Migdal effect in dark matter direct detection experiments

The elastic scattering of an atomic nucleus plays a central role in dark matter direct detection experiments. In those experiments, it is usually assumed that the atomic electrons around the nucleus of the target material immediately follow the motion of the recoil nucleus. In reality, however, it takes some time for the electrons to catch up, which results in ionization and excitation of the atoms. In previous studies, those effects are taken into account by using the so-called Migdal’s approach, in which the final state ionization/excitation are treated separately from the nuclear recoil. In this paper, we reformulate the Migdal’s approach so that the “atomic recoil” cross section is obtained coherently, where we make transparent the energy-momentum conservation and the probability conservation. We show that the final state ionization/excitation can enhance the detectability of rather light dark matter in the GeV mass range via the nuclear scattering. We also discuss the coherent neutrino-nucleus scattering, where the same effects are expected.


I. INTRODCUTION
The existence of dark matter is overwhelmingly supported by numerous cosmological and astrophysical observations on a wide range of scales.However, the nature of dark matter has not been revealed for almost a century except for its gravitational interactions.
The identification of the nature of dark matter is one of the most important challenges of modern particle physics (see [1][2][3] for review).
Among various candidates for dark matter, the weakly interacting massive particles (WIMPs) are the most extensively studied category of dark matter.The WIMPs couple to the standard model particles via interactions similar in strength to the weak nuclear force.Through the weak interaction, the WIMPs are thermally produced in the early universe, and the relic density is set as they freeze out from the thermal bath [4].The WIMPs are particularly attractive since the dark matter density does not depend on the details of the initial condition of the universe.The WIMPs are also highly motivated as they are interrelated to physics beyond the standard model such as supersymmetry (see e.g.[5]).
The ambient WIMPs can be directly detected by searching for its scattering with the atomic nuclei [6].Given a circular speed at around the Sun of 239 ± 5 km/s [7], the WIMPs recoil the nuclei elastically with a typical momentum transfer of q A ∼ 100 MeV for the target nucleus mass of m N ∼ 100 GeV. 1 The recoil signatures are detected through ionization, scintillation, and the production of heat in the detectors (see [8][9][10] for review).To date, for example, liquid xenon detectors such as LUX [11], PandaX-II [12], and XENON1T [13] have put stringent exclusion limits on the spin-independent WIMP-nucleon recoil cross section.
In those experiments, it is usually assumed that the atomic electrons around the recoil nucleus immediately follow the motion of the nucleus.However, it takes some time for the electrons to catch up, which causes ionization and excitation of the recoil atom.The ionization and the excitation result in extra electronic energy injections into the detectors. 2he importance of such effects on direct detection experiments has been pointed out [14,15].
(See also Refs.[16,17], which discuss the ionization effects in the direct detection experiments for dark matter electron scattering.) In previous studies, such effects are estimated by using the so-called Migdal's ap-proach [18,19] (see also [20]).Following [15], we call these effects the Migdal effects.In the Migdal's approach, a state of the electron cloud just after a nuclear recoil is approximated by in the rest frame of the nucleus.Here m e is the electron mass, xi the position operator of the i-th electron, v the nucleus velocity after the recoil, and |Φ ec ⟩ the state of the electron cloud before the nuclear recoil.The probability of ionization/excitation is then given by where |Φ * ec ⟩ denotes either the ionized or excited energy eigenstate of the electron cloud.In the above analysis, the final state ionization/excitation are treated separately from the nuclear recoil.Thus, the energy-momentum conservation and the probability conservation are made somewhat obscure.In this paper, we reformulate the Migdal effect so that the "atomic recoil" cross section is obtained coherently.In our reformulation, the energymomentum conservation and the probability conservation are manifest while the final state ionization/excitation are treated automatically.We also provide numerical estimates of the ionization/excitation probabilities for isolated atoms of Ar, Xe, Ge, Na, and I.
The Migdal effect should be distinguished from the ionization and the excitation in scintillation processes.The Migdal effect takes place even for a scattering of an isolated atom, while the latter occurs due to the interaction between atoms in the detectors.It should be also emphasized that the Migdal effect can ionize/excite electrons in inner orbitals, which are not expected in scintillation processes.As we will see, the ionization/excitation from the inner orbitals result in extra electronic energy injections in the keV range, which can enhance detectability of rather light dark matter in the GeV mass range.
The organization of the paper is as follows.In Sec.II, we discuss approximate energy eigenstates of an atomic state by paying particular attention to the total atomic motion.
In Sec.III, we reformulate the atomic recoil cross section with the Migdal effect by taking the energy eigenstates in Sec.II as asymptotic states.In Sec.IV, we calculate the Migdal effect with single electron wave functions.In Sec.V, we estimate the probabilities of the ionization/excitation at a nuclear recoil.In Sec.VI, we discuss implications for dark matter direct detection.In Sec.VII, we briefly discuss the Migdal effect in a coherent neutrinonucleus scattering.The final section is devoted to our conclusions and discussion.

