On the relation between Migdal effect and dark matter-electron scattering in atoms and semiconductors

A key strategy for the direct detection of sub-GeV dark matter is to search for small ionization signals. These can arise from dark matter-electron scattering or when the dark matter-nucleus scattering process is accompanied by a “Migdal” electron. We show that the theoretical descriptions of both processes are closely related, which allows for a principal mapping between dark matterelectron and dark matter-nucleus scattering rates once the dark matter interactions with matter are specified. We explore this parametric relationship for noble-liquid targets and, for the first time, provide an estimate of the “Migdal” ionization rate in semiconductors that is based on evaluating a crystal form factor that accounts for the semiconductor band structure. We also present new darkmatter-nucleus scattering limits down to dark matter masses of 500 keV using published data from XENON10, XENON100, and a SENSEI prototype Skipper-CCD. For a dark photon mediator, the dark matter-electron scattering rates dominate over the Migdal rates for dark matter masses below 100 MeV. We also provide projections for proposed experiments with xenon and silicon targets.

Introduction. Detectors searching for direct signals from dark matter (DM) are conventionally optimized to look for electroweak-scale DM that scatters elastically off atomic nuclei. The detectors typically lose sensitivity rapidly for DM masses below a few GeV, due to an inefficient energy transfer from the DM to the recoiling nucleus. However, the kinematic limitations are lifted when the DM-nucleus scattering process is accompanied by the irreducible simultaneous emission of a "bremsstrahlung" photon [1] or a "Migdal"-electron [2], or by considering alternative interactions such as DM-electron scattering [3]. In all cases the entire energy of relative motion between the atom and DM can in principle be transferred to the outgoing photon or electron. These signals have already opened a new pathway for current detectors to register DM scattering on nuclei [4][5][6][7] for DM masses below 100 MeV (driven primarily by the stronger Migdal effect), and DM scattering on electrons [8][9][10][11][12][13][14][15][16] for masses as low as 500 keV. The sensitivity to sub-GeV DM is expected to improve significantly over the next few years as new detectors with an ultralow ionization threshold are being developed [10,17]. For related and distinct directdetection ideas to probe sub-GeV DM, see e.g. .
The theoretical description of the bremsstrahlung and Migdal effect is so far exclusively tied to a picture where DM scatters on a single, isolated atom [1,2], which for inner-shell electrons should provide a correct estimate of the expected signal rate. For the outer-shell electrons, one might have some hope that the isolated atom calculations are still valid for noble-liquids, but the complicated electronic band structure for semiconductor targets definitely calls for a different approach. The results presented for noble liquids in [4,7] and for germanium in [5,6] restricted themselves to inner-shell electrons and had higher detector threshold than those needed to see a dominant signal from the outer shells. However, sev-eral experiments already have sensitivity to one or a few electrons, and a complete understanding of the Migdal and bremsstrahlung effect will maximize our sensitivity to nuclear scattering. On the other hand, DM-electron scattering has already been calculated with a detailed accounting of the band structure in [22].
In this work, our first goal is to show that the theoretical calculation of the rates for the Migdal effect and for DM-electron scattering are closely related. This observation allows us to take the first steps towards calculating the Migdal effect in semiconductors, thereby overcoming the previous limitations in the theoretical description. It allows us also to calculate the first DM-nucleus scattering limits down to DM masses of 500 keV using published SENSEI results with a single-electron threshold [12]. A second goal is to use the observed single-and few-electron events from XENON10 to calculate the Migdal effect, which provides a DM-nuclear scattering limit down to 5 MeV, lower in mass than other published constraints. We compare this limit to XENON100 and XENON1T. Finally, a third goal is to contrast the constraints from DM-electron scattering with those from the Migdal effect in DM-nuclear scattering when the DM interations with ordinary matter are mediated by a dark photon, which gives rise to both signals simultaneously. A more detailed exposition will be provided in a companion paper [49].
To appreciate the connection between the DM-electron and Migdal processes, consider the nucleus receiving a sudden three momentum transfer q in the scattering with DM. The probability for promptly ejecting an electron from an atom that was initially at rest in state |i is then found by projecting the boosted electron cloud onto the desired final state |f , Here, m e (m N ) is the mass of the electron (nucleus) and the sum runs over all electron positions x (α) . The exponential on the left hand side is indeed the electron boost operator as v N = q/m N is the velocity of the recoiling nucleus. On the right hand side, we have used the dipole approximation, noting that me m N q · x (α) 1 in all cases of interest; d f i = f |e α x (α) |i is the atom transition dipole moment. In single-electron transitions only one term will contribute in the latter sum, and d f i is to be replaced by is the coordinate of the electron undergoing the transition.
