global symmetry. For the vector operator, we consider a dark-photon-mediated model with DM-neutrino mixing. In both cases, a robust bound on the scale of the higher-dimensional operator Λ comes from requiring that DM is not overproduced in the early Universe. The absorption operator leads to the produc- Loop-induced diagrams for the leading decays of χ induced by aχΓ1νēΓ2e operator insertion, denoted by . Depending on the mediator, χ may be protected from rapid decays by various symmetries, making certain diagrams vanish without electroweak corrections. We only estimate electroweak corrections parametrically and so do not list all the diagrams. Mixing between χ and the neutrinos will induce additional decay channels. tion of DM between the time of Standard Model neutrino decoupling (T 2.3 MeV) and when the electrons leave the bath. This constrains the mediator scale, Λ, to be above the weak scale for m χ ∼ 1 keV, with stronger bounds for heavier masses. In our final results we employ the bounds computed in Ref. [82].

A. Scalar Mediator
In this section, we present a UV completion for the operator Consider a theory with a Dirac DM candidate, χ, and a scalar field, ϕ. We impose a global U (1) symmetry with charges, Q i , such that Q χ L = −Q χ R = +Q ν with ϕ and the rest of the Standard Model remaining uncharged.
Here we assume neutrinos are Majorana fermions and neglect corrections proportional to their mass. The interaction Lagrangian is We note that the symmetry is explicitly broken by electroweak interactions such that terms like ϕνP L ν will be generated at loop order; we estimate the size of such contributions below. Integrating out the scalar results in the operator that gives rise to the fermionic absorption signal in Eq. (12), where we identify the higher dimensional operator scale as, The dominant χ decay modes are shown in Table II with the operator insertion resolved by an internal ϕ line.
Without electroweak corrections, χ → νγ vanishes and so the leading decay is χ → νγγ, the rate for which is given by Importantly, the rate depends only on m χ and Λ and not on any other free parameters of the model. This makes the decay a largely irreducible constraint independent of the details of the model and a generic prediction for a scalar-mediated absorption signal. The bound on the decay rate, which is O(10 −27 sec −1 ), is computed over the mass range of interest in Ref. [83]. Other decays arise at higher loop order and are more sensitive to variations in the model parameters. The decay rate for νγ is induced by weak interactions and the precise form of the rate depends on the UV completion above the EW scale. Due to this inherent uncertainty, and difficulties in carrying out the two-loop calculation, we settle for an order of magnitude estimate for this process. In the limit that m ϕ m W , the rate is of order, Although this interaction is generated at two-loop order, it is still significant as it is not suppressed by high powers of m χ . For the limits we use the recast the bound computed in Ref. [83] (also of O(10 −27 sec −1 )) for a scalar dark matter decaying into two photons.
Decays of χ into 3 neutrinos are mediated by weak insertions. In the limit of massless neutrinos, the Lorentz structure of the scalar operator leads to the decay vanishing at one loop but it is generating at two loops leading to an effective dimension-6 operator between χ and 3 neutrinos,χP L ννP L ν with a scale, leading to a decay rate, Since the invisible decay is poorly constrained relative to visible bounds (see, e.g. Ref. [84]), it does not end up being significant in any region of the parameter space of interest.
The bounds on this model from DM decay are shown in the left panel of Fig. 4. We see that the robust bounds of the decay χ → νγγ limit the viability of observing fermionic absorption for dark matter heavier than around 50 keV.

B. Dark Photon Mediator
We now present a UV completion for the vector operator, Consider a Dirac fermion, χ, charged under a dark gauge group, U (1) , with a kinetic mixing, ε, with the Standard Model photon. For energy transfers well below the dark photon mass, m A , there is an interaction between the dark current, J µ ≡ g Xχ γ µ χ, and the electromagnetic current, J µ , At this point, χ is stable as a consequence of its U (1) charge and one cannot generate the effective operator (19). We now introduce a U (1) -charged scalar, ϕ, that breaks the symmetry but is neutral under SU (3) C × U (1) Y . Furthermore, we focus on the least constrained case where neutrinos are Dirac and ϕ couples to the righthanded component. 6 The χ mass and interaction Lagrangian is, After ϕ gets a vacuum expectation value ϕ , the neutrino mass is largely unchanged 7 , however the eigenstates shift as The χ mass remains approximately unchanged if y ϕ m χ , but the shift induces a new term in the potential. Defining θ ≡ y ϕ /m χ , the shifted Lagrangian in the limit where θ is small is, where the ellipses denote higher order θ corrections that have no bearing on the phenomenology. The fermionic absorption operator of Eq. (19) (in addition to couplings to the rest of the fermions making up the electromagnetic current) is generated via the identification Note that the third term generated in Eq. (23) couples the DM to neutrinos; the result of which is to admit additional DM decay channels which we take into consideration in what follows. The leading decay channels for the vector-mediated model are shown in Table II. Since χ mixes with the right-handed neutrino, the decay of χ → γν requires a neutrino mass insertion and can be negligibly small. The decay of χ into νγγ does not arise at 1-loop since loops of vector-like fermions with an odd number of external vector legs are zero by charge conjugation symmetry. The leading decays are thus χ → 3ν and χ → νγγγ. The decay rates for these were computed for a similar model with dark photons coupled to nucleons in [57] and we repurpose them here. Given in terms of model parameters, For setting limits on the χ → νγγγ decay, we approximate the bound with those computed in Ref. [83] for dark matter decaying to an invisible state and two photons. For setting limits on χ → 3ν decay channel, we use the results from Ref. [84] which restrict Γ −1 3ν 50 Gyr. The νγγγ decay will be present with (at least) this rate for any model that leads to a vector-mediated operator and can be considered as an irreducible decay; it may be possible to construct models which evade the 3ν channel.
In addition to decays, direct dark photon searches place constraints on the parameter space. Constraints on dark photons are well documented and have been summarized recently for visible decays in e.g. Fig. 4 of Ref. [85]. In making plots, we fix the dark photon mass and mixing angle to be m A = 20 MeV and ε = 10 −3 , leaving the remaining free parameters as g X , m X , and θ. The resulting projected sensitivity of searches for the dark photon mediator model are presented in Fig. 4. The decay and direct constraints depend on the value of the mixing angle. In Fig. 5, we show the limits for two benchmark values of θ. The shaded bands in Fig. 4 correspond to the minimal decay and direct constraints (the intersection of decay and direct bounds as θ varies) such that there exists a value of θ for which the point in the parameter space is otherwise allowed.

