11institutetext: Department of Physics, LEPP, Cornell University, Ithaca, NY
14853, USA22institutetext: Racah Institute of Physics, Hebrew University of
Jerusalem, Jerusalem 91904, Israel33institutetext: Berkeley Center for
Theoretical Physics, Department of Physics, University of California,
Berkeley, CA 94720, USA44institutetext: Theoretical Physics Group, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA55institutetext: Kavli
IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

#  Super-Resonant Dark Matter

Csaba Csáki<EMAIL_ADDRESS>1 Andrew Gomes<EMAIL_ADDRESS>2 Yonit
Hochberg<EMAIL_ADDRESS>2 Eric Kuflik
<EMAIL_ADDRESS>3 Kevin Langhoff<EMAIL_ADDRESS>3,4,5
Hitoshi MurayamaHamamatsu Professor<EMAIL_ADDRESS>
###### Abstract

We introduce Super-Resonant Dark Matter, a model of self-interacting dark
matter based on the low energy effective theory of supersymmetric QCD. The
structure of the theory ensures a resonant enhancement of the self-
interactions of the low energy mesons, since their mass ratio is set by the
number of colors and flavors. The velocity dependence of the resonantly
enhanced self-interactions allows such theories to accommodate puzzles in
small scale structure that arise from dark matter halos of different sizes.
The dark matter mass is then predicted to be around 3-4 MeV, with its
abundance set by freeze-in via a kinetically mixed dark photon.

##  1 Introduction

Dark matter (DM) is a major driving force in the fields of particle physics,
cosmology and astrophysics, and is potentially one of the best clues to
physics beyond the Standard Model (SM). The $\Lambda$CDM model of cosmology is
currently the best model we have for explaining many cosmological experiments
involving big-bang nucleosynthesis (BBN), the cosmic microwave background
(CMB) and large scale structure (LSS), which occur at scales of
$\mathcal{O}(1\text{ Mpc})$ and larger. However, several observations of small
scale structure are hard to explain with collisionless cold DM. Some of the
most studied small scale puzzles are known as ‘core vs. cusp’ CoreVsCusp1 ;
CoreVsCusp2 ; CoreVsCusp3 ; CoreVsCusp4 , ‘missing satellites’
MissingSatelites1 ; MissingSatelites2 , ‘too big to fail’ TooBigToFail1 ;
TooBigToFail2 , and the ‘diversity of rotation curves’ DiversityOfRotations .
Several proposed solutions to these small scale structure puzzles exist, such
as warm dark matter and the inclusion of dissipative baryonic processes into
the numerical simulations Baryons1996 ; Baryons1999 ; Baryons2001 ;
Baryons2002 . One such compelling notion is self-interacting dark matter
(SIDM): it has been known for some time that if dark matter self-interacts,
these self-interactions can potentially solve the small scale puzzles
Spergel_2000 ; DiversitySIDM ; SimulationsSIDM1 ; SimulationsSIDM2 .

Importantly, these small scale puzzles occur at several different length
scales, or effectively, several different velocity scales. If one assumes that
the DM self-interactions are velocity independent, then it is difficult to
explain all small scale structure puzzles with SIDM while not violating
experimental upper limits of the self-interaction strength. An alternative
path is to invoke moderate velocity dependence in the self-interaction cross
section Kaplinghat:2015aga . This allows the accommodation of constraints on
DM self-interactions set at relative velocities of
$\mathcal{O}(10^{3}-10^{4}\text{ km/s})$ while still allowing the self-
interactions to be strong enough to explain the small scale puzzles which have
velocities of $\mathcal{O}(10^{2}-10^{3}\text{ km/s})$.

A simple method for producing velocity dependent cross sections is through an
$s$-channel resonance VelocityDependence2019 . One mechanism for producing
these resonances naturally involves dark mesons that are composed of
combinations of light and heavy dark quarks. The process of interest is then
analogous to the $\Upsilon(4s)-B\bar{B}$ resonance exploited in $B-$factory
colliders VelocityDependenceHQ2020 .

In this paper we introduce a supersymmetric SIDM model, which we call Super-
Resonant Dark Matter (SRDM). The main features of the model are:

  1. 1.

SUSY: A supersymmetric dark sector that is compatible with the Minimal
Supersymmetric Standard Model (MSSM) and is UV complete in the sense of
introducing no naturalness problems. SRDM is assumed to couple to the MSSM via
a dark photon.

  2. 2.

SQCD: The UV description of the dark sector is supersymmetric QCD (SQCD),
while the low-energy effective theory is one of mesons with an Affleck-Dine-
Seiberg (ADS) superpotential ADS . Degenerate supersymmetric quark masses,
enforced via flavor symmetry, are introduced to stabilize the potential.

  3. 3.

Resonance: Like non-SUSY QCD, the model experiences chiral symmetry breaking.
The extra control over the dynamics that is provided by SUSY ensures that the
flavor singlet and adjoint mesons come in a 2:1 mass ratio at tree level. This
is a new mechanism for naturally producing resonances.

  4. 4.

Splitting: To account for small scale structure puzzles at different
velocities the above mass ratio must be shifted by an $\mathcal{O}(10^{-7})$
amount. In SRDM, this extremely small mass ratio splitting is achieved
naturally by 1-loop corrections from meson self-interactions.

  5. 5.

Predictive: Resonant SIDM models generally have three continuous parameters
which determine self-interactions and need to be fixed, giving them flat
directions in $\chi^{2}-$fits. For a fixed number of flavors, our model has
only two and picks out a specific dark matter mass and self-interaction
strength.

  6. 6.

Charged to Neutral Ratio: The relic DM is produced via freeze-in. It can be
completely neutral under the dark $U(1)$ or can have an $\mathcal{O}(1)$
fraction of charged DM, depending on the value of the dark photon coupling.

The structure of the paper is as follows. In Section 2 we describe the details
of the model and how the crucial mass ratio is generated via chiral symmetry
breaking. We also describe how the model is coupled to a spontaneously broken
$U(1)_{D}$ gauge symmetry, and how this massive dark photon is coupled in turn
to the Standard Model. Additionally, we analyze the effect of supersymmetry
breaking. In Section 3 we study the DM self-interactions and determine the
parameter space which could allow our SRDM model to account for certain small
scale structure puzzles. In Section 4 we calculate the relic abundance and
determine through which mechanisms the correct relic abundance can be
achieved. In Section 5 we summarize our results and discuss possible ways of
extending the analysis of the paper. We also include in Appendix A a
calculation of the 1-loop mass corrections and in Appendix B we discuss
anomaly matching as it relates to dark meson stability.

##  2 Model

###  2.1 Model overview

The model consists of a visible sector coupled weakly to a dark sector via a
dark photon. The visible sector is identified with the Minimal Supersymmetric
Standard Model (MSSM), into which our SM is embedded. The scale of
superpartner masses is taken sufficiently large such that all direct collider
bounds are evaded (at the price of some sub-percent level fine tuning). We
assume that a supersymmetry (SUSY) breaking sector is coupled indirectly to
the visible sector, giving rise to the soft breaking mass terms of the MSSM.
Importantly, we assume that the only coupling of the dark sector to SUSY
breaking is via the MSSM, leading to an almost completely supersymmetric dark
sector, with small calculable SUSY breaking effects that we explain in Section
2.4. This will allow us to reliably predict the spectrum of the dark sector in
a theory that has underlying strong interactions. It will also make
degeneracies in the spectrum natural. Note that the mass scales we find in the
dark sector are small compared to the scale of SUSY breaking in the MSSM.

For the dark sector, we simply use supersymmetric quantum chromodynamics
(SQCD) with $N_{c}$ colors and $N_{f}$ flavors. In the UV, the matter content
is just $N_{f}$ copies of quark and anti-quark chiral superfields transforming
in the fundamental representation of the $SU(N_{c})$ non-abelian gauge group.
The quark masses $m_{q}$ are assumed to be degenerate and below the
confinement scale, $m_{q}\ll\Lambda$. The light degrees of freedom in SUSY
gauge theories always correspond to holomorphic gauge invariants (gauge
invariants formed without the use of complex conjugation)—in the case of SQCD,
these will be the meson (and when $N_{c}\leq N_{f}$, baryon) superfields111We
will generally use the term meson to mean the meson superfield, not just the
scalar component. Similarly with the term quark.. When $N_{c}\leq N_{f}$ these
composite fields are interpreted as actual confined degrees of freedom, while
when $N_{c}>N_{f}$ (this work) the gauge group is partial broken and the
mesons are the remaining uneaten massless degrees of freedom describing the
“moduli space”. The couplings between these mesons is non-perturbative at the
confinement scale but become calculable at scales below the confinement scale,
as we will explain in detail below.

