††institutetext: Department of Physics, Korea University,
Anam-ro 145, Sungbuk-gu, Seoul 02841, Korea

# Inelastic dark matter, small scale problems, and the XENON1T excess

Seungwon Baek<EMAIL_ADDRESS>
###### Abstract

We study a generic model in which the dark sector is composed of a Majorana
dark matter $\chi_{1}$, its excited state $\chi_{2}$, both at the electroweak
scale, and a light dark photon $Z^{\prime}$ with $m_{Z^{\prime}}\sim 10^{-4}$
eV. The light $Z^{\prime}$ enhances the self-scattering elastic cross section
$\chi_{1}\chi_{1}\to\chi_{1}\chi_{1}$ enough to solve the small scale problems
in the $N$-body simulations with the cold dark matter. The dark matter
communicates with the SM via kinetic mixing parameterized by $\epsilon$. The
inelastic scattering process $\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$ followed by
the prompt decay $\chi_{2}\to\chi_{1}Z^{\prime}$ generates energetic
$Z^{\prime}$. By setting $\delta\equiv m_{\chi_{2}}-m_{\chi_{1}}\simeq 2.8$
keV and $\epsilon\sim 10^{-10}$ the excess in the electron-recoil data at the
XENON1T experiment can be explained by the dark-photoelectric effect. The
relic abundance of the dark matter can also be accommodated by the thermal
freeze-out mechanism via the annihilation
$\chi_{1}\chi_{1}(\chi_{2}\chi_{2})\to Z^{\prime}Z^{\prime}$ with the dark
gauge coupling constant $\alpha_{X}\sim 10^{-3}$.

##  1 Introduction

There are clear evidences that dark matter (DM) exists in the universe,
although its particle nature is almost unknown. Since the standard model (SM)
lacks candidate for the dark matter, new physics (NP) models are required to
incorporate DM. The standard cold dark matter (CDM) model has been very
successful in predicting the large scale structure of the universe. However,
the $N$-body simulations with CDM predict cuspy density profiles, while
observations of rotation curves of dwarf galaxies and low surface brightness
galaxies point towards flat cores. Self-interacting dark matter (SIDM) with
self-scattering cross sections, $\sigma_{\chi\chi}/m_{\chi}\sim
1\,\rm{cm^{2}/g}$, can be a possible solution to this problem Tulin:2017ara .
The constraint, $\sigma_{\chi\chi}/m_{\chi}\lesssim 0.5\,{\rm cm^{2}/g}$,
coming from collision of galaxy clusters Harvey:2015hha  can be evaded by the
SIDM’s velocity-dependent cross sections. Also the non-observation of DM in
the DM-nucleon scattering experiments can be naturally explained in the
inelastic DM models Blennow:2016gde .

In the year 2020, XENON1T collaboration has announced an excess of electron
recoil near $2-3$ keV energy Aprile:2020tmw . Although the result can be
explained by $\beta$ decays of tritium contamination, it can also be
attributed to NP contributions such as the solar axion or anomalous neutrino
magnetic moment with about 3$\sigma$ significance. However, the latter
possibilities are in strong tension with the star cooling constraints
Aprile:2020tmw . The excess may also be the result of the dark matter
scattering with electrons inside the xenon atom. In this case the signal can
give valuable information on the nature of the dark matter.

The XENON1T excess has been considered in various inelastic DM models
Harigaya:2020ckz ; Lee:2020wmh ; Bramante:2020zos ; An:2020tcg ; Chao:2020yro
; Baek:2020owl ; He:2020wjs ; Choudhury:2020xui ; Ema:2020fit ; Borah:2020jzi
; Borah:2020smw ; Aboubrahim:2020iwb ; He:2020sat ; Dutta:2021wbn . In
Baek:2020owl  we studied exothermic DM scattering on electron to explain the
anomaly. We found that both scalar and fermionic DM models can accommodate the
XENON1T excess.

In this paper we consider another possible realization of inelastic DM. As in
the fermionic DM model in Baek:2020owl  the dark sector (DS) has two Majorana
DM candidates, $\chi_{1},\chi_{2}$, with mass spltting,
$\delta=m_{\chi_{2}}-m_{\chi_{1}}$, as well as the dark photon ($Z^{\prime}$).
The current universe has only electroweak scale $\chi_{1}$ DM as opposed to
the model in Baek:2020owl 111It turns out the lifetime of $Z^{\prime}$ are
much longer than the age of the universe. However, their contribution to the
relic density is negligible due to $m_{Z^{\prime}}\ll m_{\chi}$.. The
inelastic scattering $\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$ followed by the
decay of the excited state $\chi_{2}\to\chi_{1}Z^{\prime}$ produces
$Z^{\prime}$ with energy given by the mass splitting $\delta\equiv
m_{\chi_{2}}-m_{\chi_{1}}$. The dark photon (DP) $Z^{\prime}$ whose mass is
lighter than $\delta$ is absorbed in the xenon atom, ejecting an electron,
which we may call the dark-photoelectric effect. We fix the mass splitting
$\delta=2.8$ keV to fit the XENON1T excess.

The elastic scattering $\chi_{1}\chi_{1}\to\chi_{1}\chi_{1}$ can be large
enough to solve the small scale problem, such as the cusp-core problem
mentioned above. Since its cross section is velocity-dependent, the constraint
from the galaxy clusters can also be easily evaded. The relic abundance of the
$\chi_{1}$ is obtained by the thermal freeze-out mechanism.

The paper is organised as follows. In Section 2 we introduce the model. In
Section 3 the relic density of the dark matter is calculated. In Section 4 we
show that the small scale problems of the CDM can be solved in our model. In
Section 5 we consider the dark-photoelectric effect to address the excess of
recoil electrons in the XENON1T. We conclude in Section 6.

