#  Self-Destructing Dark Matter

Yuval Grossman<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell
University, Ithaca, NY 14853, USA  Roni Harnik<EMAIL_ADDRESS>Theoretical
Physics Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA  Ofri Telem
<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell University, Ithaca, NY
14853, USA  Yue Zhang<EMAIL_ADDRESS>Theoretical Physics
Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA  Department of
Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

###### Abstract

We present Self-Destructing Dark Matter (SDDM), a new class of dark matter
models which are detectable in large neutrino detectors. In this class of
models, a component of dark matter can transition from a long-lived state to a
short-lived one by scattering off of a nucleus or an electron in the Earth.
The short-lived state then decays to Standard Model particles, generating a
dark matter signal with a visible energy of order the dark matter mass rather
than just its recoil. This leads to striking signals in large detectors with
high energy thresholds. We present a few examples of models which exhibit self
destruction, all inspired by bound state dynamics in the Standard Model. The
models under consideration exhibit a rich phenomenology, possibly featuring
events with one, two, or even three lepton pairs, each with a fixed invariant
mass and a fixed energy, as well as non-trivial directional distributions.
This motivates dedicated searches for dark matter in large underground
detectors such as Super-K, Borexino, SNO+, and DUNE.

††preprint: FERMILAB-PUB-17-505-T††preprint: NUHEP-TH-17-07

##  I Introduction

Our knowledge of the nature of dark matter (DM) is very limited. Though it
accounts for over 80% of the matter in the universe, it is often assumed that
the dark sector is minimal, consisting of a single particle that interacts
only feebly with the Standard Model (SM). This assumption, which is the most
sensible first guess, motivates a set of strategies in our search for DM. In
particular, when searching for DM directly, we try to detect the low energy
recoil of a nucleus which was struck by a slowly moving DM particle
Goodman:1984dc . DM direct detection thus focuses on detectors with thresholds
between sub keV and tens of keV.

The current search program is undoubtedly important and must be pursued
further. However, we should not let minimality be our only guide. Indeed,
considering even slightly non-minimal DM models has led to significant
variations in direct detection phenomenology TuckerSmith:2001hy ;
Finkbeiner:2007kk ; Chang:2008gd ; Bramante:2016rdh ; Chang:2008xa ;
Graham:2010ca ; Feldstein:2010su ; Essig:2010ye ; Bai:2009cd ; Pospelov:2008qx
; An:2012bs ; Chang:2009yt ; Chang:2010en ; Pospelov:2013nea ; Kumar:2011iy ,
including changes in the predicted recoil spectrum, annual modulation, and in
the manner in which the DM deposits energy in the detector. These interesting
models, however, keep direct detection phenomenology within the purview of low
threshold detectors. The reason for this is that the available energy in the
problem is at most the kinetic energy in the DM-nucleus or DM-electron system.
This energy is of order $\mu v^{2}$, where $v\sim 10^{-3}c$ is the DM
velocity, and $\mu$ is the reduced mass of the system.

In this paper, we consider a drastic departure from this picture and discuss
the possibility that DM is capable of leaving not just kinetic energy in the
detector, but rather _all_ of its mass. This can lead to DM direct detection
signals above MeV, allowing to extend the search for DM to detectors with much
higher thresholds. In particular, large existing neutrino experiments such as
Borexino and Super-Kamiokande, as well as future experiments such as SNO+ and
DUNE, can probe self destructing DM models.

A naïve idea to convert the DM rest mass to detectable signal is to consider
its down-scattering to a nearly massless state, with the difference in mass
converted to recoil energy. This is however unfeasible since it leads to a DM
particle that is cosmologically unstable by decaying to this nearly massless
state. Instead, we must consider a non-minimal setup in which an interaction
of DM with a nucleus triggers a transition from a state that is stable on
cosmological timescales to one which is very short-lived. The signal is then
not a recoil signal, but rather a subsequent decay of the short-lived state to
SM particles. A sketch of this is depicted in Figure 1.

Figure 1: A schematic picture of self-destructing DM, where DM denotes the
cosmologically long-lived state and DM’ denotes the very short-lived one.

The crucial ingredients in the story are a cosmologically long lived state and
a short lived state, with a possible transition between the two which is
induced by the interaction with the Earth. We call this class of DM models
self-destructing DM (SDDM for short).

SDDM is not the first class of DM models in which the measured energy is more
than just the DM recoil energy. In Graham:2010ca , for example, the DM down-
scatters and converts part of its rest mass to recoil energy. However, in that
model the mass splittings considered were still of the same order as the
kinetic energy. In Pospelov:2013nea  the possibility of releasing more than
the recoil energy was considered, but with signals still in the domain of low
threshold detectors. Other DM models Davoudiasl:2011fj ; Davoudiasl:2010am ;
Huang:2013xfa ; Agashe:2014yua ; Berger:2014sqa ; Kong:2014mia ;
Alhazmi:2016qcs , are similar to our scenario in that they can be probed at
neutrino detectors Kachulis:2017nci . Yet, in these models, the dynamics that
lead to signals at neutrino detector (a “removal” of a baryon from the
detector or a boosted population of DM) is different than the case of SDDM.
Lastly, we mention that the axion and other ultralight bosons that can be
fully absorbed by a detector may be considered a fully SDDM candidate, but
their lightness leads to much lower energy deposition, and thus to very
different experimental signatures Sikivie:1983ip ; Chaudhuri:2014dla ;
Hochberg:2016sqx .

The self-destructing scenario is possible thanks to the high density of matter
in the Earth, one which is unprecedented from the perspective of an incoming
DM particle. The expected number of DM interactions off of a target with
number density $n$ is roughly

$\langle N\rangle\sim n\langle\sigma v\rangle\Delta t\,,$ (1)

where $v$ is the typical relative velocity, $\sigma$ is the interaction cross
section, and $\Delta t$ is the amount of time spent in this environment. A DM
particle with $v\sim 10^{-3}c$ crosses the Earth in about 10 seconds. However,
before its short traversal through Earth, it had spent about $10^{16}$ more
time in the low density environment of the galaxy. On the other hand, the
number density of atoms in the Earth is about $10^{23}$ larger than that in
the galaxy. As a result, it is reasonable for the expected rate of transitions
to an unstable state to be much larger in one Earth-crossing than it is in the
preceding Hubble time. Of course, one should not forget that in the early
universe the DM candidate was immersed in a dense thermal bath (In particular,
the baryon number density of the universe is comparable to that of the Earth
at temperatures around MeV scale). Avoiding self-destruction of dark matter in
these early times will be a requirement for our models and will be discussed
later.

In the upcoming sections, we present three models which realize the idea of
SDDM. These ideas are intended as a proof of principle rather than as
complete, well motivated models. In each case, DM is stabilized by a different
principle that can be un-done by interactions in the Earth. None of these
models is minimal by any means. They are, however, inspired by phenomena that
are observed in the visible sector. Particles in the Standard Model find
themselves to be either short-lived or meta-stable for a variety of reasons.
In many cases different effects, forces, or interactions come into play,
balancing one against the other to produce a long- or short-lived state. We
are therefore quite liberal in allowing for several particles and
interactions. Furthermore, like the visible sector which contains several
populations of cosmic relics, we will be open to the possibility that only a
subcomponent of DM is self-destructing.

The model building challenge for SDDM is to have a dark matter component that
is very long-lived, yet capable self-annihilating once it interacts with a
nucleus in the Earth. In its long-lived state, this component needs to have a
lifetime for more than about $10^{28}$ sec to satisfy observational
constraints Arvanitaki:2008hq ; Arvanitaki:2009yb ; Essig:2013goa , if it
makes up all of DM. Even if we consider only a small sub-component of DM to
relax this bound, the state needs to live for more than the age of the
universe. In contrast, the short-lived state of our DM component has to decay
in much less than 10 seconds, its Earth-crossing time.

The paper is organized as follows: In Section II we review the first in our
class of SDDM models. In this model, some of the DM consists of excited
positronium-like bound states which are stable due to a large angular momentum
barrier. Interaction with the matter in the Earth can change the angular
momentum of the bound state, rendering it unstable. We then estimate the
signal rate of this model in large neutrino detectors and discuss
possibilities for its early Universe history. In the second model, presented
in Section III, the DM is in a metastable minimum far away from the origin of
a bound-state potential. Interaction with the Earth can shake the internal
structure and move the system to the global minimum where annihilation is very
fast. In our last example, Section IV, DM consists of dark baryons. A baryon
number violating interaction with matter causes a transition from a dark
baryon to a dark meson which can now decay in a detector. In Section V, we
discuss the phenomenological signals of our framework in large detectors
further in order to motivate more detailed studies. We conclude and provide
some further directions for exploration in Section VI.