###  2.2 The supersymmetric spectrum of resonant models

First we give a brief overview of the supersymmetric spectrum of our hidden
sector based on quarks and anti-quarks with $SU(N_{f})_{L}\times
SU(N_{f})_{R}$ flavor symmetry in the fundamental (and anti-fundamental) of
$SU(N_{c})$, with masses below the strong coupling scale. As explored in the
1990’s, the detailed form of the low-energy dynamics of these theories depends
on the exact relation between the number of flavors $N_{f}$ and the number of
colors $N_{c}$. For $N_{f}<N_{c}$, non-perturbative dynamics gives rise to an
Affleck-Dine-Seiberg (ADS) type superpotential

$\displaystyle W_{\text{ADS}}=$
$\displaystyle\left(N_{c}-N_{f}\right)\left(\frac{\Lambda_{\text{H}}^{3N_{c}-N_{f}}}{\det\widetilde{M}}\right)^{\frac{1}{N_{c}-N_{f}}}$
(1) $\displaystyle\widetilde{M}_{ij}=$ $\displaystyle Q_{i}^{a}\bar{Q}_{a\,j}$
(2)

where $\Lambda_{\text{H}}$ is the holomorphic dynamical scale222This is the
scale dynamically generated by the running of the holomorphic gauge coupling,
which appears in front of the gauge kinetic term and whose beta function is
1-loop exact. The true dynamical scale $\Lambda$ is obtained by canonically
normalizing the gauge kinetic term, and is given by the NSVZ beta function.
The two differ by an $\mathcal{O}(1)$ irrelevant for our purposes and we will
interchange them freely., $\widetilde{M}$ denotes the $N_{f}\times N_{f}$
meson matrix, $Q$ and $\bar{Q}$ are the underlying chiral quark superfields,
and $a$ is a color index while $i,j$ are flavor indices. Note that
$\widetilde{M}$ has (non-canonical) dimension $2$. In the absence of other
terms, the dependence of the potential on inverse powers of the meson field
implies the expectation values of the fields will run off to infinity.

In order to stabilize the potential, we introduce the quark mass term

$\displaystyle W_{\text{mass}}=m_{q}^{ij}Q_{i}\bar{Q}_{j}=\Tr
m_{q}\widetilde{M}$ (3)

As the quark mass increases towards the dynamical scale, the meson VEV
decreases towards the dynamical scale thereby increasing the gauge coupling at
the point where the gauge group is broken.

Just like the quark mass term in the superpotential, the quark kinetic terms
in the Kähler potential can also be written in terms of the meson fields. At
tree level it is given by Affleck:1984xz  substituting the gauge field $V$
equations of motion into the quark Kähler potential
$Q^{\dagger}e^{V}Q+\bar{Q}^{\dagger}e^{-V^{T}}\bar{Q}$ and projecting onto the
meson field

$\displaystyle
K_{\text{tree}}=2\Tr\sqrt{\widetilde{M}^{\dagger}\widetilde{M}}\ .$ (4)

We will work throughout with the quark mass far below the confinement scale.
The squark VEVs are therefore large and hence break the gauge group well above
the confinement scale. Being at weak coupling, Eq. (4) is an excellent
approximation.

In the following, we will decompose the meson matrix into its singlet and
adjoint components

$\displaystyle\widetilde{M}=\frac{1}{\sqrt{N_{f}}}S\cdot\mathbf{1}+M$ (5)

where $\Tr M=0$. Taking a supersymmetric (flavor universal) quark mass matrix
$m_{q}^{ij}=\mu\delta^{ij}$,333For readers ideologically inclined against
global symmetries, one can very weakly gauge $SU(N_{f}/2)\times
SU(N_{f}/2)\times\mathbb{Z}_{2}$ to ensure the flavor universality, as long as
the gauge coupling is small enough to avoid the annihilation of dark matter
pions into gauge bosons $\alpha\ll 10^{-5}$ which does not contradict the weak
gravity conjecture. our theory takes the form

$\displaystyle K=$ $\displaystyle
2\Tr\sqrt{\widetilde{M}^{\dagger}\widetilde{M}}$ (6) $\displaystyle W=$
$\displaystyle(N_{c}-N_{f})\Big{(}\frac{\Lambda^{3N_{c}-N_{f}}}{\det\widetilde{M}}\Big{)}^{\frac{1}{N_{c}-N_{f}}}+\sqrt{N_{f}}\mu
S$ (7)

We first consider the Kähler potential. Since we know that the superpotential
will give the singlet a VEV, let us implement this with the shift
$\widetilde{M}\rightarrow v^{2}+\widetilde{M}$, or equivalently
$S\rightarrow\sqrt{N_{f}}v^{2}+S$. Expanding around this VEV gives

$\displaystyle\begin{split}K&=2\Tr\sqrt{\widetilde{M}^{\dagger}\widetilde{M}}\\\
&=\frac{1}{2v^{2}}\Tr\widetilde{M}{{\widetilde{M}}^{\dagger}}-\frac{1}{8v^{4}}\left(\Tr\widetilde{M}^{2}{{\widetilde{M}}^{\dagger}}+\Tr\widetilde{M}\widetilde{M}^{\dagger
2}\right)+\dotsb\end{split}$ (8)

where we have discarded the sub-quadratic terms that do not contribute to the
Lagrangian. Rescaling the mesons
$\widetilde{M}\rightarrow\sqrt{2}\,v\widetilde{M}$ to get canonically
normalized fields leaves us with

$\displaystyle\begin{split}K&=\Tr\widetilde{M}{{\widetilde{M}}^{\dagger}}-\frac{1}{2\sqrt{2}\,v}\left(\Tr\widetilde{M}^{2}{{\widetilde{M}}^{\dagger}}+\Tr\widetilde{M}\widetilde{M}^{\dagger
2}\right)+\dotsb\end{split}$ (9)

We now consider the superpotential and expand the superpotential around the
meson VEV as above. We ignore the constant term. For the potential expanded
about the SUSY ground state, the term linear in $S$ must cancel, which yields
the relation

$\Lambda^{3N_{c}-N_{f}}=\mu^{N_{c}-N_{f}}v^{2N_{c}}$ (10)

For the remaining terms, with the required meson rescaling
$\widetilde{M}\rightarrow\sqrt{2}\,v\widetilde{M}$, and by employing Eq. (5)
and $\Tr\widetilde{M}^{2}=S^{2}+\Tr M^{2}$, we arrive at

$\displaystyle\begin{split}W&=\mu\left(\frac{N_{c}}{N_{c}-N_{f}}S^{2}+\Tr
M^{2}\right)\\\
&\quad-\frac{\sqrt{2}\,\mu}{3v}\left(\frac{N_{c}(2N_{c}-N_{f})}{\sqrt{N_{f}}(N_{c}-N_{f})^{2}}S^{3}+3\frac{2N_{c}-N_{f}}{\sqrt{N_{f}}(N_{c}-N_{f})}S\Tr
M^{2}+2\Tr M^{3}\right)+\dotsb\end{split}$ (11)

Notice that the adjoint meson mass is twice the quark mass, $m\equiv 2\mu$.
Additionally, we can see the tree level mass ratio of the singlet to the non-
singlet mesons

$\displaystyle r=\frac{N_{c}}{N_{c}-N_{f}}\,.$ (12)

We have also verified this result using an explicit parameterization of the
quark superfields along the $D$-flat directions.