II. ENERGY EIGENSTATES OF ATOMIC SYSTEM
As we will see in the next section, the plane wave function of a whole atomic system plays a central role to obtain the nuclear scattering cross section with the Migdal effect.In the following, we consider an isolated neutral atom consisting of a nucleus and N e electrons.
The electrons are not necessarily bounded by the Coulomb potential of the nucleus, and hence, the energy eigenstates can be ionic states with unbouded electrons.
As typical nuclear recoil energy is smaller than O(100) keV, the Hamiltonian of the system is well approximated by the non-relativistic one, ĤA ≃ p2 Here, pN and xN denote the momentum and the position operators of the nucleus with mass m N , respectively.The momentum and the position operators of the i-th electron are given by pi and xi , respectively.The Hamiltonian of the electron cloud, Ĥec , depends on the position operator of the nucleus, xN , through the interaction potential V (x i − xN ) The interaction potential also includes the interactions between the electrons.
In the coordinate representation, the energy eigen-equation is reduced to where the positions (including spinor indices) of the N e electrons are represented by {x} collectively.

A. Energy Eigenstates of an Atom at Rest
To solve Eq.( 4), let us first consider the eigenstates of Ĥec (x N ) for a given x N , Since the system is invariant under spatial translations, the energy eigenvalues do not depend on x N while the wave functions depend on x N only through {x i − x N }; The eigenstates, Φ Eec ({x − x N }), provide a complete orthogonal basis of the electron cloud for a given x N .
Next, let us show that Φ Eec well approximates an energy eigenfunction of the whole atomic system at rest, i.e., Ψ (rest) By substituting Ψ to Eq. ( 4), the energy eigen-equation results in Now, Ψ Thus, the left-hand side of Eq. ( 9) is expected to be highly suppressed, i.e., p2 for Ψ (rest) Here, we used the fact that the expectation value of the electron kinetic energy is roughly given by Therefore, we find that Ψ (rest) E A provides an approximate energy eigenstate of the whole atomic system with It should be noted that this is nothing but the Born-Oppenheimer approximation.

B. Energy Eigenstates of a Moving Atom
Once we have the energy eigenstates of an atomic system at rest, the energy eigenstates of a moving atom with a velocity v can be immediately obtained by the Galilei transformation, Here the unitary operator of the Galilei transformation is given by Under the Galilei transformation, the momentum operators are shifted by and the Hamiltonian is transformed into Here we define the nominal mass of the atom by By using Eqs.( 13), (10), and (19), we find that the boosted wave function Therefore, the boosted wave function Ψ E A provides the approximate energy eigenstate of a moving atom with energy In summary, the eigenstate of the atomic system is approximated by with It should be remembered that Ψ E A (x N , {x}) is not an eigenstate of the nucleus momentum pN .Instead, Ψ E A (x N , {x}) is an eigenstate of the momentum of the whole atom; pN Thus, p N in Eq. ( 23) parametrizes not the nucleus momentum but the eigenvalue of the total It should be also noted that the energy eigenstate in Eq. ( 23) is no more in the realm of the Born-Oppenheimer approximation for v ̸ = 0 since they are not eigenfunctions of Ĥec for a given x N .

III. MIGDAL EFFECT : FROM NUCLEAR RECOIL TO ATOMIC RECOIL
In this section, we derive the recoil cross section of the atomic system with the final state ionization/excitation.