In turn, in the calculation of the ionization probability from the direct interaction of DM with an electron at coordinate x (β) , the following matrix element enters, Here the momentum transfer q, i.e. the momentum lost by the DM particle, is directly picked up by the electron. Comparing (1) with (2) clearly exposes the similarity of Migdal-effect and DM-electron scattering. However, at the same time this also shows that the atom is probed at vastly different momentum transfers in the respective processes, a factor of ∼ 10 −3 /A softer in the Migdal case.
Migdal effect in atoms. We now establish the precise relation between DM-electron scattering and Migdal effect for isolated atoms. In an isolated atom, the bound initial-state electron wave functions are specified by their principal and orbital quantum numbers n and l. We are interested in transitions to continuum final-states that are characterized by momentum p e = √ 2m e E e and orbital quantum number l . We define the ionization probability for such a scenario as A key observation of this letter is that the electronionization probability is related to a dimensionless form factor |f ion nl (p e , q e )| 2 that was previously defined in the context of DM-electron scattering [3,8]. Concretely, we find where the form factor is evaluated at q e me m N q. Capturing the Migdal process hence becomes a question of calculating the form factor accurately. It is given by [3,8] |f ion |f ion (p e , q)| 2 holds dipole approx.  6) is valid. At higher momentum transfers, |f ion nl (pe, q)| 2 tends to peak before becoming strongly suppressed.
where R nl is the radial wavefunction of the n, l bound state orbital with shell occupancy m l ≤ 2(2l + 1), R pel is the continuum wave function of the electron with momentum p e , and j L is the spherical Bessel function of the first kind. 1 The L = 0 term does not induce a transition and is omitted in sum for clarity.
A detailed calculation of dp n,l→Ee /dE e was presented in [2] using a fully relativistic formulation. Numerical results were calculated in the dipole approximation, which in the above formulation would amount to expanding j 1 (q e r) q e r/3 and taking the L = 1 term. In [2], bound and continuum state wave functions of angular momentum j = l±s were generated with the FAC atomic code [50]. Following this approach, we find agreement with their results. Since electron energies remain in the non-relativistic domain (with small corrections for n = 1 states), a formulation where differences from spin are unresolved will capture the process to sufficient accuracy. Such approximation has been chosen in Eq. (5) where l labels two spin-degenerate states. 2 The dipole approximation in Eq. (5) results in the fol-lowing scaling-relation in the form-factor Here, q 0 is a reference momentum at which the scalingrelation holds, i.e., q 0 r e 1, where r e is the electron radius; for xenon atoms, q 0 1 keV. The scaling-relation breaks down once qr e 1 as higher multipole (L > 1) contributions in Eq. (5) become important and must be included; in our numerical results we include terms up to L = 15 using FAC-generated continuum and bound-state wave functions. The general behavior of |f ion nl (p e , q)| 2 is that it peaks at q ∼ keV before becoming strongly suppressed in the high-q region of tens of keV and above. This behavior is illustrated in Fig. 1 where the area of validity of Eq. (6) is indicated.
Cross sections. The prompt ionization resulting from DM-nuclear scattering with nuclear recoil energy E R = q 2 /2m N leads to a double-differential cross section in E R and E e . Away from kinematic endpoints, the cross-section can be factorized as the elastic cross-section times the electron ionization probability [2], The elastic DM-nucleus recoil cross section is given by where Z is the nuclear charge, f p,n are dimensionless DM couplings to the proton (neutron), and µ n is the DMnucleon reduced mass. The DM-nucleon cross section is defined asσ n ≡ µ 2 n |M n (q = q 0 )| 2 /(16πm 2 χ m 2 n ), where q 0 = αm e is a reference momentum scale. The model dependence in the DM-interaction with the nucleus is absorbed into the definition of a form factor, where |M n (q)| 2 is the matrix element for scattering on a free nucleon. In the second equality, the two given examples correspond to contact interactions and to the exchange of a vector mediator with mass m V and a reference momentum q 0 . For sub-GeV DM, the momentum transfer does not resolve the nucleus, qR N 1, and the nuclear form factor |F N (q)| 2 = 1 to good approximation.
