Atomic Responses to General Dark Matter-Electron Interactions

Atomic responses to general dark matter-electron interactions

Riccardo Catena,1, ∗ Timon Emken,1, † Nicola A. Spaldin,2, ‡ and Walter Tarantino2, § 1Chalmers University of Technology, Department of Physics, SE-412 96 Göteborg, Sweden 2Department of Materials, ETH Zürich, CH-8093 Zürich,Switzerland In the leading paradigm of modern cosmology, about 80% of our Universe’s matter content is in the form of hypothetical, as yet undetected particles. These do not emit or absorb radiation at any observable wavelengths, and therefore constitute the so-called dark matter (DM) component of the Universe. Detecting the particles forming the Milky Way DM component is one of the main challenges for astroparticle physics and basic science in general. One promising way to achieve this goal is to search for rare DM-electron interactions in low-background deep underground detectors. Key to the interpretation of this search is the response of detectors’ materials to elementary DM-electron interactions defined in terms of electron wave functions’ overlap integrals. In this work, we compute the response of atomic argon and xenon targets used in operating DM search experiments to general, so far unexplored DM-electron interactions. We find that the rate at which atoms can be ionized via DM-electron scattering can in general be expressed in terms of four independent atomic responses, three of which we identify here for the first time. We find our new atomic responses to be numerically important in a variety of cases, which we identify and investigate thoroughly using effective theory methods. We then use our atomic responses to set 90% confidence level (C.L.) exclusion limits on the strength of a wide range of DM-electron interactions from the null result of DM search experiments using argon and xenon targets.

I. INTRODUCTION experiments. In the literature, this framework is referred to as the “sub-GeV” or “light DM” scenario [11]. One of the major unsolved mysteries in modern physics Materials used in direct detection experiments which is the elusive nature of dark matter (DM) [1]. Revealed have set limits on the strength of DM-electron interac- via the gravitational pull it exerts on stars, galaxies, tions include dual-phase argon [12] and xenon [13–15] light and the Universe at large, DM is widely believed targets, as well as germanium and silicon semiconduc- to be made of hypothetical particles which have so tors [9, 16–22], and sodium iodide crystals [23]. Further- far escaped detection [2]. Detecting the particles form- more, graphene [24, 25], 3D Dirac materials [26, 27], polar ing the Milky Way DM component and understand- crystals [28], scintillators [29, 30], as well as superconduc- ing their properties in terms of particle physics mod- tors [31–33] have also been proposed recently as target els is a top priority in astroparticle physics [3]. The materials to probe DM-electron interactions. experimental technique known as direct detection will In order to interpret the results of direct detection ex- play a major role in elucidating the nature of DM in periments searching for signals of DM-electron interac- the coming years [4]. It searches for signals of DM in- tions, it is crucial to understand quantitatively how de- teractions in low background deep underground detec- tector materials respond to general DM-electron interac- tors [5,6]. Such interactions could produce observable tions. Materials’ responses to DM-electron interactions nuclear recoils via DM-nucleus scattering or induce elec- can be quantified in terms of the overlap between initial tronic transitions via DM-electron scattering in the de- (before scattering) and final (after scattering) electron tector material [7,8]. DM particles that are gravitation- wave functions. The stronger the material response, the ally bound to our galaxy do not have enough kinetic larger the expected rate of DM-electron interactions in energy to produce an observable nuclear recoil if their the target. mass is approximately below 1 GeV/c2. On the other In the pioneering works [8,9, 18], detector materials’ arXiv:1912.08204v2 [hep-ph] 19 Aug 2020 end, they have enough kinetic energy to ionize an atom responses to DM-electron interactions have been com- in a target material or induce the transition from the puted under the assumption that the amplitude for DM valence to the conduction band in a semiconductor crys- scattering by free electrons only depends on the initial tal if their mass is above about 1 MeV/c2 [9]. Motivated and final state particle momenta through the momen- 1 by recent experimental efforts in the field of astroparti- tum transferred in the scattering . While this restriction cle physics [10], here we focus on DM particles of mass significantly simplifies the calculation of the predicted in the 1 – 1000 MeV/c2 range and on their interactions DM-electron interaction rate in a given detector, it pre- with the electrons in materials used in direct detection vents computations of physical observables whenever the scattering amplitude exhibits a more general momentum

∗ [email protected] † [email protected] 1 For more recent works on the theoretical framework of so-called ‡ [email protected] spin-independent DM-electron interactions and a comparison of § [email protected] targets, we also refer to [34] and [35] respectively. 2 dependence. A more general dependence on the particle tions found in this work can only arise from a general momenta occurs in a variety of models for DM [36], in- treatment of DM-electron interactions. We will show this cluding those where DM interacts with electrons via an using an effective theory approach to DM-electron in- anapole or magnetic dipole coupling [37]. teractions. Specific examples of scenarios where the new In this paper, we calculate the response of materials response functions appear include models where DM in- used in operating direct detection experiments to a gen- teracts via an anapole or magnetic dipole coupling to eral class of nonrelativistic DM-electron interactions. We photons. build this class of nonrelativistic interactions by using These general considerations lead us to the intriguing effective theory methods developed in the context of conclusion that DM particles may represent a new type DM-nucleus scattering, e.g. Refs. [36, 38–40]. In terms of probe for the electronic structure of a sample mate- of dependence on the incoming and outgoing particle rial. At the moment, the information we can get about momenta, this class of interactions generates the most the actual electronic structure of a material is limited by general amplitude for the nonrelativistic scattering of the probes we can use. If alongside standard probes (elec- DM particles by free electrons. The predicted amplitude trons, photons, alpha particles, etc. . . ) we could use a is compatible with momentum and energy conservation, particle that interacts in a totally different way with elec- and is invariant under Galilean transformations (i.e. con- trons, we might be able to get as yet “hidden” features of stant shifts of particle velocities) and three-dimensional the electronic structures that would be missed with stan- rotations. At the same time, it depends on the incoming dard probes. While this would not change how a material and outgoing particle momenta through more than only responds to standard particles, which ultimately is what the momentum transferred. It can explicitly depend on is currently relevant for applications in the real world, it a second, independent combination of particle momenta, would help us to know more about the material, and the as well as on the DM particle and electron spin opera- more we know, the easier is to design and produce new tors. We present our results focusing on DM scattering materials with desired properties of technological rele- from electrons bound in isolated argon and xenon atoms, vance. Consequently, the detection of DM particles at since these are the targets used in leading experiments direct detection experiments might not only solve one of such as XENON1T [15] and DarkSide-50 [12]. However, the most pressing problems in astroparticle physics, but most of the expressions derived in this work also apply also open up a new window on the exploration of materi- to other target materials, such as semiconductors. They als’ responses to external probes. From this perspective, do not apply to scenarios where incoming DM particles the results of this work constitute the first step within a induce collective excitations, such as magnons or plas- far-reaching program which has the potential to open an mons [41, 42]. entirely new field of research. Within this general theoretical framework, we find that While the notion of DM as a probe for the electronic the rate of DM-induced electronic transitions in a given structure of a sample material would require good knowl- target material can be expressed in terms of four, in- edge about the local properties of the galactic DM halo, dependent material response functions. They depend on the novel material responses identified here only depend scalar and vectorial combinations of electron momenta on the materials’ electronic structure and are not degen- and wave functions. We numerically evaluate these re- erate (i.e. cannot be confused) with variations in the as- sponse functions for argon and xenon targets, and use sumed local DM velocity distribution. them to set 90% C.L. exclusion limits on the strength This work is organized as follows. In Sec.II we of DM-electron interactions from the current null result derive a general expression for the rate of DM-induced of the XENON10 [43], XENON1T [15] and DarkSide- electronic transitions in target materials which we 50 [12] experiments. We also find that the contribution then specialize to the case of general, nonrelativistic of each material response function to the rate of DM- DM-electron interactions, focusing on argon and xenon induced electronic transitions is weighted by a corre- as target materials. In Sec.III, we express the rate of sponding combination of particle momenta which is in- DM-induced electronic transitions in terms of DM and dependent of the initial and final electron wave func- atomic response functions, while in Sec.IV we apply tions. We refer to these weights as “DM response func- our results to compute physical observables for argon tions”, in analogy to previous studies in the context of and xenon targets and set limits on the strength of DM-nucleus scattering [39]. DM-electron interactions from the current null results of the XENON10, XENON1T and DarkSide-50 experi- Previous works could only identify one of the four ma- ments. We present our conclusions in Sec.V and provide terial response functions found here because they as- the details of our calculations in the appendices. In sumed that the amplitude for DM scattering by free addition, we provide the numerical code used to evaluate electrons solely depends on the momentum transfer, e.g. the atomic response functions [44].2 Ref. [18]. This previously found response function is not only generated by DM-electron scattering. It also arises when materials are probed with known particles, such as photons or electrons which interact via the electromag- 2 An archived version can be found un- netic force. In contrast, the three novel response func- der [DOI:10.5281/zenodo.3581334]. 3

χ χ χ χ χ χ E2<0 p p'=p-q p p'=p-q p p'=p-q q=p-p'

elas�c inelas�c inelas�c - - e e E2>0 k k'=k+q E1<0 E1<0

e- e- e- (b) excita�on (c) ioniza�on FIG. 1. Illustration of elastic (left) and inelastic (middle, right) DM-electron scatterings. In the inelastic case, the electron gets excited by DM, transitioning from an atomic bound state with negative energy E1 to a ﬁnal state with higher energy E2, which is either still negative for excitation (middle) or positive for ionization (right).

A. Electron transition rate Throughout this paper, we employ natural units, i.e., c = ~ = 1. We start by calculating the rate of transitions from an initial state |ii ≡ |e1, pi = |e1i⊗|pi to a final state |fi ≡ 0 0 |e2, p i = |e2i ⊗ |p i, where initial and final DM particle states are labeled by the corresponding tridimensional II. DARK MATTER-INDUCED ELECTRONIC 0 TRANSITIONS momenta, p and p , respectively, while |e1i (|e2i) denotes the initial (final) electron state. Here |e1i and |e2i are eigenstates of the electron’s energy with eigenvalue E1 In this section we derive a general expression for and E2, but not of momentum. This is illustrated in the the rate of electronic transitions induced by the scat- middle and right panel of Fig.1. Following [18], single- tering of Milky Way DM particles in target materials particle states are normalized as (Sec.IIA). This result applies to arbitrary DM-electron Z interactions (including relativistic interactions) and tar- 3 (3) 3 get materials. We then specialize our general expression hp|pi = (2π) δ (0) = d x ≡ V, (1) to the case of nonrelativistic DM-electron interactions (Sec.IIB) and to argon and xenon as target materials and, similarly, e.g., he1|e1i = V . The divergent volume (Sec.IIC). We use these results to model the response factor, V , will not appear in the expressions for physical of atoms to general DM-electron interactions in Sec.III observables. and set limits on the strength of such interactions from The first non-trivial term in the Dyson expansion of the the null results of operating direct detection experiments S-matrix element corresponding to the |ii → |fi transi- in Sec.IV. tion is

Z Z 3 Z 3 0 3 d k d k 0 0 0 = −i(2π)δ(Ef − Ei) d x he2|k ihk , p |HS(x)|p, kihk|e1i (2) (2π)3 (2π)3

