# A Dark Matter Fermionic Quantum Fluid from Standard Model Dynamics

Stephon Alexander<EMAIL_ADDRESS>Heliudson Bernardo
<EMAIL_ADDRESS>Humberto Gilmer<EMAIL_ADDRESS>Department of Physics,
Brown University, Providence, RI 02912, USA

###### Abstract

We present a model of dark matter as a superconducting fluid of Cooper pairs
of right handed neutrinos or of vector-like quarks. The superconducting dark
matter is induced by attractive channels in the Standard Model Higgs and color
sectors of the Standard Model, respectively. We show that, for each case, the
solution to the gap equation provides viable dark matter candidates for
suitable chemical potential values. The mechanism yields an ultra-light
neutrino condensate with a mass of $m_{\rm DM}\sim 10^{-19}\text{eV}$ or a
vector-like quark condensate with wide range of possible masses. Both
cosmological and particle physics constraints on the model lead to a
connection between the number of effective relativistic species $N_{\rm eff}$,
and the chemical potential and CMB temperature at the time of fermion
creation. We also find a relation between the superconducting fermion and
baryon densities, with implications for the coincidence between the dark
matter and baryon densities in standard cosmology. Given the natural eV scale
of neutrinos, this mechanism may have implications for the Hubble tension.

††preprint: APS/123-QED

##  I Introduction

The existence of dark matter (DM) is now a settled component of the
concordance model of cosmology and concrete evidence of physics beyond the
Standard Model (SM) [1]. Observations ranging from galactic rotation curves,
large scale structure, to the cosmic microwave background (CMB) radiation have
established evidence both for the existence of DM and its impact on visible
matter, and on the cosmological history of the universe [2]. However, despite
the robust evidence for its existence, the microphysical nature and identity
DM are still unknown.

In this paper we propose a minimal model of dark matter which relies on the
universality of the pairing mechanism of Bardeen-Cooper-Schrieffer (BCS)
theory (Cooper pairs) induced by attractive interations given by either the
Higgs or QCD sector of the Standard Model. The idea that the dark matter can
be a BCS condensate was first discussed by the authors [3, 4, 5], and has been
explored in other cosmological contexts [6], however its connection to the
Standard model was unexplored. In this manner, the dark matter is a stable
massive Higgs mode of a superconducting state of fermions. As we will see, the
resulting dynamics predicts a new relationship between the concordance of the
CMB temperature, $T_{\rm CMB}$ and the chemical potential of the fermions (at
the time of their creation) which form the dark matter superconductor.

Our description of the dark matter will have implications for extensions, if
any, of the Standard Model and could give indications for further experimental
detections to discriminate amongst various candidate models. Importantly, any
viable microphysical description must satisfy a few criteria. (1) It must be
dark, with a small a cross-section-to-mass ratio $\tfrac{\sigma}{M}\ll 1$ for
SM-DM processes. It is imperative that the microscopic model explain why
ordinary SM interactions are absent or small. (2) The condensate must satisfy
the observational constraints on dark matter, such as a small self-interaction
rate and a dust-like equation of state.

Beyond these basic criteria, a successful microphysical description must
further explain observational features. Both WMAP and more recently, Planck
have measured a rough parity between the dark matter energy density
$\Omega_{c}$ and baryonic energy density $\Omega_{b}$, namely $\Omega_{c}\sim
5\Omega_{b}$ [7, 8]. Classes of models exist which seek to explain this
apparent coincidence. One such class is known as asymmetric dark matter (ADM)
[9, 10, 11]. These models postulate that the present-day DM density arises due
to an asymmetry in the DM particle-antiparticle density, similar to the baryon
asymmetry, and that these were generated by similar mechanisms. This
microphysical model is contrasted with that of the WIMP dark matter [12], in
which the similarity in dark and visible matter densities must be taken as a
coincidence [9, 13, 14].

Certainly, ADM is not the only approach that invokes a common origin for
baryonic and dark matter; instead, there exists the possibility that DM is not
a separate fundamental field, but is instead a new, exotic state of visible,
Standard Model matter. Such a proposal has precedent, with previous ideas
ranging from quark nuggets, to strange matter (known as strangelets) [15, 16]
to QCD balls [17]. Under the influence of an unspecified interaction, SM
fermions may pair to form condensates that behave as scalar fields,
analogously to the Cooper pairing [18] of BCS theory [19, 20].

_A priori_ DM condensate candidates made of SM fermions and sustained by SM
interactions necessarily involve neutrinos or quarks (see discussion in
appendix A). However, since neutrinos were in thermal equilibrium at early
times and space-filling quark condensates imply electrically charged DM, we
are forced to introduce extra fermionic species to get a condensate that
satisfy cosmological constraints. We shall assume that these species are
produced in the early universe, preferentially out of equilibrium, but more
generically with a large chemical potential-to-temperature ratio. While there
are proposals in which the condensate is accomplished via a dark gauge
interaction [4, 21], in this paper, the fermion interacts via SM boson
mediation.

##  II Possible Scenarios

###  II.1 Preliminaries

In the early universe when baryogenesis occurred, the net baryon number was
established at a time $t_{b}$ giving rise to the baryon density
$\Omega_{b}(t_{b})$. We likewise assume, and will justify, that degenerate
fermionic distribution undergoes Cooper instability on its Fermi surface due
to an attractive four-fermion interaction as a result of Standard Model
dynamics; the massive Higgs mode of the resulting BCS condensate would then
function as a natural DM particle candidate. If these fermions were produced
during baryogenesis (see [22, 23, 24] for references on neutrino production),
then we would naturally obtain the scaling $\Omega_{c}\sim\Omega_{b}$.
However, as we revisit in appendix A, since SM fermions are charged and were
in thermal equilibrium at large redshifts, it is non-trivial to embed a
degenerate distribution in the cosmological history. The simplest extensions
that meeet the criteria of dark BCS condensates are right-handed neutrinos and
vector-like quark states, as explained subsequently.

The physics of the resulting fermion condensates are present in diverse
systems, ranging from condensed matter (superconductivity, helium-3) to
nuclear physics (pions). It is well-known that any attractive interaction
channel, no matter how weak, grows stronger at the Fermi surface under the
renormalization group flow [25]. For the case of a four-fermion interaction,
the corresponding operator is classically marginal but becomes relevant after
quantum corrections are taken into account [26]. At strong coupling, a gap is
formed and the non-perturbative (BCS) ground state at least spontaneously
breaks the fermion number symmetry, producing a superfluid condensate; if the
fermions are also charged under some gauge group, a superconductor is formed.
In this work, we posit that DM is such a condensate and investigate whether or
not it can arise in the SM.