Integrating Eq. (7) over E R then yields the Migdal electron spectrum, where the kinematic limits of E R are fixed by energy conservation. The cross section is then averaged over the galactic velocity distribution of DM, boosted into the laboratory frame of the detector, Here, v min is the minimum velocity such that the CM energy is larger than the total energy received by the electron, v min = 2∆E n,l /µ N ; and ∆E n,l = E e + |E n,l |, where E n,l is the binding energy, which we take from [9]. Finally, the event rate is given by dR n,l /dE e = N T (ρ dm /m χ )d σ n,l v /dE e where N T is the number of targets per kg of detector material and ρ dm 0.3 GeV/cm 3 is the local DM density. The resulting rate as a function of ∆E n,l , summed over all l for the various n-shells of xenon, is shown in Fig. 2 (left) for contact interactions (top panel) and light mediator exchange (bottom panel).
The velocity-averaged ionization cross-section can be written in a form analogous to the one used in DMelectron scattering [3] by exchanging the integration order between d 3 v and dE R (= qdq/m N ), where η(v min ) is the usual velocity average of the inverse speed, η(v min ) = 1 v Θ(v − v min ) g det , but now the minimum velocity becomes a function of both q and ∆E n,l , v min (q, ∆E n,l ) q/(2m χ ) + ∆E n,l /q, where we have made the approximation µ N m χ in the expression for v min as is applicable for sub-GeV DM. Comparing Eq. (9) to the cross-section for DM-electron scattering [3], exposes the striking similarity of both processes, with a key difference. The DM form factor is now evaluated at the scale q not q e as the electron directly picks up the entire momentum transfer. Any direct comparison between the two processes is model-dependent as it requires specifying bothσ e and σ n . Consider for concreteness the scattering mediated by a dark photon, coupling to electric charge e with a strength εe. Then, whereσ p is the DM-proton scattering cross section (f p = 1, f n = 0), α D = g 2 V /4π, where g V is the DM-dark photon coupling, and for consistency we defineσ n = (Z 2 /A 2 )σ p . In the expression for the Migdal effect, |F N (q)| 2 needs to account for the screening of the nuclear electric potential, see e.g. [46]. It is, however, of little relevance for the typically considered momentum transfers, and we will neglect it to exhibit better the parametric relations. For a dark photon mediator, the ratio of differential cross sections in Eqs. (9) and (10)  At this point, one may be tempted to exploit the dipole scaling relationship in Eq. (6) to conclude that the RHS scales as Z 2 m 2 e /m 2 N ∼ 10 −6 (Z/A) 2 implying that the rate for Migdal ionization is always much smaller than for DM-electron scattering. However, whereas the dipole approximation is always valid for the Migdal effect, since q e r e 1, kinematic arguments imply that for DMelectron scattering ∆E n,l /v max q 2µ N v max . In this region, q 1/r e and the dipole scaling breaks down. Inspection of Fig. 1 shows that at large enough momentum transfer the form factors become strongly suppressed, so that at some critical value q crit > 1/r e , |f ion nl (p e , q crit )| 2 = |f ion nl (p e , q crit e )| 2 × Z 2 (with q crit e = me m N q crit ) is certainly met. Hence, for the differential rates in Eq. (12), DMelectron scattering only dominates over Migdal scattering for q < q crit , while the Migdal effect dominates over DMelectron scattering for higher mass DM. Clearly, both effects need to be taken into account to derive accurate DM constraints. For contact interactions, the Migdal effect dominates for m χ (q crit ) 2 /(2∆E n,l ). For long-range interactions, DM-electron interactions dominate essentially for all masses as there is a bias towards lower momenta introduced by |F DM (q)| 2 .
Application to semiconductors. Until now, all studies of the Migdal effect in semiconductor targets considered isolated atoms. Using our above results and working along the lines of [22] that previously exposed the connection for DM-electron scattering between atoms and semiconductors, we are in a position to arrive at a rate equation for Migdal scattering in semiconductors. The replacement to be made in Eq. (9) to obtain the analogous cross section d σ crystal v /d ln E e is |f ion n,l (q e , E e )| 2 → 8αm 2 e E e q 3 e × |f crystal (q e , E e )| 2 , (13) where |f crystal (q e , E e )| 2 is a dimensionless crystal form factor obtained in [22,59].
