Detection of Inelastic Dark Matter via Electron Recoils in SENSEI

The low-threshold experiment SENSEI, which uses the ultralow-noise silicon Skipper-CCD to explore light dark matter from the halo, has achieved the most rigorous limitations on light DM-electron scattering cross section. In this work, we investigate the inelastic DM-electron scattering process with the SENSEI data and derive the constraints on the inelastic dark matter model with a $U(1)$ gauge boson as the mediator. Comparing with elastic scattering process, we find that the down-scattering process with the mass splitting $\delta \equiv m_{\chi_2}-m_{\chi_1}<0$ is more strongly constrained while the up-scattering process $\delta>0$ gets the weaker limits. For the down-scattering process with mass splitting $\delta \sim -5$ eV, the DM mass $m_{\chi}$ can be excluded down to as low as 0.1 MeV.


I. INTRODUCTION
Although many cosmological and astrophysical observations provide strong evidence for the existence of dark matter(DM) [1], the nature of dark matter is still elusive. Weakly Interacting Massive Particle (WIMP) with naturally correct thermal relic density is considered as a promising dark matter candidate. And yet, numerous dedicated experiments, which are aimed at directly searching for WIMP dark matter through nuclear recoil signals, have no unambiguous discoveries for WIMP [2]. This motivates a significant reconsideration of dark matter candidates such as Light Dark Matter(LDM) with mass from keV to GeV. In recent decades, the searches for light dark matter have received a great amount of attention.
In this paper, we will investigate inelastic dark matter scattering off electrons bound to semiconductors by exploiting the latest released data from the Sub-Electron-Noise Skipper-CCD Experimental Instrument (SENSEI) [38]. The SENSEI experiment, located at deep underground in the MINOS cavern at Fermi National Accelerator Laboratory(FNAL), is able to probe DM mass down to m χ ∼ 0.5 MeV by using the ultralow-noise silicon Skipper-Charge-Coupled-Devices. Given the lower binding energy of semiconductors E gap ∼ O(eV), the lower DM mass could be explored by the combination of low threshold target materials and inelastic dark matter features. The IDM-electron scattering will excite electrons from a valence band to a conduction band, resulting in the observable electron-hole pairs N e . We derive the 90% confidence level (C.L.) exclusion limits on DM-electron cross sectionσ e by using the observed number of electron-hole pairs N e from the SENSEI experimental data.
The paper is organized as follows. In Sec. II, we introduce the inelastic dark matter model with a new U (1) D gauge boson, mediating the DM particles and electrons interaction. Besides, we also evaluate the events induced by IDM-electron scattering, including the calculations of kinematical and dynamical processes as well as the crystal form factor f c (q, ∆E e ). In Sec. III we exhibit the generated events as a function of the electron deposited energy ∆E e with different DM masses m χ , mass splitting δ and DM form factors F DM . Also, we obtain the 90% C.L. bounds onσ e − m χ plane by utilizing the SENSEI exprimental data.
Finally, we draw some conclusions in Sec IV.

DUCTOR
We introduce a Dirac fermion dark matter χ in the model where the dark sector interacts with SM particles through a new mediator dark photon A µ , which is kinetically mixing with the photon [86] where is the kinetic mixing parameter, F µν F µν is the QED strength field tensor (the dark photon strength field tensor). The interactions between the dark sector are governed by the lagrangian where the covariant derivative is D µ ≡ ∂ µ + ig D A µ and the g D is the U(1) D gauge coupling.
The dark sector communicates with the standard model electromagnetic current J µ by exchanging the U (1) D gauge boson dark photon A µ . Notably, the mass splitting arises from the Majorana mass term, which may be generated from the Higgs mechanism through χ and χ c Yukawa couplings. Therefore, these dark sector interactions can decompose the Dirac fermion χ into two almost degenerate Majorana mass eigenstates where their mass splitting δ = m χ 2 − m χ 1 is much smaller than m χ . χ 1 (χ 2 ) are the ground (excited) state. Besides, the dark photon A µ can receive the mass m A through the Stueckelberg mechanism or dark Higgs.