##  2 The model

The model has a dark gauge symmetry $U(1)_{X}$ under which all the SM
particles are neutral. A $U(1)_{X}$-charged, but neutral under the SM, Dirac
fermion $\chi(x)$ is introduced. For the inelastic dark matter we study the
Lagrangian of the dark sector (DS) including the kinetic energy and kinetic
mixing terms in the form Cui:2009xq

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\mathcal{L}_{\rm
SM}-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}+{1\over
2}m_{\hat{X}}^{2}\hat{X}_{\mu}\hat{X}^{\mu}-\frac{1}{2}\hat{\epsilon}\,\hat{X}_{\mu\nu}\hat{B}^{\mu\nu}+{1\over
2}m_{\hat{Z}}^{2}\hat{Z}_{\mu}\hat{Z}^{\mu}$ (1)
$\displaystyle+\overline{\chi}(i\not{D}-m_{\chi})\chi-{\delta\over
4}(\overline{\chi^{c}}\chi+h.c.),$

where
$\hat{X}_{\mu\nu}=\partial_{\mu}\hat{X}_{\nu}-\partial_{\nu}\hat{X}_{\mu}$
($\hat{B}_{\mu\nu}=\partial_{\mu}\hat{B}_{\nu}-\partial_{\nu}\hat{B}_{\mu}$)
is the field strength tensor for the dark photon $\hat{X}_{\mu}$ (for the
$U(1)_{Y}$ gauge boson $\hat{B}_{\mu}$),
$D_{\mu}=\partial_{\mu}+ig_{X}Q_{\chi}\hat{X}_{\mu}$ is the covariant
derivative with $g_{X}$ the dark coupling constant, and we fix the dark charge
of $\chi$, $Q_{\chi}=1$. The above Lagrangian can be considered an effective
theory of the UV-complete theory given, for example, in Baek:2020owl . We
assume the kinetic mixing between the dark photon and the $U(1)_{Y}$ gauge
field is small, $\hat{\epsilon}\ll 1$. We do not specify the origin of the
dark gauge boson mass $m_{\hat{X}}$ and the $Z$-boson mass $m_{\hat{Z}}$.

The gauge fields can be written in terms of mass eigenstates: the photon
$A_{\mu}$, the SM $Z$-boson, and the physical DP $Z^{\prime}$. As we will see
in Section 5, we are interested in very small mixing parameter,
$\hat{\epsilon}\sim 10^{-10}$. So it suffices to keep only the linear terms in
$\hat{\epsilon}$ in the analysis. In this case we get Babu:1997st

$\displaystyle\hat{B}_{\mu}$ $\displaystyle=$ $\displaystyle
c_{W}A_{\mu}-s_{W}Z_{\mu}-\hat{\epsilon}c_{W}^{2}Z^{\prime}_{\mu},$
$\displaystyle\hat{W}^{3}_{\mu}$ $\displaystyle=$ $\displaystyle
s_{W}A_{\mu}+c_{W}Z_{\mu}-\hat{\epsilon}c_{W}s_{W}Z^{\prime}_{\mu},$
$\displaystyle\hat{X}_{\mu}$ $\displaystyle=$ $\displaystyle
Z^{\prime}_{\mu}+\hat{\epsilon}s_{W}Z_{\mu},$ (2)

where $c_{W}=\cos\theta_{W}(s_{W}=\sin\theta_{W})$ with $\theta_{W}$ the
Weinberg angle, and $\hat{W}^{3}_{\mu}$ is the neutral component of
$SU(2)_{L}$ gauge boson. In this approximation the gauge boson masses do not
get corrections by the mixing (2): i.e. $m_{Z}=m_{\hat{Z}}$ and
$m_{Z^{\prime}}=m_{\hat{X}}$.

The Dirac field $\chi$ splits into two Majorana mass eigenstates, $\chi_{1}$
and $\chi_{2}$, defined as Baek:2020owl

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\chi_{2}+i\chi_{1}),$ (3)
$\displaystyle\chi^{c}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\chi_{2}-i\chi_{1}),$ (4)
$\displaystyle\chi_{1}^{c}$ $\displaystyle=$ $\displaystyle\chi_{1},\ \ \
\chi_{2}^{c}=\chi_{2},$ (5)

with masses

$m_{\chi_{1},\chi_{2}}=m_{\chi}\mp yv_{\phi}\equiv
m_{\chi}\mp\frac{1}{2}\delta.$ (6)

We assume $\delta\equiv m_{\chi_{2}}-m_{\chi_{1}}>0$. Then $\chi_{1}$ becomes
the DM. The dark-gauge interactions of the DM and electron are Baek:2020owl

$\displaystyle{\cal L}$ $\displaystyle\supset$ $\displaystyle-
ig_{X}Z^{\prime}_{\mu}\overline{\chi_{2}}\gamma^{\mu}\chi_{1}-\hat{\epsilon}ec_{W}Z^{\prime}_{\mu}\overline{e}\gamma^{\mu}e.$
(7)

Note that the gauge interactions change the flavour of the dark fermions:
$\chi_{1}\leftrightarrow\chi_{2}$. In the rest of the paper we fix
$\delta=2.8$ keV, $m_{Z^{\prime}}=10^{-4}$ eV, and find $m_{\chi_{1}}$,
$\alpha_{X}(=g_{X}^{2}/4\pi)$ which can explain the DM relic abundance, the
small scale problem, and the XENON1T anomaly at the same time.

##  3 The relic abundance

In the early universe the dark sector (DS) can be in thermal equilibrium with
the SM sector via process such as $Z^{\prime}f\rightleftharpoons\gamma(Z,g)f$,
and $Z^{\prime}\gamma(Z,g)\rightleftharpoons f\overline{f}$ with $f$ a SM
fermion. In this case the Boltzmann equation for the DM number density reads
Griest:1990kh

$\displaystyle\frac{dn}{dt}+3Hn=-\sum_{i,j=1,2}\langle\sigma_{ij}v\rangle(n_{i}n_{j}-n_{i}^{\rm
eq}n_{j}^{\rm eq}),$ (8)

where $n=n_{1}+n_{2}$ with $n_{i}\equiv n_{\chi_{i}}$. Following the procedure
in Griest:1990kh , we obtain the freeze-out temperature $T_{f}$ of the DM by
solving