Leveraging the universality of these results, dark matter is identified as the
Higgs mode $\rho(x^{\mu})$ of the gap
$\bar{\Phi}\sim\Braket{\psi\psi^{\dagger}}$. Solving the self-consistent gap
equation for the fermion condensate gives rise to the emergence of
$\bar{\Phi}$ as an order parameter whose dynamics dominate in the IR (see
appendix B for details). In virialized regions, the Higgs modes form a self-
gravitating BCS condensate, which has been shown to describe dark matter halos
in [3]. The microscopic DM parameters, such as mass and self-interaction
couplings, will be given by an effective theory for the fluctuations around
$\bar{\Phi}$,

$\Phi(x^{\mu})=\left[\bar{\Phi}+\rho(x^{\mu})\right]e^{i\phi(x^{\mu})},$ (1)

where $\phi$ is the Goldstone mode of the condensate. In terms of the
canonically normalized field
$\varphi=\left(\sqrt{3}/2\pi\right)\left(\mu/\Phi\right)\rho$, the resulting
Lagrangian for the Higgs mode is given by [27, 4]

$\mathcal{L}=-\frac{1}{2}(\partial\varphi)^{2}-16\bar{\Phi}^{2}\varphi^{2}-\sum_{n\geq
2}\tilde{c}_{n}\frac{\bar{\Phi}^{2}}{\mu^{n-2}}\varphi^{n},$ (2)

where $\tilde{c}_{n}$ are order unity constants. One then identifies the DM
mass $m$ and quartic coupling $\lambda$ as

$m=4\sqrt{2}\bar{\Phi},\quad\lambda\simeq\frac{\bar{\Phi}^{2}}{\mu^{2}}.$ (3)

The energy density of the fermion fluid will be constrained by cosmological
history, specifically by the change in the effective number of relativistic
species, $\Delta N_{\text{eff}}$, which is constrained by cosmological
observations [8, 28]:

$\Delta
N_{\text{eff}}<0.17\quad(\text{CMB}+\text{BAO})\;\text{at}\;95\%\;\text{CL}.$
(4)

The shift in $N_{\text{eff}}$ during BBN is (see appendix B for details)

$\Delta
N_{\text{eff}}=\frac{8}{7}\frac{30g}{16\pi^{4}}\left(\frac{\mu(T_{\psi})}{T_{\psi}}\right)^{4}\left(\frac{g_{*S}(T_{\text{BBN}})}{g_{*S}(T_{\psi})}\right)^{4/3},$
(5)

where $T_{\psi}$ is the CMB temperature at the time of the fermion background
formation. (4) limits $\mu(T_{\psi})$ to be at most of order $T_{\psi}$:

$\frac{\mu(T_{\psi})}{T_{\psi}}\lesssim\frac{1.22}{g^{1/4}}.$ (6)

The DM self-interaction cross-section $\sigma$ is constrained to satisfy
$\sigma/m<0.1\;\text{cm}^{2}/\text{g}$ (see [29] and references therein).
Using the expression for the mass and quartic coupling (3), we have

$\frac{m}{\text{eV}}\lesssim
10^{-19}\left(\frac{\mu_{0}}{\text{eV}}\right)^{4}$ (7)

where $\mu_{0}$ represents the Fermi energy today.

The dark matter energy density can be estimated from the initial energy
density in the fermions. The energy difference before and after condensation
formation is of the order $S\Delta T$, where $S$ is the entropy of the fermion
gas and $\Delta T$ is the temperature shift between the two phases. For a
degenerate Fermi gas $S=0$, and so the energy density does not change after
condensation. Hence, the fermionic energy density $\rho_{\psi}$ is a tracer
for the dark matter energy density. There will be contributions from the
Goldstone and Higgs mode excitations, but the former redshifts away like
radiation while the latter only contributes significantly at small scales.

In appendix B, we show that the current baryonic-to-cold energy density ratio,
$\Omega_{c}\sim 5\Omega_{b}$, can be explained after assuming that the initial
fermionic background number density is of the order of the baryon asymmetry
$n_{\psi}(T_{\psi})\sim n_{B}(T_{\psi})$ and that $\mu_{\psi}\sim
10^{-3}T_{\psi}$. The former condition is a natural assumption if the fermions
are produced by the same mechanism responsible for baryogenesis, while the
latter is compatible with the bound (6).

In the following subsections, we will find a wide range of parameters
compatible with the above constraints, generating a broad spectrum of DM
masses. In general, we expect phenomenology akin to the superfluid DM (see
e.g. [30] and references therein); namely, we expect the large-scale evolution
of the DM background energy density to be described by the fermionic one,
while the scalar nature of the Higgs modes will play a leading role on
galactic and sub-galactic scales, evading constraints on fermionic DM
candidates, such as the Tremaine-Gunn bound [31].

We propose two independent scenarios that realize dark matter from fermionic
condensates: right-handed sterile neutrinos and vector-like quarks. These are
natural extensions of the SM fermion sector, with no changes in its bosonic
sector.

###  II.2 Right-handed neutrino condensate

The notion of a neutrino condensate mediated by a scalar field between left-
and right-handed neutrinos is not a new idea. However, most treatments of this
idea miss an important subtlety which opens the possible parameter space.
Yukawa interactions generically couple left- and right-handed fermion fields;
this would in turn lead to the idea that any resulting condensate should
consist of left-right (LR) bound states. However, it has been shown that the
Yukawa interaction can mediate the formation of a condensate in both the $t$
channel [32] (the expected LR condensate) and the $s$ channel [33, 34]
(leading to the possible formation of LL and RR condensates).

Given that the replusive $Z$ exchange is irrevelant at finite density, the
relevant Lagrangian for the neutrino-Higgs sector, written in terms of Weyl
spinors, is

$\mathcal{L}=-\nu^{\dagger}\bar{\sigma}\cdot\partial\nu-\bar{\nu}\sigma\cdot\partial\bar{\nu}^{\dagger}+y_{\nu}L\cdot
H\bar{\nu}+\text{H.c.}$ (8)

where $\nu$ is the left-handed (Standard Model) neutrino and $\bar{\nu}$ is
the right-handed neutrino. Upon inserting the Higgs vev and integrating out
the resulting interaction, we obtain the effective Lagrangian for Dirac
neutrinos at finite densities and below the Higgs mass scale,

$\mathcal{L}=-\bar{\psi}\left(\not{\partial}-m-\mu\gamma^{0}\right)\psi+\frac{y_{\nu}^{2}}{M_{h}^{2}}\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right),$
(9)

where $\psi^{\rm T}=(\nu,\bar{\nu}^{\dagger})$ is the Dirac spinor containing
the neutrino fields, $y_{\nu}$ is the neutrino Yukawa coupling and $M_{h}$ is
the Higgs mass. We will see that condensates containing both the same and
different chiralities are possible using this effective Lagrangian. The
resulting gap may be represented diagrammatically as in Figure 1.

Figure 1:  Diagrammatic representation of a LL or RR gap arising from the
Yukawa interaction between left- and right-handed neutrinos. The Higgs (dashed
line) induces an attractive four-fermi interaction. The grey blob denotes a
sum over neutrino ladder diagrams.

This action is equivalent to (20) with one quartic-coupling operator
$\Gamma^{1}=\mathbb{1}$ and $g^{2}/M^{2}=y_{\nu}^{2}/M_{h}^{2}$. In the
appendix C, we use a Fierz rearrangement to show that the only operators
$\mathcal{O}_{\alpha}$ corresponding to attractive channels are
$\gamma^{2}\gamma_{5}/4$ and $\gamma^{1}\gamma^{3}/4$, such that there are two
possible gaps:

$\langle\psi^{\dagger}\gamma^{2}\gamma_{5}\psi\rangle\propto\Phi_{\rm
LR},\quad\langle\psi^{\dagger}\gamma^{1}\gamma^{3}\psi\rangle\propto\Phi_{\rm
LL+RR}.$ (10)

The former corresponds to an LR condensate while the latter to an
$\text{LL}+\text{RR}$ condensate. Since the SM left-handed neutrinos were in
thermal equilibrium at early times, the LL part doesn’t contribute to the gap,
such that
$\langle\psi^{\dagger}\gamma^{1}\gamma^{3}\psi\rangle\to\langle\bar{\nu}i\sigma^{2}\bar{\nu}^{\dagger}\rangle\propto\Phi_{\rm
RR}$. Moreover, in [34], it is shown that the condensate made of neutrinos of
the same chirality is energetically favorable over $\Phi_{\rm LR}$. The
solution to the gap equation (26) is [34]

$\Phi_{\rm RR}=2\sqrt{M_{h}(\mu-
m_{\psi})}\;e^{-\frac{\pi^{2}}{2\kappa_{R}^{2}\mu\sqrt{\mu^{2}-m_{\psi}^{2}}}},$
(11)

where $\kappa_{R}^{2}\simeq y_{\nu}^{2}/M^{2}_{h}$. The exponential dependence
on the inverse coupling squared $\Phi\propto e^{-1/y_{\nu}^{2}}$ is expected
from BCS theory. Note that the condensate breaks the lepton number symmetry
spontaneously.