where HI (x)(HS(x)) is the interaction Hamiltonian tity density in the interaction (Schrödinger)picture at the Z d3k spacetime point x, and H0 is the Hamiltonian of the 1 3 |kihk| = , (3) DM-electron system with HI (x) set to zero which obeys (2π) H |ii = E |ii and H |fi = E |fi. We also used the iden- 0 i 0 f where |ki are eigenstates of the free-electron Hamilto- 4 nian. The asymptotic states |ii and |fi are eigenstates Eq. (9) back into the general S-matrix element in Eq. (2), of H0 because HI is assumed to be different from zero we obtain within a finite time window only (the so-called adiabatic 1 Z d3k hypothesis). In the nonrelativistic limit, the correspond- Sfi = (2π)δ(Ef − Ei) 3 he2|k + qihk|e1i 4mχme (2π) ing eigenvalues, denoted here by Ei and Ef , respectively, are given by × iM(k, p, k + q, p − q) , (10) where we used the nonrelativistic expression for mχ 2 p 2E 0 2E 0 2E 2E . We now introduce the electron wave Ei = mχ + me + v + E1 , (4) p k k p √ 2 ∗ 2 functions ψ1(k) = hk|e1i/ V and ψ2 (k + q) = he2|k + |mχv − q| √ Ef = mχ + me + + E2 , (5) qi/ V . Using Eq. (1) and Eq. (3), a simple calculation 2mχ shows that the wave functions ψ1 and ψ2 are normalized 0 to one. In terms of ψ1 and ψ2, Eq. (10) can be written where q = p − p is the momentum transferred in as follows, the scattering, v (v) is the initial DM particle velocity Z 3 (speed), mχ and me are the DM and electron mass, re- V d k ∗ Sfi = (2π)δ(Ef − Ei) 3 ψ2 (k + q)ψ1(k) spectively, and E1 (E2) is the energy of the initial (final) 4mχme (2π) electron bound state. Defining ∆E1→2 ≡ E2 − E1, the × iM(k, p, k + q, p − q) . (11) energy difference Ef − Ei reads 0 The probability for the transition |e1, pi → |e2, p i 2 4 2 is then given by |Sfi| /V , where we divide by q 4 Ef − Ei = ∆E1→2 + − qv cos θqv , (6) V in order to obtain unit-normalized single-particle 2mχ states from Eq. (1) (remember that Sfi has dimension [energy]−6). Consistently, the probability, (p), that a where θ is the angle between the vectors q and v, while P qv DM particle with initial momentum p and a bound elec- q = |q|. Notice that S has dimension [energy]−6, be- fi tron in the |e i state scatter to a group of final states with cause of our choice of single-particle state normalization, 1 DM momenta in the infinitesimal interval (p0, p0 + dp0) Eq. (1). and the electron in the |e i state is given by |S |2/V 4 Let us now consider the process |k, pi → |k0, p0i, where 2 fi times the number of states in the (p0 + dp0) interval, a DM particle of initial (final) momentum p (p0) scatters namely elastically off a free electron of initial (final) momentum 2 3 0 k (k0) as shown in the left panel of Fig.1. The first non- |Sfi| V d p P(p) = trivial term in the Dyson expansion of the S-matrix ele- V 4 (2π)3 ment corresponding to this process is |S |2 V d3q = fi Z V 4 (2π)3 free ˜ ˜ 3 0 0 Sfi = −i(2π)δ(Ef − Ei) d x hp , k |HS(x)|k, pi , T d3q 1 = (2π)δ(Ef − Ei) 3 2 2 (7) (2π) V 16mχme where we defined E˜ ≡ E 0 +E 0 and E˜ ≡ E +E . The 2 f k p i k p Z d3k above S-matrix element can equivalently be expressed as ∗ × 3 ψ2 (k + q)M(k, p, q)ψ1(k) (12) follows (2π) where the divergent factor T = R dt arises when squar- free 4 ˜ ˜ (3) 0 0 Sfi = i(2π) δ(Ef − Ei)δ (p + k − p − k) ing the one-dimensional Dirac delta in Eq. (11). While 1 0 0 T enters in Eq. (12) explicitly, physical observables will × p M(k, p, k , p ) , (8) 2Ep0 2Ek0 2Ek2Ep not depend on T . In order to simplify the notation, in Eq. (12) we replaced M(k, p, k + q, p − q) with where M is the amplitude for DM scattering by a free M(k, p, q). The rate per unit DM number density for the 0 0 0 0 0 electron. Compared to Eq. (4.73) of [45], Eq. (8) differs by transition |e1, pi → |e2, p i with p within (p , p + dp ) p a factor of 1/ 2Ep0 2Ek0 2Ek2Ep because of our different is therefore given by P(p)V/T . Indeed, there is just normalization of single-particle states (see Eq. (1)). By one DM particle with initial momentum p within (p0, comparing Eq. (7) with Eq. (8), we obtain the following p0 + dp0) in a phase-space element of three-dimensional identity volume V [46]. Finally, integrating P(p)V/T over the initial DM par- Z 3 0 0 3 (3) 0 0 ticle velocity distribution fχ and the transferred momen- d x hp , k |HS(x)|k, pi = −(2π) δ (p + k − p − k) tum q, and multiplying by the local DM number density, 0 0 3 M(k, p, k , p ) nχ , for the total rate of DM-induced |e1i → |e2i transi- × p , (9) 2Ep0 2Ek0 2Ek2Ep

3 where in the nonrelativistic limit 2Ep0 2Ek0 2Ek2Ep = The details of the DM density and velocity distributions are sum- 2 2 marized in AppendixE. 16mχme. By substituting the free scattering result of 5 tions we find imply that only two out of the four three-dimensional n vectors are actually independent. A convenient choice of χ 0 R1→2 = independent momenta is q = p − p , the momentum 16m2 m2 χ e transferred in the scattering, and Z d3q Z × d3vf (v)(2π)δ(E − E )|M |2 . 0 0 3 χ f i 1→2 ⊥ (p + p ) (k + k ) (2π) vel = − (15) (13) 2mχ 2me q k = v − − , (16) Here, we defined the squared electron transition ampli- 2µe me tude, |M |2, in terms of ψ , ψ and the free scattering 1→2 1 2 where µ denotes the reduced mass of the DM parti- amplitude, e cle and the electron, and v ≡ p/mχ is the incoming DM velocity. In the case of elastic DM-electron scat- Z 3 2 2 d k ∗ tering, v⊥ · q = 0 because of energy conservation. In |M1→2| ≡ ψ2 (k + q)M(k, p, q)ψ1(k) , el (2π)3 contrast, when DM scatters from electrons inelastically (14) (and initial and final electrons populate different energy ⊥ levels), vel · q 6= 0. In this case, one could define where a bar denotes an average (sum) over initial (final) spin states. ⊥ q ∆E1→2 vinel ≡ v − − 2 q , (17) In the following two sections, we motivate and ex- 2mχ q plore the implications of our assumptions for M, ψ and 1 ⊥ ψ . Here, we model M within a nonrelativistic effective such that vinel · q = 0 by construction. While M can 2 ⊥ ⊥ theory of DM-electron interactions (Sec.IIB) and use equivalently be expressed in terms of vinel or vel [36], ⊥ initial and final electron wave functions derived for iso- we find that it is more convenient to use vel as a basic variable when matching explicit models to the nonrela- lated atoms in argon and xenon targets [47] for ψ1 and tivistic effective theory built in this section (compare for ψ2 (Sec.IIC). example Eqs. (18) and (21) below). To summarize, within our effective theory the free am- B. Non-relativistic dark matter-electron plitude for nonrelativistic DM-electron scattering reads interactions 0 0 ⊥ M(k, k , p, p ) = M(q, vel ) , (18) In this section, we investigate the nonrelativistic limit where we emphasized the dependence of the amplitude ⊥ ⊥ of DM-electron interactions. Our goal is to show that the M on the vectors q and vel . In addition to q and vel , standard assumption in the context of DM direct detec- M can also depend on matrix elements of the electron tion, M = M(q), is highly restrictive. Rather than bas- and DM particle spin operators, denoted here by Sχ ing our argument on a specific model, we build a general and Se, respectively. Within the nonrelativistic effective nonrelativistic effective theory where the active degrees of theory of DM-electron interactions developed here, the freedom are electrons and DM particles. The basic sym- amplitude M does not explicitly depend on the proper- metries governing the DM-electron scattering, together ties that characterize the particle mediating the interac- with the assumption that both the DM particle and the tions between DM and electrons, such as the mediator electron are nonrelativistic, will determine the allowed mass, mmed. There are two scenarios in which our ef- interactions between the active degrees of freedom in our fective theory approach can be applied. In the first one, effective theory, and allow us to show that M = M(q) mmed is much larger than the momentum transfer in DM- is in general a too restrictive assumption. The symme- electron scattering (contact interaction). In the second tries that govern the nonrelativistic DM-electron scatter- one, the mediator is effectively mass-less (long-range in- ing are the invariance under time and space translations teraction). These are the two limiting cases that we in- (leading to energy and momentum conservation), the investigate here. For mmed ∼ |q|, M is model dependent, variance under Galilean transformations (namely, con- and some of the considerations made here do not apply. stant shifts of particle velocities) and the invariance un- In Eq. (18) we are assuming that both DM particle and der three-dimensional rotations. Galilean invariance re- electron are nonrelativistic. Indeed, Milky Way DM par- places the invariance under Lorentz boosts of relativis- ticles have a typical speed of the order of 10−3 in nat- tic theories. Let us now explore the constraints set by ural units, and the typical speed of electrons bound in these symmetries on the amplitude for nonrelativistic atoms is αZeff , where α = 1/137 and Zeff is 1 for elec- DM-electron scattering. trons in outer shells (the most interesting ones from a While the nonrelativistic scattering of Milky Way detection point of view) and larger for inner shells. No- 2 DM particles by free electrons is characterized by the tice also that for mχ ≥ 1 MeV/c the electron is the three-dimensional momenta k (k0) and p (p0) of the fastest and lightest particle in the nonrelativistic DM- incoming (outgoing) electron and DM particle, respec- electron scattering, which implies that the typical mo- tively, momentum conservation and Galilean invariance mentum transferred in the scattering is of the order of 6

q O1 = 1χ1e O11 = iSχ · 1e ` me corresponds to the limit of contact interaction, ci = 0. In q ⊥ ⊥ 2 2 O3 = iSe · × v 1χ O12 = Sχ · Se × v contrast, for m |q| the interaction between DM me el el med s ⊥ q and electrons is long-range, i.e. ci = 0. Angle brackets O4 = Sχ · Se O13 = i Sχ · v Se · el me in Eq. (19) denote matrix elements between the two- q ⊥ q ⊥ 0 0 O5 = iSχ · × v 1e O14 = i Sχ · Se · v λ s λ s me el me el dimensional spinors ξ and ξ (ξ and ξ ) associated q q h ⊥ q i O6 = Sχ · Se · O15 = iO11 Se × vel · with the initial (final) state electron and DM particle, me me me 0 0 ⊥ q ⊥ s s λ λ O7 = Se · v 1χ O17 = i · S · v 1e respectively. For example, hO1i = ξ ξ ξ ξ . An analo- el me el ⊥ q gous expansion for M was found previously in studies of O8 = Sχ · v 1e O18 = i · S · Se el me q q q nonrelativistic DM-nucleon interactions (see e.g. [36] for O9 = iSχ · Se × O19 = · S · me me me a review). q q q O10 = iSe · 1χ O20 = Se × · S · me me me It is important to mention that the distinction between contact and long-range interactions in Eq. (19) is not ⊥ TABLE I. At second order in q and at linear order in vel , Sχ this straight-forward in general. While t-channel scat- and Se, the first fourteen operators in this table are the only tering processes mediated by a scalar or vector boson three-dimensional scalars that one can build while preserving are associated with either long- or short-range interac- Galilean invariance and momentum conservation when DM tions depending on the mediator mass, mixed cases are 1 1 has spin 1/2. Here, χ and e are 2 × 2 identity matrices also frequent. For example, as we shall see in Sec.IVB, in the DM particle and electron spin space, respectively. The magnetic dipole interactions, mediated by the massless operators O , O , O and O were found to arise from 17 18 19 20 photon, correspond to a combination of two “contact” the nonrelativistic reduction of models for spin 1 DM with a s ` vector mediator. We follow the numbering used for nonrela- couplings ci and two “long-range” couplings ci . tivistic DM-nucleon interactions [40], and neglect O2, which As an illustrative example, let us compute M for DM- ⊥ is quadratic in vel , and O16, which can be expressed as a lin- electron interactions arising from the DM anapole mo- ear combination of O12 and O15. For further details, see, for ment and match the result of this calculation to the non- example, Ref. [36]. relativistic expansion in Eq. (19). The DM anapole moment is potentially important being the leading coupling between DM and photons if DM is made of Majorana Zeff αme [18]. Equation (18) can then be expanded in fermions. It generates DM-electron interactions via the ⊥ powers of |q|/me and |vel |. Each term in this expansion electromagnetic current. The Lagrangian of the model of M must be both Galilean invariant and scalar un- reads ⊥ der three-dimensional rotations. At linear order in vel and at second order in q, there are fourteen combina- 1 g µ 5 ν ⊥ L = 2 χγ γ χ ∂ Fµν , (20) tions of q, vel , Sχ and Se fulfilling the above require- 2 Λ ments when DM has spin 1/2 [38, 39, 48]. Specifically, these three-dimensional scalar combinations are opera- where χ is a Majorana fermion describing the DM par- tors in the DM/electron spin space. They correspond to ticle, g is a dimensionless coupling constant, Λ a mass the first fourteen entries in TableI. The remaining entries scale and Fµν the electromagnetic field strength ten- were found to arise from the nonrelativistic reduction of sor. The corresponding nonrelativistic scattering ampli- models for spin 1 DM coupling to nucleons via a vector tude is given by mediator [40]. We include them in TableI after replacing ( nucleon with electron operators. Based on the above con- 4eg 0 0 M = m m 2 v⊥ · ξ†s S ξs δλ λ siderations, the amplitude M can in general be expressed Λ2 χ e el χ as the linear combination 4 ) †s0 s q †λ0 λ + ge ξ Sχξ · i × ξ Seξ , (21) 2 me ⊥ X s ` qref M(q, vel ) = ci + ci 2 hOii , (19) |q| λ s i where ge is the electron g-factor, while ξ and ξ are two-component spinors for the electron and DM particle, where the reference momentum, q , is given by q ≡ ref ref respectively. The first line in Eq. (21) depends on v⊥ and αm , the interaction operators O are defined in TableI el e i is proportional to hO i, whereas the second line is linear (with i running on subsets of TableI, depending on the 8 in q and proportional to hO i. This example shows that DM particle spin), and the coefficients cs (c`) refer to 9 i i the assumption commonly made in the interpretation of contact (long-range) interactions. The case m2 |q|2 med direct detection experiments searching for DM-electron ⊥ interactions, M(q, vel ) = M(q), is likely restrictive, and ⊥ contributions to M depending on vel are in general ex- 4 The standard treatment of DM-electron interactions corresponds pected. 2 to M(q) = c1FDM(q ) hO1i, where the so-called DM form factor Let us now calculate the squared transition amplitude 2 FDM can be either 1 (contact interaction) or 1/q (long-range assuming that the expansion in Eq. (19) applies. This will interaction). lead us to introduce two atomic form factors depending 7

⊥ on ψ1, ψ2 and M. We start by noticing that the free elec- where we used vel = v − k/me − q/(2µχe). This is not tron scattering amplitude in Eq. (19) can also be written an approximation because Eq. (19) is at most linear in ⊥ as vel . Substituting Eq. (22) into the squared transition amplitude in Eq. (14), we ﬁnd ⊥ ⊥ M(q, vel ) = M(q, vel )k=0 k ⊥ + · me∇kM(q, vel )k=0 , (22) me

( 2 ⊥ 2 2 ⊥ ∗ ⊥ ∗ |M1→2| = |M(q, vel )| |f1→2(q)| + 2me< M(q, vel )f1→2(q)∇kM (q, vel ) · f1→2(q) ) 2 ⊥ 2 + me|∇kM(q, vel ) · f1→2(q)| . (23) k=0