$\displaystyle x_{f}=\log\frac{0.0382\,g_{1}m_{1}M_{\rm
pl}x_{f}^{1/2}\langle\sigma_{\rm eff}v\rangle}{g_{*}^{1/2}},$ (9)

where $x_{f}\equiv m_{1}/T_{f}$ with $m_{1}\equiv m_{\chi_{1}}$ and $M_{\rm
pl}\simeq 1.22\times 10^{19}$ GeV is the Planck mass. The effective thermal-
averaged cross section $\langle\sigma_{\rm eff}v\rangle$ is obtained by

$\displaystyle\langle\sigma_{\rm
eff}v\rangle=\sum_{i,j=1,2}\langle\sigma_{ij}v\rangle r_{i}r_{j},\quad
r_{i}=\frac{g_{i}(1+\Delta_{i})^{3/2}e^{-x_{f}\Delta_{i}}}{\sum_{i=1,2}g_{i}(1+\Delta_{i})^{3/2}e^{-x_{f}\Delta_{i}}},$
(10)

where $\Delta_{i}=(m_{i}-m_{1})/m_{1}$. Since $\Delta_{2}(=\delta/m_{1})\ll
1$, $x_{f}\Delta_{2}\ll 1$, and $\sigma_{11}\simeq\sigma_{22}\gg\sigma_{12}$
in our scenario, we can approximate

$\displaystyle\langle\sigma_{\rm eff}v\rangle\simeq{1\over
2}\langle\sigma_{11}v\rangle.$ (11)

Explicitly we get the dominant $s$-wave contributions to the DM annihilations
to be

$\displaystyle\sigma_{ii}v$ $\displaystyle\simeq$ $\displaystyle\sigma
v(\chi_{i}\chi_{i}\to
Z^{\prime}Z^{\prime})\simeq\frac{\pi\alpha_{X}^{2}}{m_{1}^{2}}+O(v^{2}),$
$\displaystyle\sigma_{ij}v$ $\displaystyle\simeq$ $\displaystyle\sigma
v(\chi_{i}\chi_{j}\to
f\overline{f})\simeq\frac{2\pi\epsilon^{2}N_{c}^{f}\alpha_{\rm
em}\alpha_{X}(2m_{1}^{2}+m_{f}^{2})(m_{1}^{2}-m_{f}^{2})^{1/2}}{9m_{1}^{5}}+O(v^{2}),$
(12)

where $i\not=j$ ($i,j=1,2$), $N_{c}^{f}$ is the color factor of $f$,
$\epsilon\equiv\hat{\epsilon}c_{W}$, and we used $m_{Z^{\prime}},\delta\ll
m_{1}$. We see that $\sigma_{ij}v$ ($i\not=j$) is $\epsilon^{2}$-suppressed
and negligible compared to $\sigma_{ii}v$.

Let us comment on a possible issue in the annihilation cross section. At high
energy, $s\to\infty$, the cross section, $\sigma(\chi_{i}\chi_{i}\to
Z^{\prime}Z^{\prime})$, behaves like

$\displaystyle\sigma(\chi_{i}\chi_{i}\to
Z^{\prime}Z^{\prime})=\frac{\alpha_{X}^{2}\pi\delta^{2}}{m_{Z^{\prime}}^{4}}+O\left(1\over
s\right),$ (13)

which violates the perturbative unitarity. So we need a UV completion of (1)
to cure this problem. For example, we can introduce Higgs field(s) coupled to
$\chi_{i}$. In this UV completion the Higgs mediated contribution to
$\chi_{i}\chi_{i}\to Z^{\prime}Z^{\prime}$ is $p$-wave, and does not change
the $s$-wave term in (12). The resulting relic density obtained from $s$-wave
contribution only from (12) has corrections of order $O(3/x_{f})\simeq
O(0.1)$. So we can take (1) as a leading effective Lagrangian for $\chi_{i}$
and $Z^{\prime}$ for a large class of microscopic theories where other NP
particles are integrated out.

By solving the Boltzmann equation (8) with the condition (9) the final relic
density from $s$-wave only in (12) is obtained to be

$\displaystyle\Omega h^{2}$ $\displaystyle\simeq\frac{2\times 1.038\times
10^{19}\,x_{f}\;{\rm GeV}^{-1}}{g_{*S}(T_{f})/g_{*}^{1/2}(T_{f})M_{\rm
pl}\langle\sigma_{11}v\rangle}$ $\displaystyle\approx 0.12\left(3.59\times
10^{-3}\over\alpha_{X}\right)^{2}\left(m_{\chi_{1}}\over 100\,{\rm
GeV}\right)^{2},$ (14)

where $g_{*S}$ and $g_{*}$ are defined in Kolb:1988aj .

##  4 The small scale problem

Given a DM with mass $m_{1}$, the value of the dark gauge coupling constant
$\alpha_{X}$ to yield the correct relic abundance can be predicted from (14).
When these two parameters $m_{1}$ and $\alpha_{X}$ and the DP mass
$m_{Z^{\prime}}$ are known, we can calculate the elastic scattering,
$\chi_{1}\chi_{1}\to\chi_{1}\chi_{1}$, and inelastic scattering,
$\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$, cross sections for a given DM relative
velocity $v$. Since for $m_{Z^{\prime}}\ll m_{1}$ the non-perturbative
Sommerfeld effect becomes significant, the perturbative calculation cannot be
applied here. Following the Ref. Blennow:2016gde , we solve the corresponding
Schrödinger equation in the CM-frame to calculate the two cross sections,

$\displaystyle\left[-\frac{\nabla^{2}}{m_{1}}+V(\vec{r})\right]\Psi(\vec{r})=\frac{k^{2}}{m_{1}}\Psi(\vec{r}),$
(15)

where $\Psi$ is the $2\times 1$ matrix wavefunction for the DM states with the
upper component for the $\chi_{1}\chi_{1}$ state and the lower component for
the $\chi_{2}\chi_{2}$ state, $\vec{r}=\vec{r}_{1}-\vec{r}_{2}$ is the
relative spatial coordinate of colliding DM particles, and
$\vec{k}=m_{1}\vec{v}/2$ is the relative momentum with $\vec{v}$ the relative
velocity. The potential is written in the $2\times 2$-matrix form,