The dark matter particles χ inelastically scatter off electrons bound to semiconductors through exchanging the dark photon A µ . The transfer momentum q is negligible relative to the electron mass m e and DM mass m χ ∼ O(MeV) since it is typically at keV scale.
Furthermore, the IDM-electron scattering cross section is approximately equal to that of elastic scattering since the mass splitting δ m χ as mentioned before. Therefore, the inelastic cross section is decomposed into the reference cross sectionσ e and DM form factor where the reference cross sectionσ e is set up by the transfer momentum q = αm e and µ is the DM-electron reduced mass. α D ≡ g 2 D /4π and α is the fine structure constant respectively. The transfer momentum q-dependent terms are absorbed in the DM form factor F DM . We study two scenarios where the heavy mediator m A αm e and the light mediator m A αm e . The DM form factor F DM = 1 is for the heavy mediator whereas is for the light mediator.
The kinematics of IDM-electron scattering, including the up-scattering (χ 1 e → χ 2 e) and down-scattering processes (χ 2 e → χ 1 e), satisfy the energy conservation where the electron deposited energy ∆E e is the energy difference between the initial and final electron energy, q is the transfer momentum and v is the velocity of the incoming dark matter. Besides, δ is positive for the up-scattering process while δ is negative for downscattering process. After simplification, the energy conservation Eq. 7 can be expressed by the following form Thus, the maximum(minimum) transfer momentum q max (q min ) can be written as with cosθ = 1, where θ is the angle between the incoming DM velocity v and the transfer momentum q. When the limit δ → 0, the maximum(minimum) transfer momentum returns to the elastic case. For a given electron deposited energy ∆E e and the transfer momentum q, the kinematically allowed minimum velocity of the incoming dark matter v min is given by In order to prevent DM from escaping the galaxy, this puts the upper limit on the minimum Earth velocity relative to the DM halo and v esc = 600 km/s is the escape velocity of the galaxy. This constraint v min ≤ v E + v esc is valid for up-scattering and down scattering processes. Additionally, in the consideration of the up-scattering process, the kinetic energy of dark matter partices E χ k ∼ 1 2 m χ v 2 should be larger than the mass splitting δ, which guarantees that the up-scattering process is kinematically allowed. This requirement also provides the upper limit on the mass splitting δ with the maximum above which the kinetic energy E χ k cannot compensate the mass splitting δ for the upscattering process. It should be noted that we derive the maximum of mass splitting δ max with q max = 18αm e and ∆E e = 0.1 eV, which are mentioned in the following numerical calculation of the crystal form factor. When evaluating the events induced by IDM-electron scattering, we consider the Standard Halo Model(SHM) [87] where the local DM velocity is described by Maxwell-Boltzmann distribution. The dependence of the generated events on different galactic dark matter velocity distributions is discussed in Refs [88,89]. Assuming that the DM velocity distribution is spherically symmetric, we take the form of the Maxwell-Boltzmann distribution in the detector rest frame with the normalization factor where v 0 = 230 km/s is the typical velocity of the halo DM. The velocity dependent integral has the following expression which is eventually expressed by a piecewise function as shown in Refs [31]. The differential event rate dR c /dE e produced by the IDM-electron scattering in a semiconductor target is determined by where the ρ χ = 0.3 GeV/cm 3 [90] is the local dark matter density, m T is the mass of target material, and f c (q, ∆E e ) is the crystal form factor for exciting an electron from a valence band to a conduction band in semiconductors. Because there exists two silicon atoms in each silicon crystal, the target mass m T = 2m Si = 52.33 GeV. In the following, we will pay more attention to calculating the crystal form factor f c (q, ∆E e ), which is related to the overlap integral of the initial and final electron wave functions. We exploit the QEdark code [31] based on Quantum ESPRESSO to numerically evaluate the crystal form factor. Because of the periodic potential in a semiconductor crystal, electrons bound to a semiconductor valence band are governed by Bloch wave functions, ψ n k (r) with the band label n and the electron momentum k in the first Brillouin Zone(BZ) with the normalization condition where V is the volume of the crystal and G is the reciprocal lattice vector. The form factor f nk→n k ,G related to electron excitation from a valence band {n, k} to a conduction band The crystal form factor as a function of transfer momentum q and the electron deposited energy ∆E e has the following expression where the crystal form factor sums over both all filled energy bands {n, k} and unfilled energy bands {n , k }, the transfer momentum q integrates over the first BZ. The factor 2π 2 (αm 2 e V cell ) −1 with the dimension of energy equals to 2.0 eV for silicon semiconductors and V cell is the volume of the unit cell. Here E n,k (E n ,k ) is the energy of level {n, k}({n , k }).