For tuned values of the chemical potential at early times, the condensate’s
Higgs mode reproduces the dark nature of DM. For $m_{\psi}\approx
0.1\,\text{eV}$ and $\mu_{\psi}\approx 1.5\times 10^{14}\;\text{GeV}$, we have
$\Phi_{\rm RR}\approx 10^{-20}\;\text{eV}$, such that the bound (B) is
compatible with (3). At later times, the cosmological bounds will continue to
be satisfied, because as $\mu$ gets closer to $m_{\psi}$, $\Phi_{\rm RR}$ is
further suppressed. Note that the gap depends mainly on the chemical potential
value.

###  II.3 Vector-Like Quark condensate

The vector-like quark scenario would involve the inclusion of new fermion
fields charged under $SU(3)_{\rm c}$ but which are otherwise singlets with
respect to the Standard Model. If we consider a scenario with three VLQ
flavors, then for a sufficiently high chemical potential $\mu\gg m_{\rm VLQ}$
we should expect the formation of a 3-color-flavor locking superconducting
phase, mediated by a single gluon exchange.

Consider the Lagrangian for VLQs at finite densities,

$\displaystyle\mathcal{L}$
$\displaystyle=-\bar{\psi}_{i}\left(\delta_{ij}\not{\partial}-m\delta_{ij}-\delta_{ij}\mu\gamma^{0}+ig\tau_{ij}^{a}\not{A}_{a}\right)\psi_{j}$
$\displaystyle\qquad\qquad+\frac{G}{2}\left(\bar{\psi}\gamma_{\mu}T_{a}\psi\right)\left(\bar{\psi}\gamma^{\mu}T_{a}\psi\right).$
(12)

This action is equivalent to is of the form (LABEL:4fermionaction) with one
quartic-coupling operator $\Gamma^{1}=\gamma_{\mu}T_{a}$ and
$g^{2}/M^{2}=\frac{G}{2}$. The color piece can be extracted through
factorization into a symmetric and antisymmetric piece; the remainder can be
reworked using the Fierz identities, yielding the operator
$\mathcal{O}_{\alpha}\propto\gamma_{0}\gamma_{2}$ corresponding to the
dominant attractive channel [35, 36, 37]. The ground state of this theory
contains a condensate of the form

$\Braket{\psi^{a\alpha}_{i{\rm L,R}}(\vec{p})\psi^{b\beta}_{j{\rm
L,R}}(-\vec{p})\epsilon_{ab}}=\pm\Delta(p^{2})\epsilon^{\alpha\beta
A}\epsilon_{ijA},$ (13)

where $\psi^{\alpha}_{i{\rm L,R}}$ denotes two-component Weyl spinors with
flavor and color indices $i$ and $\alpha$, respectively. It has been shown
that a condensate of the CFL form is the dominant ground state in three-flavor
QCD at finite densities [38, 39, 40, 41, 42]. Under this scenario, the VLQs
have a microscopic symmetry of the form $G=\text{SU}(3)_{\rm
L}\times\text{SU}(3)_{\rm R}\times\text{U}(1)_{B}$; in the CFL phase this is
broken to $G^{\prime}=SU(3)_{\rm c+L+R}\times\mathbb{Z}_{2}$. As with the
ordinary quarks, the chiral flavor symmetry is also broken, except here it is
‘locked’ to the color symmetry, necessary due to the pairing of quarks of like
helicity. Such a symmetry breaking does induce a small gluon mass. At first
glance this seems to be a fatal flaw with this scenario. However, it is only
theorized that the gluons are perfectly massless gluons; gluon masses of up to
an MeV are consistent with experimental measurements [43]. Before proceeding
further, it is worth examining whether such a condensate can form.

The dominant interaction in this model will be of single-gluon exchange within
the VLQ sector (contrast with the neutrino case where the condensate forms in
the RH neutrino sector, via a Higgs interaction with the LH neutrinos).
Single-gluon exchange diagrams lead to two different interactions, one
attractive (forming the condensate) and one repulsive (thermalizing the
quarks). The ratio of these interactions is given by
$\mathcal{M}_{a}/\mathcal{M}_{s}=(N_{c}+1)/(N_{1}-1)$ (from (A)). For
$N_{c}=3$ (as we are working with standard QCD), then the interaction rates of
the two processes $\Gamma=n\sigma v$ gives the ratio of time-scales
$t_{a}/t_{s}=1/4$. Thus the time-scale for condensate formation is shorter
than that for scattering.

Now we turn to the question of whether the condensate can form cosmologically.
In order for the interaction to be appreciable within a Hubble patch, we
expect $\Gamma/H\gg 1$. For gluon exchange dominated by modes near the Fermi
surface [4]

$\frac{\Gamma}{H}=\frac{n_{\psi}\alpha^{2}}{k^{2}_{F(\psi)}H_{\psi}}\left(\frac{a(t)}{a(t_{\psi})}\right)$
(14)

where as before the $\psi$ subscript corresponds to the quantity evaluated at
the time of the VLQ production, and $\alpha=g^{2}/4\pi$ is the QCD coupling.
If we assume most fluctuations have $k=k_{F}$, then we can estimate
$\rho_{c}\approx nk_{F}$. Furthermore, we can write an upper bound on $k_{F}$
to be $H_{\psi}$. Lastly, using the Friedmann equation we can write
$H_{\psi}=\sqrt{\rho_{{\rm tot},\psi}}/(\sqrt{3}M_{\rm Pl})$, where
$\rho_{{\rm tot},\psi}$ is the energy density of the universe at the time of
VLQ production. Putting these together we can write

$\frac{\Gamma}{H}\sim\frac{\rho_{c}}{\rho_{{\rm tot},\psi}}\frac{M_{\rm
Pl}^{2}}{H_{\psi}^{2}}\left(\frac{a(t)}{a(t_{\psi})}\right)$ (15)

Since the condensate forms deep in radiation-domination, we can write
$\rho_{c}=\rho_{\psi}\sim g\mu(T_{\psi})^{4}/8\pi^{2}$ using (35) while
$\rho_{{\rm tot},\psi}\sim\pi^{2}g_{*}(T_{\psi})T_{\psi}^{4}/30$, where
$g_{*}(T)$ is the number of degrees of freedom in radiation at temperature
$T$. Thus we arrive at

$\displaystyle\frac{\Gamma}{H}$
$\displaystyle\sim\frac{15}{4\pi^{4}}\frac{g}{g_{*}(T_{\psi})}\left(\frac{\mu(T_{\psi})}{T_{\psi}}\right)^{4}\frac{M_{\rm
Pl}^{2}}{H_{\psi}^{2}}\left(\frac{a(t)}{a(t_{\psi})}\right)$
$\displaystyle\sim
10^{84}\left(\frac{a(t)}{a(t_{\psi})}\right)\left(\frac{\mu(T_{\psi})}{T_{\psi}}\right)^{4},$
(16)

where we used $g_{*}(T_{\psi})\sim\mathcal{O}(10^{2})$ so that
$g/g_{*}(T_{\psi})\sim\mathcal{O}(10^{-2})$ and let $H_{\psi}\simeq H_{\rm
BBN}\sim 10^{-44}M_{\rm Pl}$. Hence, due to the large numerical factor in the
rightmost-hand side, the condition $\Gamma/H\gg 1$ is easily satisfied.