Since we want the singlet to provide an s-channel resonance in adjoint meson
scattering, we will require $r=2$ or $N_{c}=2N_{f}$. With the definition (12)
the superpotential becomes

$\displaystyle W=\frac{1}{2}m(rS^{2}+\Tr
M^{2})-\frac{\sqrt{2}\,\mu}{3v}\left(\frac{r(r+1)}{\sqrt{N_{f}}}S^{3}+3\frac{r+1}{\sqrt{N_{f}}}S\Tr
M^{2}+2\Tr M^{3}\right)+\dotsb$ (13)

In Appendix A we use Eq. (9) and Eq. (13) to compute the one-loop mass
corrections of the mesons. For the relevant case of $r=2$, the one-loop
corrections give

$\displaystyle r_{\text{1-loop}}=m_{S}/m\equiv
2(1-\delta)=2\left(1-\frac{\mu^{2}}{16\pi^{2}v^{2}}\frac{104+41N_{f}^{2}}{N_{f}}\log\frac{v}{2\mu}\right)$
(14)

where $m_{S}$ is the mass of the singlet and we have introduced the mass
splitting $\delta$. Note that we are being agnostic about the renormalization
scheme as we are only interested in the log running. We see that the singlet
sits below threshold so that the adjoint-adjoint scattering matrix element is
maximal at zero velocity. Finally, we will refer to the dimensionless Yukawa
coupling

$\displaystyle y\equiv\mu/v.$ (15)

While in this work we focus on the gauge group $SU(N_{c})$, the phenomenology
is qualitatively the same for the gauge groups $SO(N_{c})$ with $N_{f}$
flavors and $Sp(2N_{c})$ with $2N_{f}$ flavors. The different Dynkin indices
change the power of $\det\widetilde{M}$ in the superpotential, and thus change
Eq. (12). With $SO$ gauge groups one simply replaces $N_{c}\rightarrow
N_{c}-2$ while for $Sp$ the replacement is $N_{c}\rightarrow N_{c}+1$.

###  2.3 $U(1)_{D}$ gauge interaction

A dark photon, which we denote by $A^{\prime}$, is the means by which the SM
and dark sectors are coupled. It comes from gauging the $U(1)_{D}$ subgroup of
the flavor symmetry group. There is then a kinetic mixing between the dark
photon and the SM $U(1)_{Y}$ gauge boson. The SUSY breaking effects of this
coupling are explored in the next subsection.

In our model we take $N_{f}$ to be even and a quark charge matrix $Q$ such
that $Q^{2}\propto\mathbf{1}$. This choice is sufficient to forbid a neutral
meson from decaying to two dark photons (see Appendix B and Berlin:2018tvf ).
Therefore we have an equal number of positively and negatively charged quarks.
We also define $g_{D}$ as the $U(1)_{D}$ coupling strength, in a normalization
where charged dark mesons have charge $\pm 1$.

The dark photon gives a one-loop mass renormalization to charged mesons. In
the case of a dark photon with mass $m_{A^{\prime}}$ below the cutoff $v$ this
becomes

$\displaystyle
m_{\pm}=m_{0}\left(1+\frac{g_{D}^{2}}{8\pi^{2}}\log\frac{v}{m_{A^{\prime}}}\right)$
(16)

In this paper we will generally take $g_{D}$ to be larger than the DM self-
coupling $y$, which we will find in Section 3 to be $\sim 5\times 10^{-4}$.
This mass correction is thus larger than the the mass shift in Eq. (14) and
the charged dark matter is not expected to take part in the resonant
interaction fundamental to our model. However, the analysis can be repeated
when $g_{D}$ is small and the charged DM is approximately degenerate with the
neutral DM. This case is qualitatively very similar. Therefore, the low
velocity, resonant part of the DM self-scattering cross section is dominated
by neutral DM, while the high velocity part will be dominated by dark photon
exchange between charged DM (since the $U(1)_{D}$ coupling is greater than the
DM self-coupling).

###  2.4 Effect of SUSY breaking

Supersymmetry, while critical for producing a resonance for neutral meson
scattering, is not essential as an exact symmetry for a working model.
Moreover, SUSY breaking is a necessary result of the coupling between the
visible and hidden sectors. Since the visible sector is the MSSM with a
sizeable amount of SUSY breaking, it is inevitable that at least some of this
SUSY breaking will be transmitted to the hidden sector, even if we assume no
direct coupling between the hidden sector and dynamical SUSY breaking
sector.444In particular, we assume the gravity-mediation Alvarez-Gaume:1983drc
; Hall:1983iz  or anomaly mediation Randall:1998uk ; Giudice:1998xp  is
negligible in the hidden sector. For example, low-energy gauge mediation
Alvarez-Gaume:1981abe ; Dine:1981gu  or vector mediation Hook:2015tra ;
Hook:2018sai  models belong to this category. In this section we explain what
the leading SUSY breaking effects in the hidden sector are, some of which will
be important for the determination of the relic density and indirect detection
constraints.

The actual link to the visible sector will be as usual a small kinetic mixing
between the dark photon and $U(1)_{\text{Y}}$ gauge boson $B$ of the (MS)SM.
This kinetic mixing itself can be fully supersymmetric of the form

$\displaystyle\epsilon\int d^{2}\theta
W_{\text{D}\alpha}W_{\text{Y}}^{\alpha}+h.c.=\epsilon(-F_{\text{D}\alpha\beta}F_{\text{Y}}^{\alpha\beta}+4i\bar{\lambda}_{\text{D}}\not{\partial}\lambda_{\text{Y}}+2D_{\text{D}}D_{\text{Y}})$
(17)

where the $W^{\alpha}$s are field strength chiral superfields. Abelian vector
superfields contain a photon, a photino, and an auxiliary $D$-term. Treating
$\epsilon$ perturbatively, the largest effect on the dark spectrum comes from
the mixing of the $D$-terms given by

$\displaystyle\mathcal{L}\supset\frac{1}{2}D_{\text{D}}^{2}+D_{\text{D}}J_{\text{D}}+\frac{1}{2}D_{\text{Y}}^{2}+D_{\text{Y}}J_{\text{Y}}+2\epsilon
D_{\text{D}}D_{\text{Y}}$ (18)

where the $J$s are the usual $D$-terms $g\sum_{i}q_{i}|\phi_{i}|^{2}$ formed
from the scalar components $\phi_{i}$ of the chiral superfields charged under
the U(1)’s. After integrating out the auxiliary $D$-terms one obtains the
Lagrangian for the physical degress of freedom. In the $\epsilon=0$ case of no
mixing one just obtains the usual two independent $D$-terms (two terms
quadratic in the respective $J$s), giving rise to the expected scalar quartic
terms. However in the presence of mixing a third term is introduced, which at
linear order in $\epsilon$ is equal to $2\epsilon J_{\text{D}}J_{\text{Y}}$.
Importantly, $J_{\text{Y}}$ is quadratic in the MSSM Higgs VEV $v_{h}$ and
generates a mass term for scalar fields charged under $U(1)_{D}$,

$\displaystyle
2\epsilon\frac{g_{\text{Y}}}{2}(|H_{u}|^{2}-|H_{d}|^{2})g_{\text{D}}\sum_{i}q_{i}\phi_{i}^{*}\phi_{i}$
(19)

where the $H$s are the up- and down-Higgses of the MSSM and the sum is over
the dark degrees of freedom. Using the MSSM identities
$|H_{u}|^{2}-|H_{d}|^{2}=\frac{v_{h}^{2}}{2}\cos 2\beta$ and
$g_{\text{Y}}=\frac{e}{\cos\theta_{\text{W}}}$, this mass term is

$\displaystyle\approx\epsilon\frac{g_{\text{D}}}{e}\cos
2\beta(56\text{GeV})^{2}\sum_{i}q_{i}\phi_{i}^{*}\phi_{i}$ (20)

Note that $\epsilon$, $g_{\text{D}}$, and $\beta$ are all free parameters.
This shifts the mass-squared of charged bosons (no effect on fermions at tree
level), so that combined with the result of the previous subsection we have

$\displaystyle
m_{\pm,\text{bos}}^{2}=m_{0}^{2}\left(1+\frac{g_{D}^{2}}{4\pi^{2}}\log\frac{v}{m_{A^{\prime}}}\pm\epsilon\frac{g_{\text{D}}}{e}\cos
2\beta\frac{(56\text{GeV})^{2}}{m_{0}^{2}}\right)$ (21)

The competition between the SUSY preserving and SUSY breaking term leads to
two situations. When the SUSY preserving term dominates, we have lighter
neutral DM and heavier charged DM. When the SUSY breaking term dominates the
lightest particles in the dark sector are charged bosons, and $U(1)_{D}$
charge conservation ensures their stability. It is important in this case for
the neutral DM not to be too much heavier, otherwise it will be depleted and
unable to participate in the resonant self-interaction. For simplicity we will
therefore insist that the lightest dark particle (LDP) is neutral, whereby the
SUSY preserving term must be larger than the SUSY breaking term555This is a
somewhat conservative bound. It is possible to be above this bound if the dark
QED and SUSY breaking contributions to the charged bosons mass cancel. This
would allow the lightest charged bosons to be the SIDM as it would be
resonantly enhanced. We don’t discuss this situation further.. In fact, we are
interested in $\epsilon\approx 10^{-12}$ as we will discuss later (see Fig.
5), and hence this is indeed the case.

