# Exploring the cosmological dark matter coincidence using infrared fixed
points

Alexander C. Ritter<EMAIL_ADDRESS>(corresponding author)
Raymond R. Volkas<EMAIL_ADDRESS>ARC Centre of Excellence for Dark
Matter Particle Physics, School of Physics,
The University of Melbourne, Victoria 3010, Australia

###### Abstract

The asymmetric dark matter (ADM) paradigm is motivated by the apparent
coincidence between the cosmological mass densities of visible and dark
matter, $\Omega_{\mathrm{DM}}\simeq 5\Omega_{\mathrm{VM}}$. However, most ADM
models only relate the number densities of visible and dark matter, and do not
motivate the similarity in their particle masses. One exception is a framework
introduced by Bai and Schwaller, where the dark matter is a confined state of
a dark QCD-like gauge group, and the confinement scales of visible and dark
QCD are related by a dynamical mechanism utilising infrared fixed points of
the two gauge couplings. We build upon this framework by properly implementing
the dependence of the results on the initial conditions for the gauge
couplings in the UV. We then reassess the ability of this framework to
naturally explain the cosmological mass density coincidence, and find a
reduced number of viable models. We identify features of the viable models
that allow them to naturally relate the masses of the dark baryon and the
proton while also avoiding collider constraints on the new particle content
introduced.

##  I Introduction

Determining the particle nature of dark matter (DM) is one of the deepest
tasks facing particle physics today. This goal is hampered by the purely
gravitational nature of the observational evidence for DM, which has allowed
for a cornucopia of DM candidates to be proposed over a wide range of mass
scales [1]. To focus our model-building pursuits, it is instructive to see
what clues may lie in the existing astrophysical observations.

One such result is the similarity between the present-day cosmological mass
densities of dark and visible matter (VM), which we refer to as the
cosmological coincidence [2]:

$\Omega_{\mathrm{DM}}\simeq 5\Omega_{\mathrm{VM}},$ (1)

where $\Omega_{X}$ is the mass density $\rho_{X}$ for species $X$ divided by
the critical density $\rho_{c}$.

We take the cosmological coincidence to be a hint towards an underlying link
between the origins of the abundances of VM and DM. Such a connection is not
present in the majority of prominent DM candidates. Consider, for example, the
WIMP: it is a GeV–TeV scale thermal relic species with an abundance generated
through thermal freezeout. This stands in contrast to VM, whose number density
is due to the baryon asymmetry of universe and whose mass arises from the
confinement energy of QCD. With such distinct generation mechanisms, there is
no a priori reason why the cosmological mass densities of these species should
fall in the same order of magnitude.

The main model-building paradigm that seeks to explain the cosmological
coincidence problem is that of asymmetric dark matter (ADM) [3, 4]. In this
framework, the dark matter candidate is charged with a dark particle number
that develops an asymmetry related to the visible baryon asymmetry. Models of
ADM thus naturally generate similar number densities for VM and DM –
$n_{\mathrm{VM}}\sim n_{\mathrm{DM}}$. However, as a mass density is given by
$\Omega_{X}=n_{X}m_{X}/\rho_{c}$, to satisfactorily explain the cosmological
coincidence problem we must also motivate the similarity in the particle
masses of VM and DM – $m_{\mathrm{VM}}\sim m_{\mathrm{DM}}$. This problem is
not addressed in the majority of the ADM literature, which mostly treats the
DM mass as a free parameter and thus merely shifts the cosmological mass-
density coincidence to a particle mass coincidence.

Our goal then is to construct a theory where DM and VM naturally have similar
particle masses. By analogy with the proton, this leads us to consider DM
candidates that are confined states of a dark, QCD-like gauge group [5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], where
some mechanism is present to relate the confinement scale
$\Lambda_{\mathrm{dQCD}}$ of this gauge group to that of visible QCD,
$\Lambda_{\mathrm{QCD}}$.

Previous efforts to relate the visible and dark confinements scales have
followed one of two general directions:

  1. 1.

introduce a symmetry between the gauge groups. This generally leads to models
of mirror matter, where the mirror symmetry is either exact [5, 6, 7, 8, 9] or
judiciously broken [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] (For
other models of mirror matter, see Refs. [26, 27, 28, 29, 30, 31, 32, 33, 34,
35]).

  2. 2.

introduce new field content so that the running gauge couplings $\alpha_{s}$
and $\alpha_{d}$ approach infrared fixed points with similar magnitudes. This
has not been widely discussed in the literature, with the original idea
introduced by Bai and Schwaller [24] and expanded upon by Newstead and TerBeek
[25].

In this paper we focus on the latter approach, developing upon the framework
introduced by Bai and Schwaller by properly implementing the dependence of the
confinement scale $\Lambda_{\mathrm{dQCD}}$ on the initial values of the
running gauge couplings in the UV, $\alpha_{s}^{\mathrm{UV}}$ and
$\alpha_{d}^{\mathrm{UV}}$. We then ask how readily this approach generates
confinement scales of the same order of magnitude, and so reassess the
validity of this framework as a natural explanation of the cosmological
coincidence.

Compared to the work of Bai and Schwaller, we find that in our analysis there
are a smaller number of models that naturally generate similar confinement
scales for visible and dark QCD, where models in this framework are defined by
the new field content we introduce. This reduced set of models is due to our
definition of ‘naturalness’ now being more stringent, as it must take into
account the dependence of the dark confinement scale on the initial gauge
coupling values in the UV.

In general, we find that models whose infrared fixed points have small values
for the gauge couplings are better at generically generating similar visible
and dark confinement scales. However, we also find that these models generally
require the mass scale of the new particle content to be sub-TeV for most
selections of the initial couplings in the UV, and thus are subject to strong
collider constraints.

Looking for models where the new physics mass scale is on the order of a few
TeV, we do find a number of such models that can fairly generically generate
related visible and dark confinement scales. We identify these models as the
most promising candidates within this framework for naturally explaining the
cosmological coincidence problem.

The paper is organised at follows: in Section II we describe the dark QCD
framework of Bai and Schwaller. In Section III we describe the threshold
corrections to the running of the gauge couplings as implemented by Newstead
and TerBeek. In Section IV we analyse the effect of the values of the gauge
couplings in the UV on $\Lambda_{\mathrm{dQCD}}$. In Section V we discuss the
ability of this framework to explain the cosmological coincidence, before
presenting our results in Section VI and concluding in Section VII.

##  II The Bai-Schwaller framework

The scheme of Bai and Schwaller [24] utilises a dark QCD-like gauge group
$SU(N_{d})_{\textrm{dQCD}}$, along with a selection of new particle content
charged under $SU(3)_{\textrm{QCD}}\times SU(N_{d})_{\textrm{dQCD}}$. As in
the original paper, we only consider $N_{d}=3$, and limit the new field
content to fermions and scalars in the fundamental representations of either
one or both of the QCD gauge groups. The multiplicities of the new particles
are given in Table 1, along with their masses. In addition to the given
multiplicities, we also define $n_{f_{d}}=n_{f_{d,h}}+n_{f_{d,l}}$ as the
total number of dark fermion species that are fundamentals of
$SU(3)_{\textrm{dQCD}}$, and $n_{f_{c}}=n_{f_{c,h}}+6$ as the multiplicity of
visible fermions that are fundamentals of $SU(3)_{\textrm{QCD}}$ (including
the 6 SM quarks).