##  II High angular momentum stabilization

In this model, a component of the DM is a positronium-like bound state which
is stabilized by angular momentum conservation. In particular, the self-
annihilation of this “dark positronium” is suppressed due to a large angular
momentum barrier. An interaction with the Earth allows the dark matter to
transition to a lower angular momentum state that has a lifetime which is
exponentially shorter.

The dark sector contains light fermions $\chi$ and $\bar{\chi}$ which are
charged under two broken abelian gauge groups $U(1)_{\phi}$ and $U(1)_{V}$.
The Lagrangian for this sector takes the form

$\mathcal{L}=\bar{\chi}i\not{D}\chi-
m_{\chi}\bar{\chi}\chi-\frac{1}{4}\phi^{\mu\nu}\phi_{\mu\nu}+{1\over
2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}+\frac{1}{2}m_{V}^{2}V_{\mu}V^{\mu}-\frac{\epsilon}{2}V_{\mu\nu}F^{\mu\nu}\
,$ (2)

where

$D_{\mu}\chi=(\partial_{\mu}+ig_{\phi}\phi_{\mu}+ig_{V}V_{\mu})\chi\ ,$ (3)

and $\phi_{\mu\nu}$, $V_{\mu\nu}$, and $F_{\mu\nu}$ are the field strengths
for $\phi$, $V$ and the standard model photon respectively. For simplicity, we
assume no kinetic mixing of $\phi$ to any of the other gauge bosons. We
further define the fine-structure constants, $\alpha_{V}=g_{V}^{2}/(4\pi)$ and
$\alpha_{\phi}=g_{\phi}^{2}/(4\pi)$, and the usual
$\alpha_{\mathrm{EM}}=e^{2}/(4\pi)$.

The lighter of the two gauge bosons, $\phi$, is assumed to be light enough to
mediate a long range force which leads to $\chi\bar{\chi}$ positronium-like
bound states with $n_{*}\sim 10$ energy levels. On the other hand, $\phi$ is
taken to be too heavy to be emitted on-shell in transitions among energy
levels. These constraints give

$\frac{1}{4}\alpha_{\phi}^{2}m_{\chi}<m_{\phi}<\frac{1}{2n_{*}}\alpha_{\phi}m_{\chi}\,.$
(4)

We further assume that $V$ is too heavy to play a role in the bound state
dynamics, but is light enough to be emitted in the self-annihilation of the
$\chi\bar{\chi}$ state, thus

$\frac{1}{2}\alpha_{\phi}m_{\chi}<m_{V}\lesssim m_{\chi}\ .$ (5)

We assume that $\alpha_{V}\gtrsim\alpha_{\phi}$ so that on-shell $V$s are
produced in a sizable fraction of the annihilation processes. Our last
assumption is

$m_{V}>2m_{e}\ ,$ (6)

so $V$ could decay into $e^{+}e^{-}$ and can be detected easily.

Before proceeding to estimate lifetimes, we comment on some possible
alternative models. For example, one could consider a model with a single
vector boson from the dark sector playing the roles of both $\phi$ and $V$.
However, in this case, Eq. (6) with $\alpha_{\phi}\lesssim 10^{-2}$ and
$n_{*}\sim 10$, and the upper bound in Eq. (4) can only be reconciled when
$m_{\chi}\gg\,1\;\text{GeV}$. As we are interested in a wide range of
$m_{\chi}$, as low as the MeV scale, see Section II.2, we keep both $\phi$ and
$V$ in the following discussion. Another possibility is to allow $\phi$ to
also mix with the SM photon, a possibility which, for the most part, affect
the signal rate estimated in Section II.2. In addition, there may be other
variants of this model where either or both $V$ and $\phi$ are replaced by
light scalars or pseudoscalars. We focus here on the gauge case for
concreteness but other options are considered when the phenomenology is
discussed in Section V.

###  II.1 Bound state lifetimes

We now consider the lifetimes of the $\chi\bar{\chi}$ bound states. As we show
below, states with $\ell\gtrsim 10$ have lifetimes much longer than the
cosmological bound on DM lifetime, while $S$-wave states annihilate very
quickly, within the time it traverses the Earth.

In the vacuum, bound states could be destroyed either by direct
$\chi\bar{\chi}$ annihilation or by de-excitation to a lower-$\ell$ state. We
begin with an estimate of direct annihilation to $V$s. (In the following we
also refer to $V$ as the dark photon.) The number of $V$s in the final state
is denoted by $N_{V}$. This number is 2 for bound states that are even under
charge conjugation and 3 for odd ones. The leading decay to consider is thus
$\Psi_{n,\ell}\to VV$ or $\Psi_{n,\ell}\to VVV$. The amplitude for
annihilation into dark photons for the $\Psi_{n,\ell}$ state is proportional
to the $\ell$th derivative of the wave function at the origin. Roughly
speaking every derivative result in an extra power of $\alpha_{\phi}$ in the
amplitude. Assuming no significant phase space suppression, these decay rates
can be estimated by An:2016gad ,

$\Gamma_{n,\ell\to V^{\prime}s}\sim\left({\alpha_{\phi}\over
n}\right)^{2\ell+3}\alpha_{V}^{N_{V}}m_{\chi}\ ,$ (7)

Annihilation into $\phi\phi$ or into $\phi V$ have a similar power counting.
With $m_{\chi}\sim 1\,{\rm GeV}$, and $\alpha_{V}\sim\alpha_{\phi}\sim
10^{-2}$ we find that, for example, the inverse width of annihilation of the
$\Psi_{n=10,\ell=9}$ state is about $10^{42}$ seconds, while
$\Psi_{n=7,\ell=6}$ lives for about $10^{22}$ sec. The latter is too short to
evade bounds on the lifetime of all of DM, but is safe if that state is a
small fraction of the DM.

Next, we consider de-excitation. Since
$m_{V},m_{\phi}>\alpha_{\phi}^{2}m_{\chi}/4$, the de-excitation of
$\Psi_{n,\ell}$ to lower states cannot happen via on-shell $\phi$s or $V$s.
Rather the leading decay is to three photons through an off shell $V$ or an
off shell $\phi$, if it couples to the SM. The binding energy difference, of
order $\sim\alpha_{\phi}^{2}m_{\chi}$, is assumed to be too small to emit a
pair of $e^{+}e^{-}$ for all the parameter space under consideration. There is
also the possibility for the off-shell $V^{*}$ to turn into two neutrinos via
its kinetic mixing with the SM $Z$-boson. The partial rate, in this case, is
even more suppressed than the two or three photon final states we are
considering.

The de-excitation rate of $\Psi_{n,\ell}\to\Psi_{n-1,\ell-1}$ via off-shell
$V$ is approximately

$\displaystyle\Gamma_{n,\ell\rightarrow
n-1,\ell-1}\sim\left(\frac{\alpha_{\phi}}{n^{2}}\right)^{2}\frac{\alpha_{V}\epsilon^{2}\alpha_{\mathrm{EM}}}{m_{V}^{4}}\frac{\alpha^{3}_{EM}}{m_{e}^{8}}\left(\frac{\alpha_{\phi}^{2}m_{\chi}}{n^{3}}\right)^{13}\
,$ (8)

where the term in the first parentheses $\alpha_{\phi}/n^{2}$ is the matrix
element for dark photon radiation from the bound states, and the term in the
last parentheses $\alpha_{\phi}^{2}m_{\chi}/n^{3}$ stands for the difference
in binding energy. This dimensionful quantity is raised to the thirteenth
power in order to make up the dimension of the rate. For example, with
$m_{\chi}\sim m_{V}\sim 1\,{\rm GeV}$, $\epsilon\sim 10^{-2}$, and
$\alpha_{V}\sim\alpha_{\phi}\sim 10^{-2}$ we find that the lifetime of the
$\Psi_{n=10,\ell=9}$ state is about $10^{41}$ seconds, making it
cosmologically stable. The lifetime is even longer for smaller values of
$m_{\chi}$, or for transitions with $\Delta\ell>1$.

If $\phi$ also mixes with the photon, the bound state de-excitation rate via
off-shell $\phi$ can be estimated in a similar way, with the $V$ kinetic
mixing parameter and the dark photon mass replaced by those of $\phi$. If
$\phi$ is a scalar particle and does not mix with the SM Higgs boson, the de-
excitation is even slower because it requires at least three loops and two
powers of the kinetic mixing parameter $\epsilon$ for it to decay into two SM
photons.

