# Dark Matter in A Mirror Solution to the Strong CP Problem

Quentin Bonnefoy<EMAIL_ADDRESS>Berkeley Center for Theoretical
Physics, Department of Physics, University of California, Berkeley, CA 94720,
USA  Theoretical Physics Group, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Lawrence Hall<EMAIL_ADDRESS>Berkeley Center for
Theoretical Physics, Department of Physics, University of California,
Berkeley, CA 94720, USA  Theoretical Physics Group, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Claudio Andrea Manzari<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics, Department of Physics, University of
California, Berkeley, CA 94720, USA  Theoretical Physics Group, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Amara McCune
<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics, Department of
Physics, University of California, Berkeley, CA 94720, USA  Theoretical
Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Department of Physics, University of California, Santa Barbara, CA 93106, USA
Christiane Scherb<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics,
Department of Physics, University of California, Berkeley, CA 94720, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA

###### Abstract

We study thermal production of dark matter (DM) in a realization of the
minimal models of Ref. Bonnefoy _et al._ (2023), where parity is used to solve
the strong CP problem by transforming the entire Standard Model (SM) into a
mirror copy. Although the mirror electron $e^{\prime}$ is a good DM candidate,
its viability is mired by the presence of the mirror up-quark $u^{\prime}$,
whose abundance is intimately related to the $e^{\prime}$ abundance and must
be suppressed. This can be achieved through a sequential freeze-in mechanism,
where mirror photons are first produced from SM gluons, and then the mirror
photons produce $e^{\prime}$. After computing the details of this double
freeze-in, we discuss the allowed parameter space of the model, which lies at
the threshold of experimental observations. We find that this origin of
$e^{\prime}$ DM requires a low reheating temperature after inflation and is
consistent with the baryon asymmetry arising from leptogenesis, providing
mirror neutrinos have a significant degeneracy. Finally, we show that this
$e^{\prime}$ DM is not compatible with Higgs Parity, the simplest scheme with
exact parity, unless SM parameters deviate significantly from their central
values or the minimal model is extended.

##  I Introduction

The nature of DM and the strong CP problem are two of the most compelling
puzzles of particle physics. In the last decades, cosmological observations of
the cosmic microwave background (CMB), distant supernovae, large samples of
galaxy clusters, and baryon acoustic oscillations have firmly established a
standard cosmological model in which DM accounts for about 85% of the matter
content of the Universe, and about 27% of the global energy budget Ade _et
al._ (2016). Another puzzle stems from the observational absence of the
neutron electric dipole moment, $d_{n}<10^{-26}\;\rm{e\cdot cm}$ Pendlebury
_et al._ (2015), which constrains the amount of CP violation in the strong
interactions. (C stands for charge conjugation, P for spacetime parity and CP
for their combination.) In the QCD Lagrangian, there is only one CP violating
term ’t Hooft (1976),

$\mathcal{L}\supset\bar{\theta}_{\rm
QCD}\frac{g_{s}^{2}}{32\pi^{2}}G^{a}_{\mu\nu}\tilde{G}^{a,\mu\nu}\,,$ (1)

where $G_{a}^{\mu\nu}$ is the gluon field strength tensor, $g_{s}$ the strong
coupling constant,
$\tilde{G}_{a,\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}G_{a}^{\alpha\beta}$
and $\bar{\theta}_{\text{QCD}}$ is an angle $\in[0,2\pi]$. In the SM,
$\bar{\theta}_{\rm QCD}$ combines the bare $\theta$-angle $\theta_{\rm QCD}$
with an anomalous contribution from the quark mass matrix $M$ into a flavor-
invariant quantity, $\bar{\theta}_{\rm QCD}\equiv\theta_{\rm
QCD}+\arg\det(M)$. The constraint above results in the upper limit
$\bar{\theta}_{\rm QCD}\lesssim 10^{-10}$ Baluni (1979); Crewther _et al._
(1979); Pospelov and Ritz (2000), so that $\bar{\theta}_{\rm QCD}$ is by far
and inexplicably the smallest dimensionless parameter of the SM without any
reason. This strong CP problem is made even more surprising by the fact that
weak interactions violate CP with a phase of order unity.

Three approaches to this problem have received considerable attention in the
literature: a massless colored fermion Kaplan and Manohar (1986); Banks _et
al._ (1994); Hook (2015); Agrawal and Howe (2018) which makes
$\bar{\theta}_{\rm QCD}$ unphysical (and whose minimal incarnation is now
excluded by lattice data Aoki _et al._ (2022)), spontaneously broken P or CP
symmetries Nelson (1984); Barr (1984); Babu and Mohapatra (1989, 1990) that
fix $\bar{\theta}_{\rm QCD}=0$ in the UV and rely on its extremely suppressed
renormalization Ellis and Gaillard (1979); Valenti and Vecchi (2021); de Vries
_et al._ (2021); Valenti and Vecchi (2023); Hisano _et al._ (2023), and a
spontaneously broken global anomalous symmetry à la Peccei-Quinn Peccei and
Quinn (1977a, b) which relaxes $\bar{\theta}_{\rm QCD}$ to $0$ through the
QCD-induced dynamics of the predicted QCD axion Weinberg (1978); Wilczek
(1978). The QCD axion additionally turns out to be a natural DM candidate
Preskill _et al._ (1983); Abbott and Sikivie (1983); Dine and Fischler (1983).
(See Ref. Di Luzio _et al._ (2020) for a review on the QCD axion and a large
set of references.)

It was recognized in the 1970s already that parity might solve the strong CP
problem Beg and Tsao (1978); Mohapatra and Senjanovic (1978), and several
models have been analyzed in the literature since then (see e.g. Babu and
Mohapatra (1989, 1990); Barr _et al._ (1991); Lavoura (1997); Gu (2012);
D’Agnolo and Hook (2016); Kawamura _et al._ (2019); Hall and Harigaya (2018);
Dunsky _et al._ (2019a); Hall and Harigaya (2019); Craig _et al._ (2021); Redi
and Tesi (2023); Carrasco-Martinez _et al._ (2023)). Those models require
extending the gauge group and particle content of the SM. However, unlike the
QCD axion, the presence of a good DM candidate is not guaranteed by a suitable
choice of parameters and cosmological history, as P relates the additional
couplings and masses to the SM ones so that there is little freedom to find an
appropriate parameter space. Extra fields can be added to play the role of DM
Gu (2012); Kawamura _et al._ (2019), while within minimal models the
possibility that DM is made out of the mirror neutrinos or mirror electrons
predicted by parity has been explored in Refs. Dror _et al._ (2020); Dunsky
_et al._ (2021) and Dunsky _et al._ (2019a), respectively. For mirror
neutrinos, justifying the necessary small parameters requires extra
ingredients, while only non-thermal production mechanisms were known to work
for mirror electrons. In this work, we study whether it is possible to
thermally produce DM within a minimal setup where, as with the axion, its mass
and abundance can be computed in terms of parameters already required to solve
the strong CP problem.

Concretely, we analyze the model proposed by some of us in Ref. Bonnefoy _et
al._ (2023), based on a mirror copy of the SM. In the process, we
automatically explore other setups studied in the literature Barr _et al._
(1991); Dunsky _et al._ (2019a), to which our model reduces in the appropriate
limit. As said above, parity fixes almost all coefficients given the measured
SM parameters, so that one can genuinely scan the full parameter space.
Generically, the only DM candidate is the mirror electron $e^{\prime}$, i.e.
the new fermion paired with the SM electron by parity. We argue that
production from a dark sector in thermal equilibrium with the SM is not
allowed by experimental constraints. Moreover, the only viable thermal
production mechanism from the SM bath, that can adequately suppress the
dangerous relic abundance of mirror up quarks $u^{\prime}$, is (sequential)
freeze-in. Mirror up quarks are electrically neutral and colored and hence can
bind into fractionally-charged exotic hadrons. We perform a thorough numerical
analysis and identify the range of model parameters where one finds consistent
$e^{\prime}$ DM. This fixes all scales of the model within a few orders of
magnitude, dramatically increasing the predictivity beyond the constraints
from parity alone.

As we were completing this work, a different cosmological history for the same
model was presented in Ref. Redi and Tesi (2023), where $e^{\prime}$ DM is
obtained through freeze-out and subsequent dilution, while fractionally-
charged exotic hadrons are argued to be sufficiently suppressed. While the
bounds on such hadrons Dunsky _et al._ (2019a) are based on fluxes that have
uncertainties Dunsky _et al._ (2019b), they appear to exclude this
cosmological scenario by several orders of magnitude, calling into question
its viability.

The paper is organized as follows. In Sec. II, we summarize the relevant
features and mechanisms of the model proposed in Ref. Bonnefoy _et al._ (2023)
to solve the strong CP problem. In Sec. III, we discuss the running of masses
and gauge couplings in this model, while in Sec. IV we study the induced
kinetic mixing between the SM and mirror photon, which strongly impacts the
direct detection (DD) prospects of the model. In Sec. V we explore whether
this model provides a solution to the DM problem, and in Sec. VII we comment
on the compatibility between DM and the Higgs parity mechanism, which allows
parity to be exact instead of softly broken in the UV without adding new
degrees of freedom. Finally, we conclude in Sec. VIII. In App. A, we compute
the freeze-in distribution of mirror photons, allowing us to compute the
$e^{\prime}$ DM abundance in Sec. V.

##  II Mirror Solutions to the Strong CP problem

Here, we summarize the model presented in Ref. Bonnefoy _et al._ (2023).

The full SM gauge group is mirrored to $SU(3)\times SU(2)\times U(1)_{Y}\times
SU(3)^{\prime}\times SU(2)^{\prime}\times U(1)^{\prime}$, and so are the
fermion and Higgs fields, which come in two copies of the SM matter content.
One set of particles has the usual SM quantum numbers under $SU(3)\times
SU(2)\times U(1)_{Y}$ and is a singlet of $SU(3)^{\prime}\times
SU(2)^{\prime}\times U(1)^{\prime}$, while the converse is true for the other
set. Each of the two Higgs fields is responsible for the breaking of the
electroweak sector of its own “world”. This setup can solve the strong CP
problem when the two worlds are related via a $\mathbb{Z}_{2}$ symmetry
composed with the usual action of P, so that the (C)P-odd $\theta$ angles of
$SU(3)$ and $SU(3)^{\prime}$ satisfy the relation $\theta=-\theta^{\prime}$.
In Ref. Bonnefoy _et al._ (2023), it was shown that breaking the two $SU(3)$
gauge groups to their diagonal subgroup, which is identified with
$SU(3)_{\text{QCD}}$, provides a solution to the strong CP problem, as one
finds $\theta_{\text{QCD}}=\theta+\theta^{\prime}$. In the limit of exact or
softly broken P in the UV, the two $\theta$ angles cancel completely. In the
IR, despite P being broken, $\theta_{\text{QCD}}$ receives negligible
corrections.