A. Isolated Nuclear Recoil
Before proceeding further, let us first translate the dark matter-nucleus interaction in field theory to an interaction potential, which will be useful in the later analysis.For now, let us forget the electron cloud and take the nucleus as a free separated particle.In a relativistic field theory, the T -matrix and the invariant amplitude of a scattering process are given by Here, the plane waves of the dark matter and the nucleus are normalized by with p 0 being the relativistic energy of the particle.
As an example, let us consider a contact spin-independent interaction between a Dirac dark matter and nucleons; where M * denotes a mass parameter and g p,n are dimensionless coupling constants.In this case, the squared invariant amplitude for the nucleus scattering is given by where Z is the atomic number, A the mass number, and m DM the mass of the dark matter.
The corresponding cross section is given by σN ≃ 1 16π where µ N is the reduced mass, In the coordinate representation of quantum mechanics, the above invariant matrix element in Eq. ( 28) is reproduced by an interaction potential, with the initial and the final states Here, we normalize the initial and the final wavefunctions in conforming with the one in Eq. ( 29) with the relativistic energies approximated by their masses.
As another example, we may also consider a dark matter interaction with nucleons through an exchange of a light scalar particle, ϕ, with mass m ϕ , where y p,n,DM are Yukawa coupling constants.The invariant amplitude of the isolated nuclear scattering for each spin is given by in the non-relativistic limit.In the coordinate representation of quantum mechanics, the invariant amplitude is reproduced by adding a potential term with the initial and the final state wave functions in Eqs.(38) and (39).
In both cases, the differential cross section with respect to the nuclear recoil energy in the laboratory frame is given by 4 Here, we introduce the nuclear form factor, which is relevant for a momentum transfer q N in the tens to hundreds MeV.In the last equality, we defined σN (q N ) = 1 16π which reduces to |F A (q 2 N )| 2 × σN for the contact interaction.

B. Invariant Amplitudes with Electron Cloud
Now, let us calculate the cross section of the nuclear recoil in the presence of electron cloud.For this purpose, we consider in the coordinate representation, where ĤA is given in Eq. (3).In subsection III A, we considered the asymptotic states consist of the plane waves of dark matter and an isolated nucleus.To take into account the electron cloud, we replace the plane waves of the nucleus with the plane waves of the atomic system discussed in section II.
The initial and the final states of dark matter scattering are taken to be 4 The elastic nuclear recoil energy is related to the scattering angle in the center of the mass frame via Hereafter, we consider the initial atom at rest in the laboratory frame, v I = 0.The total energies of the initial and the final states are given by where E I,F ec are the energy eigenvalues of the initial and the final electron clouds in the rest frame, respectively.By using the energy eigenfunctions in Eq. ( 23), the T -matrix of this process is given by In the second equality, we shifted the integration variables x i and x DM by x N .
As a result, we obtain the matrix element where for v I = 0. 5 The term proportional to M denotes the nuclear recoil while the factor Z F I (q e ) denotes the transition of the electron cloud.It should be emphasized that our approach treats the nucleus and the electron cloud coherently.This treatment enables us to derive the invariant amplitude with manifest energy-momentum conservation.

C. Phase Space Integration
By noting the normalizations in Eqs. ( 23), ( 29), (see also (A1)), the differential cross section is given by 6dσ ≃ Here, we defined the physical mass of the atomic system, m A by By boosting four momentum (m A , 0, 0, 0), we obtain the four-momentum of the atomic system in an arbitrary frame.For example, the final state four-momentum is given by in the laboratory frame.
When the magnetic quantum numbers of the electrons in the initial/final states are averaged/summed, the factor |Z F I (q e )| 2 depends only on the size of q e .In this case, the differential cross section is given by Here, p I,F denotes the initial and the final state momenta in the center of the mass frame.
By using the dark matter velocity in the laboratory frame, v I DM , the initial momentum in the center of the mass frame, p I , is given by It should be noted that the scattering process is no longer elastic for E F ec ̸ = E I ec .Accordingly, the final state momentum in the center of the mass frame becomes smaller than |p I |; To satisfy |p F | > 0, there is a threshold velocity, with which |p F | is rewritten by

D. Atomic Recoil Spectrum
The atomic recoil spectrum in the laboratory frame is obtained as follows. 7The atomic recoil energy E R in the laboratory frame is given by As the momentum transfer is given by we obtain Thus, the differential cross section with respect to the atomic recoil energy is given by where Finally, the dark matter event rate for unit detector mass is given by Here, ρ DM denotes the local dark matter density 8 and f (v DM ) is the dark matter velocity distribution integrated over the directional component normalized by 9   fDM (v DM ) dv DM = 1 . (76)