$\displaystyle
V(r)=\begin{pmatrix}0&-\frac{\alpha_{X}}{r}e^{-m_{Z^{\prime}}r}\\\
-\frac{\alpha_{X}}{r}e^{-m_{Z^{\prime}}r}&2\delta\\\ \end{pmatrix}.$ (16)

As in Blennow:2016gde , we adopt the method suggested in Ershov:2011zz  to
solve the differential equation (15) numerically. The system of coupled radial
equations in (15) leads to numerical instability in a classically forbidden
region. The numerical stability is enhanced by using the modified variable
phase method presented in Ershov:2011zz . We introduce dimensionless
parameters Blennow:2016gde ,

$\displaystyle a=\frac{k}{\alpha_{X}m_{1}},\quad
b=\frac{\alpha_{X}m_{1}}{m_{Z^{\prime}}},\quad
c=\sqrt{a^{2}-{2\delta\over\alpha_{X}^{2}m_{1}}},\quad x=m_{1}\alpha_{X}r,$
(17)

in terms of which the radial part of the Schrödinger equation (15) becomes

$\displaystyle\Bigg{[}{d^{2}\over
dx^{2}}-\frac{\ell(\ell+1)}{x^{2}}+\begin{pmatrix}a^{2}&0\\\
0&c^{2}\end{pmatrix}\Bigg{]}\chi(x)=\begin{pmatrix}0&-{1\over x}e^{-x/b}\\\
-{1\over x}e^{-x/b}&0\end{pmatrix}\chi(x),$ (18)

where $\chi(x)=xR(x)$ with $\Psi(\vec{r})=R(r)Y_{\ell m}(\theta,\phi)$. As in
Ershov:2011zz  we write the solution in the form of $2\times 2$ matrix

$\displaystyle\chi_{ij}(x)=\big{(}f(p_{i}x)\delta_{ik}-h^{(+)}(p_{i}x)M_{ik}(x)\big{)}\alpha_{kj}(x),$
(19)

where $i,j,k=1,2$, the repeated indices are to be summed but not for the free
indices $i,j$. We use $f(x)=xj_{\ell}(x),h^{(+)}(x)=ixh^{(1)}_{\ell}(x)$ which
are the solutions when the potential in (18) is set to be zero and $a=c=1$.
Here $j_{\ell}(x)$ ($h_{\ell}^{(1)}(x)$) are the spherical Bessel functions
(the spherical Hankel functions of the first kind). The boundary condition
$\chi_{ij}(x=0)=0$ leads to $M_{ij}(x=0)=0$. We can write the scattering
amplitude in terms of the matrix $M(x=\infty)$, or equivalently, in terms of a
unitariy matrix $S_{\ell}\equiv 1-2iM^{\ell}(x=\infty)$ as

$\displaystyle
f(\theta)=-i\sum_{\ell=0}^{\infty}(2\ell+1)P_{\ell}(\cos\theta)\begin{pmatrix}{1\over
2k}&0\\\ 0&{1\over 2k^{\prime}}\end{pmatrix}(S_{\ell}-1),$ (20)

where $k^{\prime}=\sqrt{k^{2}-2m_{1}\delta}$.

The differential cross sections for the scattering of identical particles are
obtained by

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\xi}=\left|f(\theta)+\xi
f(\pi-\theta)\right|^{2},$ (21)

where $\xi=+1(-1)$ if the spatial wave function is symmetric (antisymmetric)
under particle exchange. Assuming the DM is unpolarized, we average over the
spin states to get

$\displaystyle\frac{d\sigma}{d\Omega}={1\over
4}\left(\frac{d\sigma}{d\Omega}\right)_{\xi=+1}+{3\over
4}\left(\frac{d\sigma}{d\Omega}\right)_{\xi=-1},$ (22)

where the first (the second) term is the contribution from the spin singlet
(triplet) state with symmetric (antisymmetric) spatial wave funtion. Then the
inelastic scattering cross section is obtained by

$\displaystyle\sigma_{\rm inel}(\chi_{1}\chi_{1}\to\chi_{2}\chi_{2})$
$\displaystyle=$ $\displaystyle{k^{\prime}\over 2k}\int\left[{1\over
4}\left|f_{21}(\theta)+f_{21}(\pi-\theta)\right|^{2}+{3\over
4}\left|f_{21}(\theta)-f_{21}(\pi-\theta)\right|^{2}\right]d\Omega$ (23)
$\displaystyle=$
$\displaystyle\frac{4\pi}{kk^{\prime}}\sum_{\ell=0}^{\infty}\zeta_{\ell}(2\ell+1)\left|M^{\ell}_{21}(x=\infty)\right|^{2},$

where $\zeta_{\ell}=1/2\,(3/2)$ for $\ell=$even (odd) and the values of
$M_{21}^{\ell}$’s are to be evaluated at $x=\infty$. For the elastic
scattering cross section which is used to solve the small scale problem, we
consider the viscosity Blennow:2016gde  and the momentum-transfer
Kahlhoefer:2017umn  cross section. The viscosity cross section is given by

$\displaystyle\sigma_{\rm el}^{V}(\chi_{1}\chi_{1}\to\chi_{1}\chi_{1})$ (24)
$\displaystyle=$ $\displaystyle{1\over 2}\int\left[{1\over
4}\left|f_{11}(\theta)+f_{11}(\pi-\theta)\right|^{2}+{3\over
4}\left|f_{11}(\theta)-f_{11}(\pi-\theta)\right|^{2}\right]\sin^{2}\theta
d\Omega$ $\displaystyle=$
$\displaystyle\frac{4\pi}{k^{2}}\sum_{\ell=0}^{\infty}\zeta_{\ell}\Bigg{[}\left|M_{11}^{\ell}\right|^{2}\frac{2(2\ell+1)(\ell^{2}+\ell-1)}{(2\ell-1)(2\ell+3)}-(M_{11}^{\ell}(M_{11}^{\ell+2})^{*}+c.c.)\frac{(\ell+1)(\ell+2)}{2\ell+3}\Bigg{]}.$

For the momentum transfer cross section it is more convenient to evaluate the
spin singlet and triplet contributions separately,