The interest of mirror solutions to the strong CP problem stems from their
simplicity and predictivity: the energy scales $v^{\prime}$ and $v_{3}$ at
which parity and $SU(3)\times SU(3)^{\prime}$ are respectively broken
represent the only two free parameters with respect to the SM, since the
Yukawa and gauge couplings are fixed by P. The mirror Higgs acquires a vacuum
expectation value (VEV) $v^{\prime}\gg v$, where $v$ denotes the SM Higgs VEV,
breaking P spontaneously. A phenomenologically-viable vacuum is achieved
either by a soft P breaking term in the tree-level scalar potential or by
radiative corrections at the scale $v^{\prime}$ Hall and Harigaya (2018). We
discuss the latter further in Sec. VII. Either way, the theory does not
address the hierarchy problem. Moreover, in this paper, we consider the
simplest $SU(3)\times SU(3)^{\prime}\to SU(3)_{\rm QCD}$ breaking mechanism,
provided by the VEV of a bifundamental scalar field $\Sigma$. Although the
solution to the strong CP problem does not depend on the specific breaking
mechanism (see Ref. Bonnefoy _et al._ (2023) for additional examples), the
phenomenology of the model is sensitive to it. This will be commented on when
needed. The constraints on the parameter space of this setup from collider
searches and from the requirement that it solves the strong CP problem are
shown in Fig. 1 of Ref. Bonnefoy _et al._ (2023). The allowed region roughly
reads

$\displaystyle\begin{split}2\cdot 10^{8}\,{\rm GeV}\lesssim v^{\prime}\lesssim
10^{14}\,{\rm GeV};\;\;\;\;v_{3}\gtrsim 5\,{\rm TeV}.\end{split}$ (2)

##  III Masses, couplings and their running

As alluded to above, parity forces the gauge and Yukawa couplings in the two
worlds to be equal at scales above $v^{\prime}$,

$g_{G^{\prime}}=g_{G}\ ,\quad y_{f^{\prime}}=y_{f}\ ,$ (3)

where $G=1,2,3$ denotes a simple factor of the SM gauge group of coupling
$g_{G}$ and similarly for the mirror gauge couplings $g_{G^{\prime}}$, while
$y_{f}$ are the SM Yukawa couplings in the up, down and charged lepton sectors
and similarly for the mirror Yukawa couplings $y_{f^{\prime}}$. The breaking
of parity will generate deviations from these relations below $v^{\prime}$ in
a calculable fashion. Knowing the precise values will be of extreme importance
for the discussion of DM below, so we discuss here the renormalization group
equations (RGEs) of our model.

For a choice of $v^{\prime}$ and $v_{3}$, the values of the parameters at all
scales and in both worlds can be found. Indeed, the solution to the RGEs is
fully fixed by the following constraints:

  1. 1.

matching to the measured SM parameters at low energies,

  2. 2.

continuity at $v_{3}$, except for

$\frac{1}{\alpha_{\text{QCD}}}=\frac{1}{\alpha_{s}}+\frac{1}{\alpha_{s}^{\prime}}$
(4)

(plus possible threshold corrections at loop level), where
$\alpha^{(\prime)}_{s}=\frac{g_{s}^{(\prime)}{}^{2}}{4\pi}$ with
$g_{s}^{(\prime)}$ the (mirror) color coupling constant above $v_{3}$,

  3. 3.

parity at $v^{\prime}$, i.e. the relations of Eq. (3).

For given $v^{\prime}$ and $v_{3}$, these constraints constitute sufficient
boundary data to fix all integration constants in the RGEs. We focus on the
one-loop RGEs Cheng _et al._ (1974) and use the values in the modified minimal
subtraction ($\rm\overline{MS}$) scheme given in Ref. Huang and Zhou (2021).
As said above, we assume that a bifundamental scalar field breaks the color
groups to their diagonal subgroup, and we include its effect on the RGEs. For
simplicity, we assume that all components of that scalar field acquire masses
of order $v_{3}$ (this can be achieved upon suitably choosing the parameters
of its scalar potential).

The most straightforward case is when the lightest colored mirror fermion,
namely the mirror up quark $u^{\prime}$, is heavy, more precisely when
$m_{u^{\prime}}\geq v_{3}$. In this case, the modification of the SM RGEs is
minimal all the way to $v^{\prime}$: one simply needs to replace
$\alpha_{\text{QCD}}\to\alpha_{s}$ and add the contribution of three flavors
of fundamental scalars to the $SU(3)$ running. One can then pick a random
value of $\alpha_{s}(v_{3})$ (which lies between111The lower bound directly
follows from Eq. (4), while the upper bound also takes into account the fact
that the mirror gauge coupling runs faster at lower energies, since the mirror
fermions are heavier. Therefore, since the two gauge couplings are equal at
$v^{\prime}$, $\alpha_{s}(v_{3})<\alpha_{s}^{\prime}(v_{3})$.
$\alpha_{\text{QCD}}(v_{3})$ and $2\alpha_{\text{QCD}}(v_{3})$), run the SM
world couplings to $v^{\prime}$, fix the mirror world parameters at
$v^{\prime}$ and run them down to $v_{3}$ to check whether the relation of Eq.
(4) holds. Spanning over all (or many) choices of $\alpha_{s}(v_{3})$, one can
identify the appropriate value.

When $m_{u^{\prime}}<v_{3}$, the situation is more intricate, since the
boundary values for the mirror quark masses depend on the SM quark masses,
which depend on the running of the QCD coupling constant, which itself depends
on the mirror quark masses and their contribution to the $\beta$ function. In
this case, one needs to numerically solve the full system of RGEs of the two
coupled mirror worlds. In practice, we implement this for all values of
$m_{u^{\prime}}/v_{3}$, and we cross-check the results with the previous
method when applicable. We checked that our numerical precision is such that
the constraints imposed by matching at $v_{3}$ and parity at $v^{\prime}$ are
satisfied at the sub-per-mille level. We illustrate some results of the
procedure in Fig. 1. In the right panel, we see that the loop correction to
the ratio of $m_{u^{\prime}}$ to $m_{e^{\prime}}$, the mirror electron
$e^{\prime}$ mass, depends on $v_{3}$ and can be noticeable, which will prove
crucial when we discuss DM.

Figure 1: $\overline{\text{MS}}$ running parameters for $v^{\prime}=10^{9}$
GeV and two choices of $v_{3}$, to which the red vertical lines correspond.
Left panel: running gauge couplings of the color groups. Right panel: ratios
of the $u^{\prime}$ and $e^{\prime}$ masses. The curves stop at
$\mu=m_{e^{\prime}}(\mu)$, and the running of $m_{u^{\prime}}$ is frozen below
$\mu=m_{u^{\prime}}(\mu)$.

The outputs of this procedure are the running gauge couplings and fermion
masses. They are used as inputs in a second step when we compute the running
kinetic mixing and Higgs quartic, which we discuss below. The RGEs of the
gauge couplings and of the Higgs potential have been studied using similar
techniques in Ref. Redi and Tesi (2023).

A further comment on the input value for $m_{u}$ is in order. In the
following, we find the DM production to be extremely sensitive to
$m_{u^{\prime}}/m_{e^{\prime}}$, whose value is obtained from the low energy
determination of the SM up-quark and electron masses, as described above. The
electron mass is known with the astonishing precision of $\sim 0.1$ ppb, while
the PDG reports $m_{u}(2\,\rm{GeV})=2.16^{+0.49}_{-0.26}\;\rm{MeV}$ Workman
_et al._ (2022). This is the largest source of uncertainty in our result, as
discussed in Sec. V. The central value is used for the right panel of Fig. 1.
Note, that a recent combination of lattice determinations of $m_{u}$ reports a
smaller uncertainty of $\sim 4\%$ Aoki _et al._ (2022), but is not without
controversy and remains under active investigation.

##  IV Kinetic Mixing

Parity is compatible with a kinetic mixing between the gauge boson of
$U(1)_{Y}$ and its mirror copy,

$\mathcal{L}\supset\epsilon F^{\mu\nu}F^{\prime}_{\mu\nu}\,.$ (5)

This term was not relevant for the solution to the strong CP problem in Ref.
Bonnefoy _et al._ (2023), but it constitutes an important source of
cosmological constraints on the parameter space of the model. Indeed, once
this term is introduced, all fermions charged under $U(1)^{\prime}$ become
charged also under $U(1)_{Y}$. If this is the case for DM, very stringent
constraints on its charge from direct detection experiments apply.
Anticipating the discussion in Sec. V, the mirror electron is the only DM
candidate of the model and kinetic mixing plays an important role in assessing
its viability.

Removing the tree-level contribution to the kinetic mixing is a reasonable
assumption. For instance, in UV completions where one $U(1)$ belongs to a
larger gauge group, the interaction term in Eq. (5) is not allowed by gauge
invariance. Therefore, it vanishes above the scale at which the two $U(1)$
gauge theories emerge. However, it will be generated at loop-level below that
scale. In the current setup, the first loop-level contribution to this term
arises at 4-loop Dunsky _et al._ (2019a), and, as described in the following,
it could be further suppressed by an appropriate arrangement of the energy
scales of the model. Nevertheless, due to the tight constraints, it is worth
studying in detail the 4-loop contribution to $\epsilon$, as it depend only on
the two free parameters of the model, $v^{\prime}$ and $v_{3}$.

The leading diagram is shown in Fig. 2. As noted in Ref. Dunsky _et al._
(2019a), the renormalization group equation of the kinetic mixing parameter
can be read off from the four-loop beta function of QCD van Ritbergen _et al._
(1997), yielding

$\displaystyle\frac{{\rm d\epsilon}}{{\rm
dln}\mu}=\frac{e^{2}g_{s}^{6}}{(4\pi)^{8}}\left(-\frac{1760}{27}+\frac{1280}{9}\zeta(3)\right)\sum_{ij}q_{i}q^{\prime}_{j}\
.$ (6)

It is clear that this contribution is only present below $v_{3}$, as above
that scale the gluons in the inner part of the diagram do not couple
simultaneously to SM and mirror quarks. Above $v_{3}$, we need higher-loop
diagrams to connect $\gamma$ and $\gamma^{\prime}$, which are further
suppressed.