IV. MIGDAL EFFECT IN SINGLE ELECTRON APPROXIMATION A. Single Electron Wave Function
In our numerical calculation, we use the electron wave function, Φ Eec , obtained by the Dirac-Hartree-Fock method, where the relativistic effects on the electron cloud are taken into account (see e.g.[25] for review).
In the Dirac-Hartree-Fock approximation, an electron state is given by a Slater determinant made up of one orbital per each electron in an averaged central potential around a nucleus.In this approximation, the energy eigenstates in Eq. ( 23) are approximated by where S Ne denotes the permutation group of degree N e .Here we explicitly show the indices of the Dirac spinor by α i = 1 • • • 4, which are encapsulated in {x} on the left-hand side.
The electron cloud consists of the N e orbitals, 10 where each orbital is specified by energy E, relativistic angular momentum κ, and magnetic quantum number m, 11 For a bounded electron, i.e.E i < 0, the state is classified by the principle quantum number, n i , while the spectrum is continuous for an unbounded electron, i.e. for E i > 0.
8 For the Burkert profile [7], for example, it is estimated to be ρ DM ≃ 0.487 +0.075 −0.088 GeV/cm 3 . 9For astrophysical uncertainties of the direct detection experiments (see e.g.[23,24]). 10The Slater determinant in Eq. ( 77) is reducible in terms of the total angular momentum of the atom. 11The value κ determines both the total angular momentum j and the orbital angular momentum ℓ via The one electron Dirac orbital ϕ α o (x i ) is given by using the two-component spherical spinors Ω κm ; See e.g.[25] for the details of the spherical spinors and the radial wave functions, P E (r) and Here, an atom is at rest and r denotes the distance between the electron and the center of the potential.The one-electron states are normalized such that In the Dirac-Hartree-Fock approximation, the electron cloud transition factor in Eq. ( 57) is rewritten by In this approximation, the transition amplitude is given by the product of the transition amplitudes between the electron orbitals.

B. Single Electron Excitation/Ionization
For an atomic recoil with a momentum transfer smaller than the hundreds MeV range, the factor |q e •x i | is expected to be small than O(1) on the atomic scale. 12Thus, we consider the Migdal effect at the leading order of q e .At the leading order of q e , only one electron can be excited/ionized, and hence, the initial and the final state configurations are where Hereafter, we assume that the initial electron cloud stays in the ground state, where all the electrons are bounded by the Coulomb potential of the nucleus.In the final electron state, o ′ k can be either a bounded or an unbounded orbital.At the leading order of q e , the electron cloud transition amplitude is reduced to At this order, electron transitions are allowed only when the orbital angular momenta of o ′ k and o k differ by one, i.e. |ℓ ′ k − ℓ k | = 1.Thus, the transition amplitude is reduced to Here, we take the quantization axis of the angular momentum corresponds to q e .The angle, θ, is the one between q e and x e .The choice of the quantization is irrelevant for the final results as we take an average/sum the magnetic quantum numbers of the electrons in the initial/final states.
In the following discussion, we only require an accuracy of O(10)% for the electron binding energies.For this accuracy, the bound state energies for given principal number and the orbital angular momentum are not distinguishable, and hence, it is useful to label the bound states by (n, ℓ).Accordingly, the transition rates are also labeled by (n, ℓ), Here, is the probability for the electrons unaffected by the nuclear recoil (see also appendix B for discussion of the probability conservation).The excitation and the ionization probabilities, p d qe and p c qe , are defined by Here, E nκ is the size of the binding energy for the bounded electron labeled by (n, κ), E e the energy of the unbounded electron, ω nℓ the occupation number of the subshell (see Tab. I), Here, It should be noted that the atomic recoil energy, E R , and the electron transition energy, ∆E, are correlated through the energy-momentum conservation; where The minimum dark matter velocity as a function of the atomic recoil energy E R for given ∆E and m DM for Ar and Xe. 2. Kinematical constraints on the plane of (E R , ∆E) for given m DM and v DM for Ar and Xe atoms.The regions below the lines are kinematically allowed.
From this expression, we find the minimum dark matter velocity for given E R and ∆E, In Fig. 1, we show the minimum velocity as a function of E R for isolated Ar and Xe atoms.
We also show the kinematically allowed region of E R and ∆E for Ar and Xe atoms for v DM = 10 −3 in Fig. 2.
It should be also noted that there is a kinematical upper limit on the electron transition energy, ∆E, for a given speed of dark matter, which is set by Eq. ( 65), In Fig. 3, we show ∆E MAX as a function of v DM .The figure shows that ∆E in the keV range is kinematically allowed for v DM > ∼ 10 −3 .It is also notable that, for ∆E = ∆E MAX , the atomic recoil energy is given by Thus the corresponding atomic recoil energy is smaller than ∆E MAX , which plays an important role on the dark matter detections as discussed in the later section.
When electrons are emitted from inner orbitals, the created core-holes are de-excited subsequently. 13The typical timescales of the de-excitation processes are of O(10) fs.Thus, the energies of the electron emission and the de-excitation are measured simultaneously, and hence, the total electronic energy released at the ionization is given by where E dex is the energy released at the de-excitation.
Accordingly, the electromagnetic energy spectrum is given by Hereafter, we simply assume that the ionization energy is released completely, that is E EM = ∆E. 14One caveat here is that E EM is not the energy of a single electron/photon but the collection of the energies of the electrons and photons emitted at the de-excitation and the ionization.Thus, the detector responses to E EM might be different from those to a single electron/photon with the same energy, although we do not take such effects into account in the following discussion.
Similarly, the excited atoms also lead to electronic energy release by de-excitation.Assuming the complete de-excitation again, we obtain the electromagnetic energy spectrum; Here ∆E nℓ→n ′ ℓ ′ is given by 13 The de-excitation proceeds through the X-ray transition, the Auger transition, or the Coster-Kronig transition (see [26,27] for review, see also [28]).For a core-hole in n > 1 states the Coster-Kroning transition dominates the de-excitation process. 14If the atom is completely isolated, the ionization and the subsequent Auger and Coster-Kronig transitions leave ionized atoms.In the medium, on the contrary, ionized atoms are also de-excited eventually.
FIG. 3. The kinematical upper limits on the electron transition energy, ∆E, as a function of the speed of dark matter for a given m DM .In the solar rest frame, typical dark matter velocities are of 500 km/s with an upper limit of around 700 km/s.