We have thus shown that high-$\ell$ excitations of the $\chi\bar{\chi}$ bound
state are cosmologically stable and are a viable candidate to be at least a
portion of DM. On the other hand, the bound states with small $n$ and $\ell$
quantum numbers can decay much faster, into $2V$ or $3V$, depending on its
charge-conjugation parity ($C=(-1)^{L+S}$) as discussed above. The amplitude
for an $S$-wave $\chi\bar{\chi}$ bound state to annihilate into $2V$ is
proportional to its wave function at the origin, and, neglecting phase space
effects, the decay rate is

$\displaystyle\begin{split}&\Gamma_{\Psi^{(+)}_{n,0}\to
2V}=\frac{\alpha_{\phi}^{3}\alpha_{V}^{2}m_{\chi}}{2n^{3}}\,,\\\
&\Gamma_{\Psi^{(-)}_{n,0}\to
3V}=\frac{2(\pi^{2}-9)\alpha_{\phi}^{3}\alpha_{V}^{3}m_{\chi}}{9\pi
n^{3}}\,.\end{split}$ (9)

Setting $m_{\chi}=1\,$GeV, $\alpha_{V}=\alpha_{\phi}=0.01$, and a DM speed of
$v\sim 10^{-3}$ we find the decay length of the ground state to be,
$v\tau_{\Psi^{(+)}_{1,0}\to 2V}\sim 4\times 10^{-7}$ cm, or
$v\tau_{\Psi^{(-)}_{1,0}\to 3V}\sim 3\times 10^{-4}$ cm, which is a prompt
decay. For an S-wave state with $n\leq 10$ the decay is prompt in view of DM
or neutrino detectors.

In conclusion, the model has the two desirable ingredients: a cosmologically
long lived state and a short lived one that can generate a signal in the
detector. The next task is to estimate the rate for transitioning from the
long to the short lived states.

###  II.2 Bound state scattering

In this subsection, we demonstrate how collisions of a high-$\ell$, long-lived
bound state $\Psi$ with a nucleus can induce a transition to a lower $\ell$
state $\Psi^{\prime}$ that decays quickly into SM particles. The amplitude for
the leading non-relativistic inelastic scattering of a proton with a bound
state

$\Psi(\vec{p}_{1})+p(\vec{p}_{2})\to\Psi^{\prime}(\vec{p}_{3})+p(\vec{p}_{4}),$
(10)

occurs via $t$-channel dark photon ($V$) exchange and it takes the form

$\displaystyle\mathcal{M}_{\Psi+p\to\Psi^{\prime}+p}=4m_{\chi}F_{D}(|\vec{q}\,|)\frac{g_{V}\epsilon
e}{|\vec{q}|^{\,2}+m_{V}^{2}}\bar{u}_{p}(\vec{p}_{4})\gamma^{0}u_{p}(\vec{p}_{2})\
.$ (11)

Here $u_{p}$ is the proton spinor, and $F_{D}(|\vec{q}\,|)$ is the form factor
for the $\Psi\to\Psi^{\prime}$ transition in the presence of a three-momentum
$\vec{q}=\vec{p}_{3}-\vec{p}_{1}$ injection,

$\displaystyle F_{D}(|\vec{q}\,|)=\int
d^{3}{\vec{x}}\,\Psi^{\prime*}(\vec{x})\Psi(\vec{x})\left[e^{{i\vec{q}\cdot\vec{x}}/{2}}-e^{{-i\vec{q}\cdot\vec{x}}/{2}}\rule{0.0pt}{11.38109pt}\right]\
.$ (12)

In the following we estimate $F_{D}(|\vec{q}\,|)$ by making the approximation
that $\Psi$ and $\Psi^{\prime}$ are Coulomb-like wave functions. Parity
invariance implies that the orbital angular quantum numbers of the initial and
final states in Eq. (12) must differ by an odd integer.

The differential scattering cross section off a nucleus (with atomic and mass
numbers $Z,A$) takes the approximate form

$\displaystyle\frac{d\sigma_{\rm
scatter}}{d|\vec{q}\,|^{2}}\simeq\frac{g_{V}^{2}\epsilon^{2}e^{2}}{4\pi
v^{2}(|\vec{q}|^{\,2}+m_{V}^{2})^{2}}\times\left|F_{D}(|\vec{q}\,|)\right|^{2}\times
Z^{2}F^{2}(|\vec{q}|)\ ,$ (13)

where $v\simeq 10^{-3}$ is the incoming DM bound state velocity. For
$F(|\vec{q}|)$, the nuclear form factor, we use the Woods-Saxon form given in
Jungman:1995df . Note that when $\Psi^{\prime}$ is a deeper bound state than
$\Psi$, both the binding energy difference and the initial kinetic energy
contribute to the momentum transfer, $\vec{q}$. In the small $v$ limit, the
upper and lower limit of $|\vec{q}|^{2}$ take the form

$\left(|\vec{q}|^{2}\right)_{\rm min}^{\rm
max}=2\mu_{\Psi^{\prime}A}(m_{\Psi}-m_{\Psi^{\prime}})\pm\mu_{\Psi^{\prime}A}v\left(\frac{m_{\Psi}}{m_{\Psi^{\prime}}}\right)\sqrt{8\mu_{\Psi^{\prime}A}(m_{\Psi}-m_{\Psi^{\prime}})}+\mathcal{O}(v^{2})\
,$ (14)

where the reduced mass $\mu_{\Psi^{\prime}A}$ is defined as
$\mu_{\Psi^{\prime}A}=m_{\Psi^{\prime}}m_{A}/(m_{\Psi^{\prime}}+m_{A})$. For
weakly bounded states, we have $m_{\Psi}\sim m_{\Psi^{\prime}}\simeq
2m_{\chi}$ and $m_{\Psi}-m_{\Psi^{\prime}}\sim m_{\chi}\alpha_{\phi}^{2}$. The
above expansion converges for $\alpha_{\phi}\gtrsim v$, i.e., when the binding
energy release, $m_{\Psi}-m_{\Psi^{\prime}}\sim m_{\chi}\alpha_{\phi}^{2}$,
dominates over the DM kinetic energy, $\sim m_{\chi}v^{2}$. To further
simplify the result, we consider the limit $|\vec{q}^{2}|\ll m_{V}^{2}$ and
$m_{\Psi}\ll m_{A}$, which is the region of interest for this study. In this
case, the scattering cross section is approximately,

$\displaystyle\sigma_{\rm
scatter}\simeq\frac{64\pi\epsilon^{2}\alpha_{\mathrm{EM}}\alpha_{V}\alpha_{\phi}m_{\chi}^{2}}{\pi
vm_{V}^{4}}\times\left|F_{D}(|\vec{q}\,|)\right|^{2}\times
Z^{2}F^{2}(|\vec{q}|)\ ,$ (15)

with $F_{D}(|\vec{q}\,|)$ and $F(|\vec{q}\,|)$ evaluated at the typical
momentum transfer, $|\vec{q}|\sim m_{\chi}\alpha_{\phi}$. Numerically,
$F(|\vec{q}|)\simeq 1$ and the typical value of $F_{D}(|\vec{q}\,|)$ is around
$10^{-2}$ for $\alpha_{\phi}=10^{-3}-10^{-2}$.

Before proceeding to estimate a signal rate, we note that if the lighter
vector boson $\phi$ were allowed to have a kinetic mixing
$\epsilon_{\phi}\phi_{\mu\nu}F^{\mu\nu}$, the scattering rate in Equation (15)
would have an additional contribution with $m_{V}$ and$\epsilon$ replaced by
$m_{\phi}$ and$\epsilon_{\phi}$. This contribution can potentially be larger
because $\phi$ is lighter. In this sense the rate estimate in the next
subsection, where $\epsilon_{\phi}$ is set to zero, is conservative.

###  II.3 Signal Rates in Neutrino Detectors

Once a high-$\ell$ state hits the Earth, it scatters and transitions to a
low-$\ell$ state which is short-lived. Generically, the short-lived state
decays promptly to two or three $V$s, both of which can have signals in a
large detector. The signal event rate depends primarily on the rate of
transitions from long- to short-lived states. It takes the familiar form

$\Gamma_{\mathrm{signal}}=N_{T}^{(\mathrm{eff})}n_{\Psi}\left<\sigma
v\right>_{\text{scatter}}\ ,$ (16)

where $N_{T}^{(\mathrm{eff})}$ is the effective number of target nuclei which
is defined below and $\left<\sigma v\right>_{\text{scatter}}$ is the velocity
average of the cross section to scatter and transition from the high to the
low-$\ell$ state.