Figure 2: 4-loop diagram contributing to kinetic mixing. Figure 3: Diagram
contributing to kinetic mixing below the lightest mirror quark mass.

Moreover, below their masses, the mirror quarks are integrated out at one-loop
and the three-loop diagram of Fig. 3 arises, where the effective vertex on the
left part connects one mirror photon to three gluons222There is no effective
operator connecting one mirror photon to any combination of two gauge fields,
due to symmetry (see the discussion on the “$X^{3}$” class of operators in
section 5 of Grzadkowski _et al._ (2010)). and is therefore suppressed by
$1/m_{u^{\prime}}^{4}$ (due to parity, $m_{u^{\prime}}$ is the smallest mass
scale among the UV fields). Dimensional analysis dictates that such
contributions must come with an extra factor
$\sim(\frac{p}{m_{u^{\prime}}})^{4}$, where $p$ is the largest momentum
flowing in the quark loops, while one retains the four-loop (one for the one-
loop matching at $\mu\sim m_{u^{\prime}}$ and three for the diagram in the IR
theory) and $e^{2}g_{S}^{6}$ suppression. Therefore, one can safely neglect
the running below $m_{u^{\prime}}$. In conclusion, 4-loop contributions to the
kinetic mixing are relevant below the scale $v_{3}$ and when there is at least
one mirror quark below that scale.

##  V Dark Matter

Given the unbroken mirror QED symmetry of the model of Ref. Bonnefoy _et al._
(2023), it is natural to expect the presence of a good DM candidate. However,
it turns out that this happens only in a specific region of parameter space
and through a non-standard production mechanism. This is the topic of this
section.

Even if little is known about DM as a particle, a good DM candidate must:

  1. 1.

Have a relic abundance today that matches the observed value
$\Omega_{DM}=\rho_{DM}/\rho_{\rm crit}=0.265(7)$, where $\rho_{\rm
crit}=8.5(1)\times 10^{-30}\,{\rm g\,cm^{-3}}$ is the critical density of the
Universe Workman _et al._ (2022),

  2. 2.

Be cosmologically stable so as to agree with current experimental observations
Audren _et al._ (2014); Baring _et al._ (2016); Mambrini _et al._ (2016);
Slatyer and Wu (2017),

  3. 3.

Become non-relativistic well before matter-radiation equality, as required by
the formation of large scale structures in the Universe Benson _et al._
(2012); Lovell _et al._ (2014); Kennedy _et al._ (2014),

  4. 4.

Have zero, or very small, electromagnetic charge, as required by searches for
stable charged particles McDermott _et al._ (2011); Sanchez-Salcedo _et al._
(2010),

  5. 5.

Have limited self-interactions, constrained by the observed DM halo profiles,
cluster collisions and the CMB spectrum Agrawal _et al._ (2017).

In the model under consideration, all mirror particles are electrically
neutral, up to kinetic mixing. Through mirror Yukawa and electroweak couplings
fixed by parity, the mirror particles have decay channels similar to their SM
counterparts, hence the only stable massive particles are the mirror electron
$e^{\prime}$ and up quark $u^{\prime}$. They cannot decay to the SM because of
the unbroken mirror electromagnetic charges that they carry. The mirror photon
remains massless and contributes to dark radiation (by an amount far below
current limits in the scheme to be described), whereas mirror neutrinos are
heavy but quickly decay to the SM unless further model building is invoked, as
we discuss below. Finally, the physical components of the bifundamental scalar
field $\Sigma$ which breaks the color groups are generically of mass $\sim
v_{3}$ and quickly decay to quarks and gluons Bai and Dobrescu (2018).

Being the lightest stable mirror fermion, moreover charged only under mirror
gauge groups, the mirror electron is the best DM candidate in this class of
models. However, in Sec. III we have seen that the $u^{\prime}$ has a mass
close to the $e^{\prime}$, therefore its abundance has to be carefully
considered. This will turn out to be of utmost importance for the viability of
a DM production mechanism, given the bounds on the $u^{\prime}$ relic density.
Another important constraint on the cosmology stems from the fact that the
potential of $\Sigma$ displays an accidental $\mathbb{Z}_{3}$ symmetry,
leading to domain walls (DWs) if the universe is reheated above the phase
transition temperature $\sim v_{3}$. (This symmetry appears to be accidental
to all mass dimensions, preventing one from introducing a bias to collapse the
domain walls.) However, over most of the parameter space, this requirement is
weaker than the one associated with the relic density of $u^{\prime}$.

###  V.1 Bounds on Mirror quarks

An extensive discussion of the cosmological history of the mirror quarks in
these models can be found in Ref. Dunsky _et al._ (2019a). The heavier mirror
quarks decay into the lighter ones, while the latter hadronize after the QCD
phase transition, forming bound states by combining with other colored
particles. Rearrangements mediated by scattering processes then lead to the
presence of two kinds of exotic bound states:

  1. 1.

hadrons made of $u^{\prime}$ and SM quarks, here named
$h^{\prime}:\;u^{\prime}qq$, $u^{\prime}u^{\prime}q$, $u^{\prime}\bar{q}$;

  2. 2.

hadrons made of three $u^{\prime}$.

The abundance of these states, relative to the abundance of $u^{\prime}$, has
been estimated in Ref. Dunsky _et al._ (2019a),

$\begin{split}Y_{u^{\prime}u^{\prime}u^{\prime}}&\simeq
Y_{u^{\prime}}\times\begin{cases}Y_{u^{\prime}}/Y_{\rm
crit}&Y_{u^{\prime}}<Y_{\rm crit}\\\ 1&Y_{u^{\prime}}>Y_{\rm
crit}\end{cases},\\\ Y_{h^{\prime}}&\simeq
Y_{u^{\prime}}\times\begin{cases}1&Y_{u^{\prime}}<Y_{\rm crit}\\\ Y_{\rm
crit}/Y_{u^{\prime}}&Y_{u^{\prime}}>Y_{\rm crit}\end{cases},\\\ Y_{\rm
crit}&\simeq 2\times 10^{-13}\;\big{(}\frac{m_{u^{\prime}}}{\rm
GeV}\big{)}^{2}\;Y_{\rm DM}\ .\end{split}$ (7)

While bound states made of only $u^{\prime}$ can in principle be a component
of DM, the abundance of $h^{\prime}$ is strongly constrained by searches for
nuclear and electric recoil at deep underground detectors Ahmed _et al._
(2010); Aprile _et al._ (2018); Essig _et al._ (2017), and by tracks of
ionizing particles in bulk matter, on earth Alvis _et al._ (2018); Ambrosio
_et al._ (2004); Kajino _et al._ (1984); Alekseev _et al._ (1983) as well as
in meteorites Jones _et al._ (1989). Again we refer the reader to the
discussion in Ref. Dunsky _et al._ (2019a).

Collider bounds on fractionally charged heavy stable states give
$m_{u^{\prime}}\gtrsim 1.5\,{\rm TeV}$ Bonnefoy _et al._ (2023), while bounds
from higher dimensional operators that can spoil the solution to the strong CP
problem imply $m_{u^{\prime}}\lesssim 10^{6}\;{\rm TeV}$. In this mass range,
one obtains $Y_{h^{\prime}}\lesssim[10^{-14}-10^{-8}]\;Y_{\rm DM}$ from the
bounds of the MACRO Ambrosio _et al._ (2004), ICRR Kajino _et al._ (1984) and
Baksan experiments Alekseev _et al._ (1983), as well as
$Y_{h^{\prime}}\lesssim m_{u^{\prime}}/{\rm GeV}\,10^{-15}\;Y_{\rm DM}$ from
searches of ionizing tracks in the Hoba meteorite Jones _et al._ (1989). The
lower bound on $m_{u^{\prime}}$ and the definition of $Y_{\rm crit}$ imply
that $Y_{\rm crit}\gtrsim 4\times 10^{-7}\;Y_{\rm DM}$, hence
$Y_{h^{\prime}}<Y_{\rm crit}$ given the bounds above. This shows that the only
viable scenario is

$Y_{u^{\prime}}<Y_{\rm crit},\quad Y_{h^{\prime}}\simeq Y_{u^{\prime}},\quad
Y_{u^{\prime}u^{\prime}u^{\prime}}\simeq\frac{Y_{u^{\prime}}^{2}}{Y_{\rm
crit}}\,.$ (8)

In the literature, it has been debated whether strongly interacting particles
can reach terrestrial experiments such as MACRO, ICRR and Baksan, because of
their interactions in the Earth’s atmosphere and crust. However, even assuming
a geometrical cross section for DM-nucleus scattering (which is the largest
possible value for such a cross section333A larger cross section can be
obtained through long range interaction or, in the case of composite DM, by
the presence of resonances or threshold bound states Digman _et al._ (2019).
Digman _et al._ (2019)), it has been shown that this is the case for neutral
DM bound states Goodman and Witten (1985) as well as charged ones Dunsky _et
al._ (2019a). The determination of the $h^{\prime}$ flux in the galactic disk
from its charged massive particle nature Dunsky _et al._ (2019b) is not
straightforward, hence the systematic errors on the bounds for
$Y_{h^{\prime}}$ are difficult to estimate. Therefore, in the following, we
show different exclusion scenarios. Nonetheless, the constraints are so
stringent that, even taking a conservative stand, they play an important role
in our results.

Concerning bound states made of only $u^{\prime}$, one finds from the
relations above that $Y_{u^{\prime}u^{\prime}u^{\prime}}\lesssim 2\times
10^{-10}\;Y_{\rm DM}$: the DM fraction of fully mirror bound states is
negligible.

###  V.2 Bounds on $e^{\prime}$ DM

Since bound states of mirror quarks are constrained to provide only an
extremely small contribution to DM, we are left with the possibility of mirror
electron DM. We remind the reader that in this model the mirror electron is
charged only under the mirror gauge groups $SU(2)^{\prime}\times
U(1)^{\prime}$ and has a mass larger than $\sim 750\,{\rm GeV}$444This limit
is obtained from the bound on the mirror quark masses which, due to parity,
implies a bound on the mirror electron mass with a mild dependence on the
parameters of the model.. Before studying the production mechanism in the
early universe, we discuss the bounds from DM direct and indirect searches.

Figure 4: Limit on $\epsilon$ vs $v^{\prime}$ from DM direct searches (blue
region) from XENON1T Aprile _et al._ (2018) (solid line), XENONnT Aprile _et
al._ (2023) (dashed line) and LUX-ZEPLIN Aalbers _et al._ (2023) (dotted line)
together with predictions for different values of $v_{3}/v^{\prime}$.