With the question of formation addressed, as found in [44] and discussed
further in [37], the CFL phase gap is given by

$\Phi\sim b\frac{\mu}{g^{5}}e^{-\frac{3\pi^{2}}{\sqrt{2}g}}$ (17)

where the prefactor $b$ has been computed in the CFL phase to be $b\sim
512\pi^{4}2^{-\frac{1}{3}}\left(\tfrac{2}{3}\right)^{\frac{5}{2}}$ [40, 45].

Compared with the neutrino case, the greater lower bound on the VLQ mass
$m_{\rm VLQ}>10^{13}\text{ eV}$ and the gap’s independence of the mediator
mass gives greater leeway in this scenario to satisfy the DM constraints
computed in appendix B. For $\mu\sim 10^{14}$ eV and $g\sim 0.23$ we find
$\Delta\sim 10^{-20}$ eV, which is compatible with (41), but other mass ranges
are possible depending mainly on the precise value of $g$.

##  III Discussion/Conclusion

In this work we have presented an analysis regarding whether dark matter
states are viable within the structure of the Standard Model. Although it is
not possible, due to precision data, to construct dark states using only
Standard Model fields, it is possible to do so within the gauge structure of
the Standard Model, by augmenting the matter sector with well-motivated
additional fermionic fields, namely the sterile neutrino and the vector-like
quark. Each scenario has its unique aspects, but much of the physics common to
both scenarios arises from the well-known physics of superconductivity.

Postulating a degenerate fermionic background, for sufficient values of
chemical potential, the formation of a condensate is inevitable given an
attractive interaction. This presents the possibility that what we have
heretofore called dark matter is simply a particular state of the fermionic
species, mediated by the known interactions of the Standard Model. In this
case, the dark matter particle would be the Higgs mode of the resulting
superconducting state, similar to how the Higgs boson detected at the LHC is
but the excitation of the background Higgs field.

The necessary chemical potential in the neutrino case is adjusted to account
for the fact that the neutrino mass is bounded from above and that the
interaction is mediated by the Higgs which is relatively heavy for such an
interaction. Due to the constraints on the neutrino mass and the Higgs mass,
the possible values for the DM particle are fixed primarily by the chemical
potential and it is thus possible in this case to generate DM masses in the
ultralight range.

Due to the higher possible VLQ mass values and the massless (or nearly so)
nature of the mediating particle, the VLQ case is not as strongly tuned and
can naturally accommodate a wide range of DM masses from the ultralight to
WIMP-scale DM.

It is worth commenting that the matter-radiation equality scale $T_{\rm
eq}\sim 0.3\;\text{eV}$ is of the order of the cosmological upper bound on the
sum of neutrino masses, $\sum_{i}m_{\nu,i}<0.13\;\text{eV}$ [8, 28]. For the
neutrino case, $\rho_{\psi}$ starts to redshift as dust when $T\sim m_{\nu}$,
i.e. just before $z_{\rm eq}$. Or, equivalently, neutrinos having an
$\mathcal{O}(10^{-1})\;\text{eV}$ mass may be the reason why $z_{\rm
eq}\sim\mathcal{O}(10^{3})$.

An important aspect left for future work is the phenomenology of this model,
specifically examining such aspects as the core-cusp problem or the baryonic
Tully-Fisher relation. For a wide range of parameters, we expect the small-
scale phenomenology of the condensate to be similar to the superfluid DM case,
thus avoiding usual astrophysical constraints on fermionic particle dark
matter candidates [30]. Other phenomena associated with superfluid DM, such as
vortices [46, 47, 48], are also possible and worth exploring in this context.
It will be important to distinguish this model from a fundamental ultralight
scalar dark matter candidate. One possibility for future work is to study the
possible neutrino oscillations induced by the background neutrino condensate
as neutrinos propagate through it. From a more theoretical perspective, one
could also investigate how the four-fermion interaction affects the fermionic
densities and get a fully-fledged expression for the condensate contribution
to the dark matter density. Another direction would be explicitly constructing
the baryogenesis model that gives origin to a degenerate fermion background
with suitable chemical potential.

###### Acknowledgements.

We thank Ian Dell’Antonio, Loukas Gouskas, Leah Jenks, Greg Landsberg, Evan
McDonough, Philip Phillips, Michael Peskin, and David Spergel for useful
discussions. We thank Jorge Sofo for inspiring SA for pursuing this project.
HBG is supported by the National Science Foundation, MPS Ascending Fellowship,
Award 2213126.

##  Appendix A Attractive channel from Standard Model dynamics and fermion
candidates

It is instructive to quickly explore all possible attractive channels of
fermions in the standard model and understand why they cannot be a viable dark
matter condensate. First, electrically charged leptons would give rise to
electrically charged Cooper pairs, so we do not consider them (moreover they
would modify the electroweak vacuum which is tightly constrained by EW
precision measurements). Therefore, with SM fermions, we have two
possibilities: quarks and neutrinos. We comment on known results on
condensates of these fermions in the following sections.

#### SM quark condensate:

Color superconductivity [49, 50, 51, 52, 53, 54] is a well-studied phenomenon
in the context of QCD at high densities. Quarks are in the fundamental of
$\text{SU}(3)_{\text{c}}$ gauge group, so a $\langle qq\rangle$ condensate
could necessarily be in a color antitriplet or a sextuplet because the tensor
product of the color group fundamental representations splits as
$\mathbf{3}\otimes\mathbf{3}=\bar{\mathbf{3}}\oplus\mathbf{6}$ [55]. Due to
asymptotic freedom [56, 57], at high densities we can consider QCD at weak
coupling [58], and single gluon exchanges generate an attractive potential
between quarks in the $\bar{\mathbf{3}}$ channel (see [37, 59] and the
discussion below). However, the specific combination of quarks in the pair
will also depend on the flavor. From the phase diagram of QCD with three
flavors [37], we expect that as we increase the chemical potential $\mu$ (at
low enough temperatures), first there is a two-flavor superconducting (2SC)
phase followed by a 3-color-flavor locking (CFL) superconducting phase as
$\mu$ gets larger than the strange quark mass [37, 59]. However, even without
increasing the chemical potential further, we already encounter a problem: the
gap in those phases will be charged under the electroweak gauge group and its
excitations will carry electric charges [60] (remarkably, in the CFL where
chiral symmetry is broken, the number of Goldstone modes are the same as in
the chiral vacuum at $\mu=0$ albeit with different masses and electric charges
[38]). If the superconductor state is space-filling111There are proposals [17,
90] that assume the state to be confined in a spatial region (“quark
clusters”). In this approach, the cluster total charge today reflects the mean
charge density of the cosmic fluid at the time of its formation. So, quark
clusters might have an overall small charge., DM would have a (large) electric
charge and thus contribute to the electroweak vacuum, changing the $W$ and $Z$
masses and messing up the electroweak precision measurements. Thus, even
without analyzing whether such pairings can happen in the early universe,
there are difficulties for a quark superconductor to play the role of dark
matter. Moreover, at finite temperature, we need a non-vanishing density of
quarks to have a gap on the Fermi surface which would be a phenomenological
catastrophe.