While the first term in Eq. (23) has already been con- C. Ionization of isolated atoms in argon and xenon sidered in previous studies, the second and third terms targets are new, and arise from our general modeling of the free electron scattering amplitude. Indeed, in the analysis of Our treatment of DM-electron scattering in target direct detection experiments searching for DM-electron materials has so far been general in terms of initial interaction signals, it is standard to assume that the DM- and final state electron wave functions, ψ1 and ψ2, re- electron free scattering amplitude M depends solely on spectively. From now onward, however, we specialize ⊥ the momentum transfer q and not on the velocity vel our results to the case of DM-induced ionization of and, therefore, on the electron’s initial momentum k. In isolated atoms. In this case the initial state (formerly this case, the scattering amplitude M(q) can be taken simply denoted by “1”) is a bound state characterized out of the integral in Eq. (13), and only the first term by the principal, angular and magnetic quantum num- in Eq. (23) contributes to the squared transition ampli- bers (n, `, m). The final state (“2”) is a free electron state 2 tude |M1→2| . The advantage of this standard assump- at large distance from the atom, but still affected by the tion is that M = M(q) implies a convenient factorization remaining ion’s presence at low distance. It is defined by of atomic physics and DM physics. Following the nota- the quantum numbers (k0, `0, m0), where k0 is the asymp- tion of Ref. [18], for M = M(q) one can define a scalar totic momentum of the electron and `0, m0 are its angu- atomic form factor, lar and magnetic quantum numbers. We obtain the to- n` tal ionization rate Rion of a full atomic orbital (n, `) by 3 Z d k summing the transition rate R1→2 over all occupied ini- f (q) = ψ∗(k + q)ψ (k) , (24) 1→2 (2π)3 2 1 tial electron states and integrating over the allowed final electron states. and obtain the transition amplitude, M1→2(q), by mul- For the initial bound electrons, the rate has to be tiplying the free scattering amplitude by the form factor summed over all values of the magnetic quantum num- f1→2(q), ber m and multiplied by 2 to account for the spin de- generacy. In order to account for all the allowed electron M1→2(q) = M(q) × f1→2(q) . (25) final states, we have to act on R1→2 with the integral operator [18] Since the momentum space wave functions ψ1 and ψ2 0 −3/2 ∞ ` Z 03 have dimension [energy] , f1→2(q) is a dimensionless V X X k d ln Ee . (27) quantity. 2 (2π)3 Besides the well-known scalar atomic form factor of `0=0 m0=−`0 Eq. (24), a new, vectorial atomic form factor appears in Here, E = k02/(2m ) is the ionized electron’s final en- Eq. (23), e e ergy, and the number of final states with asymptotic momentum between k0 and k0 + dk0 is V d3k0/(2π)3. Sum- Z d3k k ∗ marizing, the total ionization rate for the (n, `) orbital is f1→2(q) = 3 ψ2 (k + q) ψ1(k) . (26) (2π) me given by

0 The details of the evaluation of the scalar, f1→2, and ` ∞ ` Z 03 n` X X X V k vectorial, f1→2, atomic form factors are presented in Ap- R = d ln Ee R1→2 , (28) ion (2π)3 pendixB3. m=−` `0=0 m0=−`0 8 and the corresponding ﬁnal state electron ionization en- cross section from Eq. (30), i.e., the number of events per ergy spectrum by unity of ﬂux, 6

` ∞ `0 dRn` X X X V k03 dσn` 1 Z ion = . (29) ion n` 2 3 R1→2 = dq q M , (31) d ln Ee (2π) 2 2 2 ion m=−` `0=0 m0=−`0 d ln Ee 128πmχmev

The divergent factor V in Eq. (29) cancels with the 1/V such that factor arising from the normalization of ψk0m0`0 (see Ap- pendixB4). Substituting Eq. (13), we can rewrite the n` Z n` dRion 3 dσion spectrum as = nχ d v v fχ(v)Θ(v − vmin) . (32) d ln Ee d ln Ee n` dRion nχ = 2 2 d ln Ee 128π mχme 2 The squared ionization amplitude, Mn` , appearing Z Z 3 ion d v n` 2 in Eq. (30) is deﬁned as follows × dq q fχ(v)Θ(v − vmin) M , v ion

(30) 0 03 ` ∞ ` n` 2 4k X X X 2 M ≡ V |M1→2| , (33) where, following [18], we integrated over cos θqv while ion (2π)3 m=−` `0=0 m0=−`0 assuming that fχ(v) = fχ(v), and then replaced fχ(v) 5 with fχ(v) in the ﬁnal expression . In order to emphasise the connection to previous and can explicitly be expressed in terms of the amplitude ⊥ work [9, 13, 14], we can extract a diﬀerential ionization M(q, vel ),

0 ( 03 ` ∞ ` " 2 n` 2 ⊥ 2 n` 0 4k X X X ⊥ ∗ ⊥ ∗ M = |M(q, v )| f (k , q) + V 2me< M(q, v )f1→2(q)∇kM (q, v ) · f (q) ion el ion (2π)3 el el 1→2 m=−` `0=0 m0=−`0 #) 2 ⊥ 2 + me|∇kM(q, vel ) · f1→2(q)| . (34) k=0; v·q/(qv)=ξ

Here, we deﬁned ξ = ∆E1→2/(qv)+q/(2mχv). To further This ionization form factor is depicted in Fig.2 for the clarify the connection to the results and notation of pre- outer atomic orbitals of argon and xenon. vious works, in Eq. (34) we introduced the dimensionless In Eq. (30), the minimum DM speed, vmin, required to ionization form factor, deposit an energy of ∆E1→2 given a momentum transfer q is 0 03 ∞ ` ` n` 0 2 4k X X X 2 ∆E q f (k , q) = V |f1→2(q)| . 1→2 ion (2π)3 vmin = + . (36) `0=0 m=−` m0=−`0 q 2mχ (35) Finally, the wave functions of the initial and ﬁnal state electrons can be expanded in spherical harmonics,

m ψn`m(x) = Rn`(r) Y` (θ, φ) , (37) 5 This approach significantly simplifies the evaluation of Eq. (13) m0 and is justified by the fact that we are not interested in a direc- ψk0`0m0 (x) = Rk0`0 (r) Y`0 (θ, φ) . (38) tional analysis of the predicted signal. If Eq. (13) is to be used in directional analyses of energy spectra, this simplified treatment Their radial parts, Rn`(r) and Rk0`0 (r), are given in of the velocity integral should be refined along the lines discussed AppendixB4 for the case of isolated argon and xenon in Ref. [27]. atoms [47]. While these wave functions have also been 6 In the derivation of Eq. (13) we refrained from introducing a used in previous works on DM direct detection [12], we scattering cross section. Properly defining a scattering cross sec- note that they are not fully applicable to dense liquid tion between an incoming DM particle and a bound electron, not being in a momentum eigenstate, is hampered by the fact that argon and xenon systems [49]. Our neglect of the differ- the relative velocity is not well defined. Instead one would have ence in electronic structure between the isolated atoms to treat the whole atom as a multi-particle target, whose center and the liquid state makes our results conservative since of mass is indeed in a total momentum eigenstate. the electron binding energies in liquid nobles are smaller 9

102 102

2 101 Argon 3p k'[keV] 2 101 Xenon 5p k'[keV] ) 1 ) 1 -1 -1 k',q 10 k',q 10 ( ( -2 -2

3p ion 10 5p ion 10 f f | 10-3 101. | 10-3 101. 10-4 10-4 10-5 10-5 10-6 10-6 10-7 10-7 0. 0. 10-8 10 10-8 10 10-9 10-9 -10 -10 10 3p 2 10 5p 2 10-11 |fion(k',q) 10-11 |fion(k',q) nfr factor form ioniza � on factor form ioniza � on 10-12 -1. 10-12 -1. 1 101 102 103 10 1 101 102 103 10 momentum transfer q [keV] momentum transfer q [keV]

n` 0 2 FIG. 2. The ionization form factor fion(k , q) defined in Eq. (35) for the outer electron orbital of argon (left) and xenon (right). than those of the isolated atoms. Furthermore, this treat- lows: ment of the wave functions can be improved by includ- 4 ing relativistic corrections [23, 50] and many-body ef- n` 2 X n` ⊥ q n` 0 |Mion| = Ri vel , Wi (k , q) . (39) fects [51]. However, both improvements, in particular the me relativistic corrections, are expected to have limited im- i=1 pact on theoretical predictions for sub-GeV DM. Here, we define a set of DM response functions Rn` v⊥, q , which are functions of the couplings i el me n` 0 in Eq. (19), and atomic response functions Wi (k , q), which encapsulate the information on the electron’s ini- III. GENERAL DARK MATTER AND ATOMIC tial and final state wave functions.7 By means of Eq. (39), RESPONSE FUNCTIONS we are again able to disentangle the particle physics input from that of the atomic physics. However, our gen- In this section, we reformulate the expressions found eral treatment of DM-electron interactions predicts not in Sec.IIC in order to obtain compact equations which one but four atomic response functions. At this point, we are suitable for numerical applications. By doing so, we summarize the final expressions for the DM and atomic also investigate and characterize the atomic response to response functions. Their detailed derivations can be found in AppendixesA andB. DM-electron interactions of the general form considered The four DM response functions, Rn` v⊥, q , are here. We start by observing that the squared ionization i el me n` 2 amplitude, |Mion| , can in general be rewritten as fol- given by

2 2 2 2 2 2 2 n` ⊥ q 2 c3 q ⊥ 2 c3 q ⊥ c7 ⊥ 2 c10 q R1 vel , ≡ c1 + (vel ) − · vel + (vel ) + me 4 me 4 me 4 4 me ( 4 2 jχ(jχ + 1) 2 2 q 2 2 ⊥ 2 2 2 q + 3c4 + c6 + (4c8 + 2c12)(vel ) + (2c9 + 4c11 + 2c4c6) 12 me me 2 4 2 2 2 q ⊥ 2 2 q ⊥2 + 4c5 + c13 + c14 − 2c12c15 (vel ) + c15 vel me me 2 2 2 ) 2 q ⊥ q 2 ⊥ q − c15 vel · + −4c5 + 2c13c14 + 2c12c15 vel · , (40a) me me me

" 2 −2 ( −2 )# n` ⊥ q q ⊥ c7 q jχ(jχ + 1) 2 2 q 2 R2 vel , ≡ · vel − − (4c8 + 2c12) + (c13 + c14) , (40b) me me 2 me 6 me

7 This factorization is analogous to that introduced previously, scattering [39]. e.g., in Ref. [39], in the treatment of nonrelativistic DM-nucleus 10

101 10-3 100

10-1 10-4 10-2 10-5 10-3 10-6 10-4

10-5 10-7

10-3 10-5

10-6 10-4 10-7

10-8 10-5 10-9

10-6 10-10

FIG. 3. The four atomic responses for the outer atomic orbital of argon. Above the white dotted/dashed/solid line on the left plots, the minimum speed vmin in Eq. (36) exceeds the maximum speed vmax = v⊕ +vesc for a DM mass of 10 MeV/100 MeV/→ ∞, respectively (see AppendixC andE). The top left panel shows the same function as the left panel of Fig.2, but now with the ﬁnal state electron asymptotic momentum k0 on the y-axis. Note the diﬀerent color bar scales in the four panels.

2 2 2 ( 2 4) n` ⊥ q c3 q c7 jχ(jχ + 1) 2 2 2 2 2 q 2 q R3 vel , ≡ + + 4c8 + 2c12 + 4c5 + c13 + c14 − 2c12c15 + c15 , me 4 me 4 12 me me (40c) 2 ( 2 ) n` ⊥ q c3 jχ(jχ + 1) 2 2 q R4 vel , ≡ − + −4c5 − c15 + 2c12c15 + 2c13c14 , (40d) me 4 12 me

s 2 2 ` ` ∞ `0 where ci ≡ c + (q /|q| )c . They indirectly depend on 03 i ref i n` 0 4k X X X 2 the (n, `) quantum numbers through the variable ξ = W3 (k , q) ≡ V 3 |f1→2(q)| , (2π) 0 0 0 ∆E1→2/(qv) + q/(2mχv), see also AppendixC. In the m=−` ` =0 m =−` ⊥ (41c) above expressions, vel is evaluated at k = 0 due to the nonrelativistic expansion of M. The four atomic response 4k03 n` 0 W n`(k0, q) ≡ V functions Wi (k , q), which we define next in Eq. (41), 4 (2π)3 are one of the main results of this work: ` ∞ `0 2 X X X q × · f1→2(q) . (41d) me m=−` `0=0 m0=−`0 0 03 ` ∞ ` n` 0 4k X X X 2 The first atomic response function W (k , q) can be W n`(k0, q) ≡ V |f (q)| , 1 1 (2π)3 1→2 n` 0 2 m=−` `0=0 m0=−`0 identified with the ionization form factor fion(k , q) (41a) commonly used in the sub-GeV DM detection liter- n` 0 03 ature [18]. The atomic response functions Wj (k , q), n` 0 4k j = 2, 3, 4 were not considered in previous studies and W2 (k , q) ≡ V (2π)3 can only arise from a general treatment of DM-electron 0 n` n` ` ∞ ` interactions. While W4 /W1 roughly scales as particle X X X q ∗ n` n` × · f1→2(q)f1→2(q) , momenta to the power of 4, and W2,3/W1 as particle me m=−` `0=0 m0=−`0 momenta to the power of 2, numerically we find that the n` n` (41b) products Rj Wj are often of comparable magnitude for 11