The precise nature of the signal and the effective number of target nuclei
depend on the lifetime of the short lived state $\Psi^{\prime}$ and and the
mediator $V$. It is convenient to define the average decay lengths for these
two states

$\displaystyle
L_{\Psi^{\prime}}=\gamma_{\Psi^{\prime}}v_{\Psi^{\prime}}\tau_{\Psi^{\prime}}\
,\qquad L_{V}=\gamma_{V}v_{V}\tau_{V}\ ,$ (17)

where the $\tau$s are the respective lifetimes of the two particles,
$\gamma_{\Psi^{\prime}}\approx 1$ and $v_{\Psi^{\prime}}\sim 10^{-3}$ are the
typical boost and velocity of the meta-stable state, while $\gamma_{V}$ and
$v_{V}$ are the boost and velocity of $V$ which are set in any given
kinematics. In the case of $\Psi^{\prime}\to 2V$ decay,

$L_{V}=c\tau_{V}\sqrt{{m_{\Psi^{\prime}}^{2}}/{4m_{V}^{2}}-1}.$ (18)

We can now estimate the effective number of target nuclei
$N_{T}^{(\mathrm{eff})}$. If both decay lengths are shorter than the typical
size of the detector, $L_{\Psi^{\prime}},L_{V}<L_{\mathrm{det}}$ the
calculation is simple. Since neither particle can travel far the transition
scattering has most likely happened inside the detector and the number of
targets is simply the number of nuclei in the detector

$\displaystyle L_{\Psi^{\prime}},L_{V}<L_{\mathrm{det}}\,:\qquad\qquad
N_{T}^{(\mathrm{eff})}\sim n_{\mathrm{det}}\,V_{\mathrm{det}}\ ,$ (19)

where $n_{\mathrm{det}}$ and $V_{\mathrm{det}}$ are the number density of
nuclei and the volume of the detector. In this case, the coherence factor in
scattering, the $Z^{2}$ factor in Eq. (15), should be taken as that of the
detector material, be it water, liquid argonne, etc.

In the case that either of the decay lengths is larger than the typical
detector size, $L_{\Psi^{\prime}}>L_{\mathrm{det}}$ or
$L_{V}>L_{\mathrm{det}}$, the production of $V$s is likely to occur elsewhere
in the Earth. It is interesting that even in this case, the effective number
of target nuclei is parametrically the same as in the short lifetime case.
Consider, for example, the case where $L_{V}$ is much longer than the detector
but shorter than the size of the earth, while $L_{\Psi^{\prime}}$ is short. In
this case, we can assume the decay of $\Psi^{\prime}$ to $V$s occurred at the
scatter site. The effective number of targets is a sum over Earth nuclei,
weighted by the probability that a $V$ decays inside the detector given that
is originated near the nucleus in question

$\displaystyle\begin{split}L_{\mathrm{det}}<L_{V}\lesssim R_{\oplus}\,:\qquad
N_{T}^{(\mathrm{eff})}&\sim n_{\oplus}\int_{\oplus\cap
L_{V}}d^{3}\vec{r}_{T}\frac{L^{2}_{\mathrm{det}}}{4\pi|\vec{r}_{\mathrm{det}}-\vec{r}_{T}|^{2}}\frac{L_{\mathrm{det}}}{L_{V}}\sim
n_{\oplus}\,V_{\mathrm{det}}\ ,\end{split}$ (20)

where the integration limits are

$|\vec{r}_{T}|<R_{\oplus}\quad\mathrm{and}\qquad
0<|\vec{r}_{\mathrm{det}}-\vec{r}_{T}|<L_{V}$ (21)

in a coordinate system in which the origin is at the center of the Earth. Here
we have treated the earth as a sphere of constant density $n_{\oplus}$. The
fraction in the integrand is the suppression due to the small solid angle that
the detector subtends, as seen from the scatter site. In the last step, we
have used the fact this suppression factor is undone by the fact that the
number of target nuclei grows like $|\vec{r}_{\mathrm{det}}-\vec{r}_{T}|^{2}$
(which will show up as a Jacobian). This effect is similar to the signal rate
in Feldstein:2010su . Note that here we do not account for an $O(1)$
geometrical factor, which, for example, comes from the fact that there is more
Earth below the detector than above it. The case in which both $L_{V}$ and
$L_{\Psi^{\prime}}$ are larger than the detector is slightly more convolved
but parametrically similar. Note, that in cases of long lived $V$ and/or
$\Psi^{\prime}$ the coherence factor of $Z^{2}$ would be the averaged over the
rock, perhaps dominated by iron, but may get contributions from heavier
elements such as lead. In our simplified analysis we took Earth’s composition
to be $100\%$ water, which is valid up to an $\mathcal{O}(1)$ factor.

Though the signal rate does not depend strongly on $L_{V}$, the character of
the signal does. In the case of a short lived $V$, $L_{V}<L_{\mathrm{det}}$,
both $V$s can decay inside the detector and will result in _two_ back-to-back
or _three_ $e^{+}e^{-}$ (or $\mu^{+}\mu^{-}$, if kinematically allowed) pairs.
In the case where $L_{V}>L_{\mathrm{det}}$ it is more likely that just one
such pair will be in the detector. In the case where $L_{V}$ is much longer
than the underground depth of the laboratory, there will be more events coming
from below than from above. These, and other signal characteristics are
discussed in more detail in Section V.

The reach of a particular experiment depends on the detector-specific
background rate for the signal in question. Though we expect very little
background that mimics our particular signals, we did not delve into these
detector-specific details. Instead, we simply estimate the event rates in
several detectors as a function of the model parameters. In Fig. 2, we show
the parameters that lead to 100 events per year in the $\epsilon-m_{\chi}$
plane. For a weakly-coupled bound state as the SDDM, its mass is related to
that of $\chi$, $m_{\Psi}\simeq 2m_{\chi}$.

Figure 2:  Contours that give 100 DM self-destruction events per year in the
$\epsilon-m_{\chi}$ parameter space of the angular momentum model. The other
parameters are chosen to be $m_{V}=2m_{\chi}/3$, $\alpha_{V}=0.01$,
$\alpha_{\phi}=0.001$. The colorful thick curves correspond to Super-
Kamiokande (red), and DUNE (dark green). Contours for SNO+ (light green),
Borexino (orange) are shown as arrows in order to reduce the number of curves
with the understanding that they run parallel to that of Super-K. The solid,
dashed and dotted curves assume the initial DM bound state is $\Psi_{10,9}$
and comprises $10^{-9}$, $10^{-5}$ and $10^{-2}$ of the total DM relic
density, respectively. The gray curves correspond to constant decay length
contours of the dark photon $V$ (see Section V for their effect on the signal
characteristics). The light red and blue shaded regions show the existing
experimental constraints from searching for visibly-decaying dark photons and
DM direct detection (assuming $\chi$ to be the dominant DM), respectively.

The colorful thick curves correspond to the observation of this rate of
$\Psi^{\prime}$ decay events in various neutrino experiments: Super-
Kamiokande, SNO+, Borexino and DUNE. In order to reduce the clutter in the
plot, we have denoted the limits on the smaller two experiments with arrows,
with the understanding that their rate contours run parallel to those of
Super-K. The 100-event-per-year contour for $\epsilon$ is given by:

$\epsilon_{100}^{2}=\frac{100\mbox{ events}}{T_{\text{year}}\times nV\times
n_{\chi\bar{\chi}}\left<\sigma
v\right>^{(\epsilon=1)}_{\text{scatter}}\times\text{Br}\left(V\rightarrow
l^{+}l^{-}\right)}\,,$ (22)

where $nV$ is the effective number of targets, defined by the terms in
parentheses in Eqs. (19) and (20) for $L_{V}<L_{\mathrm{det}}$ and
$L_{V}>L_{\mathrm{det}}$ respectively. Our result is a rough estimate as we
have omitted the $\mathcal{O}(1)$ geometry-dependent factor for
$L_{V}>L_{\mathrm{det}}$ and we neglected the density variations in the Earth.
The curves in Fig. 2 are derived assuming that the initial DM bound state is
$\Psi_{10,9}$ and comprises $10^{-9}$, $10^{-5}$ or $10^{-2}$ of the total DM
relic density. For the final state $\Psi^{\prime}$, we consider all possible
$S$-wave bound states with $n=1,\dots,9$. (Note that this is a conservative
assumption as we did not consider the effect of any non-$S$-wave states.) The
model parameters in the plot are set to: $m_{V}=(2/3)\,m_{\chi}$,
$\alpha_{V}=0.01$ and $\alpha_{\phi}=0.001$. Since we have used
$\alpha_{\phi}\ll\alpha_{V}$ the resulting unstable bound state annihilates
predominantly into two $V$s (and then into SM particles). In the shaded
regions, we also show the existing experimental constraints from searching for
visibly-decaying dark photons Alexander:2016aln  as well as direct detection
(assuming $\chi$ to be the dominant DM). Interestingly, the signals can help
to probe a large portion of the parameter space that is not yet accessible
otherwise with the existing dark photon searches. This gives a strong
motivation for exploring the neutrino experimental data and looking for the
existence of DM bound states in nature. Although the current neutrino
detectors focus on energy ranges relevant for incoming neutrinos
Richard:2015aua ; Andringa:2015tza ; Bellini:2008mr ; Acciarri:2015uup , they
are also sensitive to higher energy ranges and might be used to probe SDDM up
to higher masses. We will return to a more detailed discussion of the
phenomenology of SDDM and some search strategies in Section V.