#### SM neutrino condensate:

At first sight, Standard Model neutrinos are promising candidates since they
are electrically neutral and do not couple with gluons. Hence,
$\langle\nu\nu\rangle$ would imply that the gap only couples with the $W$ and
$Z$ bosons. The challenge here would be to find an attractive channel in the
$\nu-\nu$ SM interactions and to explain why $\mu/T$ is large enough to allow
for the superconductor phase. Such a channel has been previously explored [62,
32] but the thermal distribution of SM neutrinos coupled with their low
chemical potential precludes the formation of any condensate at early times.

Since we wish to keep the SM bosonic content intact, let us discuss what
attractive channels can be obtained with the SM interactions alone. As we
mentioned, in color superconductivity the main attractive interaction that
gives origin to the BCS instability is the single-gluon exchange between
quarks. For quarks close to the Fermi surface at large chemical potentials,
this interaction is described by weakly coupled Yang-Mills gauge theory [58].
Motivated by this, in this section, we want to find attractive channels among
different fermionic states in the Standard Model. To do so, we recall a
generic fermion-fermion interaction via a gauge boson exchange in an SU$(N)$
gauge theory with Dirac fermions in the fundamental of the gauge group. The
amplitude for
$\psi_{i}(p_{1})+\psi_{k}(p_{3})\to\psi_{j}(p_{2})+\psi_{\ell}(p_{4})$ is
given by

$\displaystyle i\mathcal{M}$
$\displaystyle=(-g)\bar{u}_{j}(p_{2})\gamma^{\mu}T^{a}_{ji}u_{i}(p_{1})$
$\displaystyle\times\frac{-i\left(\eta_{\mu\nu}+\frac{(p_{1}-p_{2})_{\mu}(p_{1}-p_{2})_{\nu}}{M^{2}}\right)}{(p_{1}-p_{2})^{2}-M^{2}}(-g)\bar{u}_{\ell}(p_{4})\gamma^{\nu}T^{a}_{\ell
k}u_{k}(p_{3}),$ (18)

where $g$ is the gauge coupling, $u(p)$ satisfies $(i\not{p}+m)u(p)=0$, and we
consider a massive gauge boson (note that the second term in the gauge boson
propagator does not contribute to the amplitude). The Abelian case can be
easily obtained in the limit $N\to 1$ and replacing $T^{a}T^{a}\to 1$, which
gives the amplitude for the repulsive interaction between two electron-like
fermions. This limit shows that any negative contribution from the product
$T^{a}_{ji}T^{a}_{lk}$ will correspond to an attractive interaction. With the
normalization $\text{Tr}(T^{a}T^{b})=2\delta^{ab}$, the identity
$2T^{a}_{ji}T^{a}_{\ell
k}=\delta_{jk}\delta_{i\ell}-1/N\delta_{ji}\delta_{\ell k}$ implies

$\displaystyle T^{a}_{ji}T^{a}_{\ell k}$
$\displaystyle=T^{a}_{j(i}T^{a}_{|\ell|k)}+T^{a}_{j[i}T^{a}_{|\ell|k]}$
$\displaystyle=-\frac{N+1}{4N}\left(\delta_{ji}\delta_{k\ell}-\delta_{jk}\delta_{i\ell}\right)+\frac{N-1}{4N}\left(\delta_{jk}\delta_{i\ell}+\delta_{ji}\delta_{k\ell}\right),$
(19)

and so the color-antisymmetric channel $[ik]$ is attractive [37, 59]. If we
had many fermionic flavors, then the anticommutative nature of the fermions
would also force the attractive channel to be antisymmetric in the flavor
space as well. One can also have a Higgs-Yukawa interaction between the
vector-like fermions, which includes attractive chiral and non-chiral channels
[32, 33, 34].

From the discussion above, we find that the $\nu$ – $\nu$ interaction via $Z$
boson exchange is repulsive. Neutrino condensation via Higgs attractive
exchange has been explored in [32] after assuming a right-handed component,
and in [62] it was shown that there is no attractive channel for neutrino
pairing via SM gauge boson exchanges, while the $Z$ exchange may give rise to
a neutron-neutrino condensate in [63]. From the previous discussion about the
quark sector, we conclude that there cannot be an electrically neutral bound
state made entirely of SM fermions paired via single $\text{SU}(3)_{\rm
c}\times\text{SU}(2)_{\rm L}\times\text{U}(1)_{Y}$-gauge boson interactions.
We can instead consider adding extra fermions to the SM but keeping its gauge
structure intact. The following two well-motivated dark matter candidates that
are minimal extensions will be considered in this work:

#### Vector-like quark condensate:

A minimal non-chiral extension would be to consider vector-like quarks (VLQs)
neutral under the electroweak sector. Vector-like quarks, so named because
they are left-right symmetric with respect to the gauge group, as opposed to
the chiral structure of the SM quarks, are a well-studied and motivated
anomaly-free extension to the matter content of the Standard Model (see [64]
and references therein). VLQs have the benefit that their vector-like
structure allows for the existence of a VLQ mass scale which can be set
independently from the electroweak scale. Normal treatments usually set
$v_{\rm EW}<m_{\rm VLQ}\sim\mathcal{O}(1\text{ TeV})$ to explain the non-
observation of VLQs at the LHC to date [65, 66]. This freedom can be exploited
in other ways, perhaps by setting it to a small value that allows the VLQs to
condense under single-gluon exchange. The Higgs excitations of such a
condensate could then play the role of a DM candidate.

#### Higgs-Induced Right-handed Neutrino condensate:

Right-handed neutrinos are a natural extension to the Standard Model, not
least because their inclusion would restore the symmetry between quarks and
leptons [67, 68]. The inclusion of these fermions would enable a Yukawa
coupling between the left- and right-handed neutrinos, which provides the
necessary attractive channel. Furthermore, since right-handed neutrinos are
sterile relative to the rest of the SM, their cosmological evolution could
decouple from radiation and they would remain in a non-thermal distribution
that may preserve an appreciable chemical potential.

##  Appendix B Dark matter from superconductor states

Here we discuss how DM can be identified as the Higgs mode of the condensate.
We start from a generic effective theory for the interacting fermions and
discuss the particular quark and neutrino scenarios in sections (II.2) and
(II.3). The starting point is a Nambu-Jonas-Lasinio-like fermion Lagrangian
[69, 70] with an attractive four-fermion interaction of the form which is
induced from an attractive channel of fermions.