These two δ-functions are required by the energy and momentum conservation.
For numerically calculating the crystal form factor f c (q, ∆E e ), we use these methods described in Refs [31]: binning in q and ∆E e , discretization in k and cutoff in G, G shown in Appendix VI. These operations are encoded in the QEdark code. The modifications that we made to the QEdark code are the calculations of kinematic part where the mass splitting δ is encoded in those relevant functions as described in Sec. II. In Eq. 16, the differential event rate dR c /dE e is a function of the electron deposited energy ∆E e . However, the electron deposited energy cannot be directly measured by DM direct detection experiments. Instead, we should convert ∆E e to N e since the electron-hole pairs N e are detectable. Because of the conversion of ∆E e to N e involving a complicated chain of secondary scattering processes, there is no exact model describing these secondary scattering processes so far. We assume a linear response function, which is regarded as a reasonable assumption describing the true behavior where E gap = 1.2 eV is the band energy and = 3.8 eV [91] is the mean energy per electronhole pair for silicon semiconductors. And the floor function Floor[x] represents the nearest integer less than or equal to x. The first term in the linear response function represents the primary electron-hole excited by the initial IDM-electron scattering, while the second term shows the additional electron-hole pairs induced by the residual electron deposited energy.
Therefore, the observable number of electron-hole pairs is evaluated by Furthermore, before the dark matter particles arrive at the underground detectors, we should take into account the Earth shielding effect [92], containing two cases. One scenario is that the halo dark matter is mainly made of the ground states χ 1 . Before the ground states χ 1 reach the detector, they will be converted to the excited states χ 2 via up-scattering off atoms in the Earth. This terrestrial up-scattering effect is discussed in Refs [81], where the fraction of excited states created by up-scattering process only accounts for O(10 −4 ). It is much smaller than the local dark matter density ρ χ . Therefore, the excited states generated by the Earth shielding effect can be negligible. Also, we can assume another case where the halo dark matter is fully composed by the excited states χ 2 because of its long lifetime.
They will de-excite to the ground states χ 1 through down-scattering off atoms before their reaching the detectors, which results in the number density ρ χ decreasing. Given the previous scenario, it is therefore reasonable to speculate that only a small fraction of excited states χ 2 convert to the ground states through down-scattering process. The accurate calculation of the Earth shielding effect in this scenario will be delayed in the future work.

III. NUMERICAL RESULTS AND DISCUSSIONS
We show the number of events induced by the IDM-electron scattering in Fig. 1  dark photon A µ is a heavy mediator whereas F DM = (αm e /q) 2 (bottom panel) represents that the dark photon A µ is a light mediator. The events produced by two different DM masses m χ = 10 MeV, m χ = 100 MeV are shown with the blue and red lines. As shown in Fig. 1, the number of events in each electron deposited energy bin decreases with the electron deposited energy ∆E e increasing. This is because that the crystal form factor f c (q, ∆E e ) is highly suppressed by the large transfer momentum q. The large transfer momentum q indicates the large ∆E e , resulting in the less events in large ∆E e region. Note that the events mentioned here are calculated by Eq. 22 rather than the experimentally observed data. Different from the elastic DM-electron scattering process, the down-scattering process δ < 0 produces the most events while the up-scattering process δ > 0 generates the least with the same DM mass m χ , DM form factor F DM and the electron deposited energy ∆E e in each picture in Fig. 1. The DM kinetic energy not only excites electrons but also converts to the mass splitting δ for the up-scattering process. Whereas for the down-scattering process, the mass splitting δ can also contribute the extra energy to electron excitation in addition to the DM kinetic energy. Thus, the events induced by down-scattering process are more than those produced by the up-scattering process. Beside, the light dark matter results in more events in the small ∆E e region while the heavy dark matter generates more events in the large ∆E e region for the same δ and F DM . On one hand, both m χ = 10 MeV and m χ = 100 MeV have enough kinetic energy to excite electrons in the small ∆E e region. However, the generated events are enhanced by the DM mass 1/m χ as described in Eq. 16, giving rise to the more events for m χ = 10 MeV in small ∆E e range. On the other hand, although the resulting events are enhanced by 1/m χ , the light dark matter lacks enough kinetic energy to excite more electrons in the large ∆E e region. Conversely, the heavy dark matter have enough kinetic energy to induce more electron excitation in the large ∆E e region, which results in more events for m χ = 100 MeV. Additionally, the dependence of generated events on two different DM form factors F DM will be displayed with same m χ and δ. It should be noted that the larger electron deposited energy ∆E e implies the larger transfer momentum q as mentioned before. Compared with the DM form factor F DM = 1, the induced events for F DM = (αm e /q) 2 in small transfer momentum region (q < αm e ), namely small ∆E e , are large due to the produced events being enhanced by F DM while those in large transfer momentum region (q > αm e ) are relatively small because of the resulting events being highly suppressed by F DM ∼ 1/q 2 .