V. NUMERICAL ANALYSIS
In this section, we provide numerical estimates of the electron transition probabilities, p d q and p c q , for isolated Ar, Ge, Xe, Na, and I atoms.To calculate the electron wave functions, we use the Flexible Atomic Code (FAC, cFAC) [29].It is a multi-configuration Dirac-Fock (MCDF) program to calculate various atomic radiative and collisional processes.We give a brief review of the Dirac-Hartree-Fock method in appendix C. In FAC, all of the single electron wave functions, including those of excited and unboudend electrons, are calculated from a universal central potential, which is optimized for the (possible) ground state configurations.Here, V N (r) denotes the Coulomb potential from the nucleus, and α the fine-structure constant.The factor Q eff nκ provides the effective charge of the central potential for the electrons in the nκ-orbital; Here, ω nκ is the occupation number of the subshell, ρ nκ = ρ nκ,nκ , Y k nκ = Y k nκ,nκ , and where j κ is the value of j corresponding to κ, r > = max(r, r ′ ), r < = min(r, r ′ ), and is the Wigner 3j symbol.
Due to the universal central potential, the wave functions of orbitals obtained by FAC are orthogonal to each other, and hence, they can be used to calculate the transition rates in the previous sections.In our analysis, we further approximate the atomic state by a single Slater determinant of the wave functions labeled by a set of {n, κ, m}.The energy eigenvalues obtained with this approximation reproduce the measured values in Ref. [30] at the accuracy better than 20%, which is good enough for later discussion (see the energy levels in Tab.II.).

A. Transition probabilities
We show the numerical results of the transition probabilities to the order of O(q 2 e ).The ground state of the atom consists of a complex of the orbitals given in Tab I.If there are more than one energy eigenstate configurations for a given complex, we take an average of the transition probabilities for those energy eigenstate configurations.As discussed in section IV B, we consider the final states in which only one of o k 's in the initial state is replaced by an excited or an unbouded electron state.
In Fig. 4, we show the differential ionization probabilities, dp c qe /dE e , for q e = m e v F with v F = 10 −3 .In the figure, we sum all the contributions from different orbital angular momenta in the initial state, ℓ, for a given principal quantum number, n.We also sum all the possible final states for a given n.It should be noted that the spectrum shape of dp c qe /dE e does not depend on v F , and hence, the probabilities for a different q e is obtained by multiplying FIG. 4. The differential ionization probabilities as a function of the emitted electron energy, E e , for isolated Ar, Xe, Ge, Na, and I.The contributions from different ℓ's are summed.We also summed all the possible final states for a given n.The integrated probabilities are given in Tab.II.The ionization probabilities are available in ancillary files to the arXiv version (See note added before Appendix A.).The files also contain the data of C, F, and Ne.
q 2 e /(m e × 10 −3 ) 2 .The integrated probabilities are also given in Tab.II for a given initial (n, ℓ).The results show that the ionization probabilities from the inner shells can be of O(10 −2 ) for v F = 10 −3 .The ionization probabilities from the valence electrons can be even of O(10 −1 ).
As we have seen in Fig. 2, the recoil energy E R and the electron excitation energy ∆E (and hence E e ) are kinematically constrained for given m DM and v DM .Cross correlations of the ionization probabilities between E R and E e are obtained by rescaling the results in Fig. 4 by q 2 e = 2m 2 e E R /m A within the kinematically allowed region in Fig. 2. In Tab.II, we also show the excitation probabilities, p d qe (nℓ → n ′ ℓ ′ ).As the table shows, the excitation probabilities are much smaller than the ionization probabilities for a given initial n.
Before closing this section, let us comment on the probability conservation in the single electron transition.As discussed in the appendix B, the single electron transition probability satisfies, Here, p unchanged and p occupied are given in Eqs.(B6) and (B8), while the lefthand side corresponds to p ex (nℓ) in Eq. (B7) (see also Eqs. ( 89) and ( 90)).In our analysis, we numerically checked that p unchanged , p occupied , and p ex satisfy the probability conservation in Eq. ( 111). 15n Ref. [15], the probability p occupied (P 1 exc,i in Ref. [15]) is incorrectly defined, which is related to p occupied in this paper by Since p unchanged ≃ 1, p occupied | [15] is much smaller than p occupied .The ionization probability in Ref. [15] is, on the other hand, estimated by using the (incorrect) probability conservation,16 Since we find p occupied > ∼ p ex numerically, it shows that the ionization rates in Ref. [15] are overestimated.