Being coupled with the mirror photon, the $e^{\prime}$ experiences a long-
range force. However, its mass is too heavy to appreciably self-scatter and
disrupt the DM halo profile Agrawal _et al._ (2017) nor the spectrum of the
CMB. On the other hand, the kinetic mixing discussed in Sec. IV leads to an
interaction with SM particles and can produce an observable signal. The
strongest bound comes from searches for DM-nucleus scattering. The results of
the experiment XENON1T Aprile _et al._ (2018) can be recast Dunsky _et al._
(2019a) into the bound

$m_{e^{\prime}}>10^{6}\,{\rm
GeV}\bigg{(}\frac{\epsilon}{10^{-8}}\bigg{)}^{2}\,.$ (9)

Results from XENONnT Aprile _et al._ (2023) and LUX-ZEPLIN (LZ) Aalbers _et
al._ (2023) lead to similar bound, up to the respective replacement of
$10^{6}$ by $1.5\times 10^{6}$ and $3\times 10^{6}$. Since the mass of the
mirror electron depends only on the VEV of the mirror Higgs with a very good
precision, this limit translates into a bound on $\epsilon$ and $v^{\prime}$,
as shown in Fig. 4. The same figure also shows the predictions of $\epsilon$
for different values of $v_{3}$, assuming no UV kinetic mixing as discussed in
Sec. IV. As anticipated there, the smaller $v_{3}$, the weaker the bound on
$v^{\prime}$. For $v_{3}$ below the lightest mirror quark mass, the kinetic
mixing is extremely suppressed and this bound disappears. For very low values
of $v_{3}$ (above $m_{u^{\prime}}$), we find the lower limit
$v^{\prime}\gtrsim 10^{11}\,{\rm GeV}$. This complements the other bounds on
the parameters of the model. In fact, Ref. Bonnefoy _et al._ (2023) finds the
bound $v^{\prime}\lesssim 10^{14}\,{\rm GeV}$ from higher-dimensional
operators that can spoil the solution to the strong CP problem. Therefore, in
the scenario $v_{3}>m_{u^{\prime}}$, this leaves only 2-3 orders of magnitude
for $v^{\prime}$.

###  V.3 Freeze-out

The most relevant processes for the interaction of $e^{\prime}$ and
$u^{\prime}$ with the thermal bath are shown in Fig 5 when the temperature is
smaller than $v_{3}$.

Figure 5: Most relevant tree-level diagrams for the production of $e^{\prime}$
and $u^{\prime}$ in the early universe when the temperature is smaller than
$v_{3}$.

At temperatures higher than its mass, $e^{\prime}$ is efficiently produced
by/annihilated into two mirror photons. In the freeze-out scenario, the
resulting $e^{\prime}$ yield reads $Y_{e^{\prime}}\simeq Y_{\rm
DM}\bigg{(}\frac{v^{\prime}}{10^{8}\;{\rm GeV}}\bigg{)}$ and the point in
parameter space which reproduces the right DM abundance is already excluded by
collider searches, as mentioned in Sec. II. Refs. Dunsky _et al._ (2020); Redi
and Tesi (2023) studied the possibility of heavier $e^{\prime}$, whose
abundance is diluted trough the decay of the mirror neutrinos. However, we
discussed the running of the mirror fermion masses in Sec. III and we saw that
the ratio $m_{u^{\prime}}/m_{e^{\prime}}$ can be at most as large as $\sim 2$
for $m_{u^{\prime}}$ close to $v_{3}$. This implies that the abundance of
$e^{\prime}$ and $u^{\prime}$ would be similar. Indeed, for $u^{\prime}$ the
mirror photon channel is also active, but it is even dominated by the
production/annihilation through gluons or SM quarks. For a temperature above
$v_{3}$, the gluons become mirror gluons, and the bifundamental scalar
$\Sigma$ is also in equilibrium and can annihilate into a $u^{\prime}$ pair.
It then appears to us that the bounds discussed in Sec. V.1 are too strong and
prevent the freeze-out scenario from being viable. Weakening the bounds on the
$u^{\prime}$ abundance by a few orders of magnitude to account for the large
uncertainties is not sufficient to change the conclusion.

###  V.4 Freeze-in

We therefore need to assume that the mirror fermions are never in thermal
equilibrium and are produced through a freeze-in mechanism. If
$m_{u^{\prime}}>m_{e^{\prime}}\gg T$, a factor of a few in the mass leads to
an exponential suppression in the abundances,
$Y_{u^{\prime}}/Y_{e^{\prime}}\sim\exp(\frac{m_{u^{\prime}}-m_{e^{\prime}}}{T})$,
due to the Boltzmann suppression of the thermal bath particles which are
energetic enough to produce $e^{\prime}$ and $u^{\prime}$.

In the following, we assume that the reheating temperature at the end of
inflation, $T_{R}$, is the highest temperature $T_{\text{max}}$ reached in the
cosmological history of the universe and is smaller than $m_{e^{\prime}}$. We
are thus adding a new parameter to the model, whose cosmology is now
determined by three quantities: $v^{\prime},\,v_{3},\,T_{R}$. In addition, we
need to assume that the inflaton does not directly decay to mirror fermions,
or with a very reduced rate, and that it does not produce too many mirror
photons either, as those would subsequently generate a population of mirror
fermions555We note that an initial population of mirror photons is an
interesting possibility, but we choose to focus here on the minimal, purely
thermal DM production from the SM bath.. None of this happens if we assume
that the inflaton primarily decays to the SM sector, or that it mostly
produces gluons and mirror gluons. The former is not in contradiction with
exact parity due to the different electroweak VEVs in the two sectors, for
instance if the inflaton couples to matter through a parity-symmetric Higgs
portal Berezhiani _et al._ (1996). For a $P$-odd inflaton, such a coupling and
an inflaton VEV can generate a soft $P$ breaking leading to asymmetric
electroweak scales in the two sectors, even though a severe tuning is needed.
We elaborate on our assumptions regarding reheating in Sec. V.5.

In this setup, the only mirror species that can ever be in thermal equilibrium
with the SM is the mirror photon, which, below $v_{3}$, interacts with the SM
gluons through the 1-loop diagrams in Fig. 6. Again, the relevant production
processes for $e^{\prime}$ and $u^{\prime}$ below $v_{3}$ are shown in Fig. 5.
Above $v_{3}$, $\Sigma$ thermalizes gluons and mirror gluons, so that the
relevant diagrams become those of Fig. 6 and Fig. 5 where gluons are replaced
by mirror ones.

Figure 6: 1-loop portal between SM gluons and $\gamma^{\prime}$.

####  V.4.1 $\gamma^{\prime}$ in thermal equilibrium

The cross sections for the processes in Fig. 6 read

$\sigma_{gg\to\gamma^{\prime}\gamma^{\prime}}\simeq 8\times
10^{-8}\,\frac{g_{s}^{4}e^{\prime\,4}s^{3}}{m_{u^{\prime}}^{8}}+O\bigg{(}\frac{s^{4}}{m_{u^{\prime}}^{10}}\bigg{)}\,,$
(10) $\sigma_{gg\to g\gamma^{\prime}}\simeq 7.5\times
10^{-8}\,\frac{g_{s}^{6}e^{\prime\,2}s^{3}}{m_{u^{\prime}}^{8}}+O\bigg{(}\frac{s^{4}}{m_{u^{\prime}}^{10}}\bigg{)}\,,$
(11)

where $s=(p_{1}+p_{2})^{2}$ and $p_{1,2}$ are the four-momenta of the external
gluons. In the center of mass frame $s=4E_{1}E_{2}$.

Following Ref. Gondolo and Gelmini (1991), the rate for these processes
is666Note that we integrated over all energies in Eq. (12) although we used
the simple formulae of Eqs. (10)-(11) which are not valid at energies larger
than $m_{u^{\prime}}$. The exponential suppression above $E\sim T$ ensures
that the result is not noticeably affected. For instance, accounting for
resonant $u^{\prime}\bar{u}^{\prime}$ bound states formation Pancheri _et al._
(1992); Barger _et al._ (1987); Kahawala and Kats (2011); Kats and Strassler
(2012) yields an increase in the actual cross section that is comparable to
that obtained from Eqs. (10)-(11) when $s\sim m_{u^{\prime}}^{2}$. We checked
that this region and larger energies essentially do not contribute to the
result.

$\displaystyle\begin{split}\Gamma_{\gamma^{\prime}}=&n_{g}\frac{\int\sigma
v_{\rm Mol}\,e^{-E_{1}/T}e^{-E_{2}/T}d^{3}p_{1}d^{3}p_{2}}{\int
e^{-E_{1}/T}e^{-E_{2}/T}d^{3}p_{1}d^{3}p_{2}}\\\ &\simeq 1.5\times
10^{-3}\frac{(g_{s}^{4}e^{\prime\,4}+g_{s}^{6}e^{\prime\,2})\,T^{9}}{m_{u^{\prime}}^{8}}\,,\end{split}$
(12)

and the condition for having $\gamma^{\prime}$ in thermal equilibrium is

$\Gamma_{\gamma^{\prime}}\gtrsim H\,\simeq\frac{T^{2}}{M_{P}}\Rightarrow
T_{R}\gtrsim 26\,{\rm GeV}\bigg{(}\frac{m_{u^{\prime}}}{{\rm
TeV}}\bigg{)}^{8/7}\,.$ (13)

The power of $m_{u^{\prime}}$ indicates that $T_{R}$ grows faster than
$m_{u^{\prime}}$ (i.e., than $v^{\prime}$) and tends to bring mirror fermions
in thermal equilibrium, which is problematic as shown above. With this result,
it is actually impossible to find a region of parameter space which gives the
correct relic abundance for $e^{\prime}$. In fact we find

$x_{e^{\prime}}\equiv\frac{m_{e^{\prime}}}{T_{R}}\lesssim
38\,\frac{m_{e^{\prime}}}{m_{u^{\prime}}}\,\bigg{(}\frac{{\rm
TeV}}{m_{u^{\prime}}}\bigg{)}^{1/7}\,.$ (14)

As we already mentioned above, collider bounds require $m_{u^{\prime}}\gtrsim
1.5\,{\rm TeV}$, implying that $v^{\prime}\geq 2\times 10^{8}$ GeV. For such
values, we find numerically that the ratio $m_{e^{\prime}}/m_{u^{\prime}}$ is
such that $x_{e}\lesssim 16$ all over the allowed parameter space, so that we
overshoot the DM relic abundance by several order of magnitudes. (For
instance, $v^{\prime}=10^{9}$ GeV and $v_{3}=10^{5}$ GeV yield $x_{e}$ close
to the maximal value, as can be seen in Fig. 1.) This result holds even
assuming a large uncertainty $\mathcal{O}(20\%)$ on $m_{u^{\prime}}$, in which
case $x_{e^{\prime}}$ remains smaller than $\sim 22$.