101 10-3 100 10-4 10-1

-2 10 10-5 10-3 10-6 10-4

10-5 10-7

-3 10 10-5

10-6 10-4 10-7

10-8 10-5 10-9

10-6 10-10

FIG. 4. The four atomic responses for the outer atomic orbital of xenon. The white lines have the same meaning as in Fig.3. Again, the function in the top left panel was shown before in the right panel of Fig.2. However, here the final state electron asymptotic momentum k0 is on the y-axis. several DM-interaction types (see below for further de- in [52] (with the proton mass replaced by the electron tails). mass). In order to clarify the physical meaning of the new Consequently, the new atomic responses describe dis- atomic responses, let us evaluate them√ in the plane- tortions in the ionization spectrum induced by the fact wave limit, where√ ψ1(x) = exp(ik · x)/ V and ψ2(x) = that the initial state electron obeys a momentum distri- exp(ik0 · x)/ V are eigenstates of the electron momen- bution with a finite dispersion, being the electron bound tum operator with eigenvalues k and k0, respectively. In to an atom. this limit, initial and final state electron are not bound Figures3 and4 show the four atomic response func- n` to an atom and propagate as free particles. Substituting tions, Wj , j = 1,..., 4, for the 3p and 5p atomic orbitals ψ1(x) and ψ2(x) in Eqs. (41a)-(41d), we find of argon and xenon, respectively. (2π)3 |f (q)|2 = δ(3)(q + k − k0) 1→2 V IV. FIRST APPLICATION TO DIRECT q k q DETECTION · f1→2(q)f1→2(q) = − · me me me (2π)3 The nonrelativistic effective theory of DM-electron in- × δ(3)(q + k − k0) V teractions presented in the previous sections culminated 2 3 in a general expression for the electron ionization energy 2 k (2π) (3) 0 |f1→2(q)| = δ (q + k − k ) spectrum of isolated atoms due to generic DM-electron m V e interactions given by the Eqs. (30) and (39). This main 2 2 q k q result allows us to make predictions for direct searches · f1→2(q) = · me me me of sub-GeV DM particles by using an almost model- (2π)3 independent framework and a generic expression for the × δ(3)(q + k − k0) . (42) scattering amplitude in Eq. (19) in terms of effective op- V erators Oi. In the plane-wave limit, we can define a laboratory frame In the following sections, we apply this effective the- in which the initial electron is at rest and, therefore, k = ory to evaluate ionization spectra. There are a number n` 0 0. In this frame, Wj (k , q) = 0, j = 2, 3, 4 and Eq. (39) of large-scale dual-phase time-projection-chamber (TPC) reduces to the expression for the modulus squared of the detectors with xenon and argon targets, which is why we amplitude for DM scattering by a point-like proton found focus on these two elements. By re-interpreting the obser- 12

contact interac�on 104 long-range interac�on 4 1 GeV 1 GeV 10 s c =10-3 2 ℓ -3 100 MeV 6 10 100 MeV c6=10 ] ]

1 10 MeV 1 1 10 MeV

- 2 10 - yr yr -2 1

1 10 - - kg

kg -4 [ 1 [ 10 e e ion ion -6

dR 10 dR nE ln d nE ln d -8 10-2 10 10-10 Argon Argon 10-4 10-12 10-1 1 101 102 103 104 10-1 1 101 102 103 104

electron energy Ee [eV] electron energy Ee [eV]

106 1 GeV contact interac�on 104 1 GeV long-range interac�on s c =10-3 2 ℓ -3 104 100 MeV 13 10 100 MeV c13=10 ] ]

1 10 MeV 1 1 10 MeV - - yr 2 yr -2 1 10 1 10 - - kg

kg -4 [ [ 10 e e ion ion -6

dR 1 10 dR nE ln d nE ln d 10-8

10-2 10-10 Xenon Xenon 10-12 10-1 1 101 102 103 104 10-1 1 101 102 103 104

electron energy Ee [eV] electron energy Ee [eV]

FIG. 5. Selected ionization spectra for argon for contact and long-range interactions with coupling c6 (top), and for xenon for contact and long-range interactions with coupling c13 (bottom), for three different DM masses. The faint vertical lines indicate the kinematic cut-off of the spectra. vational data by XENON10 [13, 43], XENON1T [15], and GeV DM detection. The new massive gauge boson A0 is DarkSide-50 [12], we can set exclusion limits on either the called the dark photon and acts as the interaction’s medi- s ` effective coupling constants ci and ci of Eq. (19) in front ator. Its mass is usually assumed to be either much larger of individual operators, or on the parameters of specific than the typical momentum transfer qref in the scatter- DM models generating combinations of interaction oper- ing process (contact interaction), or much lower (long- ators, as in the case of DM with anapole and magnetic range interaction). By considering the nonrelativistic dipole couplings. In the latter case, one has to compute DM-electron scattering amplitude, this interaction can the elastic scattering amplitude between a DM particle be identified with O1, and the connection between the and an electron, take the nonrelativistic limit, and iden- effective couplings in Eq. (19) and fundamental parame- tify the different contributions with one or several of the ters is given by effective operators in TableI. We have already shown the example of anapole interactions, which corresponded to s 4mχmegDeε ` c1 = 2 , c1 = 0 , (43) a combination of O8 and O9, in Sec.IIB. mA0 It is also worth pointing out how the most-used in- or teraction model in the sub-GeV DM literature, i.e. the “dark photon” model, emerges in this framework. The 4m m g eε c` = χ e D , cs = 0 , (44) dark photon model extends the Standard Model of par- 1 2 1 qref ticle physics by an additional U(1) gauge group under which the DM particle is charged. Interactions between for contact and long-range interactions, respectively. the DM particle and electrically charged particles in the Here, gD is the gauge coupling of the additional “dark” Standard Model arise from a “kinetic mixing” term in U(1) gauge group, while e denotes the charge of the elec- the interaction Lagrangian between the photon and the tron. gauge boson A0 associated with the new U(1) gauge As anticipated, the scattering amplitude arising from a 0µν group [9, 53], i.e. εFµν F . Here, ε is a coupling constant given DM model can correspond to either one or several 0 0 and Fµν the A field strength tensor. This model can of the effective operators in TableI. This is why we start arguably be considered as the “standard model” of sub- by focusing on single operators, compute and study the 13

1016 1016 1014 contact interac�ons 1014

] 12 12 1 10 10 -

yr 1010 1010 1 - 108 108 kg [ 6 6 2 10 10 ) s i c 104 104 ( / 102 102 ion R 1 100 0MeV = 10 00MeV = 1000 0 MeV = 100 χ χ -2 χ -2 m m 10 m 10 1015 1015 - 1013 long range interac�ons 1013 ] 1 - 1011 1011 yr

1 9 9

- 10 10

kg 7 7

[ 10 10 2

) 5 5 l i 10 10 c

( / 103 103

ion 1 1

R 10 10 -1 -1 0MeV = 10 00MeV = 1000 0 MeV = 100 χ 10 χ 10 χ m m 10-3 m 10-3 1 3 4 5 6 7 8 9 10 11 12 13 14 15

FIG. 6. Integrated ionization spectra for individual DM-electron interaction operators divided by the corresponding coupling constant squared. We present results for argon (purple) and xenon (blue), as well as for contact interactions (top) and long- range interactions (bottom). A comparison of the different histograms shows the operators’ relative strength when coupling constants are set to the same value. As indicated for the case of the O1 operator, we assume a DM mass of 10/100/1000 MeV for the left/middle/right histograms. corresponding ionization spectra, and set exclusion limits contact interactions, whereas the right panels refer to s ` on the constants ci and ci for individual Oi. To illustrate long-range interactions. Finally, in all panels solid, long- the case of amplitudes being linear combinations of two dashed and short-dashed lines refer to different DM par- or more of the effective operators, we study three rel- ticle masses. We compute the total ionization spectrum evant exemplary cases of DM-electron interactions, the by summing the ionization spectrum for each atomic or- anapole, magnetic dipole, and electric dipole interaction. bital (n, `), Eq. (30), over the five outermost occupied These arise as leading and next-to-leading terms in an orbitals for xenon (4s,4p,4d,5s, and 5p), and all occupied effective theory expansion of the DM coupling to the or- orbitals for argon (1s, 2s, 2p, 3s, and 3p). From the overall dinary photon [37]. We briefly review these models in scale of the spectra, we can draw two conclusions. Firstly, AppendixD. we find that ionization spectra are smaller in the case of long-range interactions than for contact interactions 2 2 due to the additional qref /q factor in the scattering am- A. Individual effective operators plitude. This suppression is more severe for large DM 2 masses than for mχ around 1 MeV/c . Secondly, we find Let us consider the case where the free electron scat- that the magnitude of the ionization spectrum associated with O (and scattering on argon) is comparable with tering amplitude, M, consists of only one of the Oi op- 6 s,` that of O13 (and scattering on xenon). This is expected, erators, or, in other words, only one of the couplings ci because O6 is quadratic in q while O13 is linear in q and in Eq. (19) is different from zero. On the one hand, this ⊥ is instructive and allows us to study the impact of single vel , and at the same time scattering on argon and xenon operators in isolation. On the other hand, this situation occurs with comparable probabilities. might also arise from specific models for DM interactions, To make the last observation more precise and com- with the dark-photon model being the most obvious ex- plete, we now extend the above comparison to the 14 op- ample. erators in TableI describing spin 1/2 DM. For each From TableI, we can identify the operators that gen- of these operators, we compute the associated ioniza- n` 0 erate the three new atomic responses, i.e. Wj (k , q), tion spectrum considering scattering on both argon and ⊥ j = 2, 3, 4, by their dependence on vel . They are O3, xenon targets. Figure6 shows the integrated ioniza- O5, O7, O8, O12, O13, O14, and O15. Figure5 shows the tion spectra in argon and xenon that we find by in- total ionization spectrum, dRion/d ln Ee, for two example tegrating dRion/d ln Ee from zero up to the maximum operators, O6 (top panels) and O13 (bottom panels). In kinematically allowed energy. We present our results for the case of the O6 operator, we compute the ionization three different DM particle masses, namely 10, 100, and 2 spectrum for argon. For the O13 operator, we focus on 1000 MeV/c . By comparing the magnitude of the ion- xenon. Furthermore, the left panels in Fig.5 refer to ization spectra for a given operator, we find that the 14

Total res. 1 contact interac�on -3 long-range interac�on 107 10 Total res. 1 s -3 ℓ -3 mχ=10 MeV, c7=10 m =100 MeV, c =10 10-4 χ 15 6 res. 2 res. 3 res. 3 res. 4† ] 10 ] 1 1 - - 10-5 yr yr 1 1

- 5 10 - 10-6 kg kg [ [ e e

ion -7 104 ion 10 dR dR nE ln d nE ln d 10-8 103 10-9 Argon Argon 102 10-2 10-1 1 101 102 10-2 10-1 1 101 102 103

electron energy Ee [eV] electron energy Ee [eV]

7 Total res. 1 contact interac�on long-range interac�on 10 10-3 Total res. 1 s -3 ℓ -3 mχ=10 MeV, c7=10 mχ=100 MeV, c15=10 6 res. 2 res. 3 10-4 res. 3 res. 4† ]

10 ] 1 1 - - yr yr 10-5 1 5 1 - 10 - kg kg [ [ 10-6 e e ion 104 ion dR dR nE ln d nE ln d 10-7

103 10-8 Xenon Xenon 10-9 10-2 10-1 1 101 102 10-2 10-1 1 101 102 103

electron energy Ee [eV] electron energy Ee [eV]

n` n` FIG. 7. The four Rj Wj , j = 1,..., 4 contributions to the ionization energy spectrum (here generically denoted as “responses”) s,` s,` for the two example couplings c7 (left) and c15 (right). We present our results for argon (above) and xenon (below), as well as for contact interactions with mχ = 10 MeV (left) and long-range interactions with mχ = 100 MeV (right). A dagger † indicates negative contributions.

integrated ionization spectra in argon and xenon are al- tions [38, 39]. In that context, the equivalent of the O1 ways comparable, while the xenon ionization rate slightly and O4 operators defined here correspond to the “famil- exceeds the rate in argon in most but not all cases. iar” spin-independent (SI) and spin-dependent (SD) in- By comparing the ionization spectra of different oper- teractions – a standard benchmark in the analysis of DM- ators for a given target material, we identify O1 and O4 nucleus scattering data. Similarly to the SI/SD paradigm as the “leading-order” (LO) operators. This is expected, dominating the literature on direct DM searches via nu- since the corresponding terms in Eq. (40) are not sup- clear recoils for a long time, the dark photon model un- ⊥ pressed by powers of q or vel , and, in addition, they derlies most studies of sub-GeV DM searches nowadays. n` generate the first atomic response function W1 , which Another interesting question in the context of isolated is numerically found to be the one of largest magnitude. operators regards the relative contribution of the four The operators O7 to O12 can be regarded as “next-to- different terms in Eq. (39). Each of these terms is the leading-order” (NLO) operators, in that they are linear in product of a DM response function given in Eq. (40) and ⊥ either q or vel . Similarly, the “next-to-next-to-leading- an atomic response function given in Eq. (41). In order order” (NNLO) operators (O5, O6, O13, and O14) are to answer the above question, in Fig.7 we show the ion- ⊥ quadratic in q or in a combination of q and vel . Com- ization energy spectrum for two example operators, each pared to the LO and NLO cases, the total ionization spec- of which generates multiple atomic responses. Specifi- trum of NNLO operators is further suppressed. Lastly, cally, we show the total argon and xenon ionization spec- 3 2 the only N LO operator we include in our analysis is O15, tra for O7, contact interaction and mχ = 10 MeV/c ⊥ which is quadratic in q and linear in vel . The ioniza- (left panels), and for O15, long-range interaction and 2 tion spectrum due to this operator is the lowest and sup- mχ = 100 MeV/c (right panels). As one can see from pressed by up to ten orders of magnitude compared to O1 Fig.7, we find that in the case of contact interactions for identical couplings. of type O7 the four terms in Eq. (40) give comparable This hierarchy of operators is essentially identical contributions to the total ionization spectrum. This is to that found when classifying DM-nucleus interac- the result of a compensation between DM and atomic re- tions within the effective theory of DM-nucleon interac- sponse functions. Indeed, while the first atomic response 15