The majority of the new field content is, for simplicity, taken to exist at a
common heavy mass scale $M$, except for the $n_{f_{d,l}}$ light dark fermions.
These latter particles, which we refer to as ‘dark quarks’, have masses much
lighter than the dark confinement scale $\Lambda_{\mathrm{dQCD}}$ and are
confined into the dark baryons that serve as the DM candidate.

Field | $SU(3)_{\mathrm{QCD}}\times SU(3)_{\mathrm{dQCD}}$ | Mass | Multiplicity
---|---|---|---
Fermion | $(\mathbf{3},\mathbf{1})$ | $M$ | $n_{f_{c,h}}$
$(\mathbf{1},\mathbf{3})$ | $<\Lambda_{\mathrm{dQCD}}$ | $n_{f_{d,l}}$
$M$ | $n_{f_{d,h}}$
$(\mathbf{3},\mathbf{3})$ | $M$ | $n_{f_{j}}$
Scalar | $(\mathbf{3},\mathbf{1})$ | $M$ | $n_{s_{c}}$
$(\mathbf{1},\mathbf{3})$ | $M$ | $n_{s_{d}}$
$(\mathbf{3},\mathbf{3})$ | $M$ | $n_{s_{j}}$
Table 1: The new particle content in a model. The given multiplicities are for
Dirac fermions and complex scalars. A subscript containing $l$ ($h$) indicates
a multiplicity for light (heavy) particles in cases where this is a relevant
distinction to make.

At two-loop level, the $\beta$-functions for $g_{c}$ and $g_{d}$ are coupled,
thanks to the presence of the ‘joint’ fields charged under both
$SU(3)_{\mathrm{QCD}}$ and $SU(3)_{\mathrm{dQCD}}$. The two-loop
$\beta$-function for $g_{c}$,
$\beta_{c}(g_{c},g_{d})=\frac{dg_{c}}{d(\log(\mu))}$, is given by

$\begin{split}&\beta_{c}=\frac{g_{c}^{3}}{16\pi^{2}}\left[\frac{2}{3}\left(n_{f_{c}}+3n_{f_{j}}\right)+\frac{1}{6}\left(n_{s_{c}}+3n_{s_{j}}\right)-11\right]\\\
&+\frac{g_{c}^{5}}{(16\pi^{2})^{2}}\bigg{[}\frac{38}{3}\left(n_{f_{c}}+3n_{f_{j}}\right)+\frac{11}{3}\left(n_{s_{c}}+3n_{s_{j}}\right)-102\bigg{]}\\\
&+\frac{g_{c}^{3}g_{d}^{2}}{(16\pi^{2})^{2}}\left[8n_{f_{j}}+8n_{s_{j}}\right],\end{split}$
(2)

and the $\beta$-function for $g_{d}$,
$\beta_{d}(g_{c},g_{d})\equiv\frac{dg_{d}}{d(\log(\mu))}$, is obtained by
exchanging the indices $c\leftrightarrow d$ [36]. Note that these
$\beta$-functions are given for $N_{c}=N_{d}=3$; the $\beta$-functions for
general $N_{c}$ and $N_{d}$ can be found in Ref. [24].

Depending on the particle content, there may be nontrivial couplings
$\alpha_{s}^{*}$ and $\alpha_{d}^{*}$ for which both $\beta$-functions are
zero, where $\alpha_{s}=g_{c}^{2}/4\pi$, $\alpha_{d}=g_{d}^{2}/4\pi$, and

$\beta_{c}(g_{c}^{*},g_{d}^{*})=\beta_{d}(g_{c}^{*},g_{d}^{*})=0.$ (3)

The couplings $\alpha_{s}^{*}$ and $\alpha_{d}^{*}$ denote an infrared fixed
point (IRFP) of the renormalization group running similar to a Banks-Zaks
fixed point for a single gauge coupling [37]. The couplings at the IRFP only
depend on the multiplicities of the new particle content,
$(n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}})$;
we refer to a given selection of multiplicities as a ‘model’.

So, for a given model, if there exists a nontrivial perturbative IRFP, the
coupled gauge couplings evolve toward $\alpha_{s}^{*}$ and $\alpha_{d}^{*}$
regardless of the initial values for the gauge couplings in the UV. This fixed
point feature is broken when the new heavy fields decouple at the mass scale
$M$; below this scale, the gauge couplings run independently until they become
non-perturbative at the confinement scales $\Lambda_{\mathrm{QCD}}$ and
$\Lambda_{\mathrm{dQCD}}$.

The process to calculate $\Lambda_{\mathrm{dQCD}}$ for given model is then,
ideally, as follows:

  * •

Calculate the IRFP gauge couplings $\alpha_{s}^{*}$ and $\alpha_{d}^{*}$.

  * •

Determine $M$ by running the visible coupling up from the measured value
$\alpha_{s}(M_{Z})=$ 0.11729 [38] to $\alpha_{s}(M)=\alpha_{s}^{*}$.

  * •

Evolve the dark coupling down from $\alpha_{d}(M)=\alpha_{d}^{*}$ until the
scale $\Lambda_{\mathrm{dQCD}}$ at which it becomes large enough to trigger
confinement. As a rough perturbative condition, the Cornwall-Jackiw-Tomboulis
bound $\alpha_{s}>\pi/4$ for $N_{d}=3$ [39] is used to define the dark
confinement scale through $\alpha_{d}(\Lambda_{\mathrm{dQCD}})=\pi/4$.

So, for a given model with a nontrivial IRFP, the dark confinement scale
$\Lambda_{\mathrm{dQCD}}$ can be uniquely determined. The argument is then
that a model with related IRFP values $\alpha_{s}^{*}$ and $\alpha_{d}^{*}$
could lead to compatible confinement scales for visible and dark QCD of a
similar order of magnitude.

However, this framework relies upon a number of simplifying assumptions that
should be investigated in greater detail. The first is that the heavy fields
decouple at the mass scale $M$ without any threshold corrections, which was
addressed in Ref. [25]; we summarise their approach in the next section. The
second key assumption is that the couplings run all the way to their IRFP
values by the decoupling scale $M$ regardless of the initial coupling values
in the UV. However, this is not true in general, and in a later section we
analyse how the value of $\Lambda_{\mathrm{dQCD}}$ depends on the UV coupling
values $\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$.

##  III Threshold corrections

In MS-like mass-independent renormalization schemes, the Appelquist-Carazzone
decoupling theorem [40] does not apply in its naive form; heavy fields can
continue to influence coupling constants and $\beta$-functions at energy
scales below their masses. In these schemes there also arises the related
issue of large logarithms when working at energy scales much smaller than the
mass $M$ of the heavy fields.