The ionization rate is then given by In deriving Eq. (13) we made use of the dipole scaling relation (6). This scaling is only approximate, due to the crystal's electronic structure; there are discrete values of q e available for a given E e and a more refined computation of the crystal form-factor at low momentum is warranted and left to [49]. Here we show a proof of concept of the mapping between the ionization and crystal formfactors, and calculate the first projections on σ n from the Migdal effect in silicon. Since the dipole approximation is valid when qr e 1, where r e ∼ 1/(αm e ) in silicon, we anchor our form-factors at q 0 = 0.5αm e and take the average value of f ion in a neighborhood around q 0 . The  [52], CRESST III [53], organic liquid scintillator [54], DarkSide-50 [55], CDEX [6], XENON1T [56,57], and a recast of XENON1T data for cosmic-ray up-scatter [44]. In the bottom panel, the gray-shaded region corresponds to constraints from LUX [4] and Panda-X [58].
QEdark form-factors are calculated up to E e = 50 eV, which is sufficient for DM-electron scattering. However, from Fig. 2 (right), we see that Migdal-scattering extends out to higher values of E e and the truncation at E e = 50 eV underestimates the rate; in principle, one can combine the form-factors calculated with relativistic FAC wavefunctions, which are sufficiently accurate for E e 50 eV, to obtain a more complete spectrum [49]. Here, we show the constraints using the QEdark form factors only.
Constraints and Projections. We now apply the above results to derive new constraints and projections from direct detection experiments.
For xenon detectors, the ionization (S2) signal is obtained by folding dR/dE e with the probability to produce S2 photo-electrons (PE) given a deposited energy ∆E nl and a detection efficiency (S2), dR nl /dS2 = (S2) dE e P (S2|∆E nl )dR nl /dE e . Here, P (S2|∆E nl ) = n surv e ,ne P (S2|n surv e )P (n surv e |n e )P (n e | n e ). The number of electrons escaping the interaction point n e are assumed to follow a binomial distribution, P (n e | n e ) = binom(n e |N Q , f e ) with N Q = ∆E nl /(13.8 eV) [60,61] trials and a single event probability f e = n e /N Q ; the mean number of electrons n e is either modeled following [8] or, for ∆E nl > 0.19 keV, obtained from the measured charge yield in [62]. For P (n surv e |n e ), we assume that 80% (100%) of electrons survive the drift in XENON1T/XENON100 (XENON10). Those electrons induce scintillation at the liquid-gas interface, described by a Gaussian, P (S2|n surv e ) = gauss(S2|g 2 n e surv , σ S2 ), with a representative width σ S2 = 7 √ n surv e [63] and a gain factor g 2 = 33, 20, and 27 PE/e − for XENON1T, XENON100, and XENON10, respectively. For silicon, we use dR/dn e = dE e P (E e |n e )dR/dE e where P (E e |n e ) = δ(1+Floor[E e /3.8 eV] − n e ). We reinterpret the data and efficiencies for XENON10 [64], XENON100 [65], XENON1T [57], and SENSEI [15] to obtain the exclusion boundaries in Fig. 3. We leave to future work calculating the limits from DarkSide-50 [14], CDMS-HVeV [13], and DAMIC at SNOLAB [16]. We also show the 90% C.L. projections for a future 100 g-year silicon detector (like SENSEI) with a single-electron threshold, a dark-count of 10 6 events for n e = 1 and no background events for n e ≥ 2; and a 100 kg-year xenon detector (like LBECA) with a two-electron threshold and assuming only neutrino induced backgrounds [10,66,67]. The new constraint from XENON10 extends sensitivity to DM-nuclear scattering down to m χ ∼ 5 MeV, and future experiments will reach ∼ 500 keV. We also compare the DM-electron scattering limits with the Migdal effect, assuming a dark photon mediator, and converting the limits and projections from σ e to σ n using Eq. (11). For contact interactions, the DM-electron scattering limits are stronger for smaller DM masses, but weaker for heavier DM masses, than the Migdal limits. For light mediators, the DM-electron scattering dominates for all masses.
Conclusions. The prompt electron ionization signal that may accompany DM-nuclear scattering, "Migdal effect", allows to extend the sensitivity of noble liquid and semiconductor DM detectors into the MeV mass region. In this work, we show that the theoretical description of the process is closely related to DM-electron scattering, and set the lowest DM-mass constraints on DM-nuclear scattering using XENON10 and XENON100 data. Finally, we take the first step towards a concise formulation of the Migdal effect in semiconductors and demonstrate the future potential for both, upcoming noble liquid and semiconductor experiments.
Award 623940. J.P. is supported by the 'New Frontiers' program of the Austrian Academy of Sciences. We thank the Erwin Schrödinger International Institute for hospitality while this work was completed. T.-T.Y. also thanks the hospitality of the CERN theory department where part of this work was completed.
Note Added During the completion of this work, Ref. [68] appeared which investigates similar topics for the isolated atom case.