IV. DISCUSSION
In this work, we introduce a new class of signals where fermionic DM is absorbed by electrons and present models which give rise to such signals; these are a natural extension of Fermionic DM Absorption [56,57]. For concreteness, we have focused primarily on targets in liquid xenon and XENON1T's capabilities to discover these signals. In addition, we calculated the projected constraints for PandaX-4T, XENONnT, LZ, and DARWIN, assuming similar detector efficiencies. We have found XENON1T, with its current exposure, can probe a DM of mass 20 keV m χ 1 MeV for the dark-photonmediated model presented here, while future experiments can go down to a few keV in DM mass. Although XENON1T is not sensitive to the scalar-mediated model, near-future experiments can probe the scalar model for DM masses of about 10 keV m χ 40 keV. Nextgeneration detectors, such as DARWIN, will be able to probe orders of magnitude more parameter space in both cases. Argon-based detectors, such as DarkSide-50 [86], DarkSide-20K [87], and Argo [88], will give rise to similar constraints for comparable exposures and detector thresholds; we leave these calculations to future work.
While we focused entirely on direct detection experiments, the potentially large energy deposits of fermionic absorption open the possibility of detection in neutrino detectors, if they have energy thresholds below an MeV. A notable neutrino experiment with a sufficiently low energy threshold is the CUORE experiment [89]. Although built to search for neutrinoless double-beta decays, it can achieve a threshold of a few keV [90] with exposures comparable to XENON1T. For heavier χ, experiments such as Borexino [91] can leverage their huge exposure to search for fermionic absorption. While Borexino has an electron recoil energy threshold of ∼ 70 keV, its exposure is ∼ 817 times that of XENON1T, making it much more sensitive to absorption for heavier DM m χ 100 keV. Computing the ionization form factors of different targets would allow one to calculate the projected absorption rates in various other experiments, which is beyond the scope of this work.
Relaxing the assumption of DM stability leads to a new set of DM direct detection signals that can be probed by XENON1T and future liquid xenon experiments. This offers a new opportunity to probe DM with masses below an MeV, a region generally experimentally inaccessible for elastically-scattering DM in light of bounds from BBN, CMB, and overproduction. By contrast, fermionic DM absorption on electrons has the potential to be discovered in the current and near-future experiments.
Fermionic DM absorption by electrons will probe DM masses near their lower bound of a few keV; indeed, if DM is so light, there may be no other way to find it.  In this appendix, we build upon the derivation in App. A of [4]. The key difference between fermionic absorption cross sections and those of elastic scattering is due to the velocity-independence of the recoil energy (see [56,57] for the nuclear target case). In elastic scattering, the incoming DM velocity integral in the averaged differential cross section utilizes the energy-conserving δ function to impose a physical minimum incoming DM velocity, v min , to achieve a particular E R .
For instance, in scattering off bound electrons, the relevant piece of the differential scattering rate is (see Eq.(A 12-13) from [4]) Note here that q is not evaluated as in the fermionic absorption case, given by Eq. (6). In the second line, we used δ ∆E 1→2 + q 2 2mχ − qvcθ qv = 1 qv δ cθ qv − cθ 0 qv and noted that cθ 0 qv = ∆E1→2 qv + q 2mχv which implies v min = ∆E1→2 q + q 2mχ . The second line explains the integral form in the first line of Eq. (A 14) from [4]. In the last line, the 1/2 comes from the cθ qv integral.
By contrast, for fermionic absorption by electrons, the integral over DM's initial velocity is trivial since there is no velocity dependence in the energies at leading order: Thus, the mapping we expect from the usual scattering case is With this map, we can write down the differential ionization rate in the case of fermionic absorption. We also need to map the overall factor 1 where q = m χ + E nl B − E R and k = √ 2m e E R . This is the result quoted in Eq. (7).