All other effects of the $D$-term on the dark spectrum are subleading. The
largest additional correction is expected to come from the large SUSY breaking
bino mass $m_{\tilde{B}}$. Adding a (SUSY preserving) Stückelberg mass $m_{D}$
to the dark photon we obtain the gaugino mass matrix

$\displaystyle\begin{pmatrix}\not{p}-m_{D}&2\epsilon\not{p}\\\
2\epsilon\not{p}&\not{p}-m_{\tilde{B}}\end{pmatrix}$ (22)

This yields a corrected dark photino mass
$m_{D}(1-4\epsilon^{2}\frac{m_{D}}{m_{\tilde{B}}-m_{D}})$ at leading order in
$\epsilon$. Due to the $\epsilon^{2}$ suppression of the correction and the
expected smallness of the dark photon mass in comparison to the SM photino
mass, the effect on the dark meson spectrum is negligible. A more important
source of SUSY breaking is a dark charged meson mass correction from the
diagram in Figure 1,

$\displaystyle\Delta
m^{2}\propto\frac{\alpha_{\text{D}}}{4\pi}\epsilon^{2}m_{\tilde{B}}^{2}\log\frac{m_{\tilde{B}}^{2}}{m^{2}}$
(23)

As a loop correction at order $\epsilon^{2}$, this is negligible when compared
with the tree level $D$-term SUSY breaking.

$M$$\psi_{M}$$M$$\lambda_{D}$$\lambda_{B}$$\lambda_{D}$$\otimes$$\otimes$
Figure 1: SUSY breaking contribution from dark charged meson mass correction.

##  3 Self-Interactions

As mentioned in the introduction, one of the key features of SRDM is that it
allows for velocity dependent self-interactions capable of resolving small
scale puzzles seen in the dark matter halos of dwarf galaxies. DM halos seen
in dwarf galaxies, low surface brightness (LSB) spiral galaxies, and relaxed
galaxy clusters whose halos are appropriately spherical were studied in
Kaplinghat:2015aga . There, specific ranges of the cross section at given
average relative velocities were shown to be a potential resolution to the
small scale puzzles. This data is included in Figure 2. In this section, we
calculate the resonant self-interaction cross section and find the parameter
space of the model which is consistent with their analysis.

In our model, we will have two general types of dark matter: those charged
under $U(1)_{D}$ and those that are neutral. The charged DM, as we shall see,
will be completely negligible when it comes to self-interactions in dark
matter halos, as their dark QED mass corrections of order $\alpha_{D}$ prevent
a maximally enhanced resonant cross section. We shall also see that the
abundance of charged DM is suppressed relative to the neutral DM because of
the dark QED mass correction.

To compare the effectiveness of the SRDM model in describing small scale
puzzles in dark matter halos, we must calculate the thermally averaged cross
section $\expectationvalue{\sigma v}$, where the incoming and outgoing
particles are neutral DM. Using only neutral relic DM is an assumption that
will be justified a posteriori for a wide range of dark photon couplings.
However, for large values of $\alpha_{D}$ we will find that a sizeable
fraction of the DM is charged. Including these extra degrees of freedom and
their dark QED interactions will lead to at most $\mathcal{O}(1)$ changes in
the best fit values of $m$ and $y$.

For all of the experimental data of interest, we are well within the non-
relativistic regime. We also assume the self-interactions are strong enough
such that the velocity distribution is roughly Boltzmannian. Then the
thermally averaged cross section can be written as

$\displaystyle{}\expectationvalue{\sigma
v}\approx\frac{1}{(\sqrt{2\pi}v_{0})^{3}}\int
d^{3}v\sigma_{\text{int}}(v)ve^{-v^{2}/2v_{0}^{2}}$ (24)

where $v$ is the relative speed of the incoming particles and the mean
relative speed is given by $\expectationvalue{v}=\sqrt{8}v_{0}/\sqrt{\pi}$. We
have also defined $\sigma_{\text{int}}$ to be the average two-to-two cross
section given by

$\displaystyle\sigma_{\text{int}}(v)\approx\frac{1}{64\pi
m^{2}}\frac{1}{2!\mathcal{N}_{\text{neutral}}^{2}}\sum_{i_{1},i_{2},f_{1},f_{2}}^{\text{neutral}}|\mathcal{M}_{i_{1},i_{2}\rightarrow
f_{1}f_{2}}|^{2}$ (25)

where the indices $i_{1},i_{2},f_{1},f_{2}$ go over all neutral DM states
(including helicities), and $\mathcal{N}_{\text{neutral}}$ is the total number
of such states. We have also made the approximations $s\rightarrow 4m^{2}$,
$t\rightarrow 0$, $u\rightarrow 0$; however, we must be careful to preserve
the terms in the denominator of the form $(s-m_{S}^{2})$. We can therefore
write the cross section as

$\displaystyle\sigma_{\text{int}}(s)=\sigma_{0}+\frac{\sigma_{1}m^{2}}{s-m_{S}^{2}}+\frac{\sigma_{2}m^{4}}{(s-m_{S}^{2})^{2}}$
(26)

In the denominator, we must be a little more careful with our non-relativistic
approximation and write $s\approx 4m^{2}+m^{2}v^{2}$. Using
$m_{S}=2m(1-\delta)$ we get $s-m_{S}^{2}\approx m^{2}(8\delta+v^{2})$. Since
both $\delta$ and $v^{2}$ will be very small in the non-relativistic regime,
the final term of Eq. (26) is dominant.

Figure 2: The velocity dependence of the DM self-interaction cross section.
The red, blue, and green data points come, respectively, from the analysis of
dwarf galaxies, LSB spiral galaxies, and galaxy clusters. These are taken
directly from Kaplinghat:2015aga . There also exist error bars for
$\expectationvalue{v}$, which are relevant for the $\chi^{2}$ test in Figure
3, but these are not shown. The dashed gray lines represent the values of
$\expectationvalue{\sigma v}/m$ if $\sigma/m$ were the velocity independent
values labeled by the line. The orange, cyan, and magenta curves correspond to
the points in parameter space shown in Figure 3.

The thermally averaged cross section is given by

$\displaystyle\expectationvalue{\sigma v}\approx$
$\displaystyle\frac{16\sigma_{2}}{\pi^{2}\expectationvalue{v}^{3}}\times\left(-1-(1+\xi)e^{\xi}\text{Ei}(-\xi)\right)\quad\text{with}\quad\xi=\frac{32\delta}{\pi\expectationvalue{v}^{2}}$
(27)

where $\text{Ei}(-\xi)$ is the exponential integral function. For the purpose
of intuition, it is helpful to note the following limiting behavior

$\displaystyle\expectationvalue{\sigma v}\approx$
$\displaystyle\begin{cases}\frac{\sigma_{2}\expectationvalue{v}}{64\delta^{2}}&\expectationvalue{v}\ll\sqrt{\delta}\\\
\frac{16\sigma_{2}}{\pi^{2}\expectationvalue{v}^{3}}\left(\log\left(\frac{\pi\expectationvalue{v}^{2}}{32\delta}\right)-\gamma_{M}-1\right)&\sqrt{\delta}\ll\expectationvalue{v}\ll
1\end{cases}$ (28)

where $\gamma_{M}$ is the Euler-Mascheroni number. Using the data in Figure 2
to estimate the low- to high-velocity transition to be around
$v_{\text{trans}}\sim 500\text{ km/s}$, we can estimate that
$\delta=v_{\text{trans}}^{2}/8\sim 10^{-7}$. For $N_{f}=2$, this will imply
$y\sim 10^{-3}$.

Also, since $\sigma_{2}\propto y^{4}/m^{2}$ and $\delta\propto
y^{2}\log(y^{-1})$, we see that for fixed $N_{f}$ the thermally averaged cross
section is a function of $\expectationvalue{v}$ with only two parameters, $m$
and $y$. This is different than many other SIDM models where the mass
splitting depends on an additional continuous parameter. Using only neutral DM
external particles, $\sigma_{2}$ can be calculated by hand or with a
combination of FeynRules FeynRules1 ; FeynRules2 , FeynArts FeynArts , and
FormCalc FormCalc . The result is

$\displaystyle\sigma_{2}=\frac{8}{\pi N_{f}^{2}}\frac{y^{4}}{m^{2}}.$ (29)

The $\chi^{2}$-fits of the thermally averaged cross section for $N_{f}=2,4,10$
to the data are shown in Figure 2. The viable parameter space around the best
fit values are shown in Figure 3. The main result is that the dark matter has
a mass in the MeV range and is weakly coupled, as expected. This mass range
and coupling are also quite constrained.