In Fig. 2, we present the ratio of theoretically evaluated events R theory 1e − to the experimentally observed events R obs 1e − in the m χ − δ plane. The R theory 1e − is obtained by these given m χ , δ and F DM namely according to Eq. 22, while the R obs 1e − is shown in Tab.I. The blank area between these colored dots represents the ratio r > 10, which indicates that the theoreti- to R obs 1e − , whose range is 0 < r = R theory 1e − /R obs 1e − ≤ 10.
cally calculated events R theory 1e − is much larger than R obs 1e − . Whereas the ratio r = 0 at the right bottom blank area implies R theory 1e − = 0, which originates from two reasons. For the up-scattering process with large δ, the DM kinetic energy E χ k cannot sufficiently compensate for the large mass splitting δ, leading to the up-scattering process being kinematically forbidden. Besides, the minimum velocity of incoming DM particles v min for large δ will be larger than v esc + v E , causing the DM particles to escape the galaxy. Also, there exists the same constraint (v min ≤ v esc + v E ) for large |δ| down-scattering process. For a given DM mass m χ and large mass splitting |δ| region, the small ratio r = R theory 1e − /R obs 1e − indicates that the theoretically generated R theory 1e − is much smaller than the experimentally observed R obs 1e − for both up-scattering and down-scattering processes. This is because that for large mass splitting |δ| IDM-electron scattering process, the minimum velocity of incoming DM particles v min is so large that the theoretically evaluated events R theory 1e − are highly suppressed by the Maxwell-Boltzmann velocity distribution. Additionally, the less events R theory 1e − generated by up-scattering process with large mass splitting |δ| also simultaneously arise from that more DM kinetic energy E χ k should be transformed to the large mass splitting δ, remaining less kinetic energy to excite electrons. Note that the ratio 0 < r ≤ 1 is allowed by the observed events from SENSEI experimental data with the reference cross sectionσ e = 10 −37 cm −2 .  In Fig. 3, we utilize the different observable numbers of electrons to constrain the DMelectron scattering cross sectionσ e and mass splitting δ. The red, green and blue lines illustrate the limits from the different observed numbers of electrons (90%CL [g-day] −1 ) R obs 1e − (525.2), R obs 2e − (4.449), R obs 3e − (0.255) as shown in Tab. I. With regard to the same DM mass m χ , form factor F DM and mass splitting δ, the more observed events R obs N e − will allow the larger DM-electron scattering cross sectionσ e . In other words, the more observed events R obs N e − put weaker limits onσ e . Therefore, we can see that the least observed event R obs 3e − puts the most stringent limits onσ e − δ plane, while the most observed event R obs 1e − gives the weakest constraints in each picture in Fig. 3. As shown in Fig. 3, the constraints originating from up-scattering process will have a cutoff at the large δ where the DM kinetic energy E χ k cannot be enough transformed to the mass splitting or the corresponding minimum velocity of incoming DM particles v min > v esc + v E . In addition, we can see that the most stringent constraints occur at the mass splitting δ ∼ O(−10) eV, on either side of which the induced events are suppressed by the Maxwell-Boltzmann velocity distribution as mentioned before.