VI. EFFECTS ON DARK MATTER DIRECT DETECTION
As we have shown in the previous sections, a nuclear recoil is accompanied by the ionization and the excitation of the atom through the Migdal effect.The electric energy released by the ionization and the de-excitation can be in the keV range when the incident dark matter velocity exceeds the threshold in Eq. (65).
In this section, we discuss how those electronic energy injections affect the dark matter signals at direct detection experiments.In our analysis, we have assumed isolated atoms.
Thus, the results in the previous section are not directly applicable to the non-isolated atoms in liquid or crystals.For example, the energy levels of the valence electrons are affected by the ambient atoms by O(0.1) eV. 17 Furthermore, when the electronic band structure is formed in the medium, the excitations into the unoccupied state should be reinterpreted as transitions into the conducting band.
The ionization rates from the inner orbitals are, on the other hand, expected to be less affected by the ambient atoms.In fact, the binding energies of the inner orbital are much larger than eV, and hence, the relevant length scales for the transition factors, z e , are much smaller than the typical distance between atoms.Thus, the ionization spectrum in Eq. ( 91) can be applied rather reliably for the ones from the inner orbitals. 18In the following, we confine our arguments to liquid Xe detectors.

A. Migdal Effects on the Recoil Spectrum
In the absence of the Migdal effect, the liquid Xe detectors respond to the nuclear recoil roughly through the following steps [32][33][34].After the nuclear recoil, the electron clouds are assumed to catch up with the nucleus immediately, so that the atom remains neutral.The recoil "atom" loses its energy through scattering with adjacent atoms in the medium where the inelastic scatterings involve the ionization and excitation of the atoms.These processes continue until the scattered atoms are thermalized.Eventually, a fraction of the initial recoil energy E R is converted to the measurable electronic excitation while the rest is lost into the heat of the medium [35][36][37][38].
This should be contrasted with the electron recoils caused by incident gamma or beta rays, where the entire recoil energy is transferred to the measurable electronic excitation.
Conventionally, the calibrated nuclear recoil energy, E nr , and the electron equivalent energy, E ee , measured by the scintillation photons are related by 19 Here, L eff is defined as the ratio between E ee and E nr at zero drift field relative to 122 keV gamma rays, which is L eff ≃ 0.1 − 0.2 for E nr < ∼ 100 keV [40][41][42][43].The quantities, S e and S n , are the scintillation quenching factors of electron and nuclear recoils due to the drift electric field, E d , which are S e ≃ 0.4-1 [44] and S n ≃ 1 [45] for E d ≤ 4 kV/cm, respectively. 20  In the presence of the transition due to the Migdal effect, the nuclear recoil is accompanied by electronic energy injections in the sub-keV to the keV range.By ignoring the energy resolution of the detectors, the electron equivalent energy spectrum is given by with q nr being the conversion between E nr and E ee in Eq. (114) (see also Eqs. ( 100) and ( 101)).In the following, we include only the electronic energy injection caused by the ionizations as the excitation probabilities into the unoccupied binding energy levels are much smaller than the ionization probabilities. 21  In Fig. 5, we show the total electron equivalent energy spectrum for a given spinindependent scattering cross section of dark matter on nucleons through the contact interaction. 22Here, we assume a detector with E d = 0 kV/cm as in the single-phase experiment such as the XMASS experiment [46].We also fix q nr = L eff = 0.15.We adopt the Helm form factor [5,8].The local dark matter density is fixed to a conventional value, ρ DM = 0.3 GeV/cm 3 .The local circular velocity is also fixed to be v circ = 220 km/s with the peculiar motions of the Earth neglected. 23We also assume a Maxwell velocity distribution with the velocity dispersion, v 0 = 220 km/s, which is truncated at the Galactic escape velocity v esc = 544 km/s.
19 See e.g.Ref. [39] for the details of energy scale calibration of the liquid Xe detectors. 20For E d = 0 kV/cm, S e = S n = 1. 21Moreover, the excitation into the unoccupied binding energy levels are not well defined when the electronic band structure is formed in the medium. 22The nucleon-dark matter cross section σn is related to σN via, σN = A 2 × µ 2 N /µ 2 n × σn for g p = p n . 23The annual modulation caused by the Earth's peculiar motion can be significantly enhanced as of the inelastic nuclear scattering [47].