Figure 3: Results of a $\chi^{2}$-fit to the data from Kaplinghat:2015aga  for
Super-Resonant Dark Matter with $N_{f}=2$, $N_{f}=4$ and $N_{f}=10$. The red,
blue, and green shaded regions are the 95% confidence level regions for
fitting to the data from dwarf galaxies, LSB spiral galaxies, and galaxy
clusters, respectively. The black contours correspond to 90%, 95%, and 99%
confidence levels for the equally weighted combination of all three data sets.
The orange, cyan, and magenta dots on the $N_{f}=2$ plot correspond to the
curves shown in Figure 2. We pick some of these slightly offset from the best
fit point to show variation in Figure 2. The actual best fit points are
$(m=3.55\text{ MeV},y=5.2\times 10^{-4})$ for $N_{f}=2$,$(m=1.74\text{
MeV},y=4.4\times 10^{-4})$ for $N_{f}=4$, and $(m=0.54\text{ MeV},y=2.8\times
10^{-4})$ for $N_{f}=10$.

For example, for $N_{f}=2$ we find best fit values of $m=3.55\text{ MeV}$ and
$y=5.2\times 10^{-4}$. However, a reasonable range for these parameters is a
mass between $2.5-5$ MeV and a self-interaction coupling between $3\times
10^{-4}-10^{-3}$. The relation $\Lambda=y^{-4/5}\mu$ (see Eq. (10)) tells us
the dynamical scale of the gauge theory sits around a few GeV. Taking larger
values of $N_{f}$ while maintaining the mass splitting $\delta$ fixed, Eq.
(14) tells us that $y$ must scale as $N_{f}^{-1/2}$. To maintain
$\expectationvalue{\sigma v}/m$ the mass must therefore scale as
$N_{f}^{-4/3}$. Thus, we cannot take arbitrarily high values of $N_{f}$ due to
a lower bound on the mass from the neutral LDP condition (below).

##  4 Relic Abundance

We have so far presented a model with a natural resonance that is capable of
explaining the small scale puzzles. The final step in producing a viable dark
matter model is explaining the present-day relic abundance. Beyond this, it
must be ensured that an $\mathcal{O}(1)$ fraction of the relic DM is neutral
and can participate in the resonant interaction (for small enough $\alpha_{D}$
the charged DM can participate too).

In this section we constrain the parameters of the model by trying to produce
the correct relic abundance for both freeze-out and freeze-in scenarios, which
we describe below (detailed calculations can be found, for example, in
Essig_keV ). We find that freeze-out is excluded, while freeze-in is viable.

In both scenarios, a dark photon lighter than the dark mesons is ruled out as
the mesons would down-scatter to dark photons, which do not experience the
desired resonant self-interaction. For simplicity we will further assume
throughout that $m_{A^{\prime}}>2m$ so that an on-shell dark photon can decay
to two dark mesons.

The first mechanism we consider to produce the DM relic abundance is freeze-
out. In this scenario the SM and dark sectors of the early universe are in
thermal equilibrium. The dominant equilibrating process is pictured on the
left of Figure 4, where $\chi^{\pm}$ denotes the charged fermionic mesons and
$f^{\pm}$ SM fermions (electrons). The contributions from bosonic mesons are
subdominant because their cross section is p-wave suppressed, having an
additional factor of $v^{2}$.666One subtlety: For the range of $\epsilon$
relevant to freeze-out, the SUSY breaking mass correction of Eq. (21)
dominates SUSY preserving term. Therefore, the density of the lighter charged
bosonic DM is Boltzmann enhanced when compared with the heavier fermions. This
can overcome the p-wave suppression and thus put charged bosons in control of
the freeze-out process. However, they will be also be Boltzmann enhanced with
respect to the neutral DM and the small relative fraction of neutral DM in the
dark sector will be unable to reproduce the small scale anomalies. Note the
factor of $\epsilon^{2}$ arising from dark-SM photon mixing. As the universe
cools below the DM mass, its number density becomes Boltzmann suppressed. This
suppression eventually lowers the rate of the equilibrium process below the
Hubble rate, at which time we say that the dark sector freezes out from the SM
sector. With no more DM number changing processes, its abundance is fixed.

The parameters leading to the correct freeze-out relic abundance for $N_{f}=2$
and $m_{A^{\prime}}=3m$ are plotted in blue in Figure 5, which has been
adapted from Essig_keV . In the mass range found in Section 3 (vertical gold
band), and for $\alpha_{D}=0.5$, one finds that $\epsilon\sim 10^{-6}$. A
smaller dark photon coupling necessitates a larger $\epsilon$. Also shown is
the teal-colored $N_{\text{eff}}$ bound on the effective number of degrees of
freedom during Big Bang Nucleosynthesis (BBN). This bound lies well below the
freeze-out line in the mass range of interest and therefore eliminates the
possibility of a freeze-out scenario in our model. The freeze-out scenario is
also excluded by the aforementioned neutral LDP condition (magenta). Choosing
larger $N_{f}$ or $m_{A^{\prime}}$ does not remedy the situation.

$f^{+}$$f^{-}$$\chi^{+}$$\chi^{-}$$\otimes$Freeze-inFreeze-out$A$$A^{\prime}$

$\gamma^{*}$$\chi^{+}$$\chi^{-}$$\otimes$Freeze-in$A^{\prime}$

Figure 4: Diagrams for freeze-out and freeze-in via a dark photon.

The second possibility we explore is the freeze-in scenario. In this case the
dark and SM sectors are never in equilibrium, and the dark sector is populated
by decays from the SM. Specifically, SM particles annihilate to produce dark
meson pairs as pictured on the left of Figure 4. This process is dominated by
the contribution at temperatures of order $m_{A^{\prime}}$, when the dark
photon is on-shell. One can think of this as two-processes in which a dark
photon is produced (suppressed by $\epsilon^{2}$) and then decays
overwhelmingly to dark mesons. Plasmon decay, shown on the right of Figure 4,
also contributes to freeze-in Dvorkin:2019zdi  but is only important at lower
DM masses.

The parameters producing the correct relic abundance in the freeze-in scenario
were calculated777Note that we use the authors’ calculation of dark photon
decay to fermions, while in our model there are dark fermions and bosons.
However because freeze-in is dominated by on-shell dark photons (that
overwhelmingly decay into the dark sector), the difference is unimportant on
the scale of Figure 5. in Essig_keV  and are also plotted in green in Figure
5. The result is a viable dark matter model with $\epsilon\sim
10^{-12}\times(m_{A^{\prime}}/3m_{\chi})^{1/2}$. While only
$m_{A^{\prime}}=3m$ is plotted, there is in fact a wide range of viable dark
photon masses. The upper bound on the dark photon mass comes from the fact
that both the freeze-in and $N_{\text{eff}}$ lines in the mass range of
interest move up with dark photon mass as $\epsilon\propto
m_{A^{\prime}}^{1/2}$. However, the neutral LDP bound exhibits only weak
dependence on $m_{A^{\prime}}$. For $\alpha_{D}=0.5$ we have
$m_{A^{\prime}}\lesssim\text{TeV}$ while for smaller $\alpha_{D}$ the bound
decreases. For a number of dark meson masses, the maximum viable dark photon
mass as a function of $\alpha_{D}$ is shown in the left panel of Figure 6.

Figure 5: Constraints on kinetic mixing $\epsilon$ as a function of dark
matter mass for $N_{f}=2$ and $m_{A^{\prime}}=3m$. The blue freeze-out
scenario relic abundance line, green freeze-in relic abundance line, and teal
$N_{\text{eff}}$ bound (constraint from effective number of degrees of freedom
present during BBN) are taken from Essig_keV . In magenta, we have added the
neutral LDP bound requiring the neutral DM is lighter than the charged DM.
Note that we have assumed the largest possible SUSY breaking correction with
$|\cos 2\beta|=1$, which arises in the large $\tan\beta$ limit. The vertical
gold-shaded area indicates the region of parameter space where the SRDM model
best fits the data from Kaplinghat:2015aga . The plot exhibits very weak
$N_{f}$ dependence; in particular, the neutral LDP bound and freeze-in lines
do not depend on $N_{f}$ while the DM mass decreases for increasing $N_{f}$.