-2 10 10 101 contact interac�on long-range interac�on 2 0 -24 cm 2 10 10 -3 -30 cm 10 10

2 -1 -26 cm 10 10 2 -4 -32 cm 10 10 2 -2 -28 cm 10 10 s i ℓ i

2 2 34 cm -5 - -3 -30 cm 10 10 10 10 opigc coupling opigc coupling 2 -4 -32 cm 2 10 10 -6 -36 cm 10 10

2 s -5 -34 cm ℓ c1 XENON10 10 10 c1 XENON10 2 -7 -38 cm 10 10 s 2 ℓ c8 XENON1T -6 -36 cm c8 XENON1T 10 10

s ℓ 2 c13 DarkSide-50 2 c13 DarkSide-50 -8 -40 cm -7 -38 cm 10 10 10 10 100 101 102 103 100 101 102 103

DM mass mχ [MeV] DM mass mχ [MeV]

FIG. 8. Exclusion limits (90% C.L.) from XENON10 [13, 43](blue), XENON1T [15](dark blue) and DarkSide-50 [12](purple) on a selection of individual eﬀective couplings for both contact and long-range interactions. The dashed gray lines indicate constant reference cross sections σe, see Eq. (45). Note that the constraints on long-range interactions depend on a choice of the reference momentum transfer qref .

n` function W1 generally dominates over the second and to this day. The largest of these experiments are dual- n` third, which are in turn larger than W4 , in the case of phase time-projection-chambers (TPC) with xenon and contact interactions of type O7 we find an inverse hierar- argon targets, which is why we are focusing on these el- n` chy for the DM response functions Rj , j = 1, 2, 3, which ements in this paper. Here, we re-interpret the null re- ⊥ 2 ⊥ 2 0 sults by the xenon target experiments XENON10 [43] scale as (vel ) ,(vel ·q)me/q and q , respectively. In this respect, the operator O7 is not unique (see for example and XENON1T [15], as well as the argon experiment O8, just to name one). From this example, we conclude DarkSide-50 [12] and set 90% C.L. exclusion limits on that the four terms in Eq. (41) can in general contribute the individual effective coupling constants. The details in a comparable manner and must therefore be taken into underlying the computation of the limits for these three account in the interpretation of experimental data. experiments are summarized in AppendixE. The contribution of an individual response may also Figure8 shows our 90% C.L. exclusion limits for be negative, an example of which is shown in the right three selected interaction operators, one LO (O1), one panels of Fig.7 for the long-range ionization spectrum NLO (O8), and one NNLO operator (O13), both for 8 9 of O15. The effective operator O15 generates the DM re- contact (left) and long-range interactions (right). The sponses 1, 3, and 4. In this example, the contribution to dashed gray lines indicate constant values of a reference the total ionization spectrum from the first atomic re- DM-electron scattering cross section that we define as n` sponse function, W1 , is suppressed by the first DM re- follows, sponse function, Rn`, which renders the Rn`W n` contri- 1 1 1 2 2 bution to Eq. (41) completely negligible. In addition, µχeci n` n` σe ≡ 2 2 . (45) the contribution of R4 W4 is as anticipated negative, 16πmχme n` which originates from the sign of the c15 term in R4 in Eq. (40d). With this definition the constraints on c1 can be directly We conclude this section with an analysis of present compared with previous works which described the DM- s,` exclusion limits on the ci coupling constants from di- electron interaction using the dark photon model, see, rect detection data. Despite tremendous experimental ef- e.g., Ref. [12–14]. forts by a multitude of direct detection experiments (see Ref. [4] for a review), no conclusive DM signal has been established, and DM continues to evade direct searches 9 Due to the chosen parametrization of the scattering amplitude in Eq. (19), it is important to remember that the limits on the di- ` mensionless long-range interaction coupling constants ci depend 8 Negative contributions are marked with a dagger (†). on the choice for the reference momentum transfer qref = αme. 16

105 105

4 total

total ] ] total ] anapole 10 1 4 magne�c dipole electric dipole 1 8 9 4 11 - 1 - 1 - 1 2 -2 -2 10 g/Λ [GeV ]=10 103 /Λ[ -1]= -5 /Λ[ -1]= -6 yr g GeV 10 yr g GeV 10 yr 5 6 2 3 - 1 - 1 - 1 10-1 10 10 kg kg kg

[ 1 [ [ 10 e e e 102 10-2 1 l E dln E dln l E dln 10-1 101 / / /

-3 -2 ion 10 ion 10 ion 1 - dR 3 dR 10 dR Xenon Xenon Xenon 10-4 10-4 10-1 1 101 102 1 101 102 1 101 102

electron energy Ee [eV] electron energy Ee [eV] electron energy Ee [eV]

FIG. 9. Ionization spectra for anapole, magnetic dipole, and electric dipole interactions shown in black. We set the DM mass to 100 MeV and focus on xenon. The diﬀerent blue lines show the contributions of the individual eﬀective operator. Note that in the case of magnetic dipole interactions, the sum of the four individual contributions exceeds the total spectrum due to a negative interference between O4 and O6 in Eq. (40a).

16em2m g B. Anapole, magnetic dipole, and electric dipole c` = e χ , (47c) interactions 5 2 qref Λ 2 ` 16ememχ g As an illustration of how different combinations of c6 = − 2 . (47d) qref Λ effective operators, Oi, can arise from specific models for DM-electron interactions, we consider three examples: the anapole, magnetic dipole, and electric dipole The situation is particularly simple for electric dipole interaction. Electric and magnetic dipole interactions interactions, with the scattering amplitude given in have previously been studied in the context of direct Eq. (D6). It is linear in O11 with the single effective DM searches via electron scatterings [13, 16, 54]. These coupling, studies used different approximations and did not exploit our new atomic response functions. 16em m2 g c` = χ e . (48) In AppendixD, we provide explicit Lagrangians for 11 q2 Λ these types of interactions. Therein we also present the ref associated nonrelativistic DM-electron scattering amplitudes, which we then match on to the effective theory Figure9 shows the xenon ionization spectra for expansion in Eq. (19). In this way we can read off the anapole, magnetic dipole and electric dipole interac- connection between the Lagrangian parameters and our tions. In the three cases we assume a DM particle mass 2 effective couplings. of 100 MeVc . Different blue lines correspond to contri- Let us start with the anapole interaction. From the butions from the individual interaction operators to the amplitude in Eq. (21), we find that this model generates total ionization spectrum (shown as a black solid line). Most interesting is the spectrum of the magnetic dipole a linear combination of the O8 and O9 operators in the nonrelativistic limit. Only contact interactions are gener- interaction. While one might expect the interaction op- ated and the corresponding effective coupling constants erator O1 to dominate the spectrum (since it is one of are given by the two LO operators), its contribution is in fact negligible due to the small relative size of the four effective s g coupling constants. This can be seen e.g. by the ratio c8 = 8ememχ , (46a) Λ2 cs/cs ≈ 10−3 for m = 100 MeV/c2. g 1 4 χ cs = −8em m , (46b) Another interesting aspect of the magnetic dipole in- 9 e χ Λ2 teraction spectrum is the fact that the contribution of where g is a dimensionless coupling constant, and Λ is the single operator O4 exceeds the total result. This is the energy scale at which the anapole interaction term is explained by the interference term between O4 and O6 generated. in the first DM response function in Eq. (40a), whose ` For magnetic dipole interactions, we find that the scat- overall contribution is negative due to the sign of c6 in tering amplitude in Eq. (D5) can be expressed as a linear Eq. (47). combination of four interaction operators, namely O1, We conclude this chapter by presenting our 90% C.L. O4, O5, and O6. Here, the effective coupling constants exclusion limits on the ratio between dimensionless cou- are given by pling constant g and energy scale Λ (or Λ2) for the anapole, magnetic dipole, and electric dipole interac- s g c = 4eme , (47a) tion. Our exclusion limits are shown in Fig. 10 and 1 Λ g based on a re-interpretation of the null results reported cs = 16em , (47b) 4 χ Λ by the XENON10, XENON1T, and DarkSide-50 experi- 17

102 10-3 10-4 ] ] 1 - 1

XENON10 anapole magne�c dipole - electric dipole ]

2 1 - XENON10 DarkSide 10 GeV DarkSide 10-4 GeV 10-5 [ / Λ

GeV DarkSide [ - [ / Λ - 2 50 50 0 XENON10 / Λ 10 - 50 XENON1T XENON1T XENON1T 10-5 10-6 10-1 npl g anapole lcrcdpl g dipole electric ioeg dipole magne � c 10-2 10-6 10-7 101 102 103 101 102 103 101 102 103

DM mass mχ [MeV] DM mass mχ [MeV] DM mass mχ [MeV]

FIG. 10. Constraints (90% C.L.) on anapole, magnetic dipole, and electric dipole interactions (see AppendixD) from the null result reported by XENON10 [13, 43] (light blue), XENON1T [15](dark blue), and DarkSide-50 [12] (purple).

ments.10 For these specific DM-electron interaction mod- tron ionization energy spectrum can in general be ex- els, we find that exclusion limits from DarkSide-50 are pressed as the sum of four independent terms. Each of generically less stringent, as compared to those arising these terms is given by the product between an atomic from XENON10 and XENON1T data. While XENON10 response function, which depends on the initial and final always sets the most stringent limits for DM masses be- state electron wave functions, and a DM response func- low about 20-50 GeV/c2 (depending on the assumed in- tion, which only depends on kinematic variables, cou- teraction), the minimum coupling constant value that pling constants, and the energy gap between initial and XENON10 can exclude is comparable to the minimum final electron states. We investigated the structure and coupling probed by XENON1T only for electric dipole relative strength of the four terms contributing to the interactions. total ionization spectrum, describing under what circum- stances they can be generated in nonrelativistic DM- electron scattering processes in argon and xenon tar- V. SUMMARY gets. The effective theory approach developed in this work has proven very useful in the classification and char- We computed the response of isolated xenon and ar- acterization of the individual atomic responses and con- gon atoms to general, nonrelativistic DM-electron inter- tributions to the total ionization energy spectrum. At actions. This is a key input to the interpretation of DM- the same time, the generality of the formalism developed electron scattering data. For example, dual-phase argon here allows for straightforward applications to most spe- and xenon targets are used in direct detection experi- cific DM interaction models. ments searching for signals of nonrelativistic interactions We then used the argon and xenon atomic responses between Milky Way DM particles and electrons in the found here to compute 90% C.L. exclusion limits on the target materials. We modelled the DM-electron scatter- strength of DM-electron interactions from the null result ing by formulating a nonrelativistic effective theory of reported by the XENON10, XENON1T and DarkSide- DM-electron interactions which significantly extends the 50 direct detection experiments. DarkSide-50 is the only currently favoured framework, namely, the dark photon direct detection experiment considered here that uses an model. Within our effective theory description of DM- argon target. For contact DM-electron interactions, we electron interactions, the free amplitude for DM-electron found that XENON10 generically sets the most stringent scattering not only depends on the momentum trans- constraints for DM particle masses below O(10) MeV ferred in the scattering (as in the dark photon model), and that XENON1T is the leading experiment at larger it can also depend on a second, independent combina- masses. The DM particle mass above which XENON1T tion of particle momenta as well as on the DM particle dominates depends on the specific DM-electron interac- and electron spin operators. Quantitatively, we defined tion model. For long-range interactions, we found cases the atomic response to an external probe in terms of the in which XENON10 sets the strongest exclusion limits overlap between the initial and final state electron wave on the entire mass window explored here. Both for con- functions. tact and long-range interactions, the DarkSide-50 exper- As a first application of the atomic responses found iment currently sets 90% C.L. exclusion limits which are in this work, we computed the rate at which Milky generically less stringent than those from XENON10 and Way DM particles can ionize isolated argon and xenon XENON1T. atoms in target materials used in operating direct de- Considering the plethora of proposed DM-electron tection experiments. We found that the final state elec- scattering targets, the results presented in this work should be thought of as a first step within a far-reaching program which aims at exploring the response to general DM-electron interactions of condensed matter sys- 10 The details of these experiments can be found in AppendixE. tems used in (or proposed for) DM direct detection ex- 18 periments. Indeed, the expressions derived in Sec.IIA berg Foundation (PI, Jan Conrad). RC also acknowl- andIIB apply not just to isolated atoms, but also more edges support from an individual research grant from generally to other target materials, such as semiconduc- the Swedish Research Council, dnr. 2018-05029. NAS tor crystals and quantum materials. and WT were supported by the ETH Zurich, and by Finally, we would like to stress that only one of the four the European Research Council (ERC) under the Euro- atomic response functions computed here can arise from pean Unions Horizon 2020 research and innovation pro- standard electromagnetic interactions between external gramme grant agreement No 810451. The research pre- electrons or photons and electrons bound to target ma- sented in this paper made use of the following software terials. The remaining three atomic responses can only be packages, libraries, and tools: hankel [55], mpmath [56], generated when DM is the external probe used to inves- NumPy [57], SciPy [58], WebPlotDigitizer [59], Wolfram tigate the properties of condensed matter systems. From Mathematica [60]. Some of the computations were per- this perspective, our findings show that the detection of formed on resources provided by the Swedish National Milky Way DM particles has the potential to open up a Infrastructure for Computing (SNIC) at NSC. new window on the exploration of materials’ responses to external probes, revealing as of yet hidden properties of matter. Appendix A: Derivation of the atomic and DM response functions ACKNOWLEDGMENTS This appendix serves as a bridge between Eqs. (34) We would like to thank Rouven Essig, SinéadGrif- and (39). In particular, we identify and derive the atomic fin, Felix Kahlhoefer, Yonathan Kahn, and Tien-Tien and DM response functions given in Eqs. (40) and (41). Yu for useful comments and valuable discussions. RC This involves the computation of the three terms in and TE were supported by the Knut and Alice Wallen- Eq. (34), given more explicitly by

⊥ 2 1 1 X ⊥ 2 |M(q, vel )| = |M(q, vel )| , (A1) 2jχ + 1 2je + 1 spins

∗ ∗ 1 1 X ∗ ∗ 2me< [M∇kM · f1→2(q)f1→2(q)] = 2me< [M∇kM · f1→2(q)f1→2(q)] , (A2) 2jχ + 1 2je + 1 spins

2 ⊥ 2 1 1 X 2 ⊥ 2 me|∇kM(q, vel ) · f1→2(q)| = me|∇kM(q, vel ) · f1→2(q)| , (A3) 2jχ + 1 2je + 1 spins

where we average (sum) over initial (final) spins of the When averaging (summing) the terms in Eqs. (A1) DM particle and the electron. For the amplitude we sub- to (A3) over the initial (final) spin of the DM particle and stitute the general expression of Eq. (19), where we con- the electron, the non-vanishing terms are of the form [52], sider the first 14 operators listed in TableI.