$\mathcal{L}=-\bar{\psi}\left(\not{\partial}-m_{\psi}-\mu\gamma^{0}\right)\psi+\frac{g^{2}}{M^{2}}\left(\bar{\psi}\Gamma^{A}\psi\right)\left(\bar{\psi}\Gamma_{A}\psi\right).$
(20)

Introducing Hubbard-Stratonovich fields $\Phi_{\alpha}$ [71, 72], we can write
the theory above as one without a four-fermion interaction

$\displaystyle\mathcal{L}_{\Phi}$
$\displaystyle=-\bar{\psi}\left(\not{\partial}-m_{\psi}-\mu\gamma^{0}\right)\psi$
$\displaystyle+\Phi_{\alpha}\left(\psi^{\dagger}\mathcal{O}^{\alpha}\psi^{*}\right)+\Phi_{\alpha}^{*}\left(\psi^{\text{T}}{\mathcal{O}^{\alpha}}^{*}\psi\right)-\frac{M^{2}}{g^{2}}\Phi_{\alpha}{\Phi^{\alpha}}^{*}.$
(21)

The action corresponding to $\mathcal{L}$ can be recovered after using the
equations of motion for $\Phi_{\alpha}$ and $\Phi^{*}_{\alpha}$. Since we wish
to apply the following results for different attractive channels in different
theories, we shall write the formal dependence on $\mathcal{O}_{\alpha}$ and
the index $\alpha$ explicitly.

In the Nambu-Gorkov basis [73, 74]

$\Psi=\begin{pmatrix}\psi\\\ \psi^{*}\end{pmatrix},$ (22)

we have

$\mathcal{L}_{\Phi}=-\frac{1}{2}\bar{\Psi}S^{-1}_{F}(\Phi_{\alpha},m_{\psi},\mu)\Psi-\frac{M^{2}}{g^{2}}\Phi_{\alpha}{\Phi^{\alpha}}^{*},$
(23)

where

$\displaystyle S_{F}^{-1}(\Phi_{\alpha},m_{\psi},\mu)$
$\displaystyle=\begin{pmatrix}\gamma^{\mu}\partial_{\mu}-m_{\psi}-\mu\gamma^{0}&\Phi_{\alpha}i\gamma^{0}\mathcal{O}^{\alpha}\\\
\Phi_{\alpha}^{*}i\gamma^{0}{\mathcal{O}^{\alpha}}^{*}&i\gamma^{0}{\gamma^{\mu}}^{\text{T}}i\gamma^{0}\partial_{\mu}-m_{\psi}-\mu\gamma^{0}\end{pmatrix}$
(24)

In the mean-field approximation, an effective action for $\Phi$ is obtained
after integrating out the fermionic fields,

$\displaystyle e^{iS_{\text{eff}}(\Phi_{\alpha},\Phi_{\alpha}^{*})}$
$\displaystyle=\int\left[\mathrm{d}\bar{\Psi}\mathrm{d}\Psi\right]e^{-i\int\mathrm{d}^{4}x\left(\frac{1}{2}\bar{\Psi}S^{-1}_{F}(\Phi_{\alpha},m_{\psi},\mu)\Psi+\frac{M^{2}}{g^{2}}\Phi_{\alpha}{\Phi^{\alpha}}^{*}\right)}$
(25)

and the gap equation is

$\frac{\delta S_{\text{eff}}}{\delta\Phi_{\alpha}}=0,$ (26)

and its solution fixes the value of
$\Phi_{\alpha}=\bar{\Phi}_{\alpha}(\mu,m_{\psi},M^{2}/g^{2})$ in terms of the
parameters of the original Lagrangian. The gap $\bar{\Phi}_{\alpha}$
determines the value of the condensate:

$\langle\psi^{\text{T}}\mathcal{O}^{\alpha}\psi\rangle\propto\bar{\Phi}^{\alpha}.$
(27)

One core fact of BCS theory is that there is always a solution to the gap
equation whenever the interaction term corresponds to an attractive channel
[26, 25]. Moreover, any repulsive channel becomes irrelevant to the
computation of the gap. This can be more easily seen from a Wilsonian
perspective, in which modes away from the Fermi surface are integrated out,
and repulsive interaction terms become irrelevant at those energies [26, 25].

In this paper, we propose to identify dark matter as the Higgs mode
$\rho_{\alpha}(x^{\mu})$ of the gap. To get the microscopic DM parameters,
such as mass and self-interaction couplings, we will then need an effective
theory for fluctuations around $\bar{\Phi}_{\alpha}$:

$\Phi_{\alpha}(x^{\mu})=\left[\bar{\Phi}_{\alpha}+\rho_{\alpha}(x^{\mu})\right]e^{i\phi(x^{\mu})},$
(28)

where $\phi$ is the Goldstone mode of the condensate. This mode will be
associated to a symmetry breaking by (27), which varies for the neutrino and
vector-like quark scenarios. The effective field theory for the fluctuation
modes was studied in [27] by perturbing around the ground state of the
resulting effective action in (B). In terms of the canonically normalized
field $\varphi=\left(\sqrt{3}/2\pi\right)\left(\mu/\Phi\right)\rho$, the
Lagrangian for the Higgs mode is then [27, 4]

$\mathcal{L}=-\frac{1}{2}(\partial\varphi_{\alpha})^{2}-16\bar{\Phi}^{2}_{\alpha}\varphi_{\alpha}^{2}-\sum_{n\geq
2}\tilde{c}_{n}\frac{\bar{\Phi}_{\alpha}^{2}}{\mu^{n-2}}\varphi_{\alpha}^{n},$
(29)

where $\tilde{c}_{n}$ are order unity constants. From the Lagrangian above one
identifies the DM mass $m$ and quartic coupling $\lambda$ as

$m=4\sqrt{2}\bar{\Phi},\quad\lambda\simeq\frac{\bar{\Phi}^{2}}{\mu^{2}}.$ (30)

In both scenarios explored in section II, we introduced extra fermions on top
of the SM leptons and quarks (however, the interaction giving rise to the
four-fermion term in (20) still comes from the SM bosons). We shall assume
these fermions were produced in the early universe, possibly during lepto- or
baryogenesis. For generality, we forgo the details of production but rather
comment on constraints on the temperature and chemical potential of the
fermionic distribution. To get a Fermi surface, the fermions should be in the
degenerate limit, with chemical potential much larger than the temperature,
$\mu/T\gg 1$. Motivated by the Sakharov conditions [75] and electroweak
baryogenesis [76, 77, 78, 79], these fermions may have been produced out of
equilibrium and we can neglect the temperature corrections at leading order.

The energy density of the fermion fluid will be constrained by cosmological
history. In the standard cosmic history context, adding an extra fluid before
BBN corresponds to a change in the effective number of relativistic species,
$N_{\text{eff}}$, which is set by the relativistic particle content of the SM.
A shift $\Delta N_{\text{eff}}$ of this value is constrained by cosmological
observations [8, 28]:

$\Delta
N_{\text{eff}}<0.17\quad(\text{CMB}+\text{BAO})\;\text{at}\;95\%\;\text{CL}.$
(31)

We now use this constraint to find a maximum value for the Fermi energy of the
degenerate gas. In the limit $\mu/T\gg 1$, the number and energy densities for
a Fermi (ideal) gas are

$\displaystyle n_{\psi}$
$\displaystyle=\frac{g}{6\pi^{2}}\left(\mu^{2}-m_{\psi}^{2}\right)^{3/2},$
(32) $\displaystyle\rho_{\psi}$
$\displaystyle=\frac{g}{16\pi^{2}}\left[\mu\left(2\mu^{2}-m_{\psi}^{2}\right)\sqrt{\mu^{2}-m_{\psi}^{2}}-m_{\psi}^{4}\text{arcosh}\left(\frac{\mu}{m_{\psi}}\right)\right],$
(33)

where $g$ is the number of degrees of freedom of each fermionic particle; each
extra fermion flavor contributes a value of 2 to $g$. Note that for $\mu\gg
m_{\psi}$, we have $n_{\psi}\propto\mu^{3}$ and $\rho_{\psi}\propto\mu^{4}$.