To properly account for these problems, the decoupling is treated explicitly
by constructing an effective field theory in which the heavy fields have been
integrated out. Consistency is ensured by matching the full theory onto the
effective field theory at a matching scale $\mu_{0}$, where $\mu_{0}\sim M$ to
avoid large logarithms. This procedure leads to a ‘consistency condition’
relating the coupling constants in the full theory with the coupling constants
in the effective field theory [41, 42],

$\displaystyle\alpha_{s}^{\textrm{EFT}}(\mu_{0})=\zeta_{c}^{2}(\mu_{0},\alpha_{s}(\mu_{0}))\alpha_{s}(\mu_{0}),$
(4)
$\displaystyle\alpha_{d}^{\textrm{EFT}}(\mu_{0})=\zeta_{d}^{2}(\mu_{0},\alpha_{s}(\mu_{0}))\alpha_{d}(\mu_{0}).$
(5)

The decoupling functions $\zeta_{c}^{2}$ and $\zeta_{d}^{2}$ have been
determined to three- and four-loop order in Refs. [43, 44] for the case of
integrating out one heavy quark from QCD with $n_{f}$ flavours. Since we are
working with two-loop $\beta$-functions, we only require the one-loop
decoupling functions; in Ref. [25], the results of Ref. [43] were adapted for
integrating out the full selection of heavy field content at mass $M$, with
the one-loop decoupling functions given by

$\displaystyle\zeta_{c}^{2}(\mu,\alpha(\mu))=1-\frac{\alpha_{s}(\mu)}{6\pi}\tilde{n}_{c}\log\left(\frac{\mu^{2}}{M^{2}}\right),$
(6)
$\displaystyle\zeta_{d}^{2}(\mu,\alpha(\mu))=1-\frac{\alpha_{d}(\mu)}{6\pi}\tilde{n}_{d}\log\left(\frac{\mu^{2}}{M^{2}}\right),$
(7)

where the coefficients $\tilde{n}_{c}$ and $\tilde{n}_{d}$ account for the
degrees of freedom of the relevant heavy fields,

$\displaystyle\tilde{n}_{c}=n_{f_{c,h}}+3n_{f_{j}}+\frac{1}{4}\left(n_{s_{c}}+3n_{s_{j}}\right),$
(8)
$\displaystyle\tilde{n}_{d}=n_{f_{d,h}}+3n_{f_{j}}+\frac{1}{4}\left(n_{s_{d}}+3n_{s_{j}}\right).$
(9)

We note that, for a given $M$, the values of the couplings at low energies now
depend on the choice of decoupling scale $\mu_{0}$; this renormalization scale
dependence is a non-physical artifact of our perturbative analysis being
truncated at finite loop order. As a result, the procedure of Bai and
Schwaller (matching $\alpha_{s}(M_{Z})$ to its experimental value) no longer
uniquely determines the value of $M$ when we incorporate the threshold
corrections to the coupling constants. Instead, we obtain a relationship
between $M$ and $\mu_{0}$, and can solve for one of these values if we specify
the other.

In Ref. [25], values of $\Lambda_{\mathrm{dQCD}}$ were determined for a given
model and value of $M$. By using the consistency condition and again matching
the strong coupling to its measured value at $M_{Z}$, they solved for the
decoupling scale $\mu_{0}$, and then ran the dark coupling until
$\alpha_{d}(\Lambda_{\mathrm{dQCD}})=\pi/4$. However, this process does not
ensure that $\mu_{0}$ is on the order of $M$. We take a different approach to
implementing threshold corrections, as is detailed in the following section.

##  IV Dependence on initial UV couplings

As mentioned earlier, the framework in the previous sections assumes that the
couplings run to their IRFP values by the decoupling scale $\mu_{0}$
regardless of their initial values in the UV, $\alpha_{s}^{\mathrm{UV}}$ and
$\alpha_{d}^{\mathrm{UV}}$. However, this is not the case, as illustrated in
Fig. 1 for the model
$(n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}})=(3,3,3,3,3,2,0)$
with IRFP $(\alpha_{s}^{*},\alpha_{d}^{*})=(0.045,0.077)$, where we show the
running of a grid of couplings $(\alpha_{s},\alpha_{d})$ from the UV scale
$\Lambda_{\mathrm{UV}}=10^{19}$ GeV down to a typical infrared scale
$\Lambda_{\mathrm{IR}}=1$ TeV. While the couplings evolve towards their fixed
point values, they do not precisely reach them, and so the low energy
behaviour of the theory exhibits some dependence on the couplings in the UV.

Figure 1: Left: A grid of values for the strong coupling constant $\alpha_{s}$
and the dark coupling constant $\alpha_{d}$ at an energy scale
$\Lambda_{\mathrm{UV}}=10^{19}$ GeV. The pink star shows the infrared fixed
point $(\alpha_{s}^{*},\alpha_{d}^{*})=(0.045,0.077)$ for the model with field
multiplicities
$(n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}})=(3,3,3,3,3,2,0)$.
Right: The grid of values from the left panel has been evolved down from
$10^{19}$ GeV to an energy scale $\Lambda_{\mathrm{IR}}=10^{3}$ GeV under the
RGEs of the given model. The pink star again indicates the IRFP of the model.

Figure 2: Top: The evolution of the QCD and dQCD coupling constants for the
model
$(n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}})=(3,3,3,3,3,2,0)$
with mass scale $M=635$ GeV and initial UV couplings
$(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})=(0.15,0.25)$. The mass
scale is chosen so that $\alpha_{s}$ obtains its measured value at $M_{Z}$, as
indicated by the red square. The decoupling scale $\mu_{0}$ is shown by the
grey shaded band, where we allow it to vary between $0.5M$ and $2M$ to account
for the uncertainty introduced by threshold corrections. Below the decoupling
scale, $\alpha_{s}$ and $\alpha_{d}$ are depicted as bands, due to the
variance in the decoupling scale. The vertical blue band indicates the range
of values for the dark confinement scale $\Lambda_{\mathrm{dQCD}}$, calculated
as the energy scale where $\alpha_{d}=\pi/4$. Bottom: The same evolution,
zoomed in to the energy range between 1 GeV and $10^{5}$ GeV to show the
variance in the coupling constants below the decoupling scale more clearly.

We now account for this dependence in our calculation of
$\Lambda_{\mathrm{dQCD}}$. Consider the running of the couplings in this model
for a given mass scale $M$ and initial UV couplings
$(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})$. The couplings evolve
with the full coupled renormalization group equations until the scale
$\mu_{0}$ at which we apply the matching conditions of Eqns. 4 and 5. Unlike
the approach of Ref. [25], we explicitly ensure that
$\mu_{0}\sim\mathcal{O}(M)$; following the approach of Refs. [44, 45], we vary
$\mu_{0}$ between $0.5M$ and $2M$ and treat the variance in the low energy
running of the couplings as a theoretical uncertainty due to working at finite
loop-order.111Refs. [44, 45] apply this approach to the running of
$\alpha_{s}$ across quark thresholds in the SM between $m_{\tau}$ and $M_{Z}$.
For a quark of mass $m_{q}$, they vary the decoupling scale between $0.7m_{q}$
and $3m_{q}$; however, since this process is intended to minimise large
logarithms, we chose to vary the decoupling scale between multiplicatively
symmetric values around $M$. Regardless of the specific choice made, note that
this is merely a simple, arbitrary condition which roughly captures the
uncertainty in the decoupling of the heavy fields. Indeed, other more
sophisticated schemes have been developed to minimise the effect of unphysical
renormalization scale dependence in threshold corrections (see for example
Refs. [46, 47, 48]) but these are beyond the scope of this analysis. This then
leads to an uncertainty in the value of $\Lambda_{\mathrm{dQCD}}$. This
evolution is depicted in Fig. 2, where the blue band shows the possible range
of $\Lambda_{\mathrm{dQCD}}$ values for this given model, mass scale $M$ and
initial UV couplings.