###  II.4 Bound state production in the early universe

To complete our picture of angular momentum stabilized dark matter, we discuss
some possibilities for the production of high-$\ell$ bound states in the early
universe. As we will see, the cosmology in this model is challenging and might
involve a non-thermal production or other mechanisms. The difficulty arises
from the constraint Eq. (4), which implies that the mediator $\phi$ is heavier
than the Rydberg energy, $m_{\phi}>(\alpha_{\phi}^{2}m_{\chi})/4$. This
implies that even at zero temperature, an incoming mediator has the potential
to dissociate the $\chi\bar{\chi}$ bound states. This fact strongly restricts
the production mechanisms of $\chi,\bar{\chi}$ bound states in the early
universe. We now discuss several possible production mechanisms. We leave a
more dedicated study of these possibilities for a future work.

Thermal freeze out. Consider the case where $\chi$, $\bar{\chi}$, and $\phi$
are in thermal equilibrium with each other in the early universe. For
simplicity, we assume that the dark sector has the same temperature as the SM
sector. Around $T\sim m_{\chi}/25$, the $\chi$ and $\bar{\chi}$ particles
freeze out by annihilating into $\phi$s and/or $V$s. Assuming
$\alpha_{\phi}\sim\alpha_{V}$, their relic abundances satisfy

$\Omega_{\chi}=\Omega_{\bar{\chi}}\sim
10^{-4}\left(\frac{\alpha_{\phi}}{0.01}\right)^{-2}\left(\frac{m_{\chi}}{1\,\rm
GeV}\right),\qquad\frac{n_{\phi}}{n_{\chi}}\sim
10^{14}\left(\frac{\alpha_{\phi}}{0.01}\right)^{2}\left(\frac{m_{\chi}}{1\,\rm
GeV}\right)^{-1}\ .$ (23)

After freeze out, $\chi,\bar{\chi}$ bound states could be formed through the
process $\chi+\bar{\chi}\to\left(\chi\bar{\chi}\right)+\phi$, where
$\left(\chi\bar{\chi}\right)$ is a $\chi\bar{\chi}$ bound state. Since the
formation rate is smaller than the Hubble expansion, the resulting bound state
relic abundance is given by

$\Omega_{BS}=\Omega_{\chi}\frac{R_{BSF}}{H},\qquad
R_{BSF}=n_{\chi}\sigma_{BSF}\ ,$ (24)

where $\sigma_{BSF}$ is the bound state formation cross section. On the other
hand, a bound state could be dissociated by encountering a $\phi$ particle,
$\left(\chi\bar{\chi}\right)+\phi\to\chi+\bar{\chi}$. The dissociation cross
section $\sigma_{DIS}$ is related to $\sigma_{BSF}$ via crossing symmetry. For
$T\gtrsim m_{\phi}$, the two are related by
$\sigma_{DIS}\sim(m_{\chi}/T)^{3/2}\sigma_{BSF}$ Wise:2014jva . Note that
because the $\phi$ mass is greater than the binding energies, see Eq. (4), we
must rely on the kinetic energies of $\chi,\bar{\chi}$, dictated by the
temperature $T$, for bound state formation to occur. The dissociation rate for
each bound state is then

$R_{DIS}=n_{\phi}\sigma
v_{DIS}\sim\left(\frac{n_{\phi}}{n_{\chi}}\right)\left(\frac{m_{\chi}}{T}\right)^{\frac{3}{2}}R_{BSF}\
.$ (25)

From Eqs. (23) and (24), we learn that the probability for a bound state to
dissociate satisfies,

$P_{DIS}=\frac{R_{DIS}}{H}\gtrsim
10^{8}\left(\frac{m_{\chi}}{T}\right)^{\frac{3}{2}}\left(\frac{\alpha_{\phi}}{0.01}\right)^{4}\left(\frac{m_{\chi}}{1\,\rm
GeV}\right)^{-2}\ .$ (26)

The above result implies that all bound states are quickly destroyed within a
Hubble time as soon as they are produced.

This is a problem since we would like $\Omega_{BS}>10^{-10}$ for the SDDM to
be of experimental interest (see Fig. 2). This problem could be cured if we
further extend the dark sector with light states which $\phi$ can annihilate
or decay into. The coupling of these light states to $\phi$ has to be large
enough to deplete the $\phi$ population and prevent bound-state dissociation.
Its coupling to the $\chi$, has to be small enough to not dissociate the bound
states themselves.

$\chi$ as asymmetric DM. Since $\chi$ is much lighter than the weak scale, the
thermal production scenario cannot accommodate a situation where $\chi$
comprises all the DM relic abundance. It is then tempting to consider the
production of $\chi$ in an asymmetric scenario. Additionally, in such a
scenario one can avoid cosmological constraints on DM annihilation such as the
CMB Madhavacheril:2013cna . The problem with this scenario is that the number
of $\bar{\chi}$ is exponentially suppressed at late time Graesser:2011wi ;
Gelmini:2013awa  because the $\chi\bar{\chi}$ annihilation cross section is
very large. The exponential suppression of $\bar{\chi}$ would make it even
harder to form any $\chi\bar{\chi}$ bound states in the early universe.

Non-thermal history. The thermal freeze out picture we discussed above does
not work in its minimal form mainly because it produces many more $\phi$
particles than $\chi$ and $\bar{\chi}$. This problem could be alleviated if
the dark sector particles are produced non-thermally, with a suppressed
$n_{\phi}/n_{\chi}$ ratio compared to Eq. (23). Non-thermal production
mechanisms include the freeze-in of $\chi$ and $\bar{\chi}$ through a small
kinetic mixing between $V$ and/or $\phi$ and the SM photon, or the late decay
of some heavier dark sector states. The relic abundance of high-$\ell$ bound
states in these cases is model dependent but it seems plausible that they can
still be probed in neutrino detectors.

Additional confining dark force. Another way of efficiently producing the
high-$\ell$ bound states is to replace $\phi$ by a confining dark force. This
implies that $\chi$ is colored under a non-abelian dark gauge interaction. In
this case, there are no free $\chi$ and $\bar{\chi}$ particles at distances
larger than the inverse of the dark confinement scale. In that case, it is
likely for the high-$\ell$ $\chi\bar{\chi}$ states to form efficiently
Kang:2006yd . If the dark confinement scale of this gauge interaction is very
small, the low-level states are to a good approximation given by a $1/r$
potential. Then, the self-destructing DM scattering, which involves short-
distance physics, and the experimental detection discussions in section II.2
still remain valid.

Late production of bound states. In addition to the production in the early
universe, we can consider the more recent production of $\chi\bar{\chi}$ bound
states by a late time binding process of relic $\chi$ and $\bar{\chi}$. The
$\phi$ particles are sufficiently diluted in the universe today so they no
longer play a role in dissociation. Because $m_{\phi}$ is larger than the
Rydberg energy, the $\chi$ and $\bar{\chi}$ in the initial state must have
large enough velocities. One might imagine more efficient bound state
formation inside neutron stars where $\chi$ and $\bar{\chi}$ are
gravitationally accelerated. The formed bound states could be liberated if the
neutron star belongs to a binary system which inspirals and finally merges
TheLIGOScientific:2017qsa . This picture then ties the bound state formation
rate to the neutron star merger rate.

##  III Tunneling stabilization

We now consider a different mechanism to generate the meta-stable state of DM
required for the self-destructing phenomenology. The idea is that the long-
lived DM state is protected by a potential barrier which keeps the
constituents apart, requiring exponentially suppressed tunneling to self
annihilate. The same bound state can scatter into a short-lived bound state,
where the constituents are localized closer together. This transition can
happen through excitation to a de-localized state and a decay to the localized
state near the origin.