In an Friedmann-Lemaître-Robertson-Walker (FLRW) background, the chemical
potential dependence of the scale factor can be obtained from particle number
conservation. Using the conservation of the cosmic fluid’s entropy to write
scale factor dependence in terms of temperature, we have (see [80] for
details)

$\mu(T)=m_{\psi}\sqrt{1+\left(\frac{g_{*S}(T)}{g_{*S}(T_{\psi})}\right)^{2/3}\left(\frac{\mu(T_{\psi})^{2}-m_{\psi}^{2}}{T_{\psi}^{2}}\right)\left(\frac{T}{m_{\psi}}\right)^{2}},$
(34)

where $g_{*S}(T)$ is the effective number of degrees of freedom in entropy at
temperature $T$ and $T_{\psi}$ is the temperature of the cosmic fluid when the
non-thermal fermions were created. The evolution of the fermionic energy
density is given by $\rho_{\psi}(T)=\rho_{\psi}(\mu(T))$. At early times, when
$T\gg m_{\psi}$, $\rho_{\psi}$ redshifts as radiation,

$\rho_{\psi}(T)\simeq\frac{g}{8\pi^{2}}\left(\frac{\mu(T_{\psi})}{T_{\psi}}\right)^{4}\left(\frac{g_{*S}(T)}{g_{*S}(T_{\psi})}\right)^{4/3}T^{4},$
(35)

where we assumed $\mu(T_{\psi})\gg m_{\psi}$. After the temperature of the
cosmic radiation drops below $m_{\psi}$, $\rho_{\psi}\propto T^{3}$ and we
have a dust-like equation of state.

The shift in $N_{\text{eff}}$ during BBN is then [81]

$\Delta
N_{\text{eff}}=\frac{8}{7}\frac{30g}{16\pi^{4}}\left(\frac{\mu(T_{\psi})}{T_{\psi}}\right)^{4}\left(\frac{g_{*S}(T_{\text{BBN}})}{g_{*S}(T_{\psi})}\right)^{4/3}.$
(36)

As long as $\Delta N_{\text{eff}}$ is less than the upper bound in (31), the
mild cosmic history modification due to the presence of the fermion fluid
satisfies the current cosmological constraints. Hence, (31) implies that
$\mu(T_{\psi})$ should be at most of order $T_{\psi}$:

$\frac{\mu(T_{\psi})}{T_{\psi}}\lesssim\frac{1.22}{g^{1/4}}.$ (37)

The DM self-interaction cross-section $\sigma$ is constrained to satisfy
$\sigma/m<0.1\;\text{cm}^{2}/\text{g}$ (see [29] and references therein).
Using the expression for the mass and quartic coupling (30), we have

$\frac{m}{\text{eV}}\lesssim
10^{-19}\left(\frac{\mu_{0}}{\text{eV}}\right)^{4}$ (38)

where $\mu_{0}$ represents the Fermi energy today. There is a range of
possible DM masses depending on the chemical potential right after the fermion
production. From (34), we have

$\displaystyle\frac{m}{\text{eV}}$ $\displaystyle\lesssim
10^{-19}\left(\frac{m_{\psi}}{\text{eV}}\right)^{4}\times$
$\displaystyle\left[1+\left(\frac{g_{*S}(T_{0})}{g_{*S}(T_{\psi})}\right)^{2/3}\left(\frac{\mu(T_{\psi})^{2}-m_{\psi}^{2}}{T_{\psi}^{2}}\right)\left(\frac{T_{0}}{m_{\psi}}\right)^{2}\right]^{2}$
(39)

Using the constraint (37) with $\mu(T_{\psi})\gg m_{\psi}$, we get

$\frac{m}{\text{eV}}\lesssim
10^{-19}\left(\frac{m_{\psi}}{\text{eV}}\right)^{4}\left[1+\frac{1.22}{g^{1/4}}\left(\frac{g_{*S}(T_{0})}{g_{*S}(T_{\psi})}\right)^{2/3}\left(\frac{T_{0}}{m_{\psi}}\right)^{2}\right]^{2},$
(40)

which for $T_{0}\ll m_{\psi}$ gives

$\frac{m}{\text{eV}}\lesssim
10^{-19}\left(\frac{m_{\psi}}{\text{eV}}\right)^{4}.$ (41)

Hence, for a wide range of fermion masses, this model satisfies the self-
interaction constraints provided DM is ultralight [82, 83] and consequently
the DM phenomenology resembles the one for ultralight axions (see [84, 85, 86]
and references therein). The neutrino scenario has a smaller mass,
$m_{\psi}\sim 0.1\;\text{eV}$, compared to the vector-like quark case in which
$m_{\psi}>10^{13}\;\text{eV}$ [65, 66]. However, from the specific form of the
gap for each case, the DM mass can take on a wider range of values,
corresponding to anything from ultralight DM to WIMPs [87] (and references
therein), all depending primarily on the chemical potential (for the neutrino
case) and the gauge coupling (for the VLQs case), as we see in section II. The
ultralight DM region is phenomenologically interesting because it would
reproduce many aspects of superfluid dark matter (see e.g. [88] and references
therein).

It also worth commenting on the DM density. In this model the DM particle
consists of the Higgs mode excitations of the background superconducting
state. The density of such modes is labelled by $\rho_{h}$; the density of the
background fermions comprising this state is $\rho_{\psi}$ as given in (32).
The total dark matter density will thus be a combination of these two
$\rho_{c}\sim\rho_{\psi}+\rho_{h}$. This expression will be dominated by
$\rho_{\psi}$ so we can expect $\rho_{c}\sim\rho_{\psi}$. Thus any bounds on
$\rho_{\psi}$ from cosmic evolution will translate into bounds on $\rho_{c}$
and thus to $\Omega_{c}$.

We can estimate which values of $\mu_{\psi}$ can explain the coincidence
$\Omega_{c}\sim 5\Omega_{b}$ after assuming that the number density of
fermions right after they are created $n_{\psi}(T_{\psi})$ is of the same
order of the net baryon number density $n_{b}=\eta n_{\gamma}$ at that time,
where $\eta$ is the baryon-to-photon ratio and $n_{\gamma}$ is the photon
number density. The present baryonic density parameter is related to
$\eta(T_{0})$ as

$\Omega_{b}h^{2}\simeq 3.66\times 10^{7}\eta,$ (42)

and BBN an CMB data implies $\eta(T_{0})\simeq 6\times 10^{-10}$ [8, 28, 89].
So, the baryonic density can be explained as long as we reproduce the observed
value for $\eta(T_{0})$. From

$n_{b}(T_{\psi})=\xi_{\psi}(T_{\psi})n_{\psi}(T_{\psi}),$ (43)

and the expression for the fermionic number density in (32), we get

$\left(\frac{\mu_{\psi}^{2}-m_{\psi}^{2}}{T^{2}_{\psi}}\right)^{3/2}=\frac{12\zeta(3)}{g}\frac{\eta(T_{\psi})}{\xi_{\psi}}.$
(44)

From

$\eta=\frac{n_{\text{B}}}{n_{\gamma}}=\frac{\pi^{4}}{45\zeta(3)}g_{*S}(T)\frac{n_{\text{B}}}{s},$
(45)

where $s$ is the radiation entropy density, we have

$\left(\frac{\mu_{\psi}^{2}-m_{\psi}^{2}}{T^{2}_{\psi}}\right)^{3/2}=\frac{12\zeta(3)}{g}\frac{g_{*S}(T_{\psi})}{g_{*S}(T_{0})}\frac{\eta(T_{0})}{\xi_{\psi}}.$
(46)

For large $\mu_{\psi}$ and $g_{*S}(T_{\psi})\sim\mathcal{O}(10^{2})$,

$\frac{\mu_{\psi}}{T_{\psi}}\simeq\frac{1}{(g\xi_{\psi})^{1/3}}6\times
10^{-3}.$ (47)

As $\xi_{\psi},\,g\sim\mathcal{O}(1)$ we conclude that the relation between
the baryonic and cold matter densities is reproduced provided $\mu_{\psi}$ is
around 3 orders of magnitude smaller than the cosmic radiation temperature
when the fermions are created. Note that (47) and (37) are compatible; hence,
under the assumptions of this section, a degenerate gas of fermions in the
early universe can indeed explain the $\Omega_{c}\sim 5\Omega_{b}$ coincidence
today.