The process for calculating $\Lambda_{\mathrm{dQCD}}$ for a given model is now
as follows:

  * •

We first specify the UV couplings
$(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})$; we then determine the
required decoupling scale $\mu_{0}$ by solving Eqn. 4 where
$\alpha_{s}^{\mathrm{EFT}}(\mu_{0})$ is given by running the visible coupling
up from $\alpha_{s}^{\mathrm{EFT}}(M_{Z})=$ 0.11729 and $\alpha_{s}(\mu_{0})$
is given by running
$(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})\rightarrow(\alpha_{s}(\mu_{0}),\alpha_{d}(\mu_{0}))$.

  * •

However, this does not uniquely specify $\mu_{0}$, as the decoupling function
$\zeta_{c}^{2}$ depends on $M$. Since $\zeta_{c}^{2}$ depends on $M$ through
the factor $\log(\mu^{2}/M^{2})$, we can solve for $\mu_{0}$ (and also for
$M$) by specifying a value for $\mu_{0}/M$.

  * •

To account for the uncertainty introduced by threshold corrections, we vary
the value of $\mu_{0}/M$ between 0.5 and 2. This results in a range of values
for $M$ for the given pair of initial UV couplings.

  * •

We then calculate $\Lambda_{\mathrm{dQCD}}$ as usual by running
$\alpha_{d}^{\mathrm{EFT}}(\mu_{0})$ to
$\alpha_{d}^{\mathrm{EFT}}(\Lambda_{\mathrm{dQCD}})=\pi/4$. Since there is a
range of possible values for $M$ (and thus $\mu_{0}$), we also obtain a range
of values for $\Lambda_{\mathrm{dQCD}}$.

We note that this process will not work for all pairs of UV couplings, as we
require the low energy running of $\alpha_{s}$ to match with experiment. This
matching is performed by solving for the decoupling scale such that
$\alpha_{s}$ reaches its measured value at $M_{Z}$, and for some initial UV
couplings, there will be no value of the new physics mass scale $M$ between
$M_{\mathrm{Pl}}$ and $M_{Z}$ for which this occurs.

With this process specified, we are interested in how
$\Lambda_{\mathrm{dQCD}}$ depends on the initial UV couplings for a given
model. As an example, in Fig. 3 we plot contours of $\Lambda_{\mathrm{dQCD}}$
on axes of $\alpha_{s}^{\mathrm{UV}}$ against $\alpha_{d}^{\mathrm{UV}}$,
where we work with perturbative couplings between 0 and 0.3 to ensure that we
can trust our two-loop beta-function calculations222We discuss this choice for
the range of initial UV couplings in greater detail in Sec. V.1.. We see that
$\Lambda_{\mathrm{dQCD}}$ is larger for small values of
$\alpha_{s}^{\mathrm{UV}}$, and is smaller for small values of
$\alpha_{d}^{\mathrm{UV}}$. Note that the contours are in fact shaded bands to
account for the uncertainty in $\Lambda_{\mathrm{dQCD}}$; however, the
threshold corrections do not have a drastic effect, as the bands are fairly
narrow.

In Fig. 3 we also show contours for the required new physics mass scale $M$,
which increases as $\alpha_{s}^{\mathrm{UV}}$ decreases. In the red hatched
areas there are no allowed values of $M$ for which $\alpha_{s}$ has the
correct running at low energy.

Figure 3: For the model
$(n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}})=(3,3,3,3,3,2,0)$
we show contours for the dark confinement scale $\Lambda_{\mathrm{dQCD}}$
(blue) and for the new physics scale $M$ (red) on axes of the initial UV
couplings $\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$. The width
of these contours is due to the variance in the decoupling scale $\mu_{0}$
between $0.5M$ and $2M$ to account for the uncertainty introduced by threshold
corrections. In the red hatched regions there are no allowed values of $M$ for
which $\alpha_{s}$ has the correct running at low energy. The pink star shows
the infrared fixed point of this model. The green hatched region is the
‘asymptotically free’ region, where $SU(3)_{\mathrm{QCD}}$ and
$SU(3)_{\mathrm{dQCD}}$ both exhibit asymptotic freedom.

###  IV.1 Asymptotic freedom

In Fig. 3 we have also included a star and a green hatched region. The star
indicates the coupling values at the IRFP, and the green hatched region shows
the ‘asymptotically free’ region: that is, the region of the perturbative
$(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})$ parameter space for
which $SU(3)_{\mathrm{QCD}}$ and $SU(3)_{\mathrm{dQCD}}$ both exhibit
asymptotic freedom.

It is an interesting consequence of the existence of a nontrivial IRFP with
coupled $\beta$-functions that the asymptotic freedom of the theory depends
upon the values of the two gauge couplings. The intuition for this observation
is as follows: as the energy scale approaches the IR, the gauge couplings
approach their IRFP values, which we can also think of as points in the
$(\alpha_{s},\alpha_{d})$ parameter space running toward the IRFP. So, as the
energy scale approaches the UV, all points in the parameter space evolve away
from the IRFP. There is a UV fixed point at the origin,
$\alpha_{s}=\alpha_{d}=0$; points that are roughly ‘closer’ to the origin than
the IRFP evolve towards this fixed point and thus both visible and dark QCD
will exhibit asymptotic freedom. Those points ‘further away’ from the origin
than the IRFP instead diverge toward large couplings and do not exhibit
asymptotic freedom.

To see this region more concretely, we treat the $\beta$-functions as a vector
field describing the flow of the couplings with increasing energy scale and
plot its streamlines in Fig. 4. We can clearly see the qualitative behaviour
described in the previous paragraph, along with the edge of the asymptotically
free region as shown in Fig. 3.

Figure 4: A streamline plot for the vector field defined by the coupled
$\beta$-functions for $\alpha_{s}$ and $\alpha_{d}$ for the model
$(n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}})=(3,3,3,3,3,2,0)$,
where the green arrows indicate the change in the coupling constants with
increasing energy scale $\mu$. The shaded green region is the ‘asymptotically
free’ region, corresponding to the green hatched region in Fig. 3. Within this
region, both coupling constants approach zero as the energy scale increases.
The pink star shows the infrared fixed point of this model.

We note that although we have identified the region of
$(\alpha_{s},\alpha_{d})$ parameter space for which the gauge groups are not
asymptotically free, we do not discount this region. Since we work with
$\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$ below 0.3, we only
consider running in which the couplings are perturbative up to the Planck
scale. Thus, dynamics at energies above the Planck scale may change the
running again and regain asymptotic freedom or create a nonzero UV fixed point
as per the “asymptotic safety” idea [49, 50].