$\displaystyle\sigma_{\rm el}^{T}(\chi_{1}\chi_{1}\to\chi_{1}\chi_{1})$
$\displaystyle=$ $\displaystyle{1\over 4}\sigma_{\rm el}^{T,{\rm
Singlet}}+{3\over 4}\sigma_{\rm el}^{T,{\rm Triptlet}},$ (25)

with

$\displaystyle\sigma_{\rm el}^{T,{\rm Singlet}}$ $\displaystyle=$
$\displaystyle{1\over
2}\int\left|f_{11}(\theta)+f_{11}(\pi-\theta)\right|^{2}(1-|\cos\theta|)d\Omega$
$\displaystyle=$ $\displaystyle\frac{8\pi}{k^{2}}\sum_{\ell,\ell^{\prime}={\rm
even}}(2\ell^{\prime}+1)M_{11}^{\ell}{M_{11}^{\ell^{\prime}}}^{*}\left[\delta_{\ell\ell^{\prime}}-\left((\ell+1)f_{\ell^{\prime},\ell+1}+\ell
f_{\ell^{\prime},\ell-1}\right)\right],$ $\displaystyle\sigma_{\rm el}^{T,{\rm
Triplet}}$ $\displaystyle=$ $\displaystyle{1\over
2}\int\left|f_{11}(\theta)-f_{11}(\pi-\theta)\right|^{2}(1-|\cos\theta|)d\Omega$
(26) $\displaystyle=$
$\displaystyle\frac{8\pi}{k^{2}}\sum_{\ell,\ell^{\prime}={\rm
odd}}(2\ell^{\prime}+1)M_{11}^{\ell}{M_{11}^{\ell^{\prime}}}^{*}\left[\delta_{\ell\ell^{\prime}}-\left((\ell+1)f_{\ell+1,\ell^{\prime}}+\ell
f_{\ell-1,\ell^{\prime}}\right)\right],$

where

$\displaystyle
f_{\ell,\ell^{\prime}}=\frac{(-1)^{(\ell+\ell^{\prime}+1)/2}\ell!\ell^{\prime}!}{2^{\ell+\ell^{\prime}-1}(\ell-\ell^{\prime})(\ell+\ell^{\prime}+1)\left[\left(\ell\over
2\right)!\right]^{2}\left[\left(\ell^{\prime}-1\over
2\right)!\right]^{2}},\quad\text{with $\ell=$even, $\ell^{\prime}=$odd}.$ (27)

To obtain $M^{\ell}(x=\infty)$ we first transform $M$-matrix to $U$-matrix
Ershov:2011zz ,

$\displaystyle
U_{ij}(x)=f(p_{i})h^{(+)}(p_{i})\delta_{ij}-h^{(+)}(p_{i})M_{ij}(x)h^{(+)}(p_{j}),$
(28)

which gives numerically more stable solutions. From (18) we get a coupled
first-order differential equation for $U(x)$,

$\displaystyle U^{\prime}_{ij}(x)$ $\displaystyle=$ $\displaystyle
p_{i}\delta_{ij}+p_{i}{g^{\prime}(p_{i}x)\over
g(p_{i}x)}U_{ij}(x)+U_{ij}(x){g^{\prime}(p_{j}x)\over
g(p_{j}x)}p_{j}-U_{ik}(x){1\over p_{k}}\hat{V}_{kl}U_{lj}(x),$ (29)

where

$\displaystyle\hat{V}(x)=\begin{pmatrix}0&-{1\over x}e^{-x/b}\\\ -{1\over
x}e^{-x/b}&0\end{pmatrix}.$ (30)

The $S$-matrix is related to the $U$-matrix as

$\displaystyle
S_{ij}=\frac{h^{(-)}(p_{i}x)}{h^{(+)}(p_{i}x)}\delta_{ij}+2i\frac{1}{h^{(+)}(p_{i}x)}U_{ij}(x)\frac{1}{h^{(+)}(p_{j}x)},$
(31)

where $h^{(-)}(x)=-ixh^{(2)}_{\ell}(x)$, $h^{(2)}_{\ell}(x)$ the spherical
Hankel function of the second kind, and we take $x\to\infty$. We solved (29)
numerically with initial condition

$\displaystyle
U_{ij}(x_{0})=f(p_{i}x_{0})h^{(+)}(p_{i}x_{0})\delta_{ij}\simeq\frac{p_{i}x_{0}}{2\ell+1}\delta_{ij},$
(32)

where we take $x_{0}=0.01$. The results are not very sensitive to the value of
$x_{0}$ as long as $x_{0}\ll 1$. When we integrate (29) to large $x$,
$x=x_{\infty}(\gg 1)$, we take reasonably large value of $x_{\infty}$ in such
a way that not only the $S$-matrix obtained in (31) keeps unitarity (when
$k,k^{\prime}>0$) but also $U(x_{\infty})$ converges to a constant value.