One can notice that for up-scattering process (δ > 0) with same DM form factor F DM and mass splitting δ, the heavy dark matter provides the stronger constraints on cross sectionσ e while the light dark matter puts weaker limits onσ e . After overcoming the mass splitting δ, the heavy dark matter has more kinetic energy left to generate more events R theory N e − , so this will lead to the more stringent limits onσ e . Contrarily, with regard to down-scattering process (δ < 0), the light dark matter gives stronger restrictions onσ e while the heavy dark matter provides the weaker limits onσ e . For down-scattering process, due to the mass splitting contribution to extra energy to excite electrons, both the heavy and light dark matter have enough energy to excite all the electrons. However, the produced events in each    tering with different mass splitting δ in the inelastic dark matter model. The two upper pictures indicate the up-scattering process, whereas the two bottom pictures represent the down-scattering process. As we can see in Fig. 4, compared with the elastic DM-electron scattering process(the green solid line), the down-scattering process δ < 0 gets the stronger constraints onσ e due to its resulting in more events R theory N e − as mentioned before. Whereas the up-scattering process provides the relatively weaker restrictions onσ e because the part of the DM kinetic energy E χ k will be converted to mass splitting δ, rather than absolutely transformed to the electron deposited energy ∆E e . This will generate the less events R theory N e − . Besides, the constraints become weaker with the mass splitting δ increasing for up-scattering process since more DM kinetic energy E χ k has to be converted the mass splitting δ, which leads to the less observable events. However, the limits are more stringent with the mass splitting |δ| increasing for down-scattering process. This is because that large mass splitting |δ| implies that more extra energy induces the more observable events. But, the resulting events are also highly suppressed by the Maxwell-Boltzmann velocity distribution since large mass splitting |δ| also indicates large v min as mentioned before. The number of generated events R theory N e − depends on the competition between the large δ enhancement and Maxwell-Boltzmann velocity distribution suppression. Therefore, we take the mass splitting δ = −3, −5 eV for down-scattering as a benchmark point from Fig 3. Besides, XENON1T excess can be accounted for by the inelastic dark matter with the mass splitting δ ∼ 2 − 3 keV located at the peak of the electron recoil energy spectrum excess. The down-scattering process with |δ| ∼ 2 − 3 keV, on the other hand, gives rise to less events because of the Maxwell-Boltzmann velocity distribution suppression. Therefore, the constraints from the down-scattering process with |δ| ∼ 2−3 keV are weaker than those shown in Fig. 4.

IV. CONCLUSION
Given the current status of searching for WIMP dark matter, the searches for the light dark matter have attracted a great amount of attention. Light dark matter can reside in some well-motivated models such as the inelastic dark matter model. In this work, we have studied the IDM-electron scattering in silicon semiconductors due to their lower binding energy. Furthermore, the SENSEI experiment with the ultralow-noise silicon Skipper-CCD has given strong limits on cross sectionσ e . We utilize the latest released data from SENSEI experiment to constrain the cross sectionσ e in the inelastic dark matter model. With regard to IDM-electron scattering, the SENSEI experiment gives the stronger(weaker) constraints on cross sectionσ e for down-scattering (up-scattering) process. Especially, the SENSEI experiment can detect the DM mass down to 0.1 MeV for down-scattering process with the mass splitting δ ∼ −5 eV.
where q i (∆E j ) is the central value of i-th q bin(j-th energy bin). The range of ∆E e is from 0.1 eV to 50 eV with 500 bins at intervals δE e = 0.1 eV, while the range of q is from 0.02 αm e to 18 αm e with 900 bins at intervals δq = 0.02αm e . In addition to binning in q and ∆E e , the reciprocal lattice vectors G follows the cutoff requirement resulting in a fundamental cutoff on transfer momentum q ≤ √ 2m e E cut , where E cut is the plane-wave energy cutoff. We take the same E cut = 70 Ry value as mentioned in Refs [31]. Furthermore, the numerical calculation also requires replacing the k-integral with a discretization in k.
where the 243 representative k-points with the corresponding weightings ω k are used in the sum of k and the weightings ω k satisfy the condition k ω k = 2.