##  V Explaining the cosmological coincidence problem

Now that we have accounted for the dependence of $\Lambda_{\mathrm{dQCD}}$ on
the initial UV couplings, in this section we reassess whether this framework
offers a natural solution to the cosmological coincidence problem.

To ‘solve’ the cosmological coincidence problem, we mean that we wish our
theory to naturally generate a mass density $\Omega_{\mathrm{DM}}$ on the
order of $\Omega_{\mathrm{VM}}$. In this work we focus only on explaining the
similarity between the particle masses, $m_{\mathrm{VM}}\sim m_{\mathrm{DM}}$,
and do not specify the ADM mechanism through which the number densities are
related.333For an example of an ADM mechanism that can be implemented in this
framework, see the leptogenesis model of Ref. [24]. We assume that the ADM
mechanism generates similar number densities $n_{\mathrm{VM}}\sim
n_{\mathrm{DM}}$, and also assume that the relationship between the baryon
mass and confinement scale is the same for both visible and dark QCD. Then, to
satisfy Eqn. 1, we need a dark confinement scale $\Lambda_{\mathrm{dQCD}}\sim
1$ GeV. We let this vary by a factor of 5 to approximately account for
different dynamical completions that would be expected to produce a range of
reasonable values for the $n_{\mathrm{VM}}/n_{\mathrm{DM}}$ ratio. So, we
define a ‘valid’ value of $\Lambda_{\mathrm{dQCD}}$ to be between 0.2 GeV and
5 GeV.

We now need to determine how ‘natural’ this framework is; that is, how readily
it generates valid values for $\Lambda_{\mathrm{dQCD}}$. We consider the
theory to provide a natural explanation of the coincidence problem if one
obtains a valid $\Lambda_{\mathrm{dQCD}}$ value for a large proportion of the
available parameter space, as we do not wish for the cosmological coincidence
to arise from any fine-tuning of the parameters.

Of course, an ambiguity arises in the notion of ‘available’ parameter space.
The parameters we vary are the multiplicities of the field content and the
initial values of the couplings in the UV; clearly, we must place some
restrictions on these parameters when searching for valid
$\Lambda_{\mathrm{dQCD}}$ values, both to provide a finite space in which to
look and to avoid considering unphysical theories. So, the question we ask of
the theory is as follows: _given a certain set of assumptions, is it a
surprise that we obtain a valid value for the dark confinement scale?_

In this section we do not argue for a unique set of such assumptions; rather,
we discuss a number of assumptions one could make, and give freedom to the
reader to place their own set of requirements on the theory when assessing its
validity. We then present a set of results given only a minimal set of
assumptions, and highlight features of the theory that affect how readily it
obtains a valid dark confinement scale.

###  V.1 Assessing the validity of a given model

We start this discussion by considering a given model, where we fix the
multiplicities of the new fields as given in Table 1. This specifies the IRFP
and the beta-functions, but leaves us free to vary the initial UV couplings
$\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$. The question then
is what assumptions we place on the selection of these initial UV couplings;
in particular, what range of values do we select our couplings from, and how
do we randomly select coupling values from this range?

To answer this question requires knowledge of the inaccessible dynamics above
the Planck scale that generate the values of these parameters at the UV scale.
We must remain agnostic to these dynamics, and so the simplest assumption we
apply is that the gauge couplings are perturbative at the UV scale – that is,
both $\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$ lie between 0
and 1.

However, as seen in Figs. 1 and 3, we have chosen to work with initial UV
couplings that lie between 0 and 0.3. This is done to ensure the perturbative
validity of our two-loop $\beta$-function calculations. From Eqn. 2, and
recalling that $\alpha=g^{2}/4\pi$, we see that the individual one- and two-
loop non-gluonic terms that contribute to the $\beta$-functions – which we
call $\beta^{(1)}$ and $\beta^{(2)}$, respectively – are of the form

$\beta^{(1)}\sim\frac{\alpha^{2}n}{2\pi},\quad\beta^{(2)}\sim
20\frac{\alpha^{3}n}{8\pi^{2}},$ (10)

where $n$ is a multiplicity, and the factor of 20 accounts for the relative
size of the numerical coefficients for the one- and two-loop terms, due to
various group-theoretic factors. Then, to ensure that the individual two-loop
terms are smaller than the individual one-loop terms implies the rough
condition $\alpha_{s},\alpha_{d}<0.3$. From now on, we refer to gauge
couplings smaller than 0.3 as ‘perturbative’ couplings, in the stricter sense
that both the couplings themselves are perturbative, and that the
$\beta$-functions governing their RGE running admit a perturbative analysis.

Other restrictions we could place on $\alpha_{s}^{\mathrm{UV}}$ and
$\alpha_{d}^{\mathrm{UV}}$ depend upon how strongly we require a given model
to replicate our universe. For example, one could discount points in the red
hatched regions of Fig. 3 where there is no value of the new physics scale $M$
for which the low energy running of $\alpha_{s}$ matches with experiment.
However, one could argue that a given model is equally theoretically valid
with any perturbative values for the initial UV couplings; the fact that the
model fails this phenomenological test for certain initial UV couplings is
just one way in which it is unable to obtain a valid value for the dark
confinement scale. Put another way, the difference is between asking the
following questions:

  1. 1.

“Given that our model has perturbative values for $\alpha_{s}^{\mathrm{UV}}$
and $\alpha_{d}^{\mathrm{UV}}$, how likely is it that we replicate the low-
energy running of $\alpha_{s}$ and also obtain a valid value for the dark
confinement scale?”

  2. 2.

“Given that our model has perturbative values for $\alpha_{s}^{\mathrm{UV}}$
and $\alpha_{d}^{\mathrm{UV}}$ and replicates the low-energy running of
$\alpha_{s}$, how likely is it that we obtain a valid value for the dark
confinement scale?”

For the results we present in Sec. VI, we take the first of these approaches
when analysing a given model.

Another possible assumption of this type is to only consider UV couplings for
which the required value of $M$ is on the order of a TeV or greater, as new
sub-TeV colour-charged particles would be produced readily at colliders.

For a given model, one could also choose to only work with UV couplings for
which the model is truly asymptotically free. However – as discussed in Sec.
IV.1 – given that we only consider $\alpha_{s}^{\mathrm{UV}}$ and
$\alpha_{d}^{\mathrm{UV}}$ with values smaller than 0.3, then there will be no
Landau poles below the Planck scale. Above the Planck scale, unknown physics
could lead these models to regain asymptotic freedom or achieve asymptotic
safety, so it is valid to consider such points.

The last point to consider here is the process by which we randomly select
values for $\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$ from a
specified range. This again requires knowledge of physics above the Planck
scale; given our lack of knowledge, we make the simple assumption that the
couplings are selected uniformly randomly between 0 and 0.3 for each coupling.

