Spin-dependent sub-GeV Inelastic Dark Matter-electron scattering and Migdal eect I. Velocity Independent Operator Jiwei Li1Liangliang Su1yLei Wu1zand Bin Zhu2x

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Spin-dependent sub-GeV Inelastic Dark Matter-electron

scattering and Migdal eﬀect: (I). Velocity Independent Operator

Jiwei Li,1, ∗Liangliang Su,1, †Lei Wu,1, ‡and Bin Zhu2, §

1Department of Physics and Institute of Theoretical Physics,

Nanjing Normal University, Nanjing, 210023, China

2Department of Physics, Yantai University, Yantai 264005, China

Abstract

The ionization signal provide an important avenue of detecting light dark matter. In this work,

we consider the sub-GeV inelastic dark matter and use the non-relativistic eﬀective ﬁeld theory

(NR-EFT) to derive the constraints on the spin-dependent DM-electron scattering and DM-nucleus

Migdal scattering. Since the recoil electron spectrum of sub-GeV DM is sensitive to tails of galactic

DM velocity distributions, we also compare the bounds on corresponding scattering cross sections

in Tsallis, Empirical and standard halo models. With the XENON1T data, we ﬁnd that the ex-

clusion limits of the DM-proton/neutron and DM-electron scattering cross sections for exothermic

inelastic DM are much stronger that those for the endothermic inelastic DM. Each limits of the

endothermic inelastic DM can diﬀer by an order of magnitude at most in three considered DM

velocity distributions.

∗ljw@njnu.edu.cn

†liangliangsu@njnu.edu.cn

‡leiwu@njnu.edu.cn

§zhubin@mail.nankai.edu.cn

arXiv:2210.15474v2 [hep-ph] 12 Apr 2023

CONTENTS

I. Introduction 3

II. Dark Matter Velocity Distribution Function 4

III. Inelastic Dark Matter-Nucleus Migdal Scattering 8

A. Calculations 8

B. Numerical Results and Discussions 16

IV. Inelastic Dark Matter-Electron Scattering 23

A. Calculations 23

B. Numerical Results and Discussions 26

V. Conclusion 30

VI. Acknowledgements 31

References 31

I. INTRODUCTION

Numerous astronomical and cosmological observations have provided evidence for the

existence of dark matter (DM) in the universe. However, besides its gravitational interaction,

other physical properties of DM remain mystery. From the perspective of particle physics,

dark matter may be made up of a hypothetical particle that is still undetected. Among

the various conjectures, the weakly interacting massive particles (WIMPs) have been widely

studied in the various experiments.

Direct detection that attempts to discern signals induced by DM at extremely low back-

grounds has made great eﬀorts in the past few years [1–13]. However, there is no any

evidence of WIMP dark matter in the typical mass range. This strongly motivates the

search for sub-GeV dark matter [9–32]. While the low momentum transfer of sub-GeV DM

can not produce the observable nuclear recoil signal in the conventional detectors. With

the improvements of direct detection experiments, we can access to the low mass DM by

using the ionization events. Such signals can arise from the scattering of electrons with

DM [5,10–14,18,19,27–35], and the secondary eﬀects in the DM-nuclear interactions,

such as the Migdal scattering [9,20–26,36–41]. There have been many studies on DM-

electron scattering to date. For instance, in the context of elastic scattering, various op-

erators for spin-dependent (SD) interactions are discussed in Ref. [42] in an eﬀective ﬁeld

theory (EFT). The inelastic dark matter (iDM) model [43–58], originally used to explain the

DAMA anomaly, has also been used to study DM-electron scattering with spin-independent

interactions to explain the XENON1T excess [34,57–59]. Dent et al. [24] showed some en-

lightening results on the Migdal eﬀect of inelastic dark matter scattering with nuclei through

the spin-independent (SI) interaction. However, there is still much scope for discussion of

iDM-electron/Migdal scattering via SD interactions.

In this paper, we will study the ionization signals of sub-GeV inelastic dark matter (iDM),

including Migdal eﬀect and DM-electron scattering. Given the current strong constraints

on the spin-independent (SI) cross section, we calculate the spin-dependent (SD) iDM-

nucleus/electron scattering. We consider the Lagrangian density Lint ⊃¯χ0γµγ5χ¯

Nγµγ5N

for the axial-vector interaction of DM χwith the standard model particle Nand derive the

operator O4=~

Sχ·~

SN; this type of SD interaction is the only one in the leading order that

is not suppressed by momentum transfer ~q. For some models, the SD interaction may still

dominate, e.g. the scattering cross section for a Dirac DM particle interacting through its

anomalous magnetic dipole moment, where the SD-like part (dipole-dipole) dominates in

certain parameter space [60]. Or when the DM is the Majorana fermion or a real vector

boson, the SD interaction can naturally dominate (but is not always guaranteed) [61]. In the

future, if a signal associated with SD is observed, it will rule out the spinless DM particles

by and large.