FIG. 5.The differential event rates expected at the single-phase experiments with the liquid Xe target.The black lines show the nuclear recoil (NR) spectrum without ionization.(The NR spectrum with respect to E det and the one with respect to E R differ by a factor of 1/q nr .)The green, blue, and pink lines show the ionization rates from n = 3, 4, and 5, respectively.Here, we do not take the energy resolution into account.Since we apply the estimations for the isolated atoms, the ionization spectrum from the valence electrons, i.e. n = 5, are not reliable.
The figures show that the electronic energy from the ionizations can be larger than the maximum value of the (electron equivalent) nuclear recoil energy for a rather light dark matter.As discussed in the previous section, the shape of the energy spectrum of the electronic injections is not sensitive to the incident dark matter velocity as long as they are kinematically allowed.The nuclear recoil energy, on the other hand, depends on the dark matter velocity, which is suppressed for light dark matter.These features can be seen from the figures; the electronic energy from the ionizations are less sensitive to the dark matter mass, while the maximum nuclear recoil energy is sensitive.
Given a typical threshold of the electron equivalent energy of the liquid Xe detectors of about a keV ee , the ionization from n = 3 provides a new detection channel for rather light dark matter.For example, with exposures of about 10 5 kg•days, a few hundred events are expected for m DM ≃ 500 MeV and σn ≃ 10 −36 cm 2 in the liquid Xe detectors. 24Similarly, a few signal events are expected for m DM ≃ 5 GeV for σn ≃ 10 −42 cm 2 for the same exposure.
It should be noted that those signals are eliminated as background events in the conventional analysis of the dual-phase experiments.In those experiments, one needs different analyses to cover such signals (see e.g.[48,49]).
For heavier dark matter, m DM > O(10) GeV, on the other hand, the Migdal effect is submerged below the conventional nuclear recoil spectrum, and hence, does not affect the detector sensitivities.In principle, the additional electronic energy injections affect the socalled S2/S1 ratio in the dual-phase detectors.This is because the numbers of direct excitons N ex and direct ionizations N i are different for the atomic recoil and the electronic recoil. 25owever, such effects should have been taken into account by the detector calibration by the neutron sources for a given momentum transfer.
Finally, let us compare the Migdal effect with the final state photon emission of the nuclear scattering [51] (see also [52]).Similarly to the Migdal effect, the final state emission is also a universal effect and irreducible.The expected rates are, however, subdominant compared with the Migdal effect for E R in the keV range [53].The Migdal effect should also be distinguished from the photon emissions in the inelastic nuclear scatterings which require a larger momentum transfer [21,22].

VII. MIGDAL EFFECTS IN COHERENT NEUTRINO-NUCLEUS SCATTERING
As another application, let us briefly discuss the Migdal effects in the coherent neutrinonucleus scattering (CνNS).In a similar manner to the dark matter scattering cross section, the differential cross section of the coherent neutrino-nucleus scattering with the Migdal effect is given by Here, σ CνN S denotes the coherent neutrino-nucleus scattering [54][55][56], with G F , sin θ W , N , Q W being the Fermi constant, the weak mixing angle, the number of neutrons N = A − Z, and the weak charge of the nucleus respectively.As in the previous section, the magnetic quantum numbers of the electrons in the initial/final states are averaged/summed.For given E ν and ∆E, the recoil energy is constrained in where the maximal recoil energy corresponds to the back-to-back scattering.
In Fig. 6, we show the total electron equivalent energy spectra for the CνNS for the pp and the 8 B solar neutrinos.Here, we take the central values of the neutrino fluxes of the SFII-GS98 model given in Ref. [57].The single-phase liquid Xe detectors are assumed as in the previous section with q nr = 0.15.The figures show that the nuclear recoil signal without the Migdal effect is below the energy thresholds of the current liquid Xe detectors, i.e.O(1) keV.The signal of the ionization from n I = 3 is, on the other hand, above the energy threshold, a few events of which are expected with exposures of about 10 ton•years. 26