Given the simplest of these assumptions, we define the ‘validity fraction’
$\epsilon_{v}$ to be the fraction of the UV coupling parameter space –
$0<\\{\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}}\\}<0.3$ – that lies
between the 0.2 GeV and 5 GeV contours for $\Lambda_{\mathrm{dQCD}}$. This is
roughly the likelihood that a given model will obtain a valid value of
$\Lambda_{\mathrm{dQCD}}$ given a random choice of perturbative initial UV
couplings. As an approximate condition, we consider models with large values
of $\epsilon_{v}$ to be those that explain the cosmological coincidence
problem. The definition of a ‘large’ value of $\epsilon_{v}$ is, of course,
subjective to some extent, and we shall see what constitutes the largest
possible values of $\epsilon_{v}$ when we present results in Sec. VI.

###  V.2 Assessing the validity of the framework

Now that we have discussed the assumptions one could make to assess the
validity of a given model, we can look at how readily the overall framework
can serve as an explanation of the coincidence problem. That is, within the
space of all possible models (as defined by the multiplicities of the new
fields we introduce), how many of them provide viable explanations of the
cosmological coincidence problem? To answer this question, we must look at the
assumptions we choose to restrict this space of all possible models.

In Ref. [25], the space of all possible models is set by calculating a
‘maximum multiplicity’ for each new field; these are determined by the
requirement that the one-loop terms for $\beta_{c}$ and $\beta_{d}$ in Eqn. 2
remain negative.444In Ref. [25], this condition is chosen to ensure that the
gauge groups are QCD-like. In our work, we often work with gauge couplings
where the models are not asymptotically free, so it is less obvious that we
should also require negative one-loop terms in our $\beta$-functions. However,
the existence of an IRFP ensures that there is an asymptotically free region
surrounding the origin – as shown in Fig. 4 – and to have asymptotic freedom
for $\alpha_{s}$ and $\alpha_{s}$ close to zero requires that the one-loop
terms in the $\beta$-functions are negative. For each new field, its maximum
multiplicity is calculated assuming that all other new field multiplicities
are zero; the resulting values are given in Table 2.

Multiplicity | $n_{f_{c,h}}$ | $n_{f_{d}}$ | $n_{f_{j}}$ | $n_{s_{c}}$ | $n_{s_{d}}$ | $n_{s_{j}}$
---|---|---|---|---|---|---
Maximum value | 10 | 16 | 3 | 42 | 66 | 14
Table 2: The maximum multiplicities for each new field. These are set by the
requirement that the one-loop terms for the $\beta$-functions are negative,
which is necessary for the existence of an infrared fixed point.

Within this total space of possible models, we can apply further restrictions
on the set of models we consider. The first, of course, is to consider only
those models with a perturbative infrared fixed point. We could also consider
only those models whose IRFP satisfies $\alpha_{s}^{\star}<\alpha_{s}(M_{Z})$
and $\alpha_{d}^{\star}<\pi/4$. Before accounting for the dependence of the
dark confinement scale on the initial UV couplings, this was a necessary
condition to apply to a given model to even obtain a dark confinement scale;
since the couplings were assumed to take their IRFP values at the decoupling
scale, these values must be lower than the values they need to reach at low
energy after decoupling. However, with our alteration to the framework, there
will be initial UV coupling values for which a dark confinement scale can be
calculated, even for models whose IRFP values $\alpha_{s}^{\star}$ and
$\alpha_{d}^{\star}$ are larger than $\alpha_{s}(M_{Z})$ and $\pi/4$,
respectively; thus, we are not required to apply this restriction to the
models we consider.

##  VI Results

In this section we present results using a minimal set of the assumptions
discussed in the previous section. We focus on finding models that can viably
explain the coincidence problem, and identifying the features of these models
that allow them to do so 555While this section focuses on identifying viable
models within our expanded framework, in Appendix A we present results for the
seven benchmark models from the original work of Bai and Schwaller; this
serves as a point of comparison between the new and old analyses.. To identify
these viable models, we use the validity fraction $\epsilon_{v}$ as defined in
Sec. V.1.

The entire space of possible models, as limited by the maximum field
multiplicities of Table 2, is too large to search through efficiently. We
begin this section by instead looking in detail at a limited set of models -
those that have at most three copies of each new field, as specified by the
field identifications in Table 1. We also require that $n_{f_{d,l}}\geq 1$, so
that there is at least one light dark quark species to confine into the dark
baryons. The choice of at most three copies of each field is only to make the
set of models more amenable to study; however, these models also would present
an easier task if one wished to do further model-building and detailed
phenomenology with one of them in particular.

###  VI.1 Analysing models with $n_{i}\leq 3$

There are 12,288 total models of this type. Of these, 1354 have an infrared
fixed point and 155 have an IRFP for which both couplings are perturbative;
recall that we consider a gauge coupling to be perturbative if it is smaller
than 0.3, as discussed in Sec. V.1. The values of $\epsilon_{v}$ for this set
of 155 models are shown in Fig. 5. These are no models for which
$\epsilon_{v}$ is greater than 0.4, and there is a tendency towards smaller
values of $\epsilon_{v}$.

We consider models with larger values of $\epsilon_{v}$ to be those that are
best able to explain the cosmological coincidence problem; we thus are
interested in the features of the models with the largest values of
$\epsilon_{v}$, and so in Fig. 6 we show the 12 models for which
$\epsilon_{v}>0.35$. Roughly, these are the models for which there is at least
a 1 in 3 chance that a randomly chosen pair of perturbative initial UV
couplings will generate similar values of the visible and dark confinement
scales.

Figure 5: A histogram of the validity fractions for the set of 155 models that
have at most three of each new field and a perturbative IRFP. Figure 6: Each
subplot shows the results for a given model on axes of the initial UV
couplings, where the plot features are similar to those in Fig. 3. In blue we
show the contours for $\Lambda_{\mathrm{dQCD}}=$ 0.2 GeV (right-most contour)
and 5 GeV (left-most contour); the validity fraction $\epsilon_{v}$ is
calculated as the proportion of the parameter space between these two
contours. In red we show the contour for $M$ = 1 TeV, and note that $M>1$ TeV
to the left of the red contour; we also show the red hatched regions where
there are no valid values for $M$. The green hatched region indicates the
‘asymptotically free’ region, and the pink star shows the IRFP of the model.
The 12 models shown are those with at most 3 of each new field, a perturbative
IRFP, and a validity fraction $\epsilon_{v}>0.35$. The title of each subplot
gives the field multiplicities of the given model:
$n_{f_{c,h}},n_{f_{d,l}},n_{f_{d,h}},n_{f_{j}},n_{s_{c}},n_{s_{d}},n_{s_{j}}$.
Figure 7: Each star shows the infrared fixed point of a model, sized by the
validity fraction $\epsilon_{v}$ for that model. The set of models shown in
this plot are the 155 models with at most 3 of each new field and a
perturbative infrared fixed point. The pink stars correspond to the 12 models
with $\epsilon_{v}>0.35$ that are shown in Fig. 6. The black dashed line is at
$\alpha_{s}(M_{Z})=$ 0.11729.