####  V.4.2 $\gamma^{\prime}$ out of equilibrium

To reduce the abundance of $e^{\prime}$ and even further of $u^{\prime}$,
$T_{R}$ needs to be lower than the limit in Eq. (13). This implies that also
$\gamma^{\prime}$ are out of equilibrium and DM is produced via a sequential
freeze-in process Hambye _et al._ (2019); Bélanger _et al._ (2020). In App. A
we describe how to solve the Boltzmann equation for the momentum distribution
of the mirror photons. We were able to solve it analytically in the limit
where the number of degrees of freedom is constant and for high-energy mirror
photons (i.e. $E_{\gamma^{\prime}}\gg T$). These are consistent assumptions:
the SM degrees of freedom are all in the bath until much lower temperatures
than $T_{R}$ where most of the $e^{\prime}$ are produced, and making
$e^{\prime}$ (such that $m_{e^{\prime}}\gg T_{R}$ for freeze-in) requires
highly energetic photons. Soft photons do not contribute to this process since
they need to scatter with a very energetic one, whose number density is
extremely suppressed. We checked the consistency of this argument, as detailed
in App. A.

With the mirror photon distribution, one can then solve the Boltzmann equation
for $e^{\prime}$ and $u^{\prime}$. Fig. 7 and Fig. 8 show our results in the
$v^{\prime}-T_{R}/v^{\prime}$ plane, for two benchmark cases: when
$v_{3}>v^{\prime}$, and when $v_{3}$ has the smallest value allowed by
collider searches ($v_{3}=5\,{\rm TeV}$), respectively. We discuss the
extrapolation of intermediate cases later on, while for $v_{3}>v^{\prime}$ the
result is independent of $v_{3}$. (We fixed it to be $10\,v^{\prime}$ in our
code to generate Fig. 7. The actual value of $v_{3}>v^{\prime}$ only matters
for kinetic mixing, whose RG running kicks in below $v_{3}$ as explained in
Sec. IV.) The points which provide the right yield of $e^{\prime}$ are shown
with a solid orange line. The blue solid and dotted lines show the
experimental bounds on $u^{\prime}$, discussed in Sec. V.1. Given the
uncertainty on these bounds, we show two benchmarks:
$Y_{h^{\prime}}<10^{-8}\,Y_{\rm DM}$ (solid blue) and
$Y_{h^{\prime}}<10^{-12}\,Y_{\rm DM}$ (dotted blue). For comparison, the
region where the mirror photons are in equilibrium is shown in green,
confirming that it is incompatible with $e^{\prime}$ DM. Moving to the right
of the plot, the ratio $T_{R}/v^{\prime}$ increases as well as the abundances
of both $e^{\prime}$ and $u^{\prime}$. A few values of
$m_{u^{\prime}}/m_{e^{\prime}}$ are also shown along the DM line, from which
one sees that the conservative bounds on the $u^{\prime}$ abundance give the
constraint $m_{u^{\prime}}/m_{e^{\prime}}\gtrsim 1.6-1.7$ (with a small
dependence on the value of $v_{3}$) to get a viable DM model. In the two
figures, this translates to $v^{\prime}\lesssim 5\times 10^{10}\,{\rm GeV}$
and $T_{R}\lesssim 5\,{\rm TeV}$. Such a low reheating temperature raises the
question of the baryogenesis mechanism at play in the early universe. We
discuss this point in Sec. VI.

Figure 7: Values of $T_{R}$ and $v^{\prime}$ reproducing the correct DM relic
abundance for $e^{\prime}$ (solid orange line) in the scenario
$v_{3}>v^{\prime}$. Requiring $Y_{h^{\prime}}<10^{-8}\,Y_{\rm DM}$
($Y_{h^{\prime}}<10^{-12}\,Y_{\rm DM}$) rules out the blue region to the right
of the solid blue (dotted blue) line. The region where the mirror photons
distribution is thermal is shown in green. Dashed lines are analogous to solid
ones, but using a 20% higher value for $m_{u}$. Figure 8: Values of $T_{R}$
and $v^{\prime}$ reproducing the correct DM relic abundance for $e^{\prime}$
(solid orange line) in the scenario $v_{3}=5\,{\rm TeV}$. Requiring
$Y_{h^{\prime}}<10^{-8}\,Y_{\rm DM}$ ($Y_{h^{\prime}}<10^{-12}\,Y_{\rm DM}$)
rules out the blue region to the right of the solid blue (dotted blue) line.
The region where the mirror photons distribution is thermal is shown in green.
Dashed lines are analogous to solid ones, but using a 20% higher value for
$m_{u}$. The gray region is such that $T_{R}>v_{3}$ and DWs are expected to
form in the $\Sigma$ field.

Anticipating the results of the numerical analysis for all values of
$v^{\prime}$ and $v_{3}$, the prediction for the kinetic mixing in Fig. 4 then
suggests that models with $v_{3}\gtrsim m_{u^{\prime}}$ do not contain a good
DM candidate produced by freeze-in. This applies in particular to the models
of Refs. Barr _et al._ (1991); Dunsky _et al._ (2019a) (which can be obtained
from ours in the $v_{3}\to\infty$ limit), and it can be seen in Fig. 9, which
shows the ratio $m_{u^{\prime}}/m_{e^{\prime}}$ as a function of $v_{3}$ and
$v^{\prime}$. The observed trend is understood as follows: the ratio is larger
for smaller $m_{u^{\prime}}$, i.e. smaller $v^{\prime}$, as the $u^{\prime}$
feels the effect of a larger strong coupling constant which grows in the IR.
Similarly, the smaller $v_{3}$, the larger its effect on the running of
$m_{u^{\prime}}$ since strong couplings are typically larger above $v_{3}$, as
illustrated in Fig. 1. This effect competes with the fact that larger
couplings make $m_{u}$ run faster, and that the RG running of the strong
couplings above $v_{3}$ is steeper than the one below. Numerically, we find
that the ratio $m_{u^{\prime}}/m_{e^{\prime}}$ is maximized for points where
$m_{u^{\prime}}\sim v_{3}$, corresponding to the peaks in the contour plot of
Fig. 9. The bound from DM direct detection is also shown as a shaded region.
It arises from the kinetic mixing between $\gamma$ and $\gamma^{\prime}$ as
discussed in Sec. IV and prevents the possibility of having DM for large
values of $v_{3}$.

Figure 9: Ratio $m_{u^{\prime}}/m_{e^{\prime}}$ as a function of $v^{\prime}$
and $v_{3}$. In the shaded region, $e^{\prime}$ DM is excluded by the
irreducible kinetic mixing and its impact on direct detection signals at
Xenon1T (darker region) and LZ (lighter region).

We can make the trend displayed in Fig. 9 sharper by running our RGEs and
Boltzmann codes for all values of $v^{\prime}$ and $v_{3}$. The result is
shown in Fig. 10, where we therefore identify a precise region of parameter
space where the model explains the observed DM abundance while satisfying all
other constraints:

$\boxed{\begin{array}[]{c}\text{\bf Parameters for Viable DM}\vspace{0.1cm}\\\
\;\;\;10^{9}\,{\rm GeV}\lesssim v^{\prime}\lesssim 10^{11}\,{\rm
GeV}\,,\vspace{0.1cm}\\\ \;\;\;5\times 10^{3}\,{\rm GeV}\lesssim v_{3}\lesssim
10^{-5}\;v^{\prime}\,\,,\vspace{0.1cm}\\\ T_{R}\simeq
10^{-7}\,v^{\prime}\,.\end{array}}$ (15)

In this parity solution to the strong CP problem, the mirror electron is a
good DM candidate with a mass in the range $[5\,\rm{TeV},\,250\,\rm{TeV}]$. We
stress that the requirement of DM greatly reduces the large ranges for
$v^{\prime}$ and $v_{3}$ that solve the strong CP problem, shown in Eq. (2).

We also investigated the impact of a large error, $\mathcal{O}(20\%)$, on the
IR value of $m_{u}$. As commented in Sec. III, this is the major source of
uncertainty in our result. The results of our numerical analysis are
represented by the dashed blue lines in Figs. 7 and 8. These lines can be
roughly reproduced upon rescaling the bounds (solid blue lines) through a
$20\%$ shift in $T_{R}/v^{\prime}$. (This amounts to assuming a
$\mathcal{O}(20\%)$ shift on $m_{u^{\prime}}$.) Note however that the DM line
shifts to slightly higher $T_{R}$ (a $\sim 5\%$ effect) (dashed orange lines),
due to the dependence of the mirror photon production cross section on
$m_{u^{\prime}}$. Numerically, we find that increasing $m_{{u}^{\prime}}$ by
$20\%$ extends the $e^{\prime}$ DM region to $v^{\prime}\lesssim 2\times
10^{11}\,\rm{GeV}$ and $T_{R}\lesssim 25\,\rm{TeV}$. In addition, the lower
bound on the ratio $m_{u^{\prime}}/m_{e^{\prime}}$ becomes larger as the bound
on $Y_{u^{\prime}}$ becomes stronger. However, for low $v_{3}$ most of this
region is excluded by the presence of $\Sigma$ domain walls.

On the other hand, lowering $m_{u}$ by $20\%$ gives stronger bounds, as
expected, constraining $v^{\prime}$ between $10^{9}\,\rm{GeV}$ and
$10^{10}\,\rm{GeV}$ and $T_{R}$ below $10^{5}\,\rm{GeV}$. Overall, this
doesn’t change the main picture of our result. The summary of results in the
$(v_{3},v^{\prime})$ plane can be found in Fig. 10 below.

Finally, a non-thermal population of mirror photons remains until today, but
it is far beyond observational prospects. Being more suppressed than thermal
at $T_{R}$ and being subsequently diluted by the SM thresholds, it gives a
very small contribution to dark radiation, $\Delta N_{\text{eff}}\leq 7\times
10^{-6}$, which is saturated for the smallest reheating temperatures that
allow $e^{\prime}$ DM.

###  V.5 Comments on Inflation and Reheating

Given the non trivial set of constraints which make DM viable in this model,
we briefly comment on their impact on cosmological inflation and the
subsequent reheating period. The following conditions are assumed in the
predictive scenario for $e^{\prime}$ production presented above.