Figure 6: Left: Maximum value of the dark photon mass as a function of the
dark photon coupling $\alpha_{D}$, for several different values of the dark
meson mass. Lower meson masses are applicable for a great number of flavors
$N_{f}$. Right: The ratio of charged to neutral DM density for $N_{f}=2$ after
mutual decoupling, as a function of $\alpha_{D}$. A smaller coupling leads to
a smaller charged mass splitting and therefore a more resonant self-
interaction. As a result, a greater amount of Boltzmann suppression is
required before the DM decouples.

There is one important subtlety to take care of in the freeze-in scenario. The
dark photon decays only to charged DM, which is heavier than the neutral DM
and does not participate in the resonance. We must therefore ensure that the
dark sector is at one point in equilibrium with itself so that the charged DM
can partially down-scatter to neutral DM. How much charged DM remains will
depend on when the dark sector leaves equilibrium with itself. We make these
estimates below.

Assuming a dark photon of mass $m_{A^{\prime}}>2m$, the dark matter is
produced with temperature888Properly we should say energy and not temperature.
However, we check that the dark sector comes into equilibrium with itself and
so use the terms interchangeably. $T_{\rm DM}\sim T\sim m_{A^{\prime}}/2$,
where $T$ is the SM sector temperature. Redshifting in a radiation dominated
era means that $p\propto T$, so that while the DM remains relativistic we have
$T_{\rm DM}\sim T$. Once the temperature crosses the DM mass threshold,
$v\approx p/m\sim T/m$. Therefore $T_{\rm DM}\sim mv^{2}\sim T^{2}/m$. From
Eq. (26) we have $\sigma\approx\sigma_{2}/v^{4}$ (before the charged meson
mass splitting becomes relevant—what we call ‘saturation of the resonance’)
and thus

$\expectationvalue{\sigma
v}\propto\frac{\sigma_{2}}{\expectationvalue{v}^{3}}\propto T^{-3}$ (30)

On the other hand, the dark matter number density at temperature $T$ after the
dark matter has frozen in goes like

$n(T)=\frac{\Omega_{{\rm DM},0}\,\rho_{c,0}}{m}\frac{s(T)}{s_{0}}\propto
T^{3}$ (31)

where $\Omega_{{\rm DM},0}$ is the present-day energy fraction of dark matter,
$\rho_{c,0}$ is the critical energy density, and $s$ is entropy density.
Therefore, until the resonance saturates, the scattering rate in the dark
sector $\Gamma=n\expectationvalue{\sigma v}$ is temperature independent. We
have checked that for all $\alpha_{D}\lesssim 0.5$ this exceeds the Hubble
rate, which decreases like $T^{2}$, at some temperature before the resonance
saturates. The neutral and charged DM do indeed equilibrate.

The resonance saturates once $mv^{2}\sim 8(m_{\pm}-m_{0})$ (recall the
argument of Section 3) at which point the cross section no longer increases as
the velocity decreases. At this time the Boltzmann suppression of the charged
DM abundance relative to neutral DM abundance begins. Therefore, the charged
to neutral annihilation rate will decrease exponentially with decreasing
temperature. Once $\Gamma$ falls below the Hubble rate at dark matter
temperature $T_{D,\text{freezeout}}$, the charged DM is frozen at relative
abundance $\exp(-(m_{\pm}-m_{0})/T_{D,\text{freezeout}})$, shown in the right
panel of Figure 6. For $\alpha_{D}\lesssim 0.01$ the charged DM forms a small
fraction of the total relic DM. This justifies a posteriori not including the
charged DM interactions in the self-interaction calculation for a wide range
of dark photon couplings.

##  5 Conclusion

We have proposed a SIDM model based upon supersymmetric QCD. Due to SUSY and
flavor symmetry, the low energy effective theory exhibits a natural 2:1 mass
ratio leading to a velocity dependent self-interaction cross section. Fitting
to self-interaction constraints and small scale puzzles fixes a weak self-
coupling and a DM mass in the few MeV range. We find that while freeze-out is
excluded, the freeze-in scenario can reproduce the DM relic abundance. For a
dark photon with a mass just above the DM mass, we find a kinetic mixing of
$\epsilon\sim 10^{-12}$, while with a heavier dark photon $\epsilon$ can be a
few orders of magnitude larger. Finally, $\alpha_{D}$ controls the fraction of
charged DM, which is $\mathcal{O}(1)$ for large $\alpha_{D}$ and exponentially
suppressed for small $\alpha_{D}$.

###### Acknowledgements.

AG and KL thank Simon Knapen for useful discussions. CC and AG are supported
in part by the NSF grant PHY-2014071. CC, AG and EK are supported in part by
the US-Israeli Binational Science Foundation grant 2020220. AG is also
supported in part by the NSERC PGS-D fellowship. YH is supported in part by
the Israel Science Foundation (grant No. 1112/17), by the Azrieli Foundation
and by an ERC STG grant (grant No. 101040019). YH and HM are also supported in
part by the US-Israeli Binational Science Foundation (grant No. 2018140). EK
is supported by the US-Israeli Binational Science Foundation (grants 2016153
and 2020220), the Israel Science Foundation (grant No. 1111/17), and by the
I-CORE Program of the Planning and Budgeting Committee (grant No. 1937/12). HM
is supported in part by the Director, Office of Science, Office of High Energy
Physics of the U.S. Department of Energy under the Contract No. DE-
AC02-05CH11231, by the NSF grant PHY-1915314, by the JSPS Grant-in-Aid for
Scientific Research JP20K03942, MEXT Grant-in-Aid for Transformative Research
Areas (A) JP20H05850, JP20A203, by WPI, MEXT, Japan, the Institute for AI and
Beyond of the University of Tokyo, and Hamamatsu Photonics. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon Europe research and innovation programme (grant agreement No.
101040019). Funded by the European Union. Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union. The European Union cannot be held responsible for them.

## Appendix

##  Appendix A 1-Loop Mass Ratio Calculation

In this Appendix we present the details of the one loop adjoint and singlet
meson mass corrections. Recall that the UV $SU(N_{c})$ gauge theory is broken
to $SU(N_{c}-N_{f})$ by the squark VEVs $v$. At this scale the gauge bosons
corresponding to the broken generators eat some of the quark superfields, with
the remaining low energy degrees of freedoms being the meson superfields.
Therefore, the cutoff of the low energy meson theory is $v$, modulo order one
corrections.

We calculate superfield renormalizations via fermion self-energy diagrams. The
leading correction comes from the diagrams pictured below in Figure 7 and 8.

[layered layout, horizontal=a to d,scale = 0.8] a[particle = $S$] –[fermion] b
[dot] – [blue,anti fermion,edge label’ = $M$] c [dot] –[fermion] d [particle =
$S$], c – [blue, charged scalar, half right, looseness=1.5,edge label’ = $M$]
b, ;

[layered layout, horizontal=a to d,scale = 0.8] a[particle = $S$] –[fermion] b
[dot] – [anti fermion,edge label’ = $S$] c [dot] –[fermion] d [particle =
$S$], c – [ charged scalar, half right, looseness=1.5,edge label’ = $S$] b, ;

Figure 7: Diagrams contributing to the mass correction of the singlet.

[layered layout, horizontal=a to d,scale = 0.8] a[blue, particle = $M$]
–[blue,fermion] b [dot] – [blue,anti fermion,edge label’ = $M$] c [dot]
–[blue,fermion] d [blue, particle = $M$], c – [blue, charged scalar, half
right, looseness=1.5,edge label’ = $M$] b, ;

[layered layout, horizontal=a to d,scale = 0.8] a[blue,particle = $M$]
–[blue,fermion] b [dot] – [anti fermion,edge label’ = $S$] c [dot]
–[blue,fermion] d [blue,particle = $M$], c – [blue, charged scalar, half
right, looseness=1.5,edge label’ = $M$] b, ;

[layered layout, horizontal=a to d,scale = 0.8] a[blue,particle = $M$]
–[blue,fermion] b [dot] – [anti fermion,edge label’ = $S$] c [dot]
–[blue,fermion] d [blue,particle = $M$], c – [blue, charged scalar, half
right, looseness=1.5,edge label’ = $M$] b, ;

Figure 8: Diagrams contributing to the mass correction of the adjoint.