Looking at the $\Lambda_{\mathrm{dQCD}}$ contours for these 12 models, we can
first see why there may be no models for which $\epsilon_{v}$ is greater than
0.4. When the initial UV couplings are too dissimilar, we do not obtain a dark
confinement scale similar to the visible confinement scale; for smaller
$\alpha_{s}^{\mathrm{UV}}$ and larger $\alpha_{d}^{\mathrm{UV}}$,
$\Lambda_{\mathrm{dQCD}}$ is too large, and for smaller
$\alpha_{d}^{\mathrm{UV}}$ and larger $\alpha_{s}^{\mathrm{UV}}$,
$\Lambda_{\mathrm{dQCD}}$ is too small. This makes sense – to obtain similar
confinement scales for the two gauge groups, their couplings must be similar
at the decoupling scale, and so cannot be too different in the UV.

The next observation we make is that all 12 models have small fixed points,
with both $\alpha_{s}^{\star}$ and $\alpha_{d}^{\star}$ less than 0.1. This
correlation between smaller IRFP values and larger validity fractions is clear
in Fig. 7, where we plot the IRFPs for the set of 155 models, sized by the
value of $\epsilon_{v}$. The 12 models with $\epsilon_{v}>0.35$ are shown in
pink, and cluster in a region centred on
$\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}}\sim 0.05$.

We can understand easily why larger $\epsilon_{v}$ would correlate with
smaller $\alpha_{s}^{\star}$. At the decoupling scale, $\alpha_{s}(\mu_{0})$
must be smaller than $\alpha_{s}(M_{Z})$ (whose value is inidicated in Fig. 7
with a black dashed line) in order to have the correct low-energy running. So,
for this to occur in a decent proportion of the UV coupling parameter space,
the couplings must evolve towards a fixed point where $\alpha_{s}^{\star}$ is
smaller than $\alpha_{s}(M_{Z})$. Conversely, when $\alpha_{s}^{\star}$ is
greater than $\alpha_{s}(M_{Z})$, then for many pairs of initial UV couplings
with larger values of $\alpha_{s}^{\mathrm{UV}}$ there is no mass scale $M$
for which $\alpha_{s}$ has the correct low-energy running. This can be seen in
Fig. 6 for the models with $\alpha_{s}^{\star}$ closest to 0.1, and is clearer
in Fig. 8, where we show the models with the four largest values of
$\alpha_{s}^{\star}$.

Given that small $\alpha_{s}^{\star}$ correlates with larger $\epsilon_{v}$,
we can also understand heuristically why $\alpha_{d}^{\star}$ should also be
small in these cases. As mentioned earlier, for the visible and dark
confinement scales to be of a similar order of magnitude, then $\alpha_{s}$
and $\alpha_{d}$ should generally have similar values at the decoupling scale;
to achieve this for a decent proportion of the UV coupling parameter space,
the couplings should thus be evolving towards a fixed point with similar
values for $\alpha_{s}^{\star}$ and $\alpha_{d}^{\star}$.

The last observation we make here is about the size of the new physics scale
in these models. In Fig. 6, to the right of the red contour in each plot we
have $M<$ 1 TeV. This would imply a plethora of new, coloured sub-TeV
particles, and the model at these parameter points would be ruled out by
collider constraints. So, although within our selection of 155 models we have
found 12 with a decent validity fraction $\epsilon_{v}>0.35$, they would be
ruled out by experiment for the majority of their valid parameter space.

Now that we have identified some features of models that correlate with larger
values of $\epsilon_{v}$, we wish to see if we can find models that have a
sufficiently large new physics scale $M$ to avoid strong collider constraints
in addition to having a large validity fraction.

###  VI.2 Searching for models with larger $M$

To find models that have larger scales of new physics, we need to know how $M$
depends upon the selection of field multiplicities.

Recall that $M$ is determined by the requirement that we replicate the low-
energy running of $\alpha_{s}$: using the consistency condition of Eqn. 4, we
solve for the decoupling scale $\mu_{0}$ at which $\alpha_{s}(\mu_{0})$
running under the $\beta$-functions of the full model matches onto
$\alpha_{s}^{\mathrm{EFT}}(\mu_{0})$ given by its SM running from
$\alpha_{s}^{\mathrm{EFT}}(M_{Z})=0.11729$. By specifying a value of
$\mu_{0}/M\in(0.5,2)$ we can then solve for $M$.

The low-energy SM running of $\alpha_{s}^{\mathrm{EFT}}$ is fixed; so, to
increase the value of $M$ for a given set of initial UV couplings, we need
$\alpha_{s}$ to run more strongly under the $\beta$-functions of the full
model. In general, increasing the field multiplicities will produce stronger
running, as the coefficients of the terms in the $\beta$-functions increase.
Since we need the coefficients of the one-loop terms to be negative in order
to have an IRFP, the strongest running will occur when the coefficient of the
one-loop term is negative with a small magnitude, and the coefficients of the
two-loops terms are positive with a larger magnitude666Note that although the
magnitude of the one-loop term will be smaller than the two-loop terms, this
is due to a cancellation between the matter terms and gluon terms in the one-
loop coefficient. The individual one-loop terms will still be smaller than the
individual two-loop terms for perturbative $\alpha<0.3$, as discussed in Sec.
V.1.. This also correlates with the IRFP couplings $\alpha_{s}^{\star}$ and
$\alpha_{d}^{\star}$ having small values, as the $\beta$-functions will be
zero when, roughly, the $\text{one-loop
coefficient}\times\alpha^{2}\sim\text{two-loop coefficient}\times\alpha^{3}$.

From Eqn. 2, we see that increasing the multiplicities of the jointly-charged
fields – $n_{f_{j}}$ and $n_{s_{j}}$ – increases the coefficients of all terms
in the $\beta$-functions. Since $n_{f_{j}}$ can be no larger than 3 in order
to keep the one-loop terms negative, we consider models with larger
$n_{s_{j}}$ when searching for models with a larger new physics scale $M$. In
particular, we look at all models with $n_{s_{j}}\geq 10$, where the maximum
multiplicity for each field is given in Table 2.

Figure 8: The results for the four models with the largest values of
$\alpha_{s}^{\star}$ from the set of models with at most 3 of each new field
and a perturbative IRFP. The features of each subplot are the same as those in
Fig. 6. Note that a larger value of $\alpha_{s}^{\star}$ correlates with a
larger proportion of the parameter space that has no valid value for $M$ (the
red hatched regions).

We note that in Table 2 there is only an upper limit on the total number of
dark fermions $n_{f_{d}}$, and not on the specific multiplicities of the light
and heavy dark fermions, $n_{f_{d,l}}$ and $n_{f_{d,h}}$; this is because the
$\beta$-functions of the full model only depend on $n_{f_{d}}$. The new
physics scale $M$ and the decoupling scale $\mu_{0}$ also only depend on
$n_{f_{d}}$, as they only depend upon the SM running of $\alpha_{s}$ in
addition to the RGE evolution using the full $\beta$-functions. So, for the
purposes of searching for models with larger values of $M$, we only specify
the value of $n_{f_{d}}$.