  * •

At the end of reheating, when the universe enters a period of radiation
domination, only the SM sector is populated and in thermal equilibrium at a
temperature $T_{R}$ in the range [0.1 - 5] TeV. Higher $T_{R}$ would not
generate an appropriate $e^{\prime}$ relic density (see Figs. 7 and 8).

  * •

$T_{R}\leq{\cal O}(0.1)\,m_{e^{\prime}}$ and $m_{\phi}\lesssim
m_{e^{\prime}}$, where $m_{\phi}$ is the mass of the inflaton. The former
implies that our freeze-in calculation is applicable and that we do not
freeze-out (hence overproduce) $e^{\prime}$ and $u^{\prime}$. The latter
prevents the inflaton from directly decaying to $e^{\prime}$ and $u^{\prime}$,
or its high-energy decay products from scattering and producing $e^{\prime}$
or $u^{\prime}$.

  * •

During reheating, when the universe is dominated by the energy density of the
inflaton field which is transferred to radiation, the temperature cannot be
larger than $T_{R}$ to avoid populating the mirror sector777Calling
$T_{\text{max}}$ the maximum temperature reached by the plasma during
reheating, one may wonder if a qualitatively similar DM production could be
performed if $m_{e^{\prime}}>T_{\text{max}}>T_{R}$. In that case, $e^{\prime}$
freeze-in happens during (inflaton) matter domination, and a large enough
$e^{\prime}$ relic density needs to be frozen-in in order to compensate the
subsequent dilution. This implies that the Boltzmann factor
$e^{-m_{e^{\prime}}/T_{\text{max}}}$ should be increased with respect to the
$e^{-m_{e^{\prime}}/T_{R}}$ evaluated in Sec. V.4. However, the correlated
$e^{-m_{u^{\prime}}/T_{\text{max}}}$ will also be increased, leading to
$u^{\prime}$ overproduction. In summary, $e^{\prime}$ freeze-in at
$T_{\text{max}}>T_{R}$ does not work in this model..

Furthermore, we require $T_{R}<v_{3}$ to avoid domain walls for $\Sigma$. It
is clear that the possibilities for a viable inflationary model are strongly
constrained.

A thorough analysis of inflation and reheating models is beyond the scope of
this work; here we just note a few promising possibilities. The requirement
that the maximal temperature reached during reheating is $T_{R}$ is fulfilled
in the limit of instantaneous reheating, i.e. when the universe transitions
from inflation to a phase of radiation domination in less than a Hubble time.
For instance, this is achieved if the inflaton potential makes an almost
discontinuous change from being very flat to be very steep. Alternatively, one
can deal with a smoother reheating if the temperature of the SM bath increases
(or stays constant) throughout this phase, reaching a maximum at $T_{R}$.
Scenarios of this kind, using dissipative processes other than decays, have
been discussed in Ref. Co _et al._ (2020).

##  VI Mirror neutrinos and leptogenesis

Finally, we discuss mirror neutrinos, which are electrically neutral. In our
minimal model, the neutrino masses must arise from Weinberg-type operators.
Due to parity, one finds two independent coupling matrices
$x_{\nu},x^{\prime}_{\nu}$, the former being symmetric and the latter
hermitian Dror _et al._ (2020),

$\displaystyle{\cal L}_{\nu}$
$\displaystyle=\frac{x_{\nu,ij}}{2\Lambda}(HL_{i})(HL_{j})+\frac{x^{*}_{\nu,ij}}{2\Lambda}(H^{\prime}L^{\prime}_{i})(H^{\prime}L^{\prime}_{j})$
(16)
$\displaystyle+\frac{x^{\prime}_{\nu,ij}}{\Lambda}(HL_{i})(H^{\prime}L^{\prime}_{j})\
.$

Below $v^{\prime}$, $H^{\prime}$ is frozen to its vev and the mirror neutrinos
acquire a mass matrix, $m_{\nu^{\prime}}$, and Yukawa coupling matrix to $LH$,
$y_{\nu}$, of

$m_{\nu^{\prime}}=x^{*}_{\nu}\frac{v^{\prime\,2}}{\Lambda},\hskip
36.135pty_{\nu}=x^{\prime}_{\nu}\frac{v^{\prime}}{\Lambda}\,.$ (17)

It is convenient to work in a basis where $m_{\nu^{\prime}}$ is real and
diagonal. Below the electroweak scale a mass matrix arises also for the SM
neutrinos:

$m_{\nu}=v^{2}\left(\frac{m_{\nu^{\prime}}}{v^{\prime}{}^{2}}-y_{\nu}\,m_{\nu^{\prime}}^{-1}\,y_{\nu}{}^{T}\right)\,.$
(18)

The spontaneous breaking of parity at scale $v^{\prime}$ leads to a “direct”
neutrino mass term proportional to $m_{\nu^{\prime}}$ as well as a
conventional “seesaw” mass term proportional to $1/m_{\nu^{\prime}}$. If
$x_{\nu,ij}$ and $x^{\prime}_{\nu,ij}$ are comparable, as expected for example
from approximate flavor symmetries, then the direct and seesaw contributions
to the light neutrino masses will also be comparable, so that neutrino physics
differs from that of just adding right-handed neutrinos to the SM.

With $e^{\prime}$ DM from sequential freeze-in, the cosmological effects of
the mirror neutrinos are highly dependent on the coupling matrices
$x,x^{\prime}$ and the scale $\Lambda$. For example, if the matrix elements of
$x,x^{\prime}$ are order unity, and $v^{\prime}$ is of order $10^{10}$ GeV,
the observed neutrino masses require the scale $\Lambda$ to be of order
$10^{15}$ GeV. In this case the mirror neutrinos have masses of order $10^{5}$
GeV, well above the reheating temperature of $10^{3}$ GeV, so the mirror
neutrinos are not made in the thermal bath and play no cosmological role.

For other parameters the mirror neutrinos are light enough to be produced at
reheating, and they decay via the Yukawa coupling $y_{\nu}\,\nu^{\prime}LH$ to
$LH$ with decay rate

$\Gamma_{\nu_{i}^{\prime}\to\text{SM}}=\frac{1}{8\pi}\left(y_{\nu}{}^{\dagger}y_{\nu}\right)_{ii}m_{\nu_{i}^{\prime}}\
.$ (19)

Could such decays lead to the cosmological baryon asymmetry via thermal
leptogenesis Fukugita and Yanagida (1986)? If there is no degeneracy among
mirror neutrinos, the answer is no: for this case Ref. Carrasco-Martinez _et
al._ (2023) finds that thermal leptogenesis requires $v^{\prime}>10^{12}$ GeV
to avoid fine tuning in the SM neutrino masses; Fig. 10 shows this is
inconsistent with $e^{\prime}$ DM, even allowing for a large uncertainty in
the up quark mass. Furthermore, thermal leptogenesis without $\nu^{\prime}$
degeneracy requires the lightest $\nu^{\prime}$ to be heavier than $10^{9}$
GeV, many orders of magnitude above $T_{R}$.

However, it is well known that degeneracy among $\nu^{\prime}$ produces the
observed baryon asymmetry for much lower values of $m_{\nu^{\prime}}$ Luty
(1992). In this case, in theories with neutrino masses arising from (16),
leptogenesis can occur for lower values of $v^{\prime}$: Fig. 5 of Ref.
Carrasco-Martinez _et al._ (2023) shows that degeneracies in the range of
$10^{-3}-10^{-6}$, resulting from approximate flavor symmetries, gives
successful leptogenesis throughout the allowed range of
$v^{\prime}\sim(10^{9}-10^{11})$ GeV required by $e^{\prime}$ DM.

An important feature of $e^{\prime}$ DM from sequential freeze-in, relevant
for leptogenesis, is that it requires a very low reheat temperature,
$T_{R}\sim 10^{-7}v^{\prime}\sim(10^{2}-10^{4})$ GeV. Interestingly, this is
above the electroweak scale, so any lepton asymmetry produced can be processed
to a baryon asymmetry via electroweak sphalerons. A key question is the size
of $m_{\nu_{1}^{\prime}}$ relative to $T_{R}$. If $m_{\nu_{1}^{\prime}}\gg
T_{R}$, thermal production of $\nu^{\prime}_{1}$, which occurs via the inverse
of the decay process (19), will be highly Boltzmann suppressed, leading to a
negligible lepton asymmetry. On the other hand, if
$m_{\nu_{1}^{\prime}}\lesssim T_{R}$ the produced lepton asymmetry will be
strongly erased by rescatterings involving $\nu^{\prime}_{2}$, which is
degenerate with $\nu^{\prime}_{1}$. Avoiding such strong washout requires
reducing the Yukawa coupling coupling of $\nu^{\prime}_{2}$ to the point that,
at these low values of $m_{\nu_{1}^{\prime}}$, the production of the asymmetry
in $\nu^{\prime}_{1}$ decays is too small, unless it is boosted by a
degeneracy of at least $10^{-8}$. This exceeds the natural limit of degeneracy
in this theory, $10^{-6}$, set by radiative corrections from the tau coupling
Carrasco-Martinez _et al._ (2023).

Thus the only possibility is that $\nu^{\prime}_{1,2}$ have masses close to
$T_{R}$, but sufficiently above that a near thermal abundance of
$\nu^{\prime}_{1}$ can be produced at reheating, while rescattering via
virtual $\nu^{\prime}_{2}$ leads to little washout of the asymmetry. We
conclude that thermal leptogenesis may occur in the same minimal theory where
$e^{\prime}$ from sequential freeze in accounts for dark matter, but only if
$\nu^{\prime}_{1,2}$ are highly degenerate with mass several times larger than
$T_{R}\sim 10^{-7}v^{\prime}\sim(10^{2}-10^{4})$ GeV.

Other possibilities for leptogenesis exist. The effective theory below
$v^{\prime}$ may contain the coupling $\phi\nu^{\prime}\nu^{\prime}$, allowing
the inflaton $\phi$ to decay to mirror neutrinos as well as SM particles. Non-
thermal leptogenesis then occurs in $\nu^{\prime}$ decays before they
thermalize. Even though strong washout may be avoided by having
$m_{\nu^{\prime}}\gg T_{R}$, some degeneracy among $\nu^{\prime}$ is still
required for a sufficient baryon asymmetry. If the mirror neutrino is the
lightest mirror fermion, inflaton decays to other mirror fermions may be
kinematically forbidden, so that our previous calculations of the $e^{\prime}$
and $u^{\prime}$ abundance still applies. Another possibility is to augment
the SM sector with gauge single fermions N and, by parity, SM′ is augmented by
$N^{\prime}$. In this case thermal leptogenesis can result purely in the SM +
$N$ sector via $N$ decay, as in conventional minimal leptogenesis. The low
reheat temperature again requires significant $N$ degeneracy, but this is much
less constrained by radiative corrections. The breaking of parity by
$v^{\prime}$ can lead to $N^{\prime}$ coupling to $\nu^{\prime}$ in pseudo-
Dirac states much heavier than the reheat temperature.