A key ingredient in this tunneling stabilization mechanism is the existence of
a metastable minimum far from the origin, so that the tunneling probability is
exponentially small. This happens in nature usually due to an interplay
between several forces, both attractive and repulsive. Consider for example
the states that can be made of two deuterium atoms. The two atoms can be
assembled into a di-deuterium molecule, $D_{2}$. The two nuclei of $D_{2}$
can, with a very long lifetime, tunnel through the Coulomb barrier to form a
Helium-4 atom. Similarly, some common isotopes of table salt, NaCl, can tunnel
to become a Nickel atom since they have the same number of protons, neutrons,
and electrons. Though these examples are inspiring, they do not quite fulfill
our requirements since excited molecular states of $D_{2}$ are still very
unlikely to fuse into helium. Perhaps real-world examples which are closer to
what self-destructing DM requires are super cooled systems in crystals and
liquids supercool . 111For further inspiration, click here $\href
https://www.youtube.com/watch?v=fXZrzMPdBNo$.

To illustrate the main features of tunneling stabilization, we consider a
different kind of $\chi\bar{\chi}$ bound state, one which is bounded not by a
Coulomb mediator but by a potential $V(r)$ such that $r$ is the relative
distance between the $\chi$ and $\bar{\chi}$. We remain agnostic as to the
origin of this potential, but one can imagine an effective potential due to
several mediators. Furthermore, we consider the case in which $V(r)$ has a
global minimum at $r=0$ and a local minimum at $r=R$, with the two separated
by a potential barrier as shown in Fig. 3.

Figure 3: An illustration of a two minima potential as described in Section
III.

This potential is reminiscent of a molecular potential, but with the coulomb
barrier truncated by hand. We call a potential of this sort a two minima
potential. This potential has three kinds of states:

  * •

Long lifetime states centered around the local minimum at $r=R$.

  * •

Short lifetime states centered around the global minimum at $r=0$.

  * •

Excited states above the barrier which are de-localized within the full
potential well.

Just like the angular momentum case, the $S$-wave decay rate of a
$\chi\bar{\chi}$ state is proportional to its wavefunction squared at the
origin

$\displaystyle\Gamma_{\chi\bar{\chi}}=4\pi\,\frac{\alpha^{2}_{V}}{m^{2}_{\chi}}\,{\left|\psi_{\chi\bar{\chi}}(0)\right|}^{2}\,.$
(27)

The states localized at $r=0$ have significant support at the origin and so
their lifetime is very short. In contrast to these, the states localized at
$r=R$ have an exponentially small support at the origin and so their lifetime
is very long. Additionally, the overlap of the long and short lifetime
eigenfunctions is negligible, and so is the rate of spontaneous emission from
one to the other. The excited states are unsuppressed at both $r=0$ and $r=R$
so they quickly either annihilate or spontaneously emit a dark photon and
relax to a lower state. When the depth of the global minimum is much larger
than the local minimum, the number of short lifetime states is much larger
than the number of long lifetime states ($N_{\text{Short}}\gg
N_{\text{Long}}$). In that limit, the excited state decays predominantly to
short lifetime states, which annihilate instantaneously.

For our SDDM scenario, we envision a fraction of DM is made up of the long-
lived state with $\langle r\rangle\sim R$. An interaction of this state with a
nucleus on Earth can excite it to the delocalized state, which then proceeds
to decay and annihilate as discussed above. The annihilation products, assumed
to be dark photons, can then decay in a detector leading to a signal. In the
Appendix, we show a toy example which we used to estimate the various rates
more quantitatively. We find that the lifetimes of the long, short and excited
states scale as

$\displaystyle\frac{\tau_{\text{Short}}}{\tau_{\text{Long}}}\sim\frac{\tau_{\text{Excited}}}{\tau_{\text{Long}}}\sim
e^{-2\sqrt{m_{\chi}h_{b}}R}\,,$ (28)

where $h_{b}$ is the height of the potential barrier.

In this paper, we do not construct a full model which yields a potential
similar to the one in Fig. 3. A working model, however, would need to have
both a significant barrier, say with $\sqrt{h_{b}}R\sim 20$, and excited bound
states that can live classically above the barrier. Molecules found in nature
have the former, but not the latter.

While we were unable to construct a full model, we do not see any fundamental
reason that we will not be able to. We can speculate about ways to modify the
molecular picture to make it viable to our model. One way is to add a
confining potential that will make the probability to decay into the short
lived state significantly. Another idea is to have droplets of a super-cooled
dark fluid which make up a part of DM.

##  IV Symmetry stabilization

As a third class of models, we consider bound states in an asymmetric DM
scenario, with the DM charged under baryon number. The self-destructing DM
picture can be realized when the bound state interacts with SM nuclei.

More specifically, consider a Dirac fermion $\chi$ which carries the baryon
number $q$, with $q\neq\pm 1,\pm 1/2$. The exchange of a scalar dark force
carrier $\phi$ (with zero baryon number) allows both $\chi,\chi$ and
$\chi,\bar{\chi}$ to form bound states. Because the $(\chi\chi)$ bound state
is stable, we can assume it comprises a significant fraction or even makes up
all of the total DM relic abundance today.222With a scalar dark force, it is
possible for a large number of $\chi$ particles to form stable bound states
(also called nuggets) Wise:2014jva ; Wise:2014ola ; Gresham:2017zqi . In
principle, a similar phenomenon as discussed below could also apply to
nuggets. For simplicity, we only consider two-body bound states here. We also
introduce another fermion $\eta$ which carries non-zero baryon number $1+2q$.
The assigned baryon quantum numbers above allow a dimension 6 effective
interaction between $\chi,\eta$ and the neutron $n$,

$\displaystyle\mathcal{L}_{int}=\frac{(\overline{\chi^{c}}\chi)(\bar{\eta}n)}{\Lambda^{2}}+{\rm
h.c.}\ .$ (29)

If $2m_{\chi}<m_{\eta}+m_{p}$, the $(\chi\chi)$ bound state is stable.
However, when it reaches the detector, the above interaction triggers the
following reaction,

$\displaystyle(\chi\chi)+n\to(\chi\bar{\chi})+\eta\ ,$ (30)

as shown by the Feynman diagram in Fig. 4 (left). At the “partonic level” in
the dark sector, one of the $\chi$s participates in the reaction
$\chi+n\to\bar{\chi}+\eta$. The other $\chi$ is a spectator, and it becomes
bounded with the $\bar{\chi}$ in the final state into a $(\chi\bar{\chi})$
bound state. After this reaction, a neutron is removed from the target and the
final state particle $\eta$ can escape away without further interaction. We
consider $m_{\eta}\approx m_{n}$, and then the cross section for the
scattering of a $(\chi\chi)$ ground state into a $(\chi\bar{\chi})$ ground
state is,

$\displaystyle\sigma_{(\chi\chi)+n\to(\chi\bar{\chi})+\eta}\simeq\frac{\mu_{\chi
n}^{2}}{\pi\Lambda^{4}}\ ,$ (31)

where $\mu_{\chi n}=m_{\chi}m_{n}/(m_{\chi}+m_{n})$, and we make the
approximation that the dark form factor, in analogy to Eq. (12), is of order
1, when both $(\chi\chi)$ and $(\chi\bar{\chi})$ are in the ground states.

Figure 4: Left: Feynman diagram for the scattering process
$(\chi\chi)+n\to(\chi\bar{\chi})+\eta$ in the model described in Section IV.
Right: A possible way of destructing the $(\chi\bar{\chi})$ bound state after
the scattering.

Like the other two scenarios we discuss before, the most striking signal is
generated due to the destruction of the unstable $(\chi\bar{\chi})$ state. It
can self-annihilate into SM particles via mediators (e.g., a pair of dark
photons, see Fig. 4 (right)) between the two sectors. The present data from
neutrino and DM experiments can be interpreted as constraints on the cutoff
scale $\Lambda$. As an estimate, if the $(\chi\chi)$ bound state comprises all
the DM relic abundance, having 100 events per year from
$(\chi\chi)\to(\chi\bar{\chi})$ scattering and the $(\chi\bar{\chi})$
destruction inside the Super-K or DUNE detectors corresponds to a cutoff scale

$\displaystyle\Lambda\simeq 50\,{\rm TeV}\left(\frac{m_{\chi}}{1\,{\rm
GeV}}\right)^{1/4}\ .$ (32)

An additional signature predicted in this model besides DM self destruction is
the removal of a neutron from a target nucleus. The resulting nucleus is often
unstable which leads to additional hadronic activities inside the detectors.