| BP1 | BP2 | BP3
---|---|---|---
$m_{\chi_{1}}\;(\rm GeV)$ | $70$ | $100$ | $120$
$\alpha_{X}$ | $2.51\times 10^{-3}$ | $3.59\times 10^{-3}$ | $4.31\times 10^{-3}$
$\sigma_{\rm inel}\;(\rm pb)$ | $3.93\times 10^{14}$ | $1.78\times 10^{14}$ | $5.71\times 10^{13}$
$\sigma^{V}_{\rm el}/m_{\chi_{1}}\;(\rm cm^{2}/g)$ | $10.6(1.95,1.86\times 10^{-3})$ | $3.19(0.871,1.38\times 10^{-3})$ | $1.49(0.270,8.12\times 10^{-4})$
$\sigma^{T}_{\rm el}/m_{\chi_{1}}\;(\rm cm^{2}/g)$ | $7.63(1.22,1.12\times 10^{-3})$ | $2.14(0.645,5.88\times 10^{-4})$ | $0.962(0.191,1.74\times 10^{-4})$
$\epsilon$ | $5.39\times 10^{-10}$ | $1.14\times 10^{-9}$ | $2.02\times 10^{-9}$
Table 1:  The results for the three benchmark masses $m_{\chi}=70,100,120$
GeV. In the 2nd line the required dark gauge coupling constant $\alpha_{X}$ to
give the correct relic abundance, $\Omega_{\rm DM}h^{2}=0.119$, is given. In
the 3rd–5th lines, the predictions of the cross sections for the inelastic
scattering, $\sigma_{\rm inel}(\chi_{1}\chi_{1}\to\chi_{2}\chi_{2})$, and
elastic scattering, $\sigma_{\rm
el}^{V(T)}(\chi_{1}\chi_{1}\to\chi_{1}\chi_{1})/m_{\chi_{1}}$, are listed. We
have fixed the DM velocity $v\approx 220\,{\rm km/s}$ in the Milky Way for the
inelastic scattering calculation which is relevant for the XENON1T experiment.
For the elastic cross sections, we used the DM velocity $v\approx
30(220,3000)\,{\rm km/s}$ which corresponds to the dwarf galaxies (the Milky
Way, the bullet cluster). In the last line we show the value of kinetic mixing
parameter ($\epsilon=\hat{\epsilon}c_{W}$) which explains the XENON1T excess.
We fixed the other parameters: $\delta=2.8\,{\rm keV}$,
$m_{Z^{\prime}}=10^{-4}\,{\rm eV}$.

The predictions for the elastic and inelastic DM annihilation cross sections
for benchmark DM masses $m_{\chi_{1}}=70,100,120$ (GeV) are shown in Table 1
along with other results. To calculate the cross sections we have fixed,
$v=30,220,3000\,{\rm km/s}$, corresponding to a typical DM velocity at the
dwarf galaxies, the Milky Way, and the clusters of galaxies, respectively. For
the inelastic scattering we show the results only with $v=220\,{\rm km/s}$
which is relevant for the XENON1T experiment. We can see that the inelastic
cross sections have the correct values to solve the small scale problems. The
momentum-transfer cross sections $\sigma_{\rm el}^{T}$ are smaller than the
viscosity cross sections $\sigma_{\rm el}^{V}$ as can be expected from the
suppression of $1-|\cos\theta|$ compared to $\sin^{2}\theta$. But they are
similar in size and either of them can be used to measure the effect of the
elastic scattering. The elastic cross sections are also highly velocity-
dependent and can evade the constraints from the Milky Way and the galaxy
clusters such as the bullet cluster. We note that the results in Table 1 are
not very sensitive to $m_{Z^{\prime}}$ as long as $m_{Z^{\prime}}\ll\delta$.

Some comments are in order. For $v=30,220\,{\rm km/s}$, we sum only up to
$l\lesssim 30$ in (23), (24) and (26) because $M_{ij}^{\ell}\to 0$ rapidly
beyond $\ell\sim 30$. For $v=3000\,{\rm km/s}$ the solution does not decrease
easily as $\ell$ increases and the unitarity of $S$-matrix begins to be
violated by ${\cal O}(1)$ when $\ell\gtrsim 60$, and we stop near $\ell\sim
60$ to keep the unitarity. So the cross sections for $v=3000\,{\rm km/s}$ in
Table 1 are expected to have ${\cal O}(1)$ errors. To make sure the elastic
cross sections are suppressed to satisfy the constraints for this DM velocity
we cross-checked the elastic cross sections using the Born approximation which
is a good approximation in this regime. The elastic scattering occurs at the
second order of the Born expansion. The result, $\sigma^{V}_{\rm
el}/m_{\chi_{1}}=1.37\times 10^{-3},2.26\times 10^{-3},1.88\times
10^{-3}\,{\rm cm^{2}/g}$(for $m_{\chi_{1}}=70,100,120\,{\rm GeV}$,
respectively), indeed shows that they are small enough to satisfy the
constraint $\sigma_{\rm el}/m_{\chi_{1}}\lesssim 0.5\,{\rm cm^{2}/g}$.

##  5 The XENON1T excess

Now the excited state $\chi_{2}$ produced by the inelastic scattering decays
promptly back into $\chi_{1}$ and $Z^{\prime}$,
$\chi_{2}\to\chi_{1}Z^{\prime}$, with 100% branching ratio. Since the mass
difference $\delta=m_{\chi_{2}}-m_{\chi}$ is fixed to be $2.8\,{\rm keV}$, the
energy of $Z^{\prime}$ is also fixed to be that of the $\delta$. The
relativistic $Z^{\prime}$ is absorbed in a xenon atom in the XENON1T
experiment and ejects an electron with energy close to $2.8\,{\rm keV}$ via a
photoelectric-like effect.

The flux of $Z^{\prime}$ per unit energy within the solid angle
$\Delta\Omega$, coming from the inelastic scattering
$\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$ and the subsequent decay
$\chi_{2}\to\chi_{1}Z^{\prime}$, is obtained by

$\displaystyle\frac{d\Phi_{Z^{\prime}}}{dE_{Z^{\prime}}}=\Delta\Omega\,\frac{r_{\odot}}{8\pi}\,\frac{dN_{Z^{\prime}}}{dE_{Z^{\prime}}}\,\left(\frac{\rho_{\odot}}{m_{\chi_{1}}}\right)^{2}\,\langle\sigma_{\rm
inel}v\rangle\,\bar{J},$ (33)

where $r_{\odot}\simeq 8.33$ kpc is the distance from the Earth to the
galactic center (GC), $\rho_{\odot}\simeq 0.3\,{\rm GeV/cm^{3}}$ is the local
DM density, the $J$-factor,
$\bar{J}=\int\frac{d\Omega}{\Delta\Omega}\int_{l.o.s.}\frac{ds}{r_{\odot}}\left(\rho(r)\over\rho_{\odot}\right)^{2}$,
is the line-of-sight integration of the DM density squared, and the energy
spectrum is given by
$\frac{dN_{Z^{\prime}}}{dE_{Z^{\prime}}}=2\delta(E_{Z^{\prime}}-\delta)$ in
our model. Considering the $Z^{\prime}$-flux from the full sky, i.e.
$\Delta\Omega=4\pi$, we get $\bar{J}\simeq 2.20$ by using the cored isothermal
DM density profile Jimenez:2002vy ; Ng:2013xha :