The one-loop renormalization from any one particular diagram is of the form

$\displaystyle\delta
Z_{1-loop}=-\frac{C\mu^{2}}{16\pi^{2}v^{2}}\int_{0}^{1}dz(1-z)\log\frac{-z(1-z)M_{0}^{2}+(1-z)M_{\text{bos}}^{2}+zM_{\text{ferm}}^{2}}{v^{2}}$
(32)

Where $C$ is a Lagrangian and group theory dependent prefactor.

We must find the Yukawa couplings of the theory to find the values of the
prefactors. We use Eq. (9) to find the Christoffel symbols of the Kähler
metric to be at leading order

$\displaystyle\Gamma^{i}_{jk}=-\frac{1}{2\sqrt{2}\,v}\Tr
T_{i}\\{T_{j},T_{k}\\}$ (33)

With this the Lagrangian’s Yukawa term is

$\displaystyle\begin{split}\mathcal{L}_{\text{Yukawa}}&=-\frac{1}{2}\left(\frac{\partial^{2}W}{\partial\widetilde{M}_{j}\widetilde{M}_{k}}-\Gamma^{i}_{jk}\frac{\partial
W}{\partial\widetilde{M}_{i}}\right)\psi_{j}\psi_{k}+h.c\\\
&=\frac{1}{2}\frac{\sqrt{2}\,\mu}{v}\left(\frac{r(2r+1)}{\sqrt{N_{f}}}S\psi_{S}^{2}+\frac{r+2}{\sqrt{N_{f}}}S\Tr\psi_{M}^{2}+2\frac{2r+1}{\sqrt{N_{f}}}\psi_{S}\Tr
M\psi_{M}+3\Tr M\psi_{M}^{2}\right)+h.c.\end{split}$ (34)

The prefactors, dependent on the representation of the external, internal
scalar, and internal fermion respectively, are found to be

$\displaystyle C_{S,S,S}=\frac{2r^{2}(2r+1)^{2}}{N_{f}}\underset{r\rightarrow
2}{\longrightarrow}\frac{200}{N_{f}}$ (35) $\displaystyle
C_{S,M,M}=\frac{2(N_{f}^{2}-1)(2r+1)^{2}}{N_{f}}\underset{r\rightarrow
2}{\longrightarrow}\frac{50(N_{f}^{2}-1)}{N_{f}}$ (36) $\displaystyle
C_{M,M,M}=9\frac{N_{f}^{2}-4}{N_{f}}$ (37) $\displaystyle
C_{M,S,M}=\frac{2(r+2)^{2}}{N_{f}}\underset{r\rightarrow
2}{\longrightarrow}\frac{32}{N_{f}}$ (38) $\displaystyle
C_{M,M,S}=\frac{2(2r+1)^{2}}{N_{f}}\underset{r\rightarrow
2}{\longrightarrow}\frac{50}{N_{f}}$ (39)

Given that the Kähler potential and superpotential are expansions in powers of
$v^{-1}$ while the cutoff is $v$, higher derivative operators can also
contribute to the one-loop at $\mathcal{O}(\mu^{2}/v^{2})$. However, treating
$m$ as a spurion of the explicitly broken chiral symmetry, one can show that
these contributions come without the $\log(v/m)$ and are therefore
subdominant.

Putting this all together gives the 1-loop correction to the correction to the
mass ratio to be

$\displaystyle\delta_{\text{1-loop}}=$
$\displaystyle-\frac{(C_{SSS}+C_{SMM})-(C_{MMM}+C_{MSM}+C_{MMS})}{16\pi^{2}}\frac{\mu^{2}}{v^{2}}\log\left(\frac{v}{2\mu}\right)+\text{finite}$
$\displaystyle=$
$\displaystyle-\frac{104+41N_{f}^{2}}{16\pi^{2}N_{f}}\frac{\mu^{2}}{v^{2}}\log\left(\frac{v}{2\mu}\right)+\cdots$

Here we expanded in $\mu/v\ll 1$, which is shown to be a consistent
approximation in the region of parameter space fitting the self-interaction
results of section 3.

##  Appendix B Anomaly matching for neutral meson decay

In this Appendix we discuss the conditions required to avoid neutral meson
decay. We find the same condition as was found in a non-SUSY context by
Berlin:2018tvf . The decay of neutral mesons in the SRDM model parallels the
decay of SM neutral pions. Even without the weak force, the decay
$\pi^{0}\rightarrow\gamma\gamma$ is not forbidden by $U(1)_{EM}$ charge
conservation, and is in fact realized via anomalies. The process comes from
the Wess-Zumino-Witten (WZW) term $~{}\pi^{0}F\widetilde{F}$ in the pion
effective Lagrangian. In the pion effective theory of the IR, a chiral
rotation effects a shift $\pi^{0}\rightarrow\pi^{0}+\alpha$. Thus the WZW term
ensures that the chiral anomaly produced by fermions (quarks) in the UV is
matched in the IR where there are no fermions. The story in our model changes
only slightly with the addition of SUSY where the IR mesons now have fermionic
components.

We introduce a dark photon $U(1)_{D}$ that couples, in the UV theory, to the
quark superfields with diagonal $N_{f}\times N_{f}$ charge matrix $Q$, which
in the SM would be $\text{diag}(+\frac{2}{3},-\frac{1}{3})$. Maintaining a
vector-like theory, the anti-quarks couple with matrix $-Q$. Let us perform an
axial rotation using any diagonal generator $T$ of $SU(N_{f})_{A}\times
U(1)_{R}$ (note in the SM we have no $U(1)_{R}$ so that $T$ must be the
traceless $\sigma^{3}$).999We exclude $U(1)_{A}$ because it carries the
$U(1)_{A}SU(N_{c})^{2}$ anomaly. This acts on the quarks and anti-quarks of
the UV theory. The $U(1)_{T}U(1)_{D}^{2}$ anomaly in the UV is $2\Tr(TQ^{2})$.
The same quantity in the IR is

$\begin{split}\Tr_{\text{M}}T_{\text{M}}Q_{\text{M}}^{2}&=\sum_{ij}(T_{i}+T_{j})(Q_{i}-Q_{j})^{2}=2\sum_{ij}(T_{i}Q_{i}^{2}-2T_{i}Q_{i}Q_{j}+T_{i}Q_{j}^{2})\\\
&=2(N_{f}\Tr(TQ^{2})-2\Tr(TQ)\Tr Q+\Tr T\Tr Q^{2})\end{split}$ (40)

The difference between the UV and IR anomalies must be made up by a WZW term
that transforms at linear order as

$\sim 2((1-N_{f})\Tr(TQ^{2})+2\Tr(TQ)\Tr Q-\Tr T\Tr
Q^{2})F_{\alpha\beta}\widetilde{F}^{\alpha\beta}$ (41)

The meson field without the VEV subtracted off,
$\widetilde{M}^{\prime}=v^{2}\mathbf{1}+\widetilde{M}$, transforms as
$\widetilde{M}^{\prime}\rightarrow
e^{iT}\widetilde{M}^{\prime}e^{iT}\approx\widetilde{M}^{\prime}+2iv^{2}T$.
Instead of calculating the full WZW term, we simply note that to reproduce the
anomaly shift in Eq. (41), at linear order in $\widetilde{M}^{\prime}$ we can
have101010Note that these coefficients do not necessarily match those of Eq.
(41) because terms higher order in $\widetilde{M}^{\prime}$ also transform at
linear order in $T$.

$\sim\left(c_{1}\Tr(\widetilde{M}^{\prime}Q^{2})+c_{2}\Tr(\widetilde{M}^{\prime}Q)\Tr
Q+c_{3}\Tr\widetilde{M}^{\prime}\Tr
Q^{2}\right)(W_{\alpha}W^{\alpha}|_{\theta^{2}})$ (42)

Generically these terms would give rise to decays of non-singlet neutral
mesons, via off-shell dark photons, to SM particles. To avoid this, Eq. (42)
should vanish. It is sufficient to require either that $Q\propto\mathbf{1}$
(mesons are uncharged under $U(1)_{D}$), or that $Q^{2}\propto\mathbf{1}$ and
$\Tr Q=0$ as noted in Berlin:2018tvf . We assume the latter condition holds
throughout the paper.