##  VII Higgs Parity

Figure 10: In the DM region, there exists a $T_{R}$ for which $e^{\prime}$
constitutes all of DM while the $u^{\prime}$ relics pass the experimental
constraints. The solid, dashed and dotted curves assume that those constraints
take the form $Y_{h^{\prime}}<xY_{\text{DM}}$ with
$x=10^{-8},10^{-10},10^{-12}$ respectively. The dark (light) gray-shaded
region is excluded by direct detection searches for $e^{\prime}$ DM by Xenon1T
Aprile _et al._ (2018) (LZ Aalbers _et al._ (2023)), and the black-shaded one
by the presence of $\Sigma$ domain walls. In the Higgs parity region, the
Higgs quartic coupling vanishes at the scale $v^{\prime}$ for some values of
$\alpha_{S}$ and $m_{t}$ within their $2\sigma$ contours at $m_{Z}$. The
dashed red line assumes central values for $\alpha_{S}$ and $m_{t}$, the solid
line that they saturate their $2\sigma$ contours (in the direction of large
$m_{t}$ and small $\alpha_{S}$), and the dotted one that $m_{t}$ takes its
central value while $\alpha_{S}$ is increased to saturate its $2\sigma$
contour.

Models that implement a parity solution to the strong CP problem break parity
either explicitly, via a soft breaking term in the potential, or
spontaneously, with a vacuum having $v^{\prime}\gg v$ stabilized by Coleman-
Weinberg radiative corrections Hall and Harigaya (2018). The latter mechanism,
dubbed “Higgs Parity”, explains why the SM quartic coupling vanishes when
evolved to a high energy scale, and implies that the new physics at this scale
is that of parity restoration.

As described in Refs. Dunsky _et al._ (2019a, 2020), the tree-level parity
symmetric potential for $H$ and $H^{\prime}$ has an important feature: in the
limit that $v\ll v^{\prime}$ is imposed, an accidental $U(4)$ global symmetry
emerges, with $\mathcal{H}=(H,H^{\prime})$ forming a fundamental
representation so that an $SU(2)\times SU(2)^{\prime}$ subgroup is gauged.
When $\mathcal{H}$ gets a VEV, $\langle\mathcal{H}\rangle=(0,0,0,v^{\prime})$,
the $U(4)$ is spontaneously broken to a $U(3)$, $H^{\prime}$ acquires a mass
and at tree-level the SM Higgs arises as a massless Nambu-Goldstone boson.
However, radiative corrections to the scalar potential (the leading
contribution coming from the SM and mirror top quarks) break explicitly the
$U(4)$ global symmetry, giving radiative contributions to the SM Higgs mass
and quartic coupling. The large hierarchy $v^{\prime}/v$ results mainly from
fine-tuning and the SM Higgs quartic coupling $\lambda$ at the scale
$v^{\prime}$ takes a small value. At lower energies, quantum corrections
within the SM renormalize $\lambda$ so that it grows logarithmically. In this
section, we discuss whether the condition $\lambda(v^{\prime})\simeq-0.001$
Dunsky _et al._ (2019a) is compatible with the parameter space giving rise to
$e^{\prime}$ DM.

The leading contributions to the RGE for $\lambda$ below the scale
$v^{\prime}$ are

$\displaystyle\frac{d\lambda}{d\text{ln}\mu}=$
$\displaystyle\frac{24\lambda^{2}+12\lambda
y_{t}^{2}-6y_{t}^{4}}{16\pi^{2}}-\frac{\lambda\left(3\alpha_{1}+9\alpha_{2}\right)}{4\pi}$
(20)
$\displaystyle+\frac{3}{8}\alpha_{1}^{2}+\frac{9}{8}\alpha_{2}^{2}+\frac{3}{4}\alpha_{1}\alpha_{2}\
\ ,$

where $y_{t}$ is the top Yukawa. The couplings $y_{t},\alpha_{1}$ and
$\alpha_{2}$ are computed at all scales, for any given $v_{3}$ and
$v^{\prime}$, as discussed in Sec. III. Hence, starting with the low energy
value of $\lambda$, known from the Higgs boson mass, its value at higher
energies is computed in terms of $v_{3}$ and $v^{\prime}$. The input
parameters, in the $\rm\overline{MS}$ scheme, that we use here are Huang and
Zhou (2021)

$\begin{split}\lambda(m_{Z})&=0.13947\pm 0.00045\,,\\\
\alpha_{S}(m_{Z})&=0.1179\pm 0.0009\,,\\\ m_{t}(m_{Z})&=(168.26\pm
0.75)\,\mbox{GeV}.\end{split}$ (21)

The region of the ($v_{3},v^{\prime}$) plane consistent at $2\sigma$ with
$\lambda(v^{\prime})=0$ is shown in red in Fig. 10. That it shuts off for
small $v_{3}$ is understood as follows. Low $v_{3}$ increases the strong
coupling constants at a given scale, thereby enhancing the running of $y_{t}$,
making it decrease faster in the UV than in the SM, hence reducing its impact
on $\lambda$, which ends up not crossing zero when $y_{t}$ runs too fast. One
can see that, for the central values in Eq. (21), the condition
$\lambda(v^{\prime})=0$ can be satisfied only for $v_{3}\gtrsim 10^{10}\,{\rm
GeV}$ and $v^{\prime}\gtrsim 6\times 10^{11}\,{\rm GeV}$. Varying the input
parameters within their $2\sigma$ uncertainty, $\lambda(v^{\prime})=0$ can be
obtained with $v_{3}$ as low as $\approx 10^{7}\,{\rm GeV}$. These conditions
for Higgs Parity are however incompatible with the results obtained for
$e^{\prime}$ as DM candidate, using the central value of $m_{u}$, as shown by
the blue region of Fig. 10. (Compare with Eq. (15)). As already discussed, a
value for $m_{u}$ larger by $20\%$ would weaken the bounds in Fig. 7 and Fig.
8, which is however not sufficient to reconcile Higgs parity and $e^{\prime}$
DM, as shown by the green region of Fig. 10. Higgs Parity in this model would
therefore require that measured SM parameters deviate significantly from their
central values. Fig. 10 neglects the threshold corrections to the quartic at
$v^{\prime}$, but it is found to be very small and negative Dunsky _et al._
(2019a), making the situation slightly worse for Higgs parity.

A non-minimal theory, with additional heavy fermion states coupled to the
Higgs, with mass well below $v^{\prime}$, would allow $\lambda(v^{\prime})=0$
to be consistent with $e^{\prime}$ DM from sequential freeze-in. Thus parity
could be spontaneously broken by the radiative Higgs Parity mechanism in non-
minimal theories. Froggatt-Nielsen type theories of flavor contain such heavy
fermions; if their masses are well below $v^{\prime}$ the scale of spontaneous
flavor symmetry breaking is relatively low.

##  VIII Conclusion

We have shown that certain models based on Parity solutions to the strong CP
problem have a DM candidate already embedded in their particle spectrum.
Having as a benchmark the model detailed in Ref. Bonnefoy _et al._ (2023),
where the full gauge group of the SM is copied. We have discussed the
parameters of the model, stressing that Parity leaves little freedom, making
the model very predictive. There are two free parameters in addition to the SM
ones: the scale at which parity is broken, $v^{\prime}$, which is also the
scale of mirror electroweak symmetry breaking, and the scale $v_{3}$ at which
the two color group break to their diagonal subgroup. We computed the impact
of broken parity on the RGEs of the model, and we then studied the unavoidable
kinetic mixing between the SM $U(1)_{Y}$ gauge group and its mirror copy,
which plays a relevant role for DM direct detection.

We identified the mirror electron $e^{\prime}$ as a good DM candidate, while
the mirror up quark $u^{\prime}$ can form fractionally charged bound states
with SM quarks after QCD confinement, being therefore excluded by several
experimental searches. The closeness in mass of $e^{\prime}$ and $u^{\prime}$
strongly determines the DM production mechanism. We find that production of
$e^{\prime}$ DM from the SM bath, with a sufficiently suppressed $u^{\prime}$
abundance, can occur via a sequential freeze-in mechanism through an out-of-
equilibrium bath of mirror photons. This can occur only in the blue wedge-
shaped region in the $(v_{3},v^{\prime})$ plane of Fig. 10. Hence, the mass of
the mirror electron is in the range $[5-250]$ TeV and the $SU(3)\times
SU(3)^{\prime}$ breaking scale is in the range $[5-500]$ TeV. At any point in
this blue wedge region, the reheating temperature must be close to
$10^{-7}v^{\prime}$, and hence is predicted to be low, in the range $[0.1-5]$
TeV. We noted that knowing the mass of the up quark with good precision is
crucial to make a robust prediction, therefore we commented on the possibility
that the precision of lattice determinations is underestimated.

The blue wedge-shaped region of Fig. 10 for $e^{\prime}$ DM is not large and
has several observational signals associated with it. Near the vertical edge
of the wedge, at $v_{3}=5$ TeV, there are colored states associated with
$SU(3)\times SU(3)^{\prime}$ breaking that may be accessible to future
collider experiments, as discussed in Bonnefoy _et al._ (2023). Close to the
long sloped edge of the wedge, $e^{\prime}$ DM may be discovered by direct
detection, via kinetic mixing of our photon with the mirror photon. Higher in
the wedge $v^{\prime}$ increases and the $u^{\prime}/e^{\prime}$ mass ratio
falls; the abundance of the fractionally charged hadrons $h$, containing
$u^{\prime}$ increases, leading to signals of this exotic DM component as
discussed in Dunsky _et al._ (2019b). Finally, throughout the wedge,
$e^{\prime}$ DM is self interacting, with a long-range mirror electromagnetic
force that is precisely predicted, and this may lead to future observational
signals in large scale structure.

The low reheating temperature requires a late production of the cosmological
baryon asymmetry. The theory satisfies two key requirements for leptogenesis:
heavy neutral fermions ($\nu^{\prime}$) with Yukawa couplings to SM leptons,
and a reheat temperature above the electroweak phase transition. Generating
sufficient baryons at such late times requires $\nu^{\prime}$ degeneracy, to
enhance the asymmetry, and $\nu^{\prime}$ masses somewhat larger than the
reheat temperature.