0 ⊥ Here, s(s ) is the initial (ﬁnal) spin’s index of either the vel , which occurs frequently, does not vanish since we are electron or DM particle, and A and B are two general not considering an elastic scattering process. vectors. All cross terms which are linear in Si vanish. It is important to keep in mind that the scalar product q · 19

1. Response 1 the evaluation of the electron and DM spin sums, we make use of the identities of Eq. (A4). We find We evaluate the spin sums in Eq. (A1) for the scattering amplitude including the first 14 effective operators. For

2 2 2 2 2 ∗ 2 ⊥ 2 2 |c3| q ⊥ |c7| ⊥ |c10| q =(c7c10) q ⊥ |M(q, vel )| = |c1| + × vel + vel + + · vel 4 me 4 4 me 2 me ( 2 4 jχ(jχ + 1) 2 2 ∗ q ⊥ 2 q 2 2 ⊥2 + 3|c4| + 4|c5| − 2<(c12c15) × vel + |c6| + 4|c8| + 2|c12| vel 12 me me 2 2 2 2 2 2 ∗ q 2 2 q ⊥2 2 q q ⊥ + 2|c9| + 4|c11| + 2<(c4c6) + |c13| + |c14| vel + |c15| × vel me me me me 2 ∗ ∗ ∗ ∗ ∗ ∗ q + 2 =(c4c13) + =(c4c14) + 4=(c8c11) + 2=(c9c12) + (=(c6c13) + =(c6c14)) me ) ∗ q ⊥ q ⊥ + <(c13c14) · vel · vel . (A5) me me

∗ For real couplings (ci = ci for all i), the squared amplitude simpliﬁes to

2 2 2 2 2 2 2 ⊥ 2 2 c3 q ⊥ 2 c3 q ⊥ c7 ⊥ 2 c10 q |M(q, vel )| = c1 + (vel ) − · vel + (vel ) + 4 me 4 me 4 4 me ( 4 2 jχ(jχ + 1) 2 2 q 2 2 ⊥ 2 2 2 q + 3c4 + c6 + (4c8 + 2c12)(vel ) + (2c9 + 4c11 + 2c4c6) 12 me me 2 4 2 2 2 q ⊥ 2 2 q ⊥2 + 4c5 + c13 + c14 − 2c12c15 (vel ) + c15 vel me me 2 2 2 ) 2 q ⊥ q 2 ⊥ q − c15 vel · + −4c5 + 2c13c14 + 2c12c15 vel · . (A6) me me me

The square of the amplitude can be identified as the first as we saw in Eq. (41a). DM response function, n` ⊥ q ⊥ 2 R1 vel , = |M(q, vel )| , (A7) me 2. Response 2 which was given in Eq. (40a) since the first atomic response function is of the form By using the spin-sum identities of Eqs. (A4), the second n` 0 2 W1 (k , q) ∝ |f1→2(q)| , (A8) term given by Eq. (A2) results in

" 2 2 ! # ∗ |c3| q ⊥ q q ⊥ |c7| ⊥ 1 q ⊥ ∗ ∗ 2me< [M∇kM · A] = · vel − vel − vel · <(A) + × vel · =((c3c7 + c3c7)A) 2 me me me 2 2 me ( 2! 2 ! 1 q ∗ jχ(jχ + 1) 2 2 q q ⊥ q q ⊥ + · =(c7c10A) + 4|c5| + |c15| · vel − vel 2 me 6 me me me me 2! 2 2 2 2 q ⊥ q ∗ q ∗ − 4|c8| + 2|c12| + (|c13| + |c14| ) vel · <(A) − · =(c4c13A) − · =(c4c14A) me me me 20

2 2 q ⊥ ∗ ∗ q q ∗ q q ∗ q ∗ + 4 × vel · =((c5c8 + c5c8)A) − · =(c6c13A) − · =(c6c14A) + 2 · =(c9c12A) me me me me me me q ∗ q ⊥ ∗ ∗ q ⊥ ∗ ∗ + 4 · =(c11c8A) − × vel · =((c12c13 + c12c13)A) + × vel · =((c12c14 + c12c14)A) me me me 2 ! q ⊥ q q ⊥ ∗ ∗ q ⊥ q ∗ ∗ − · vel − vel · <((c12c15 + c12c15)A) − · vel · <((c13c14 + c13c14)A) me me me me me 2 ) q q ⊥ ∗ ∗ − × vel · =((c14c15 + c14c15)A) . (A9) me me

∗ To keep the expressions more manageable, we introduced the vector A ≡ f1→2(q)f1→2(q). For real couplings and a real vector A, this simpliﬁes to

∗ 2me< [M∇kM · A] = " 2 ( 2)# q c3 q ⊥ jχ(jχ + 1) q ⊥ 2 2 q = · A · vel + · vel 4c5 − 2(c12c15 + c13c14) + c15 me 2 me 6 me me

" 2 2 2 ( 2 4 ⊥ c3 q c7 jχ(jχ + 1) 2 q 2 q 2 2 + vel · A − − + − 4c5 − c15 − 4c8 − 2c12 2 me 2 6 me me 2 2 )# 2 2 q q − (c13 + c14) + 2c12c15 . (A10) me me

The first line corresponds to the second response in DM response function, Eq. (39), since we recognize the second atomic response function, " 2 −2 n` ⊥ q q ⊥ c7 q 0 R v , = · v − ` ∞ ` 2 el m m el 2 m n` 0 X X X q e e e W2 (k , q) ∝ · A . (A11) ( −2 )# 0 0 0 me m=−` ` =0 m =−` jχ(jχ + 1) 2 2 q 2 − (4c8 + 2c12) + (c13 + c14) . 0 P 6 me Indeed, the vector A ≡ mm0 A is real which justifies our assumption above. (A13) Furthermore, it is not obvious that the second term in ⊥ Eq. (A10), which is proportional to (vel · A), becomes This is the expression we presented in Eq. (40b) of this part of the second DM response function as well. The paper. Note that the c , c , and c terms as well as reason for this is the fact that the vector A0 is (anti- 3 5 15 the c12c15 interference term cancel due to the use of )parallel to q, which allows us to write Eq. (A12). −2 3. Response 3 and 4 ⊥ 0 q 0 q q ⊥ vel · A = · A · vel . (A12) me me me Lastly, we evaluate the third term in a similar fashion. Hence, we obtain the final expression for the second Substituting the 14 first operators into Eq. (A3), we find

2 2 2 ! 2 2 2 ⊥ 2 |c3| q |c7| 2 |c3| q me|∇kM(q, vel ) · f1→2(q)| = + |f1→2(q)| − · f1→2(q) 4 me 4 4 me ( " 2 4# jχ(jχ + 1) 2 2 2 2 ∗ q 2 2 2 q + |f1→2(q)| (4|c5| + |c13| + |c14| − 2<(c12c15)) + 4|c8| + 2|c12| + |c15| 12 me me 2 " 2 # q 2 2 q ∗ ∗ + · f1→2(q) −4|c5| − |c15| + 2<(c12c15) + 2<(c13c14) me me 21

" 2#) q ∗ ∗ ∗ ∗ ∗ ∗ q + · i (f1→2(q) × f1→2(q)) 8<(c3c7) + 8<(c5c8) − 2<(c12c13) + 2<(c12c14) − 8<(c14c15) . me me (A14)

For real couplings, this becomes

" 2 2 2 ( 2 4)# 2 c3 q c7 jχ(jχ + 1) 2 2 2 2 2 q 2 q = |f1→2(q)| + + 4c8 + 2c12 + 4c5 + c13 + c14 − 2c12c15 + c15 4 me 4 12 me me

2 " 2 ( 2 )# q c3 jχ(jχ + 1) 2 2 q + · f1→2(q) − + −4c5 − c15 + 2c12c15 + 2c13c14 me 4 12 me " ( 2)# q ∗ jχ(jχ + 1) q + · i (f1→2(q) × f1→2(q)) 8c3c7 + 8c5c8 − 2c12c13 + 2c12c14 − 8c14c15 . (A15) me 12 me

From the deﬁnition of the third and fourth atomic re- Appendix B: Evaluation of the atomic response sponse functions in Eqs. (41c) and (41d), functions

1. Useful identities ` ∞ `0 n` 0 X X X 2 W (k , q) ∝ |f1→2(q)| , (A16) 3 This chapter contains a number of mathematical iden- m=−` `0=0 m0=−`0 tities necessary for the evaluation of the atomic response ` ∞ `0 2 n` 0 X X X q functions. W4 (k , q) ∝ · f1→2(q) , (A17) me a. Spherical harmonic addition theorem The spher- m=−` `0=0 m0=−`0 m ical harmonics, Y` (θ, φ) are given by

m m imφ m we can directly assign the ﬁrst and second line of Y` (θ, φ) = N` e P` (cos θ) , (B1) Eq. (A15) to the third and fourth DM response func- m where P` (cos θ) are the associated Legendre polynomi- tions, m als, and N` is a normalization constant ensuring that R m0 ∗ m dΩ (Y`0 ) Y` = δ``0 δmm0 . The normalization constant 2 2 2 n` ⊥ q c3 q c7 jχ(jχ + 1) is given by R3 vel , ≡ + + me 4 me 4 12 s ( 2` + 1 (` − m)! 2 N m ≡ . (B2) 2 2 2 2 2 q ` × 4c8 + 2c12 + 4c5 + c13 + c14 − 2c12c15 4π (` + m)! me ) The spherical harmonic addition theorem states that for q 4 + c2 , (A18) two given unit vectors deﬁned by the spherical angle co- 15 m e ordinates (θ1, φ1) and (θ2, φ2), the sum over all values q c2 j (j + 1) of m yields [61] Rn` v⊥, ≡ − 3 + χ χ 4 el m 4 12 e ` 2` + 1 ( 2 ) X m m∗ Y` (θ1, φ1)Y` (θ2, φ2) = P`(cos ω) . (B3a) 2 2 q 4π × − 4c5 − c15 + 2c12c15 + 2c13c14 . (A19) m=−` me

Here, P`(x) is a Legendre polynomial, and ω is the angle These two response functions were presented in the paper between the two vectors given by as Eqs. (40c) and (40d). cos ω = cos θ1 cos θ2 + sin θ1 sin θ2 cos(φ1 − φ2) . (B3b) The third line of Eq. (A15) does not contribute to ion- P ∗ ik·x ization rates, since the vector mm0 (f1→2(q) × f1→2(q)) b. Plane-wave expansion A plane-wave e can be vanishes. expressed as a linear combination of spherical waves [62],

∞ ik·x X ` This completes the derivation of the atomic and DM re- e = i (2` + 1)j`(kr)P`(cos ω) , (B4a) sponse functions. `=0 22 where ω is the angle between k and x. Using the spherical where the VSH are defined following the conventions of harmonic addition theorem in Eq. (B3), we can re-write Barrera et al. [63], this equation as m m Y` (θ, φ) ≡ ˆr Y` (θ, φ) , (B10a) ∞ +` m m X ` X m∗ m Ψ` (θ, φ) ≡ r ∇Y` (θ, φ) . (B10b) = 4π i j`(kr) Y` (θk, φk) Y` (θx, φx) . (B4b) `=0 m=−` The Cartesian components of the VSHs can themselves be expanded in spherical harmonics with coefficients c. Identities with the Wigner 3j-symbol The angular closely related to Clebsch-Gordan coefficients [62], integral of the product of three spherical harmonics can be expressed in terms of the Wigner 3j-symbol [61], `+1 m+1 X X (i) (Ym(θ, φ)) = c Y mˆ (θ, φ) , (B11) Z ` i `ˆmˆ `ˆ dΩ Y m1 (θ, φ)Y m2 (θ, φ)Y m3 (θ, φ) = `ˆ=`−1 mˆ =m−1 `1 `2 `3 r (2` + 1)(2` + 1)(2` + 1) with the non-vanishing coefficients = 1 2 3 4π A− (1) (2) −1 ` ` ` ` ` ` c`−1 m−1 = −ic`−1 m−1 = − , × 1 2 3 1 2 3 . (B5) 2p(2` − 1)(2` + 1) 0 0 0 m1 m2 m3 A+ c(1) = −ic(2) = −1 , The 3j-symbols satisfy the following orthogonality rela- `+1 m−1 `+1 m−1 2p(2` + 3)(2` + 1) tion [62], A− c(1) = ic(2) = 1 , 0 `−1 m+1 `−1 m+1 p X j1 j2 j j1 j2 j 2 (2` − 1)(2` + 1) (2j + 1) 0 = m1 m2 m m1 m2 m + m1,m2 (1) (2) A1 c`+1 m+1 = ic`+1 m+1 = − p , = δjj0 δmm0 ∆(j1, j2, j) , (B6) 2 (2` + 3)(2` + 1) − (3) A where ∆(j , j , j) is equal to 1 if (j , j , j) satisfy the c = 0 , 1 2 1 2 `−1 m p2(2` − 1)(2` + 1) triangular condition, i.e., |j1 − j2| ≤ j3 ≤ j1 + j2, and zero otherwise. A+ c(3) = − 0 . `+1 m p2(2` + 3)(2` + 1)