Note that it may also be possible to come up with non-trivial charge
configurations, depending on the coefficients $c_{i}$, that conspire to make
the non-singlet terms of Eq. (42) disappear. However, for large enough
$g_{D}$, the breaking of the quark degeneracy could be sizeable and could lead
to the destruction of the resonance present in the meson spectrum.

## References

  * (1) R.A. Flores and J.R. Primack, _Observational and theoretical constraints on singular dark matter halos_ , _Astrophys. J. Lett._ 427 (1994) L1 [astro-ph/9402004].
  * (2) B. Moore, _Evidence against dissipation-less dark matter from observations of galaxy haloes_ , _Nature_ (1994) .
  * (3) J.F. Navarro, C.S. Frenk and S.D.M. White, _The Structure of cold dark matter halos_ , _Astrophys. J._ 462 (1996) 563 [astro-ph/9508025].
  * (4) B. Moore, T.R. Quinn, F. Governato, J. Stadel and G. Lake, _Cold collapse and the core catastrophe_ , _Mon. Not. Roy. Astron. Soc._ 310 (1999) 1147 [astro-ph/9903164].
  * (5) B. Moore, S. Ghigna, F. Governato, G. Lake, T.R. Quinn, J. Stadel et al., _Dark matter substructure within galactic halos_ , _Astrophys. J. Lett._ 524 (1999) L19 [astro-ph/9907411].
  * (6) A.A. Klypin, A.V. Kravtsov, O. Valenzuela and F. Prada, _Where are the missing Galactic satellites?_ , _Astrophys. J._ 522 (1999) 82 [astro-ph/9901240].
  * (7) M. Boylan-Kolchin, J.S. Bullock and M. Kaplinghat, _Too big to fail? The puzzling darkness of massive Milky Way subhaloes_ , _Mon. Not. Roy. Astron. Soc._ 415 (2011) L40 [1103.0007].
  * (8) M. Boylan-Kolchin, J.S. Bullock and M. Kaplinghat, _The Milky Way’s bright satellites as an apparent failure of LCDM_ , _Mon. Not. Roy. Astron. Soc._ 422 (2012) 1203 [1111.2048].
  * (9) K.A. Oman et al., _The unexpected diversity of dwarf galaxy rotation curves_ , _Mon. Not. Roy. Astron. Soc._ 452 (2015) 3650 [1504.01437].
  * (10) J.F. Navarro, V.R. Eke and C.S. Frenk, _The cores of dwarf galaxy halos_ , _Mon. Not. Roy. Astron. Soc._ 283 (1996) L72 [astro-ph/9610187].
  * (11) S. Gelato and J. Sommer-Larsen, _On ddo154 and cold dark matter halo profiles_ , _Mon. Not. Roy. Astron. Soc._ 303 (1999) 321 [astro-ph/9806289].
  * (12) J. Binney, O. Gerhard and J. Silk, _The Dark matter problem in disk galaxies_ , _Mon. Not. Roy. Astron. Soc._ 321 (2001) 471 [astro-ph/0003199].
  * (13) O.Y. Gnedin and H. Zhao, _Maximum feedback and dark matter profiles of dwarf galaxies_ , _Mon. Not. Roy. Astron. Soc._ 333 (2002) 299 [astro-ph/0108108].
  * (14) D.N. Spergel and P.J. Steinhardt, _Observational evidence for selfinteracting cold dark matter_ , _Phys. Rev. Lett._ 84 (2000) 3760 [astro-ph/9909386].
  * (15) A. Kamada, M. Kaplinghat, A.B. Pace and H.-B. Yu, _How the Self-Interacting Dark Matter Model Explains the Diverse Galactic Rotation Curves_ , _Phys. Rev. Lett._ 119 (2017) 111102 [1611.02716].
  * (16) M. Rocha, A.H.G. Peter, J.S. Bullock, M. Kaplinghat, S. Garrison-Kimmel, J. Onorbe et al., _Cosmological Simulations with Self-Interacting Dark Matter I: Constant Density Cores and Substructure_ , _Mon. Not. Roy. Astron. Soc._ 430 (2013) 81 [1208.3025].
  * (17) A.H.G. Peter, M. Rocha, J.S. Bullock and M. Kaplinghat, _Cosmological Simulations with Self-Interacting Dark Matter II: Halo Shapes vs. Observations_ , _Mon. Not. Roy. Astron. Soc._ 430 (2013) 105 [1208.3026].
  * (18) M. Kaplinghat, S. Tulin and H.-B. Yu, _Dark Matter Halos as Particle Colliders: Unified Solution to Small-Scale Structure Puzzles from Dwarfs to Clusters_ , _Phys. Rev. Lett._ 116 (2016) 041302 [1508.03339].
  * (19) X. Chu, C. Garcia-Cely and H. Murayama, _Velocity Dependence from Resonant Self-Interacting Dark Matter_ , _Phys. Rev. Lett._ 122 (2019) 071103 [1810.04709].
  * (20) Y.-D. Tsai, R. McGehee and H. Murayama, _Resonant Self-Interacting Dark Matter from Dark QCD_ , _Phys. Rev. Lett._ 128 (2022) 172001 [2008.08608].
  * (21) I. Affleck, M. Dine and N. Seiberg, _Dynamical Supersymmetry Breaking in Supersymmetric QCD_ , _Nucl. Phys. B_ 241 (1984) 493.
  * (22) I. Affleck, M. Dine and N. Seiberg, _Dynamical Supersymmetry Breaking in Four-Dimensions and Its Phenomenological Implications_ , _Nucl. Phys. B_ 256 (1985) 557.
  * (23) A. Berlin, N. Blinov, S. Gori, P. Schuster and N. Toro, _Cosmology and Accelerator Tests of Strongly Interacting Dark Matter_ , _Phys. Rev. D_ 97 (2018) 055033 [1801.05805].
  * (24) L. Alvarez-Gaume, J. Polchinski and M.B. Wise, _Minimal Low-Energy Supergravity_ , _Nucl. Phys. B_ 221 (1983) 495.
  * (25) L.J. Hall, J.D. Lykken and S. Weinberg, _Supergravity as the Messenger of Supersymmetry Breaking_ , _Phys. Rev. D_ 27 (1983) 2359.
  * (26) L. Randall and R. Sundrum, _Out of this world supersymmetry breaking_ , _Nucl. Phys. B_ 557 (1999) 79 [hep-th/9810155].
  * (27) G.F. Giudice, M.A. Luty, H. Murayama and R. Rattazzi, _Gaugino mass without singlets_ , _JHEP_ 12 (1998) 027 [hep-ph/9810442].
  * (28) L. Alvarez-Gaume, M. Claudson and M.B. Wise, _Low-Energy Supersymmetry_ , _Nucl. Phys. B_ 207 (1982) 96.
  * (29) M. Dine and W. Fischler, _A Phenomenological Model of Particle Physics Based on Supersymmetry_ , _Phys. Lett. B_ 110 (1982) 227.
  * (30) A. Hook and H. Murayama, _Low-energy Supersymmetry Breaking Without the Gravitino Problem_ , _Phys. Rev. D_ 92 (2015) 015004 [1503.04880].
  * (31) A. Hook, R. McGehee and H. Murayama, _Cosmologically Viable Low-energy Supersymmetry Breaking_ , _Phys. Rev. D_ 98 (2018) 115036 [1801.10160].
  * (32) A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, _FeynRules 2.0 - A complete toolbox for tree-level phenomenology_ , _Comput. Phys. Commun._ 185 (2014) 2250 [1310.1921].
  * (33) N.D. Christensen and C. Duhr, _FeynRules - Feynman rules made easy_ , _Comput. Phys. Commun._ 180 (2009) 1614 [0806.4194].
  * (34) T. Hahn, _Generating Feynman diagrams and amplitudes with FeynArts 3_ , _Comput. Phys. Commun._ 140 (2001) 418 [hep-ph/0012260].
  * (35) T. Hahn and M. Perez-Victoria, _Automatized one loop calculations in four-dimensions and D-dimensions_ , _Comput. Phys. Commun._ 118 (1999) 153 [hep-ph/9807565].
  * (36) J.H. Chang, R. Essig and A. Reinert, _Light(ly)-coupled Dark Matter in the keV Range: Freeze-In and Constraints_ , _JHEP_ 03 (2021) 141 [1911.03389].
  * (37) C. Dvorkin, T. Lin and K. Schutz, _Making dark matter out of light: freeze-in from plasma effects_ , _Phys. Rev. D_ 99 (2019) 115009 [1902.08623].