Using this definition of a model, there are 10,141,120 models for which
$n_{s_{j}}>10$, of which 120,015 have a perturbative fixed point. To have
sufficiently strong running of the gauge couplings, we then look for models
whose one-loop coefficients for the $\beta$-functions are negative with a
small magnitude, where we take that to mean that the coefficient is between
$-0.1$ and 0; there are 188 such models. For all of these models, the scale of
new physics $M$ is greater than 1 TeV for at least $80\%$ of the
$\alpha_{s}^{\mathrm{UV}}$-$\alpha_{d}^{\mathrm{UV}}$ parameter space; that
is, the value of $M$ is large enough to avoid strict collider constraints in
the majority of parameter space.

So, having found this set of models, we now wish to check which values of
$\epsilon_{v}$ they can obtain. If we find models that have a decently large
value of $\epsilon_{v}$, and also have sufficiently large new physics scales
in the majority of parameter space, then these models could serve to naturally
explain the cosmological coincidence problem.

For each of the 188 models, we have only specified $n_{f_{d}}$; so, for a
given model we are free to choose $n_{f_{d,l}}\leq n_{f_{d}}$, setting
$n_{f_{d,h}}=n_{f_{d}}-n_{f_{d,l}}$. We also ensure that $n_{f_{d,l}}>0$ so
that there is at least one light dark fermion species to confine into dark
baryons.

The choice of $n_{f_{d,l}}$ will change the value of the dark confinement
scale $\Lambda_{\mathrm{dQCD}}$ for a given pair of initial UV couplings, as
$\Lambda_{\mathrm{dQCD}}$ depends on the low-energy running of $\alpha_{d}$
below the decoupling scale. The $\Lambda_{\mathrm{dQCD}}$ contours, and thus
the validity fraction $\epsilon_{v}$, then also depend on $n_{f_{d,l}}$. So,
for each of the 188 models, we choose $n_{f_{d,l}}$ such that the value of
$\epsilon_{v}$ is maximised. We show these values of $\epsilon_{v}$ in Fig. 9.

Figure 9: A histogram of the validity fractions for the set of 188 models that
have 10 or more joint scalars, and one-loop coefficients for the
$\beta$-functions between $-0.1$ and 0. Each of these models has a mass scale
$M>1$ TeV for at least 80% of the UV coupling parameter space.

Most of these models have values for $\epsilon_{v}$ above 0.3, with the
largest values just over 0.32. We show 4 models with $\epsilon_{v}>0.32$ in
Fig. 10. While these values of $\epsilon_{v}$ are not quite as large as those
for the best models in Sec. VI.1, a validity fraction of $\sim 0.3$ still
corresponds to a decent proportion of the initial UV coupling parameter space
for which the dark confinement scale is on the order of the visible
confinement scale. These models also have $M>1$ TeV for every point in the
viable region of parameter space, and so manage to potentially provide an
explanation for the cosmological coincidence problem while avoiding stringent
collider constraints.

Figure 10: The results for four models with $\epsilon_{v}>0.32$. These are
some of the models with the largest viability fraction in the set of 188
models shown in Fig. 9. The features of each subplot are the same as those in
Fig. 6. Note that for each model, at least 80% of the parameter space has
$M>1$ TeV, where $M=1$ TeV is shown by the red contour.

##  VII Conclusions

In this paper we reassessed the feasibility of explaining the cosmological
coincidence problem by utilising infrared fixed points in a dark QCD
framework. In the original work of Bai and Schwaller, the confinement scales
of dark and visible QCD are related by a dynamical mechanism depending only on
the value of the gauge couplings at the IRFP and the mass scale of the new
particle content introduced. We extended this framework by incorporating the
dependence of these results on the initial values of the gauge couplings in
the UV.

To assess whether models within this framework are able to explain the
cosmological coincidence problem, we needed to carefully define our notion of
how ‘naturally’ a given model produces visible and dark confinement scales of
a similar order of magnitude. With the dark confinement scale now depending on
the initial UV couplings, we analysed each model by looking at the proportion
of UV coupling parameter space for which one obtains a valid value for the
confinement scale: denoted the ‘validity fraction’ $\epsilon_{v}$.

We began by looking at a limited set of models where we introduce at most
three of each new field. Of these 12,288 models, 155 have a perturbative IRFP.
The maximum validity fraction for these models is $\sim 0.4$, and there are 12
with $\epsilon_{v}>0.35$. The models with larger validity fractions generally
have smaller IRFPs.

While these models have a decently sized proportion of UV coupling parameter
space for which the dark and visible confinement scales are related, the scale
of new physics is below 1 TeV for most of this parameter space. These new
coloured sub-TeV fields face strict collider constraints. To see if models in
this framework can avoid these constraints, we identified a set of 188 models
for which at least 80% of the UV coupling parameter space has $M>1$ TeV; all
these models have 10 or more scalars charged under both visible and dark QCD.
Of these models, most have $\epsilon_{v}\sim 0.3$.

In conclusion, there do exist models within this framework that can
potentially provide a natural explanation for the cosmological coincidence
problem. While many of these models introduce new sub-TeV coloured particles,
there is a well-defined set of models that have a new physics mass scale high
enough to avoid these collider constraints. In future work, these models could
be candidates for more detailed model building, in order to incorporate a
definite baryogenesis mechanism to generate the asymmetries in visible and
dark matter, and to perform a proper phenomenological analysis.

Figure 11: The results for the seven benchmark models identified in the
original paper by Bai and Schwaller. The features of each subplot are the same
as those in Fig. 6. The results presented in the original paper correspond to
our results when choosing the initial UV couplings to have their IRFP values
(the point in each subplot identified by the pink star).

## Acknowledgements

We would like to thank Jayden Newstead and Michael Schmidt for useful
conversations. ACR was supported by an Australian Government Research Training
Program Scholarship. This research was supported by the Australian Government
through the Australian Research Council Centre of Excellence for Dark Matter
Particle Physics (CDM, CE200100008).

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##  Appendix A The benchmark models from Bai and Schwaller

In Fig. 11 we plot the results for the seven benchmark models chosen by Bai
and Schwaller. Table 2 of the original paper gives values of $M$ and $m_{D}$
(the dark baryon mass) for each model; in our analysis, these correspond to
$M$ and $\Lambda_{\mathrm{dQCD}}$ calculated for initial UV couplings at their
IRFP values,
$(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})=(\alpha_{s}^{\star},\alpha_{d}^{\star})$,
and with $m_{D}\approx 1.5\Lambda_{\mathrm{dQCD}}$. In Fig. 11, these are the
values for $M$ and $\Lambda_{\mathrm{dQCD}}$ at the point indicated by the
pink star. We see that models C, E, and G have valid values of
$\Lambda_{\mathrm{dQCD}}$ for initial UV couplings at their IRFP values; these
correspond to dark baryon masses of 0.32 GeV, 3.5 GeV, and 1.2 GeV
respectively. Most of the models have $\epsilon_{v}\sim 0.25$, with Model B
having the largest validity fraction of $\epsilon_{v}=0.300\pm 0.009$. While
this is a decently large validity fraction, as with the results in Sec. VI.1,
most of the valid region of parameter space for these models has a new physics
scale below 1 TeV (the initial UV couplings that lie to the right of the red
contour).