$\displaystyle\rho(r)$
$\displaystyle=\rho_{\odot}\left[\frac{r}{r_{\odot}}\right]^{-\gamma}\left[\frac{1+(r_{\odot}/r_{s})^{\alpha}}{1+(r/r_{s})^{\alpha}}\right]^{\frac{\beta-\gamma}{\alpha}},$
(34)

where $\\{\alpha,\beta,\gamma,r_{s}\\}=\\{2,2,0,3.5\,{\rm kpc}\\}$222We obtain
$\bar{J}=2.97$ for the NFW profile with
$\\{\alpha,\beta,\gamma,r_{s}\\}=\\{1,3,1,20\,{\rm kpc}\\}$.. Then the
differential event rate of the dark-photoelectric effect per ton of the xenon
target per year can be written as Chiang:2020hgb

$\displaystyle\frac{dR}{dE_{e}}$ $\displaystyle=\int
dE_{Z^{\prime}}\,\frac{d\Phi_{Z^{\prime}}}{dE_{Z^{\prime}}}\,\frac{\sigma_{Z^{\prime}}(E_{Z^{\prime}})}{m_{Xe}}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(E_{e}-E_{Z^{\prime}})^{2}}{2\sigma^{2}}}\epsilon(E_{e}),$
(35)

where $E_{e}$ is the emitted electron energy, $m_{Xe}$ is the mass of a xenon
atom, and
$\sigma_{Z^{\prime}}(E_{Z^{\prime}})=\epsilon^{2}\sigma_{\gamma}(E_{Z^{\prime}})$
is the dark-photoelectric cross section of the xenon atom at the energy
$E_{Z^{\prime}}$. The Gaussian function simulates the smearing effect of the
electron energy by the detector resolution with Aprile:2020tmw

$\displaystyle\frac{\sigma}{E_{e}}=\frac{a}{\sqrt{E_{e}/{\rm keV}}}+b,$ (36)

where $a=0.3171\pm 0.0065$ and $b=0.0015\pm 0.0002$. The function
$\epsilon(E_{e})$ is the total detector efficiency reported in Aprile:2020tmw
. We obtain $\sigma_{\gamma}(2.8\,{\rm keV})\simeq 2.0\times 10^{5}\,{\rm
barn}$ XCOM . To fit the XENON1T data we find the required values of
$\epsilon$ values are $\epsilon=5.39\times 10^{-10},1.14\times
10^{-9},2.02\times 10^{-9}$ for $m_{\chi_{1}}=70,100,120$ GeV, respectively.

In Fig. 1 we show the resulting differential event rate (solid red curve) for
a benchmark point, $m_{\chi_{1}}=100\,{\rm GeV}$, $\alpha_{X}=3.59\times
10^{-3}$, $\epsilon=1.14\times 10^{-9}$, $\delta=2.8\,{\rm keV}$, and
$m_{Z^{\prime}}=10^{-4}\,{\rm eV}$. The dashed curve is the contribution of
the NP signal only. The experimental data and the background blue curve are
extracted from Aprile:2020tmw .

These values of $\epsilon$ are consistent with the current experimental
constraints Jaeckel:2013ija ; An:2013yua . The null observation of the dark
matter at the DM-nucleon scattering experiments can be explained by the
$\epsilon^{2}$-suppressed cross section of the relevant scattering
$\chi_{1}q\to\chi_{2}q$ and also by the inelasticity of the scattering.

Figure 1: The differential event rate (solid red curve) for a benchmark point,
$m_{\chi_{1}}=100\,{\rm GeV}$, $\alpha_{X}=3.59\times 10^{-3}$,
$\epsilon=1.14\times 10^{-9}$, $\delta=2.8\,{\rm keV}$, and
$m_{Z^{\prime}}=10^{-4}\,{\rm eV}$. The dashed curve is the contribution of
the NP signal only. The experimental data and the background blue curve are
extracted from Aprile:2020tmw .

##  6 Conclusions

Although the existence of dark matter is well-established, its particle nature
is almost unknown. The small scale problem and the recent observation of the
excess in electron-recoil at XENON1T experiment may reveal the nature of dark
matter. We studied a dark matter model which can address these issues while
explaining its abundance in the universe.

In the model the Majorana dark matter candidate $\chi_{1}$ has its excited
partner $\chi_{2}$ with mass difference $\delta\simeq 2.8$ keV. The dark mass
is about $100$ GeV and it can explain the current relic density by the thermal
freeze-out mechanism whose main annihilation process is
$\chi_{1}\chi_{1}(\chi_{2}\chi_{2})\to Z^{\prime}Z^{\prime}$. We find that the
necessary dark gauge coupling is $\alpha_{X}\sim 10^{-3}$.

The light $Z^{\prime}$ can also mediate (in)elastic scattering
$\chi_{1}\chi_{1}\to\chi_{1(2)}\chi_{1(2)}$. We solve the Schrödinger equation
numerically to calculate the cross sections. We get the elastic cross section
large enough to explain the small scale problems $\sigma_{\rm
el}/m_{\chi_{1}}\sim 1\;{\rm cm^{2}/g}$ .

The dark sector can communicate with the SM sector through kinetic mixing
parameterized by $\epsilon\sim 10^{-10}$. In the current universe the rate for
the up-scattering $\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$ followed by
$\chi_{2}\to\chi_{1}Z^{\prime}$ can be enhanced by the small $Z^{\prime}$ mass
$m_{Z^{\prime}}\ll\delta$. The energetic $Z^{\prime}$ is absorbed by the xenon
atom at the XENON1T detector via the mechanism similar to the photoelectric
effect. The above mentioned values of $\epsilon$ and $\delta$ can explain the
spectrum and the excess event rate observed by the XENON1T.

###### Acknowledgements.

This work was supported in part by the National Research Foundation of
Korea(NRF) grant funded by the Korea government(MSIT), Grant No.
NRF-2018R1A2A3075605.