2. Gradient of a wave function Here, we used the notation of [62],

+ p In order to compute the vectorial atomic form factor A1 = (` + m + 1)(` + m + 2) , (B12a) in Eq. (B18), we need to evaluate the gradient of a + p A0 = − 2(` + m + 1)(` − m + 1) , (B12b) wave function ψnlm(x) as deﬁned in spherical coordi- + p nates (r, θ, φ) by Eq. (37) which is given by A−1 = (` − m + 1)(` − m + 2) , (B12c) − p A1 = (` − m − 1)(` − m) , (B12d) dRn` m ∇ψnlm(x) = Y` (θ, φ) ˆr + − p dr A0 = 2(` + m)(` − m) , (B12e) m m Rn`(r) dY` 1 dY` − p + θˆ + φˆ , (B7) A−1 = (` + m − 1)(` + m) . (B12f) r dθ sin θ dφ Similarly, the second VSH can be expanded as with the unit vectors `+1 m+1     X X (i) sin θ cos φ cos θ cos φ (Ψm(θ, φ)) = d Y mˆ (θ, φ) , (B13) ˆ ` i `ˆmˆ `ˆ ˆr = sin θ sin φ, θ = cos θ sin φ , `ˆ=`−1 mˆ =m−1 cos θ − sin θ with coeﬃcients, − sin φ ˆ φ =  cos φ  . (B8) (` + 1)A− 0 d(1) = −id(2) = − −1 , `−1 m−1 `−1 m−1 2p(2` + 1)(2` − 1) + This gradient can be expressed in terms of vector spher- (1) (2) ` A−1 ical harmonics (VSH), d`+1 m−1 = −id`+1 m−1 = − , 2p(2` + 1)(2` + 3) − dRn` m Rn`(r) m (1) (2) (` + 1)A1 ∇ψnlm(x) = Y (θ, φ) + Ψ (θ, φ) , (B9) d = id = , dr ` r ` `−1 m+1 `−1 m+1 2p(2` + 1)(2` − 1) 23

+ (1) (2) ` A a. Scalar atomic form factor Focussing on the d = id = 1 , `+1 m+1 `+1 m+1 2p(2` + 1)(2` + 3) scalar atomic form factor for now, we replace the expo- ix·q − nential e with the plane-wave expansion of Eq. (B4). (3) (` + 1)A0 d`−1 m = p , Z 2(2` + 1)(2` − 1) 3 ∗ m0∗ m f1→2(q) = d x Rk0`0 (r)Y`0 (θ, φ)Rn`(r)Y` (θ, φ) + (3) ` A0 d`+1 m = p . ∞ +L 2(2` + 1)(2` + 3) X L X M∗ M × 4π i jL(qr) YL (θq, φq) YL (θ, φ) L=0 M=−L Eqs. (B11) and (B13) allow us to the express the gra- ∞ L dient of the wave function ψlmn in a compact expansion X L X M∗ = 4π i I1(q)YL (θq, φq) in spherical harmonics, L=0 M=−L Z m0∗ m M 3 `+1 m+1 × dΩ Y 0 (θ, φ)Y (θ, φ)Y (θ, φ) , (B20) X X X ` ` L ∇ψnlm(x) = ei i=1 `ˆ=`−1 mˆ =m−1 where we absorbed the radial integral into dR R (r) c(i) n` + d(i) n` Y mˆ (θ, φ) (B14) Z ˆ ˆ `ˆ 2 ∗ `mˆ dr `mˆ r I1(q) ≡ dr r Rk0`0 (r)Rn`(r)jL(qr) . (B21) where ei,(i = 1, 2, 3) are the three Cartesian unit vectors. This integral is the spherical Bessel transform ∗ of Rk0`0 (r)Rn`(r), i.e. the product of the radial components of the final and initial wave functions. For its evaluation, we refer to AppendixB5. 3. The atomic form factors The integral over three spherical harmonics in the last expression can be re-written in terms of the Wigner 3j- In Eqs. (24) and (26), we introduced the scalar and symbols via Eq. (B5), vectorial atomic form factors. In this section, we describe 0 the details of evaluting these form factors. √ `+` +L X L X M∗ First, we denote the initial (‘1’) and final (‘2’) state f1→2(q) = 4π i I1(q) YL (θq, φq) electron wave functions by their respective quantum L=|`−`0| M=−L 0 0 0 0 numbers (n, `, m) and (k , ` , m ). Consequently, the 0 ` ` L × (−1)m p(2` + 1)(2`0 + 1)(2L + 1) atomic form factors take the form 0 0 0 0 Z 3 ` ` L d k ∗ × 0 . (B22) f (q) = ψ 0 0 0 (k + q)ψ (k) , (B15) m −m M 1→2 (2π)3 k ` m nlm Z 3 d k ∗ k At this point, we can already evaluate the first atomic f (q) = ψ 0 0 0 (k + q) ψ (k) . (B16) n` 1→2 3 k ` m nlm response W given by Eq. (41a), which essentially given (2π) me 1 by the squared modulus of the scalar atomic form factor 0 Moving to position space via a Fourier transformation, summed over m, m , we find ` `0 X X 2 X X L L0 2 Z |f1→2(q)| = 4π i (−i) |I1(q)| 3 ∗ ix·q f1→2(q) = d x ψk0`0m0 (x)e ψn`m(x) , (B17) m=−` m0=−`0 mm0 LL0 X M∗ M 0 Z × Y (θq, φq)Y 0 (θq, φq) 3 ∗ ix·q i∇ L L f1→2(q) = d x ψk0`0m0 (x)e ψn`m(x) . (B18) M,M 0 me × (2` + 1)(2`0 + 1)p(2L + 1)(2L0 + 1) In Eqs. (37) and (38) we expressed the initial and final ` `0 L ` `0 L0 × state electron wave functions in terms of spherical coor- 0 0 0 0 0 0 dinates x(r, θ, φ), where the angular dependence is given ` `0 L ` `0 L0 by the spherical harmonics, × . (B23) m −m0 M m −m0 M 0 ψ (x) = R (r) Y m(θ, φ) . (B19) n`m n` ` The orthogonality of the Wigner 3j-symbols, Eq. (B6), allows us to sum over the L0 and M 0 indexes, We postpone the discussion of the radial components of the wave functions to AppendixB4. and start with the ` `0 evaluation of the scalar atomic form factor f . After- X X 2 X 2 X M∗ 1→2 |f1→2(q)| = 4π I1(q) YL (θq, φq) wards, we discuss the vectorial atomic form factor f1→2. m=−` m0=−`0 L M 24

M 0 × YL (θq, φq)(2` + 1)(2` + 1) where we defined two new radial integrals, 0 2 Z ` ` L 2 ∗ dRn` × . (B24) I (q) ≡ dr r j (qr)R 0 0 (r) , (B29) 0 0 0 2 L k ` dr Z ∗ Lastly we apply the spherical harmonic addition theorem I3(q) ≡ dr rjL(qr)Rk0`0 (r)Rn`(r) , (B30) given in Eq. (B3), which will require extra attention in AppendixB5. ` `0 `+`0 We again apply Eq. (B5) to explicitly perform the an- X X 2 X 0 |f1→2(q)| = (2` + 1)(2` + 1) gular integral, m=−` m0=−`0 L=|`−`0| √ i 4π 3 `+1 m+1 ∞ L 0 2 X X X X X M∗ 2 ` ` L f1→2(q) = eî YL (θq, φq) × (2L + 1) |I1(q)| . (B25) me 0 0 0 i=1 `ˆ=`−1 mˆ =m−1 L=0 M=−L q L m0 Note that the dependence on the momentum trans- × i (−1) (2`ˆ+ 1)(2`0 + 1)(2L + 1) fer’s direction has disappeared. The explicit form of `ˆ `0 L `ˆ `0 L n` 0 × the first atomic response W1 (q, k ) for an atomic or- 0 0 0 mˆ −m0 M bital (n, `) given by Eq. (41a) (or equivalently the ‘ion- h i n,` 2 × c(i) I (q) + d(i) I (q) . (B31) ization form factor |fion | defined in Eq. (35)) is obtained `ˆmˆ 2 `ˆmˆ 3 by adding the remainder of the final state phase space11, see Eq. (27), It can be useful to evaluate the above expression in a particular coordinate frame, namely the one in which the ∞ `+`0 z-axis points towards q, such that θ = 0. Then we 4k03 X X q W n`(q, k0) = V (2` + 1)(2`0 + 1) can use Y M (θ = 0, φ ) = p(2L + 1)/(4π)δ and sum 1 (2π)3 L q q M0 `0=0 L=|`−`0| over M, 2 0 ˆ 0 2 ` ` L 3 `+1 m+1 `+` × (2L + 1)|I1(q)| . (B26) i X X X X L 0 0 0 f1→2(q) = eî i me i=1 `ˆ=`−1 mˆ =m−1 L=|`ˆ−`0| In order to evaluate the three new atomic responses we 0 q need to follow similar steps for the vectorial atomic form (−1)m (2L + 1) (2`0 + 1)(2`ˆ+ 1) factor as for the scalar one. ˆ 0 ˆ 0 b. Vectorial atomic form factor We start from ` ` L ` ` L × 0 Eq. (B18) and substitute the gradient of the initial wave 0 0 0 mˆ −m 0 h i function using Eq. (B14), × c(i) I (q) + d(i) I (q) . (B32) `ˆmˆ 2 `ˆmˆ 3 3 `+1 m+1 Z i X X X 3 ∗ m0∗ We explicitly checked that this choice of coordinate frame f1→2(q) = ei d xRk0`0 (r)Y`0 (θ, φ) me has no impact on the final values of the atomic response i=1 `ˆ=`−1 mˆ =m−1 functions. dR R (r) With the scalar and vectorial atomic form factors given × eix·q c(i) n` + d(i) n` Y mˆ (θ, φ) . `ˆmˆ dr `ˆmˆ r `ˆ in the forms of Eqs. (B22) and (B32), it is possible to eval- (B27) uate the three new atomic response functions given by the Eqs. (41b) to (41d). However, before doing so, we have As for the scalar atomic form factor, the exponential in to specify the radial components of the initial and final the above equation can be expressed in terms of spherical state electron wave functions without which we cannot harmonics via Eq. (B4), evaluate the radial integrals defined in Eqs. (B21), (B29), and (B30). 3 `+1 m+1 ∞ L 4πi X X X X X M∗ f1→2(q) = eî YL (θq, φq) me i=1 `ˆ=`−1 mˆ =m−1 L=0 M=−L 4. Initial and final state wave functions Z L m0∗ M mˆ × i dΩ Y 0 (θ, φ)Y (θ, φ)Y (θ, φ) ` L `ˆ For the radial part of the initial state wave func- h i tion ψn`m(x), we assume Roothaan-Hartree-Fock (RHF) × c(i) I (q) + d(i) I (q) , (B28) `ˆmˆ 2 `ˆmˆ 3 ground state wave functions expressed as linear combinations of Slater-type orbitals,