As a necessary condition for this scenario to work, the $\eta$ particle mass
must lie in the following window,

$\displaystyle
m_{n}-2m_{\chi}+BE_{(\chi\chi)}<m_{\eta}<m_{n}+KE+BE_{(\chi\chi)}-BE_{(\chi\bar{\chi})}\
,$ (33)

where $m_{n}$ is the effective neutron mass inside the nucleus,
$BE_{(\chi\chi),(\chi\bar{\chi})}$ is the binding energy for the
$(\chi\chi),(\chi\bar{\chi})$ bound states, respectively, and $KE$ is the
kinetic energy of the incoming $(\chi\chi)$ bound state that enters the
detector. The upper bound of the above window is derived for the process
$(\chi\chi)+n\to(\chi\bar{\chi})+\eta$ to be kinematically allowed, and the
lower bound is from the requirement of neutron stability against the decay
$n\to\eta+(\bar{\chi}\bar{\chi})$. In addition, if $m_{\eta}\ll m_{n}$, the
final state $\chi,\bar{\chi}$ particles would carry too large a relative
momentum for them to remain bounded together. This limitation could be relaxed
if the dark force for binding the $(\chi\bar{\chi})$ and $(\chi\chi)$ states
is not from the exchange of a scalar $\phi$ but a confining non-abelian
($SU(2)$) dark gauge interaction. In this case, there are no free
$\chi,\bar{\chi}$ particles and bound states of them always form.

The idea of DM eliminating a neutron and rendering a nucleon unstable has been
discussed previously in Davoudiasl:2010am ; Davoudiasl:2011fj ; Huang:2013xfa
. However, in those models, the energy release is completely due to the
destruction of a nucleon, yielding a signal very similar to the usual nucleon
decay. In contrast, in the model discussed here, as well as the two SDDM
models presented in the previous sections, the visible energy is released from
$\chi\bar{\chi}$ annihilation into mediators and then into pairs of
$e^{+}e^{-}$ or $\mu^{+}\mu^{-}$. These striking signals are not currently
looked at. Furthermore, in these scenarios, it is possible to reconstruct the
masses of the DM particle as well as the mediator from the self destruction
final states.

##  V Experimental signatures and Model independent searches

Each one of the different SDDM scenarios presented above has its own unique
signature. Still, these different experimental signatures share certain broad
brush features. Some of the phenomenology was discussed in the context of the
angular momentum model in Section II.3. In this section, we discuss the signal
characteristics more broadly. Our aim is not to perform a detailed study of a
specific model but rather, to discuss the general properties of the possible
signals and suggest a set of relatively model-independent searches that an
experiment can perform.

We assume that the couplings between the two sectors are very small. In that
limit, the DM annihilates into mediators that eventually decay to SM particles
rather than into the Standard Model directly. The signal characteristics
depend on several parameters of the model, and we discuss a few of them below.

Event Rate: The rate of events in a detector is discussed in detailed for a
specific model in Section II.3 but will be repeated here for completeness. The
rate depends on the scattering cross section $\sigma_{\mathrm{scatter}}$ which
induces the transition from a long-lived state, $\Psi$, to a short-lived one,
$\Psi^{\prime}$. As discussed previously, the rate does not depend strongly on
the mediator decay length $L_{V}\equiv\gamma_{V}v_{V}\tau_{V}$, with
$\tau_{V}$ the mediator lifetime. If the mediator is short-lived,
$L_{V}<L_{\mathrm{det}}$ the scattering takes place inside the detector, but
if it is long-lived it can also happen in the Earth, outside the detector.
However, as long as the mediator decay length is within the size of the Earth,
the signal rate is largely independent of the mediator lifetime. This is
because the $1/r^{2}$ fall-off in the flux emanating from a particular scatter
site is compensated by the $r^{2}$ growth in the amount of volume to scatter
on. As a result, up to order one geometric factors

$\mbox{Rate}\sim nV_{\mathrm{det}}\,n_{\Psi}\langle\sigma
v\rangle_{\mathrm{scatter}}\ ,$ (34)

where $V_{\mathrm{det}}$ is the volume of the detector, $n$ is the nuclear
number density either in the detector or in Earth, depending on whether the
mediator is long or short lived. Also, $n_{\chi\bar{\chi}}$ is the number
density of long-lived bound states, $\langle\sigma
v\rangle_{\mathrm{scatter}}$ is the velocity averaged cross section to
transition from the long lived DM state to its short lived counterpart. Note
that the atomic number of the scattering nucleus may be different for $L_{V}$
larger or smaller than the detector size, which may affect the coherence
factor in the cross section.

Particle ID: The mass, spin, and couplings of the mediator determine its decay
branching fractions. A well motivated mediator is a dark photon which couples
to the SM via kinetic mixing. In this case, the mediator has a sizable
branching fraction to decay to electrons as well as to muons, charged pions or
heavy flavor (whenever the mediator is above the relevant thresholds). A
scalar or pseudo-scalar mediator could have a sizable branching fraction to
the heaviest particle to which it is kinematically allowed. Di-photons are
also an interesting final state, particularly for light (pseudo)scalars.

Event Multiplicity:  The decay length of the mediators in the lab frame is

$L_{\mathrm{V}}=c\tau_{V}\sqrt{\frac{E_{V}^{2}}{m_{V}^{2}}-1}\ .$ (35)

For $\Psi^{\prime}\to 2V$ decay, $E_{V}=m_{\Psi^{\prime}}/2$ while for
$\Psi^{\prime}\to 3V$ decay, there is a spectrum for $E_{V}$ between $0$ and
$m_{\Psi^{\prime}}/2$. If this decay length is shorter than the size of the
detector, there can be a sizable fraction of the events in which both
mediators decay inside the detector. The signature, in this case, is two
simultaneous pairs of leptons with opposite reconstructed velocities (since
the decaying bound state is non-relativistic), consistent with a single vertex
where the bound state decay took place. This is schematically shown in the
left panel of Figure 5. Such events are expected to have an extremely low
background.

Event Kinematics: Each event consists of decays of on-shell mediators produced
via the annihilation $\chi\bar{\chi}\to 2V$ or $3V$ inside the short-lived
bound state $\Psi^{\prime}$. Consequently, we expect a double or triple lepton
pair-production. In the case of $\Psi^{\prime}\to 2V$ decay, each pair has
energy which is half of the dark bound state mass and an invariant mass of
$m_{V}$,

$E_{\mathrm{pair}}=\frac{m_{\Psi^{\prime}}}{2}\qquad\mbox{and}\qquad
m_{\mathrm{pair}}=m_{V}\,.$ (36)

Opening Angle: As the mediators in all of our scenarios are produced boosted,
their decay products can’t be back-to-back. When the decay products are
relativistic, the typical opening angle between the, say, $\ell^{+}$ and
$\ell^{-}$ in each pair roughly satisfies

$\cos\theta_{\ell^{+}\ell^{-}}\sim 1-\frac{8m_{V}^{2}}{m_{\Psi^{\prime}}^{2}}\
,$ (37)

in the case of $\Psi^{\prime}\to 2V$ decay. As long as $m_{V}$ is not too
small, detectors with good angular resolution will be able to distinguish the
pair from a single energetic particle. On the other hand, as long as the
mediator is not too close to half the bound state mass, the pairs will not be
back-to-back and the detector will be able to reconstruct the direction of the
mediator velocity.

Figure 5: A schematic illustration of the expected signals for various decay
lengths of the mediator $V$. In the case of $\Psi^{\prime}\to 2V$ decay, each
black arrow represents a pair of SM particles with a total energy of
$m_{\mathrm{\Psi^{\prime}}}/2$, an invariant mass of $m_{V}$, and a total
momentum pointing in the arrow direction. A double arrow represents two such
pairs back-to-back.

Event Directionality: When the decay length of the mediator is much larger
than the detector, each event will only consist of a single pair. This region
of parameter space can be further categorized based on the directional
distribution of the mediator velocity. If the decay length is shorter than the
depth of the underground laboratory, the events will be isotropic. If on the
other hand, the decay length is larger than the laboratory depth, the
reconstructed velocities would be primarily coming from below. These two
possibilities are also shown schematically in Figure 5.

Associated Signals: The transition from the long-lived to the short-lived
state could also lead to a visible signal. In this case, if $L_{V}$ is smaller
than the detector size, both the transition and the decay of the mediator(s)
can be visible. For example, in the model described in section IV, the
transition from the long-lived to the short-lived state was associated with
the removal of a neutron from a target nucleus which would lead to an
additional energy deposit. So long as this process, as well as the decay time
of the short live state, are prompt, the mediator velocity would then point
back to the transition point. This class of events can also have very low
backgrounds.