## References

  * (1) S. Tulin and H.-B. Yu, _Dark Matter Self-interactions and Small Scale Structure_ , _Phys. Rept._ 730 (2018) 1 [1705.02358].
  * (2) D. Harvey, R. Massey, T. Kitching, A. Taylor and E. Tittley, _The non-gravitational interactions of dark matter in colliding galaxy clusters_ , _Science_ 347 (2015) 1462 [1503.07675].
  * (3) M. Blennow, S. Clementz and J. Herrero-Garcia, _Self-interacting inelastic dark matter: A viable solution to the small scale structure problems_ , _JCAP_ 03 (2017) 048 [1612.06681].
  * (4) XENON collaboration, _Observation of Excess Electronic Recoil Events in XENON1T_ , _Phys. Rev. D_ 102 (2020) 072004 [2006.09721].
  * (5) K. Harigaya, Y. Nakai and M. Suzuki, _Inelastic Dark Matter Electron Scattering and the XENON1T Excess_ , _Phys. Lett. B_ 809 (2020) 135729 [2006.11938].
  * (6) H.M. Lee, _Exothermic dark matter for XENON1T excess_ , _JHEP_ 01 (2021) 019 [2006.13183].
  * (7) J. Bramante and N. Song, _Electric But Not Eclectic: Thermal Relic Dark Matter for the XENON1T Excess_ , _Phys. Rev. Lett._ 125 (2020) 161805 [2006.14089].
  * (8) H. An and D. Yang, _Direct detection of freeze-in inelastic dark matter_ , 2006.15672.
  * (9) W. Chao, Y. Gao and M.j. Jin, _Pseudo-Dirac Dark Matter in XENON1T_ , 2006.16145.
  * (10) S. Baek, J. Kim and P. Ko, _XENON1T excess in local $Z_{2}$ DM models with light dark sector_, _Phys. Lett. B_ 810 (2020) 135848 [2006.16876].
  * (11) H.-J. He, Y.-C. Wang and J. Zheng, _EFT Approach of Inelastic Dark Matter for Xenon Electron Recoil Detection_ , _JCAP_ 01 (2021) 042 [2007.04963].
  * (12) D. Choudhury, S. Maharana, D. Sachdeva and V. Sahdev, _Dark matter, muon anomalous magnetic moment, and the XENON1T excess_ , _Phys. Rev. D_ 103 (2021) 015006 [2007.08205].
  * (13) Y. Ema, F. Sala and R. Sato, _Dark matter models for the 511 keV galactic line predict keV electron recoils on Earth_ , _Eur. Phys. J. C_ 81 (2021) 129 [2007.09105].
  * (14) D. Borah, S. Mahapatra, D. Nanda and N. Sahu, _Inelastic fermion dark matter origin of xenon1t excess with muon $(g-2)$ and light neutrino mass_, _Phys. Lett. B_ 811 (2020) 135933 [2007.10754].
  * (15) D. Borah, S. Mahapatra and N. Sahu, _Connecting Low scale Seesaw for Neutrino Mass and Inelastic sub-GeV Dark Matter with Abelian Gauge Symmetry_ , 2009.06294.
  * (16) A. Aboubrahim, M. Klasen and P. Nath, _Xenon-1T excess as a possible signal of a sub-GeV hidden sector dark matter_ , 2011.08053.
  * (17) H.-J. He, Y.-C. Wang and J. Zheng, _GeV Scale Inelastic Dark Matter with Dark Photon Mediator via Direct Detection and Cosmological/Laboratory Constraints_ , 2012.05891.
  * (18) M. Dutta, S. Mahapatra, D. Borah and N. Sahu, _Self-interacting Inelastic Dark Matter in the light of XENON1T excess_ , 2101.06472.
  * (19) Y. Cui, D.E. Morrissey, D. Poland and L. Randall, _Candidates for Inelastic Dark Matter_ , _JHEP_ 05 (2009) 076 [0901.0557].
  * (20) K.S. Babu, C.F. Kolda and J. March-Russell, _Implications of generalized Z - Z-prime mixing_ , _Phys. Rev. D_ 57 (1998) 6788 [hep-ph/9710441].
  * (21) K. Griest and D. Seckel, _Three exceptions in the calculation of relic abundances_ , _Phys. Rev. D_ 43 (1991) 3191.
  * (22) E.W. Kolb and M.S. Turner, _The Early Universe_ , Westview Press (1990).
  * (23) S. Ershov, J. Vaagen and M. Zhukov, _Modified variable phase method for the solution of coupled radial Schrodinger equations_ , _Phys. Rev. C_ 84 (2011) 064308.
  * (24) F. Kahlhoefer, K. Schmidt-Hoberg and S. Wild, _Dark matter self-interactions from a general spin-0 mediator_ , _JCAP_ 08 (2017) 003 [1704.02149].
  * (25) R. Jimenez, L. Verde and S. Oh, _Dark halo properties from rotation curves_ , _Mon. Not. Roy. Astron. Soc._ 339 (2003) 243 [astro-ph/0201352].
  * (26) K.C.Y. Ng, R. Laha, S. Campbell, S. Horiuchi, B. Dasgupta, K. Murase et al., _Resolving small-scale dark matter structures using multisource indirect detection_ , _Phys. Rev. D_ 89 (2014) 083001 [1310.1915].
  * (27) C.-W. Chiang and B.-Q. Lu, _Evidence of a simple dark sector from XENON1T excess_ , _Phys. Rev. D_ 102 (2020) 123006 [2007.06401].
  * (28) M. Berger, J. Hubbell, S. Seltzer, J. Chang, J. Coursey, R. Sukumar et al., “XCOM: Photon cross sections database.” http://www.nist.gov/pml/data/xcom/index.cfm, 2010.  https://dx.doi.org/10.18434/T48G6X.
  * (29) J. Jaeckel, _A force beyond the Standard Model - Status of the quest for hidden photons_ , _Frascati Phys. Ser._ 56 (2012) 172 [1303.1821].
  * (30) H. An, M. Pospelov and J. Pradler, _Dark Matter Detectors as Dark Photon Helioscopes_ , _Phys. Rev. Lett._ 111 (2013) 041302 [1304.3461].