On the other hand, the velocity distribution function (VDF) of the local DM halo can

have a non-negligible impact on the direct detection [62–66]. In particular, the electron recoil

spectrum is sensitive to the high-velocity tail of the DM halo. As a benchmark distribution,

the Standard Halo Model (SHM) is usually adopted [67], however, it still can not accurately

describe the distribution of DM in the Galaxy [68]. This motivates other alternative halo

models for the VDF [69], such as Tsallis and Empirical models. We will also discuss their

impacts on the exclusion limits of iDM-nucleus/electron scattering.

The paper is structured as follows. In Sec. II, we compare the velocity distribution

functions for three models: the Standard Halo Model, the Tsallis model and the empirical

model. In Sec. III and Sec. IV, we investigate the ionization rates of the spin-dependent

scattering of the inelastic dark matter with the nucleus and the electron targets, respectively.

With the available data, we obtain the exclusion limit for spin-dependent inelastic dark

matter-nucleus Migdal/electron scattering in three velocity distribution models. Finally, we

draw the conclusions in Sec. V.

II. DARK MATTER VELOCITY DISTRIBUTION FUNCTION

In the DM direct detections, the astrophysical properties of the local DM halo distribu-

tion, such as local DM density, mean DM velocity, etc., can signiﬁcantly change the sensi-

tivity. In particular, the electron spectrum is exceptionally sensitive to the high-velocity tail

of the local velocity distribution of dark matter [66,69,70]. The most popular and widely

used standard halo model (SHM) in DM direct detection experimental analysis, which as-

sumes DM particles are in an isothermal sphere and obey the isotropic Maxwell-Boltzmann

velocity distribution function (VDF). Although its simple analytical form is appealing [68],

this model cannot adequately explain the distribution of DM particles in the Galaxy. Con-

sequently, it is important to investigate diﬀerent velocity distribution models to substitute

for the halo model. Based on the work in Ref. [69], this paper also introduces two additional

velocity distribution models: Tsallis Model and an Empirical Model. We will discuss the

eﬀects on DM-Target scattering caused by diﬀerent VDF models.

In the rest frame of the Galaxy, the SHM is given by

fSHM(~v)∝









e−|~v|2/v2

0|~v| ≤ vesc

0|~v|> vesc.

(1)

The escape speed of the galaxy limits the speed of DM particles gravitationally bound to our

galaxy, so a physical cut-oﬀ point is set at the local escape speed vesc, with v0as the circular

velocity at the Solar position [62]. The rotation curve in this model will be asymptotically

ﬂat at large r(i.e. the distance from the centre of the Galaxy), and v0is usually regarded as

the value of the curve at this point. In the laboratory frame it has the following analytical

forms

fSHM(~v) = 1

Ke−|~v+~vE|2/v2

0Θ(vesc − |~v +~vE|),(2)

where vEis the Earth’s Galactic velocity. The velocity distribution of the SHM is truncated

at the escape speed vesc through the Heaviside function Θ, with the normalization coeﬃcient

K=v3

0π3

2Erf(vesc

)−2πvesc

exp −v2

esc

0 (3)

that results from Rf(~v) d3~v = 1.

The features of the local VDFs derived from DM cosmological simulations that include

baryonic physics are largely consistent with the SHM; however, several studies [62,69,71–74]

using data from DM-only simulations reveal a signiﬁcant deviation from the overall trends

manifested by the relevant local VDFs compared to the SHM. These simulations show that,

especially in the high-velocity tail of the distribution, diﬀerent features with the SHM will

appear. One point worth making is that although adding baryons to the simulation makes

the process more complex, it is nevertheless essential to restore the possible real universe.

Next, we discuss some alternative models in which the VDF of the Tsallis Model (Tsa) [75]

can be considered more compatible with the numerical results of N−body simulations that

include baryons [76,77]. According to the statistical results of Tsallis, the deﬁnition of

standard Boltzmann-Gibbs entropy is extended by introducing the entropy index qs, as

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Spin-dependentsub-GeVInelasticDarkMatter-electronscatteringandMigdaleect:(I).VelocityIndependentOperatorJiweiLi,1,LiangliangSu,1,yLeiWu,1,zandBinZhu2,x1DepartmentofPhysicsandInstituteofTheoreticalPhysics,NanjingNormalUniversity,Nanjing,210023,China2DepartmentofPhysics,YantaiUniversity,Yantai264005...

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Spin-dependent sub-GeV Inelastic Dark Matter-electron scattering and Migdal eect I. Velocity Independent Operator Jiwei Li1Liangliang Su1yLei Wu1zand Bin Zhu2x

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