Simplified Model Searches: Having discussed the signal characteristics, we
finally suggest simplified model searches that large detectors can perform. Of
course, picking a model and presenting limits in the spirit of Figure 2 can
certainly be useful, in the sense that a comparison to other searches can be
made. However, a more model independent presentation of the limits, which can
then be translated to any model, may also be useful. This is common practice
in collider searches, where a limit on a particular final state, say a di-
lepton resonance, is presented as a limit on a cross section times a branching
fraction rather as a limit on couplings. Other examples are searches for
simplified models of, say, supersymmetry, where results are shown as limits on
the cross section in the stop-neutralino mass plane using a contour or a color
scale.

In the case of SDDM, the signal consists of one or two lepton pairs. The
invariant mass of the pairs is set by $m_{V}$ and the energy is set by
$m_{\Psi^{\prime}}$, see equation (36). A useful way to present results is
thus to show the limit on the event rate per unit volume in the
$m_{V}$-$m_{\Psi^{\prime}}$ plane. To further sophisticate the analysis, this
should be repeated for three different assumptions about the decay length
$L_{V}$ which correspond to different angular distributions and multiplicities
as shown in Figure 5, since each of these options can have a different
background rate. In the same way that LHC resonances are searched for in a
variety of final states, here too, searches can be carried out for pairs of
electrons, muons, photons, pions, etc.

We conclude that the framework of self-destructing DM has a rich
phenomenology. The predicted events can have very particular characteristics
which in many cases have very low backgrounds. The background rates and
rejection efficiencies can be studied systematically for current and future
detectors on a phenomenological basis, varying the model parameters
$m_{\Psi^{\prime}}$, $m_{V}$ and $\tau_{V}$ as well as the decay products
(electrons, muons, etc).

##  VI Conclusions

The search for DM is currently at the frontiers of high energy physics. In
recent years, great efforts have been made in the direct detection of dark
matter, aiming at recoil energies around the keV scale or even lower. This is
based on the assumption that the available energy in each dark matter
scattering is no larger than its kinetic energy. In this paper, we have
proposed a novel class of DM models called self-destructing Dark Matter
(SDDM), in which the scattering of a DM with the detector (or the Earth)
induces its decay to SM particles. The striking new feature of this class of
models is the conversion of the entire rest mass of the SDDM to a detectable
signal, rather than just its recoil energy. We have demonstrated how large
neutrino detectors such as Super-K and DUNE could be at the frontier of the
search for SDDM. Additionally, we presented three concrete realizations of
SDDM, all of which are based on the DM being a bound state of some dark
interaction, with qualitatively different stabilization mechanisms. Finally,
we briefly described the broad brush features of the expected detector signals
in the SDDM scenario.

There are several directions for further study. More detailed studies of the
background to SDDM in large neutrino detectors are required, as well as a
closer look at its production in the early universe. Though outside the scope
of the current paper, SDDM also has potentially interesting indirect
signatures. The most striking one is DM decay in the Sun or in Jupiter. In
traditional DM models, the DM may be captured in the core of the Sun, yielding
an annihilation signal. In our model, however, the DM can simply “self-
destruct” in the Sun in a similar manner to Earth. This could yield, e.g.,
gamma ray or $e^{+}e^{-}$ signals for indirect detection experiments
Schuster:2009fc ; Feng:2016ijc . Additional model dependent indirect signals
could arise from DM self destruction induced by DM-DM scattering in the
galactic center or in extragalactic sources.

## ACKNOWLEDGEMENTS

We thank Kaustubh Agashe, Zackaria Chacko, André de Gouvêa, Shmuel Nussinov,
Surjeet Rajendran, Harikrishnan Ramani, Kate Scholberg, and Yael Shadmi for
useful discussions. This work is supported in part by the DOE grant DE-
SC0010143 and in part by the NSF grant PHY-1719877. Fermilab is operated by
Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the
United States Department of Energy.

##  Appendix A Estimation of Rates in the tunneling Model

To estimate the relevant rates in the tunneling stabilization model discussed
in section III, we focus on an idealized potential model

$\displaystyle V(r)=\begin{cases}-h_{0}&r<w_{0}\\\
-h_{R}+h_{b}&w_{0}<r<w_{0}+R\\\ -h_{R}&w_{0}+R<r<w_{0}+R+w_{R}\\\ \end{cases}$
(38)

The results derived in this section, however are quite generic for more
realistic two minima potentials with a wide barrier. The potential Eq. (38) is
depicted in Fig. 6 (left) for $h_{0}=50,h_{R}=15,w_{0}=w_{R}=1$ and
$h_{b}=8,R=30$, where $h_{0},h_{R},h_{b}$ are in units of $m_{\chi}$ and
$w_{0},w_{R},R$ are in units of $m_{\chi}^{-1}$.

We denote the height and width of the barrier by $h_{b}$ and $R$,
respectively. The height and width of the potential wells are denoted
$h_{0},w_{0}$ and $h_{R},w_{R}$ for the $r=0$ and $r=R$ minima. The typical
wavefunctions of the relevant states are shown in Fig. 6 (right), and their
values near the origin are then,

$\displaystyle{\left|\Psi_{\text{Long}}(0)\right|}^{2}\sim\frac{1}{w_{R}}\,e^{-2\sqrt{h_{b}}R}\,,\hskip
28.45274pt{\left|\Psi_{\text{Short}}(0)\right|}^{2}\sim\frac{1}{w_{0}}\,,\hskip
28.45274pt{\left|\Psi_{\text{Excited}}(0)\right|}^{2}\sim\frac{1}{R}\ .$ (39)

As a result, we have

$\displaystyle\frac{\tau_{\text{Short}}}{\tau_{\text{Long}}}\sim\frac{\tau_{\text{Excited}}}{\tau_{\text{Long}}}\sim
e^{-2\sqrt{h_{b}}R}\ .$ (40)

For $2\sqrt{h_{b}}R\sim 40$, the long lifetime states could be cosmologically
stable while the short lifetime ones decay in a second. The rate for
spontaneous emission $\Psi_{\text{Long}}\rightarrow\Psi_{\text{Short}}$ is

$\displaystyle\Gamma^{SE}_{L\rightarrow S}=\frac{4\alpha_{V}(\Delta
E)^{3}}{3}\,\left|\int\,d^{3}r\,\Psi^{*}_{\text{Short}}(r)\,\vec{r}\,\Psi_{\text{Long}}(r)\right|^{2}\sim
e^{-2\sqrt{h_{b}}R}\,,$ (41)

and so the long lifetime states are cosmologically stable. In the above
estimate, we assume the energy difference $\Delta E$ is large enough for a
dark photon to be radiated on-shell in this model.

Figure 6: An illustration of tunneling Stabilization. Left: The potential and
energy eigenvalues. Right: Sample wavefunctions. The coordinate $r$ is in unit
of $m_{\chi}^{-1}$, the potential energy $V$ is in unit of $m_{\chi}$ and the
bound state wavefunctions are in unit of $m_{\chi}^{3/2}$. Figure 7: Lifetimes
for states in the two minima potential as function of the barrier width, for
$m_{\chi}=0.5\text{GeV},\,\alpha_{V}=10^{-2}$.

The cross section for a long lifetime state to hit the Earth and up scatter to
an excited state is given by Eqs. (12)-(13). The form factor
$F_{D}(|\vec{q}\,|)$ is given by integral between $\Psi_{\text{Long}}$ and
$\Psi_{\text{Excited}}$ which, given a large enough momentum transfer, is not
exponentially suppressed. Once in the excited state, the total rate for
spontaneous emission to short lifetime states is

$\displaystyle\Gamma^{SE}_{E\rightarrow S}=\sum_{i}\,\frac{4\alpha_{V}(\Delta
E_{i})^{3}}{3}\,\left|\int\,d^{3}r\,\Psi^{*}_{\text{Short,i}}(r)\,\vec{r}\,\Psi_{\text{Excited}}(r)\right|^{2}\
,$ (42)

which is also free from the exponential suppression. For $N_{\text{Short}}\gg
N_{\text{Long}}$, the excited states transit exclusively to short lifetime
states. A quantitative analysis of the rates in this model is given in Fig. 7
for the simple two minima potential Eq. (38), with
$m_{\chi}=0.5\text{GeV},\,\alpha_{V}=10^{-2}$. As we can see in the plot, the
only lifetimes that depend significantly on $R$ are the lifetimes of the long
lifetime states to annihilate or spontaneously emit a dark photon and transfer
to a short lifetime state. This is because the tunneling suppression for these
states depends exponentially on $R$. For $R\gtrsim 12\,m^{-1}_{\chi},$ the
long lifetime states are cosmologically stable, while the short lifetime
states annihilate in roughly a picosecond. Interestingly, for this particular
configuration, the excited states tend to annihilate rather than spontaneously
emit to either the short or long lifetime states. This is, of course, a
completely viable option from the point of view of self-destructing
phenomenology.

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