0 nj`+1/2 −3/2 X (2Zj`) Rn`(r) = a Cj`n 11 Regarding the numerical evaluation of the sum over `0, we sum 0 r 0 0 j 2n ! up contributions up to `max = 7. j` 25

n0 −1 r j` r Below, we outline two of the methods we used to solve × exp −Zj` . (B33) this integral. The first one is exact, but requires the nu- a0 a0 merical integration of an oscillating function, the second 0 The coefficients Cj`n, Zj`, and nj` are tabulated for the one is approximate, but fully analytical. atomic orbitals of argon and xenon in [47] together with The python code we used for the evaluation of the ra- n` dial integrals, the atomic form factors, and the atomic their respective binding energies EB . Furthermore, a0 denotes the Bohr radius. response functions is publicly available [44]. The final electron state is described by a positive en- a. Numerical solution One of the most straight- ergy continuum solution of the Schrödingerequation with forward ways to evaluate the radial integrals is the use a hydrogenic potential −Zeff /r [12, 64]. of specific numerical integration methods of numerical libraries like e.g. NumPy [57], or computer algebra sys- q πZeff iZ 0 2 0 eff 2k a0 tems such as Wolfram Mathematica [60]. The numerical 3/2 Γ ` + 1 − 0 e (2π) 0 `0 π k a0 integration method “DoubleExponential” of the Wolfram Rk0`0 (r) = √ (2k r) V (2`0 + 1)! language yielded reliable results. However, for large val- −ik0r 0 iZeff 0 0 ues of the momentum transfer, the method becomes crit- × e 1F1 ` + 1 + 0 , 2` + 2, 2ik r . (B34) ically slow. k a0 It is usually beneficial for the performance of the nu- Here, 1F1 (a, b, z) is the so-called Kummer’s function, or merical integration to split up the integration domain, confluent hypergeometric function of the first kind. Note ∞ Z ∞ X Z (i+1)∆r that both radial wave functions are√ purely real. It should dr f(r) = dr f(r) (B39) also be noted that the factor of 1/ V involving the vol- 0 i=0 i∆r ume always cancels due to the application of the integral operator in Eq. (27), or equivalently with the factor of V with ∆r = a0 and truncate the sum at a finite i after the in Eqs. (41a) to (41d). result has converged to a desired tolerance. Since the in- The factor Zeff is determined by matching the binding tegrand in our case is highly oscillatory and this method n` energy EB of the ionized orbital [13, 14, 20], does not ensure that we integrate from one root to the next, it is possible that an accidental cancellation of pos- 2 Zn` itive and negative contributions over one sub-interval of En` =! 13.6 eV eff , B n2 the domain can be misinterpreted as convergence. To r avoid this subtle problem, we always verify the series’ En` n` B convergence on two consecutive terms. ⇒ Zeff = × n . (B35) 13.6 eV b. Analytic solution The second method effectively With the definition of the radial parts of the wave func- replaces the integral with an infinite sum, which, if it tions in place, we can continue with the evaluation of the converges sufficiently fast, yields accurate and prompt radial integrals. results. This method worked best when applied to large values of q, i.e., exactly where the numerical integration is problematic. 5. Numerical evaluation of the radial integrals The Kummer function (or confluent hypergeometric function of the first kind), which appears in Eq. (B34), It is fair to say that the largest obstacle to the evalua- can be written as a power series, tion of the scalar and vectorial atomic form factors is the ∞ ∞ X a(s) zs X Γ(s + a)Γ(b) zs computation of the radial integrals12 F (a, b, z) = = , (B40) 1 1 b(s) s! Γ(a)Γ(s + b) s! Z ∞ s=0 s=0 2 ∗ I1(q) = dr r jL(qr)Rk0`0 (r)Rn`(r) , (B36) (s) 0 with x ≡ x(x + 1) ... (x + s − 1) being the rising facto- Z ∞ rial. In combination with the analytic form of the initial 2 ∗ dRn` I2(q) = dr r jL(qr)R 0 0 (r) , (B37) wave function in Eq. (B33), we find that the above radial k ` dr 0 integrals are infinite sums of simpler integrals of the form Z ∞ ∗ I3(q) = dr rjL(qr)Rk0`0 (r)Rn`(r) . (B38) Z α 0 IL(q, α, β) = dr r exp(−βr)jL(qr) , (B41) The highly oscillatory behavior of the spherical Bessel function can hinder the success of standard numerical where the values of the parameters α and β depend on integration methods, especially for large values of q. whether we evaluate I1(q), I2(q) or I3(q). Importantly, this integral can be solved analytically (provided that <(L + α) > −1), √ 12 πqL Γ(L + α + 1) These types of integrals are known in the mathematical literature I (q, α, β) = as spherical Bessel and Hankel transforms. L L+1 α+L+1 3 2 β Γ L + 2 26

2 02 (L + α + 1) (L + α + 2) 3 q electron ionization with ∆E1→2 = EB + k /(2me) can × 2F1 , ,L + , − 2 2 2 2 β be obtained by setting cos θqv = 1, (B42) 02 0 EB + k /(2me) q vmin(k , q) = + . (C2) where the hypergeometric function 2F1 (a, b, c, z) appears q 2mχ which is defined via Since the local DM velocity distribution of the galactic ∞ halo is assumed to have a maximum speed vesc, DM parti- X a(s)b(s) zs F (a, b, c, z) ≡ . (B43) cles are only able to ionize a bound electron with binding 2 1 c(s) s! s=0 energy EB if vmin < vmax = vesc + v⊕. Thereby, we find the range of momentum transfers which can contribute Provided that the sum in Eq. (B40) converges after a to the ionization, finite number of terms, we can compute the radial in- q 2 2 tegrals analytically. For large q the convergence occurs qmin = mχvmax − mχvmax − 2mχEB , (C3) quickly. For lower values, we run into numerical precision EB problems, since we have to include very large numbers of = , for mχ → ∞ , (C4) terms of (B40). In this case, the numerical approach de- vmax q 2 2 scribed in the previous section performs better. qmax = mχvmax + mχvmax − 2mχEB . (C5) As an example, by using the analytical method described here, the first radial integral, I1(q), can now be From the requirement that vmin < vmax = vesc + v⊕, written as follows: we can also derive the maximum final momentum of the electron for a given momentum transfer q, ∞ 0 `0 X X 4π(2k ) iZeff πZeff s I1(q) = Γ l + 1 − e 2ka0 2 (2`0 + 1)! ka 0 q s=0 j 0 k < 2me vmaxq − + EB , (C6) 2mχ (n0 +1/2) C (2Z ) jl Γ(s + a)Γ(b) (2ik0)s × jln jl (n0 +1/2)q which were shown as white lines in Figs.3 and4. a jl (2n0 )! Γ(s + b)Γ(a) s! ⊥ 0 jl Finally, there are a few identities involving vel , which √ are necessary for the evaluation of the DM response func- πqL γ(L + α + 1) × tions in the Eqs. (40). We can write v⊥ as 2L+1βα+L+1 Γ(L + 3/2) el q k L + α + 1 L + α + 2 q2 v⊥ = v − − . (C7) × F , ,L + 3/2, − , el 2µ m 2 1 2 2 β2 e e ⊥ (B44) Hence the square of the vel (with k = 0) is given by 2 where we defined ⊥ 2 2 q mχ − me ∆E1→2 (vel ) |k=0 = v + 2 − . (C8) 4µe me + mχ µe Zjl α = 1 + `0 + n0 + s , β = + ik0 , (B45) We will also need the scalar product with the momentum jl a 0 transfer q, 0 iZeff 0 a = l + 1 + 0 , b = 2` + 1 . (B46) 2 2 k a0 ⊥ q q (vel · q)|k=0 = ∆E1→2 + − 2mχ 2µe The sum can be truncated when the series converges to q2 a given tolerance, provided that it converges sufficiently = ∆E1→2 − . (C9) quickly. 2me For the last two relations, we used energy conservation via Eq. (C1). The dependence on the binding energies Appendix C: Scattering kinematics via ∆E1→2 also explains why the DM response func- n` tions Ri carry the atomic quantum numbers (n`) as In this appendix, we summarize a few basic kinematic indices. relations for DM-electron scatterings. The initial and final energies of the inelastic scattering process are given in Eqs. (4) and (5). By the conservation Appendix D: Anapole, magnetic and electric dipole of energy one can show that dark matter

q2 The anapole, magnetic dipole and electric dipole DM v · q = ∆E1→2 + . (C1) 2mχ models are described by the following interaction La- grangians From this expression, it is clear that for a given momen- g µ 5 ν Lanapole = χγ γ χ ∂ Fµν , (D1) tum transfer q, the minimum DM speed necessary for an 2Λ2 27

g µν Lmagnetic = ψσ ψ Fµν , (D2) the DM particles of the galactic halo, which first enters Λ Eq. (13). The local DM number density is simply given g µν 5 Lelectric = iψσ γ ψ Fµν , (D3) by Λ ρχ where χ (ψ) is a Majorana (Dirac) spinor describing the nχ = , (E1) DM particle, g a dimensionless coupling constant and Λ mχ a mass scale. For simplicity, we use the same notation for 3 the coupling constants of the three interactions, but they where we set ρχ = 0.4 GeV/cm [65]. For the veloc- can in principle be different. The Lagrangians in Eq. (D3) ity distribution, we use the standard halo model (SHM) induce a DM-electron coupling via the electromagnetic which approximates the distribution by a truncated ν µν ν current, J , and Maxwell equations, ∂µF = eJ , where Maxwell-Boltzmann distribution that has been boosted ψe is the four component electron spinor, Fµν = ∂µAν − into the Earth’s rest frame, ν ∂ν Aµ is the photon field strength tensor, and A the 2 photon field. 1 (v + v⊕) f (v) = exp − χ 3/2 3 2 Taking the nonrelativistic limit of the free field solution Nescπ v0 v0 of the equations of motion for ψ (or χ), and using Gor- × Θ(v − |v + v |) . (E2) don’s identity to express the matrix element of the cur- esc ⊕ rent J ν between single particle electron states in terms of √ Here, N ≡ erf(v /v ) − 2(v /v ) exp(−v2 /v2)/ π electromagnetic form factors, at leading order we find the esc esc 0 esc 0 esc 0 is a normalization constant, and the standard choices following amplitudes for DM scattering by free electrons −1 for the other parameters are v0 = 220km sec for the −1 ( Sun’s circular velocity [66], and vesc = 544km sec 4eg 0 0 ⊥ †s s λ λ for the galactic escape velocity [67]. For the speed of Manapole = 2 mχme 2 vel · ξ Sχξ δ Λ the Earth/the observer in the galactic rest frame, we −1 ) choose v⊕ ≈ 244km sec . 0 q 0 + g ξ†s S ξs · i × ξ†λ S ξλ , In all cases, we have to translate the energy spectrum e χ m e e given in Eq. (30) into the spectrum as a function of the (D4) number of final electrons ne, ( eg s0s λ0λ n` Z n` Mmagnetic = 4meδ δ dRion dRion Λ = dEe P(ne|Ee) . (E3) dne dEe 16mχme ⊥ †s0 s λ0λ + iq · vel × ξ Sχξ δ q2 In modeling the probability P(ne|Ee) for ne electrons to " reach the gas phase of the TPC given an initial electron 8gemχ †s0 s †λ0 λ energy of Ee, we follow [13, 14]. − 2 q · ξ Sχξ q · ξ Seξ q For the three experiments, we obtain the 90% C.L. #) exclusion limits by applying Poisson statistics to each 2 †s0 s †λ0 λ − q ξ Sχξ · ξ Seξ , (D5) event bin independently.

eg 16mχme †s0 s λ0λ Melectric = iq · ξ Sχξ δ . (D6) Λ q2 1. XENON10 and XENON1T Here, the notation is the same one used in the main body of the paper. Matching these expressions on to the effec- XENON10 XENON1T tive theory expansion in Eq. (19), one can express the bin [S2] obs. events bin [S2] obs. events effective coupling constants given in Sec.IVB in terms [14,41) 126 [150,200) 8 of g and Λ. In the numerical calculations, we set ge = 2. [41,68) 60 [200,250) 7 [68,95) 12 [250,300) 2 [95,122) 3 [300,350) 1 Appendix E: Details of direct detection and [122,149) 2 -- exclusion limits [149,176) 0 -- [176,203) 2 -- The exclusion limits presented in this paper are based on a re-interpretation of published data and results by TABLE II. Events at XENON10 (left) and XENON1T (right) observed in the given S2 bins. DarkSide-50 [12], XENON10 [43], and XENON1T [15]. In this section, we briefly summarize the necessary details to compute the exclusion limits. For XENON10 and XENON1T, the number of ob- An essential input for the prediction of ionization served signals given in bins of the ionization signal (S2) rates is the local density and velocity distribution of are available and listed in TableII. It is necessary to have 28 the spectrum in terms of S2, i.e. the number of photo- exposure E is given by electrons (PEs) in the photomultiplier tubes, 2 E = R π × ∆z × ρXe × ∆t , (E6) ∞ dRn` X dRn` ion = ε(S2) P(S2|n ) ion . (E4) where the target’s radius is R = 47.9cm, the height of dS2 e dn the search’s volume is ∆z = 20cm (corresponding to n =1 e e z ∈ [−30cm, −10cm]), the density of liquid xenon is −3 taken to be ρXe = 3.1g cm , and the time of the search For a given number of electrons ne, we assume that the data ∆t = 180.7 days [15]. This yields an exposure of resulting number of PEs follows a Gaussian distribution √ ∼ 80755 kg days. with mean neg2 and width neσS2 [14, 68, 69], Compared to the official XENON1T limits on the standard DM-electron scattering cross section (corresponding √ P(S2|ne) = Gauss(S2|neg2, neσS2) . (E5) to our effective coupling c1), our constraints are weaker by a factor of a few and therefore conservative, which can potentially be explained by the additional background The used parameters for the distribution’s mean and subtraction the XENON collaboration performs. width are the secondary-scintillation gain factor g2 = 27(33) and the associated width factor σS2 = 6.7(7) for XENON10 (XENON1T)[14, 15, 70, 71]. 2. DarkSide-50 For XENON10, the binned observed numbers of events are given on the left hand side of TableII, which corre- Just like XENON10 and XENON1T, DarkSide is a spond to an exposure of 15 kg days. The efficiency ε(S2) dual-phase TPC. However, instead of xenon, it uses ar- is given by the product of a flat cut efficiency of 92% [43] gon as target [20]. The DarkSide-50 experiment had an multiplied by the trigger efficiency given in Fig. 1 of [13]. exposure of 6786 kg days, and a threshold of ne = 3 In the case of XENON1T, we extracted the number of electrons. The number of observed events we base our observed events passing all the cuts from Fig. 3 of [15]. limits on have been extracted from Fig. 3 of [20]. The They are listed on the right side of TableII. The ef- official constraints on the standard DM-electron interac- ficiency ε(S2) is the product of a flat efficiency of 93% tion (corresponding to the effective coupling c1) turn out and the different efficiencies given in Fig. 2 of [15]13. The to be stronger than our own limits by a factor of a few.

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