Finally, we discussed whether the mechanism of Higgs Parity, which provides
the spontaneous breaking of exact parity, can be realised in these models and
is compatible with the parameters leading to a good DM candidate. We showed
that the current central values for $m_{t}$ and $\alpha_{S}$ clearly exclude
this possibility. Stretching these values within their $2\sigma$ confidence
intervals gets one closer to the region of parameter space where $e^{\prime}$
can be DM, but overlap also requires a large uncertainty in the up quark mass.
If parity is broken spontaneously, either SM parameters are far from their
central values, the Higgs Parity theory contains couplings of the Higgs to
exotic fermions, or the breaking occurs first in some other sector of the
theory and appears as soft breaking in the electroweak sector.

## Acknowledgments

We thank the members of the Berkeley Center for Theoretical Physics for
several discussions, as well as David Dunsky, Keisuke Harigaya, Soubhik Kumar
and Keith Olive for discussions on dark matter direct detection, inflation and
reheating. This work is supported by the Office of High Energy Physics of the
U.S. Department of Energy under contract DE-AC02-05CH11231 and by the NSF
grant PHY-2210390. CS acknowledges additional support through the Alexander
von Humboldt Foundation.

##  Appendix A Freeze-in of $\gamma^{\prime}$

As discussed in Sec. V.3 and Sec. V.4, a thermal population of mirror photon
is not consistent with the mirror electron being DM. If the latter is produced
via a freeze-out mechanism, it leads to an overabundance of mirror up-quarks,
forbidden by direct searches. If the $e^{\prime}$ is produced via freeze-in,
it leads to a yield larger than the observed one for DM. Therefore, we instead
considered a frozen-in population of $\gamma^{\prime}$ that re-scatters into
$e^{\prime}$.

Computing the yield of $e^{\prime}$ through this mechanism then requires
tracking the distribution $f_{\gamma^{\prime}}(t,E)$ of mirror photons as they
are produced from gluons thorugh the processes of Fig. 6. Neglecting all terms
proportional to $f_{\gamma^{\prime}}$ at leading order, the Boltzmann equation
reads

$\frac{\partial f_{\gamma^{\prime}}}{\partial t}-HE\frac{\partial
f_{\gamma^{\prime}}}{\partial E}=\frac{1}{E}\int
d\Pi_{1}d\Pi_{2}d\Pi_{3}\left(2\pi\right)^{4}\delta^{(4)}({\mathbf{p}}_{1}+{\mathbf{p}}_{2}-{\mathbf{p}}_{3}-{\mathbf{p}}_{\gamma^{\prime}})\absolutevalue{{\cal
M}_{12\to 3\gamma^{\prime}}}^{2}f_{1}(t,p_{1})f_{2}(t,p_{2})$ (22)

with $d\Pi_{i}\equiv\frac{d^{3}\vec{p}_{i}}{(2\pi)^{3}2E_{i}}$ and
$p\equiv|\vec{p}|$. In our case, the particles labeled by $1,2$ are SM gluons
in thermal equilibrium, and $3$ is a mirror photon or SM gluon in the final
state. Anticipating that the mirror photons will create heavy $e^{\prime}$, we
focus on high energy mirror photons, and hence on high energy gluons. In
addition, since the gluons are in thermal equilibrium, we can approximate
$f_{1,2}(t,p)\sim e^{-\absolutevalue{p}/T}$ in the comoving frame. Then, via
momentum conservation, we find

$f_{1}(t,p_{1})f_{2}(t,p_{2})=e^{-(p_{3}+E)/T}\ .$ (23)

Consequently, we can first compute

$\int
d\Pi_{1}d\Pi_{2}\left(2\pi\right)^{4}\delta^{(4)}({\mathbf{p}}_{1}+{\mathbf{p}}_{2}-{\mathbf{p}}_{3}-{\mathbf{p}}_{\gamma^{\prime}})\absolutevalue{{\cal
M}_{12\to 3\gamma^{\prime}}}^{2},$ (24)

which is Lorentz invariant. In the center of mass frame, it reads

$\int d\cos\theta\frac{1}{16\pi}\absolutevalue{{\cal
M}\left(s,t=-\frac{s}{2}\left(1+\cos\theta\right)\right)}^{2}\ .$ (25)

In the present case, we have computed the cross sections to be,

$\displaystyle\absolutevalue{{\cal
M}_{gg\to\gamma^{\prime}\gamma^{\prime}}}^{2}=\frac{e^{\prime
4}g_{s}^{4}\left(145586s^{4}+516433s^{3}t+725757s^{2}t^{2}+450254st^{3}+141256t^{4}\right)}{114307200\pi^{4}m_{u^{\prime}}^{8}}$
(26) $\displaystyle\absolutevalue{{\cal M}_{gg\to
g\gamma^{\prime}}}^{2}=\frac{e^{\prime
2}g_{s}^{6}\left(145586s^{4}+516433s^{3}t+725757s^{2}t^{2}+450254st^{3}+141256t^{4}\right)}{121927680\pi^{4}m_{u^{\prime}}^{8}}\
.$

We integrate over $\cos\theta$ and express the result in terms of $s$ in order
to have the expression in any Lorentz frame, including the comoving frame
where we know the gluon distribution:

$gg\to\gamma^{\prime}\gamma^{\prime}\ :\frac{224881e^{\prime
4}g_{s}^{4}s^{4}}{4572288000\pi^{5}m_{u^{\prime}}^{8}}\ ,\qquad gg\to
g\gamma^{\prime}\ :\quad\frac{224881e^{\prime
2}g_{s}^{6}s^{4}}{4877107200\pi^{5}m_{u^{\prime}}^{8}}\ .$ (27)

Performing the last integration over $\vec{p}_{3}$, we obtain

$\int
d\Pi_{3}s^{4}e^{-(p_{3}+E)/T}=\frac{2E^{4}e^{-E/T}}{\left(2\pi\right)^{2}}\int
dp_{3}d\cos\theta\,p_{3}^{5}\left(1-\cos\theta\right)^{4}e^{-(p_{3}+E)/T}=\frac{1536E^{4}T^{6}e^{-E/T}}{\left(2\pi\right)^{2}}\
,$ (28)

where we have used the rotational invariance of the system to align
$\vec{p}_{\gamma^{\prime}}$ with the z-axis. Summing over the two production
channels, we find

$\frac{\partial f_{\gamma^{\prime}}}{\partial t}-HE\frac{\partial
f_{\gamma^{\prime}}}{\partial E}=\frac{224881e^{\prime
2}g_{s}^{4}\left(16e^{\prime
2}+15g_{s}^{2}\right)T^{6}E^{3}e^{-\frac{E}{T}}}{190512000\pi^{7}m_{u^{\prime}}^{8}}.$
(29)

We then wish to manipulate this equation into a numerically-friendly form.
First, we convert the time derivative into a temperature derivative using
$s=\frac{2\pi^{2}}{45}h_{\rm eff}T^{3}$,

$\frac{\partial{f_{\gamma^{\prime}}}}{\partial
t}=\frac{\partial{f_{\gamma^{\prime}}}}{\partial T}\frac{\partial T}{\partial
s}\frac{\partial s}{\partial
t}=-\bigg{(}\frac{8\pi^{3}}{90}\bigg{)}^{1/2}\frac{T^{3}}{M_{\rm
pl}}\frac{h_{\rm
eff}(T)}{g_{*}^{1/2}(T)}\frac{\partial{f_{\gamma^{\prime}}}}{\partial T}\,,$
(30)

where $g_{*}^{1/2}\equiv\frac{h_{\rm eff}}{g_{\rm
eff}^{1/2}}\left(1+\frac{T}{3h_{\rm eff}}\frac{dh_{\rm eff}}{dT}\right)$, and
we write the Hubble parameter in terms of temperature,

$H(T)=\left(\frac{\pi^{3}g_{\rm eff}(T)}{90}\right)^{1/2}\frac{T^{2}}{M_{P}}.$
(31)

The equation becomes

$T\frac{\partial{f_{\gamma^{\prime}}}}{\partial
T}+\alpha_{1}(T)E\frac{\partial f_{\gamma^{\prime}}}{\partial E}\\\
=-\alpha_{2}(T)T^{4}E^{3}\,e^{-E/T},$ (32)

where we define $\alpha_{1}(T)=\left(\frac{g_{\rm eff}(T)g_{*}(T)}{8h_{\rm
eff}^{2}(T)}\right)^{1/2}$ and $\alpha_{2}(T)=\frac{224881e^{\prime
2}g_{s}^{4}\left(16e^{\prime
2}+15g_{s}^{2}\right)}{190512000\pi^{7}m_{u^{\prime}}^{8}}\frac{g_{*}^{1/2}(T)}{h_{\rm
eff}(T)}\sqrt{\frac{90}{8\pi^{3}}}M_{P}$ for simplicity. Because we know that
the freeze-in is UV-dominated, we can take $\alpha_{1}$ and $\alpha_{2}$ to be
constant, giving us the analytical solution,

$\begin{split}f(T,E)=\frac{\alpha_{2}E^{3}T^{-3\alpha_{1}}}{\alpha_{1}-1}&\left[T^{3\alpha_{1}+4}\left(\frac{E}{T}\right)^{\frac{3\alpha_{1}+4}{1-\alpha_{1}}}\Gamma\left(\frac{3\alpha_{1}+4}{\alpha_{1}-1},\frac{E}{T}\right)\right.\\\
&\qquad\left.-T_{\rm RH}^{3\alpha_{1}+4}\left(E\,T_{\rm
RH}^{\alpha_{1}-1}T^{-\alpha_{1}}\right)^{\frac{3\alpha_{1}+4}{1-\alpha_{1}}}\Gamma\left(\frac{3\alpha_{1}+4}{\alpha_{1}-1},T^{-\alpha_{1}}E\,T_{\rm
RH}^{\alpha_{1}-1}\right)\right]\,.\end{split}$ (33)

where $\Gamma$ is the so-called incomplete gamma function. In our Boltzmann
codes, we are using the above formula even for energies larger than
$m_{u^{\prime}}$, although Eqs. (26) only hold for $s\leq m_{u^{\prime}}^{2}$.
However, as in footnote 6, energies larger than $m_{u^{\prime}}$ essentially
do not affect the $\gamma^{\prime}$ distribution and the resulting
$e^{\prime}$ relic abundance.

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