VIII. CONCLUSIONS AND DISCUSSION
In this paper, we reformulated the Migdal effect at the nuclear recoil caused by a dark matter scattering and a coherent neutrino-nuclear scattering.In our formalism, we take the 26 More detailed study including background estimation will be given elsewhere.FIG. 6.The differential event rates for expected for the coherent neutrino-nucleus scattering for the pp and the 8 B solar neutrinos.The single-phase liquid Xe detectors are assumed as in the previous section with q nr = 0.15.The black lines show the nuclear recoil (NR) spectrum without the ionizations.(The NR spectrum with respect to E det and the one with respect to E R differ by a factor of 1/q nr .)Since we apply the estimations for the isolated atoms, the ionization spectrum from the valence electrons are not reliable.
plane waves of the whole atomic system as the asymptotic states for the scattering process.
The coherent treatment of the electron cloud makes the energy-momentum conservation and the probability conservation transparent.We also provide numerical estimates of the ionization and the excitation rates for isolated atoms of Ar, Xe, Ge, Na, and I by using the Flexible Atomic Code (FAC, cFAC) [29].
We also applied the results for the dark matter direct detections by taking a liquid Xe detector as an example.We showed that the ionization signals through the Migdal effect provide new detection channels for light dark matter with a mass in the GeV range.Since such signals are eliminated as background events in the conventional analysis of the dualphase experiments, different analyses are required to cover such signals.For rather heavy dark matter, on the other hand, the Migdal effects are submerged below the conventional atomic recoil spectrum.
In our analysis, we have not studied detailed detector responses nor the precise treatment of the Migdal effects of the non-isolated atoms in liquid or crystals.For more precise estimation, detailed detector simulations are imperative.In particular, it is important to study detector responses to the energy released by the de-excitation of the core-hole.More theoretical efforts are also required for precise estimation of the Migdal effect in the medium.
We also note that it is an interesting future work to discuss whether the Migdal effect affects the direct detection experiments that mainly aim the electron recoil [58][59][60][61][62][63][64][65][66].As the Migdal effect provides the electronic signals via the nuclear recoil, dark matter without electron recoil can be searched for by those experiments.
Eq. (A13) is reduced to where we have shifted the integration variable x i by x N .By remembering p N + N e q e = p A , (A16) we find where we have used q e ′ = q e for p ′ A = p A .Finally, by using the orthogonality of the electron orbitals, we confirm that (A18) at the leading order of q e .Here we use the notation in the appendix A. Accordingly, the excitation probability is given by Let us define probabilities of the single electron transitions, To the order of O(q 2 e ), the total probabilities can be expressed by using the single electron transition probabilities, P unchanged + P ex ≃ i (p unchanged (k) + p ex (k) + p occupied (k)) .
(B10) Thus, we find that Eq. (B9) guarantees the probability conservation in terms of the single electron transition probabilities.It should be emphasized that p occupied plays an important role for the conservation of the probability.As is clear from the above argument, however, p occupied is a part of P unchanged , and hence, it does not contribute to P ex up to O(q 3 e ).
where h is the same as h j but operates on a single electron state.Here, the expectation values in the coordinate representation are give by Since the non-local potential is numerically demanding, the Slater approximation is often adopted to localize the potential [67,68].However, since it has incorrect asymptotic behavior, FAC uses an improved potential given in Eq. (105).From Eq. (C11), we finally which is solved iteratively for the ground state.As for the excited states and the unbouded states, the single electron wave functions are obtained by solving Eqs.(C13) and (C14) by using the potential in Eq. ( 105), which is iteratively obtained for the ground state.

TABLE I .
The number of electrons in a shell for the ground state configurations.
C. Ionization Spectrum at the Leading OrderBy combining Eqs.(75) and (88), we find that the ionized electron spectrum from an initial orbital o k associated is given by

TABLE II .
The excitation probabilities for a given initial state (n, ℓ).P →n ′ ℓ ′ is defined byP →n ′ ℓ ′ ≡ p d qe (nℓ → n ′ ℓ ′ ).The probabilities not shown in this table are forbidden or negligibly small.The integrated ionization probabilities are also shown in the rightmost column.The averaged binding energies of the initial and the final orbitals are also shown which are obtained by FAC.