Since the particulars, from the form of the gap to the allowed chemical
potentials and associated densities, are different for the neutrino and
vector-like quark scenarios, the consistency of the values for $\Phi$ and $m$
individually for each fermion case are covered in the body of the text.

##  Appendix C Fierz rearrangement

We want to express the bilinear squared $(\bar{\psi}\psi)^{2}$ as a product of
$\psi^{\dagger}\psi^{\dagger}$ and $\psi\psi$ to make particle-particle
pairing in the mean field approximation. We have

$(\bar{\psi}\psi)^{2}=(\psi^{\dagger}i\gamma^{0}\psi)(\psi^{\dagger}i\gamma^{0}\psi)=\sum_{A,B}^{16}\xi_{AB}(\psi^{\dagger}\Theta^{A}\psi^{*})(\psi^{\text{T}}\bar{\Theta}^{B}\psi),$
(48)

where $\Theta^{A}$ are the elements of the Clifford algebra basis and only the
six antisymmetric among those contribute to the summation. Typically, one
works on the basis

$\displaystyle\Gamma^{A}$
$\displaystyle=\\{\mathbb{1},\gamma^{\mu},\gamma^{\mu\nu},\gamma^{\mu\nu\rho},\gamma^{\mu\nu\rho\sigma}\\},$
$\displaystyle\Gamma_{A}$
$\displaystyle=\\{\mathbb{1},\gamma_{\mu},\gamma_{\nu\mu},\gamma_{\rho\nu\mu},\gamma_{\sigma\rho\nu\mu}\\},\quad\sigma>\rho>\nu>\mu,$
(49)

where $\gamma^{\mu_{1}\cdots\mu_{r}}=\gamma^{[\mu_{1}}\dots\gamma^{\mu_{r}]}$.
These matrices have normalization such that

$\text{tr}\left(\Gamma^{A}\Gamma_{B}\right)=4\delta^{A}_{B}.$ (50)

Note that $\\{\Gamma^{A}\\}$ is the basis for the vector times dual-vector
representation. This can be seen from the fact that $\bar{\psi}\Gamma^{A}\chi$
is a scalar. On the other hand, $\Theta^{A}$ ($\bar{\Theta}^{B}$) denotes the
basis elements in the vector-vector (dual-vector times dual-vector)
representation, such that
$\bar{\chi}\Theta^{A}\bar{\psi}^{\text{T}}=(\psi^{\text{T}}\bar{\Theta}^{A}\chi)^{\dagger}$
is a scalar. This relation also implies that
$\bar{\Theta}^{A}=(i\gamma^{0})^{*}{\Theta^{A}}^{\dagger}(i\gamma^{0})^{*}$.
The basis in both representations are related by $\Theta^{A}=\Gamma^{A}C$,
where $C$ is the unitary charge conjugation matrix. Such a matrix enters in
the symmetry property of $\Gamma^{A}$:

${\Gamma^{A}}^{\text{T}}=t_{0}t_{r}C\Gamma^{A}C^{-1},$ (51)

where $t_{r}=\pm 1$ depending on the rank of $\Gamma^{A}$. Choosing
$t_{0}=1=-t_{1}$, we have $t_{2}=-1$ and $t_{3}=1=t_{4}$. In a representation
in which ${\gamma^{\mu}}^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}$
(hermitian representation) we have

${\gamma^{\mu}}^{*}=-t_{0}t_{1}B\gamma^{\mu}B^{-1},\quad B=it_{0}C\gamma^{0}.$
(52)

Using this result, it is straightforward to show that

${\Gamma^{A}}^{\dagger}=t_{1}t_{r}\gamma^{0}\Gamma^{A}\gamma^{0},$ (53)

and hence

$\bar{\Theta}^{A}=-t_{0}t_{r(A)}C^{\dagger}\Gamma^{A}.$ (54)

One can show that, with the choice of basis above, we have the normalization

$\text{tr}\left(\Theta^{A}\bar{\Theta}_{B}\right)=-t_{0}t_{r(B)}\text{tr}\left(\Gamma^{A}\Gamma_{B}\right)=-t_{0}t_{r(B)}4\delta^{A}_{B}.$
(55)

To get the typical particle-antiparticle Fierz rearrangement, we start with
the identity

$\left(\Gamma^{C}\right)_{ij}\left(\Gamma^{D}\right)_{k\ell}=\sum_{A,B}C^{CD}_{AB}\left(\Gamma^{A}\right)_{i\ell}\left(\Gamma^{B}\right)_{kj}$
(56)

and multiply it by $\left(\Gamma_{E}\right)_{jk}\left(\Gamma_{F}\right)_{li}$
to find

$C^{CD}_{EF}=\frac{1}{16}\text{tr}\left(\Gamma^{C}\Gamma_{E}\Gamma^{D}\Gamma_{F}\right).$
(57)

Similarly, to get the particle-particle rearrangement, we start from

$\left(\Gamma^{A}\right)_{ij}\left(\Gamma^{B}\right)_{kl}=\sum_{C,D}D^{AB}_{CD}\left(\Theta^{C}\right)_{il}\left(\bar{\Theta}^{D}\right)_{kj}.$
(58)

To find

$D^{AB}_{FE}=\frac{1}{16}\text{tr}\left(\Gamma^{A}\Theta_{E}{\Gamma^{B}}^{\text{T}}{\bar{\Theta}}_{F}\right).$
(59)

So, from the relation between $\Theta$ and $\Gamma$, we have

$\displaystyle D^{AB}_{FE}$
$\displaystyle=-\frac{1}{16}t_{r(B)}t_{r(F)}\text{tr}\left(\Gamma^{A}\Gamma_{E}\Gamma^{B}\Gamma_{F}\right)$
$\displaystyle=-t_{r(B)}t_{r(F)}C^{AB}_{EF}.$ (60)

A glance at (48) reveals that $\xi_{AB}=-D^{00}_{AB}=-t_{r(F)}C^{00}_{BA}$. We
now compute $C^{00}_{BA}$. Firstly, note that the trace of an odd number of
$\gamma^{\mu}$ matrix vanishes and so will the trace of $\Gamma^{A}$’s with
unmatched rank. Thus $C^{00}_{BA}\propto\delta_{BA}$. To compute the traces in
(57), it is convenient to write the basis as
$\Gamma^{A}=\\{\mathbb{1},\gamma^{\mu},\gamma^{0i},\gamma^{ij},\gamma^{\mu}\gamma_{5},\gamma_{5}\\}$
with $i<j$, where $\gamma_{5}=-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$.
Note that for $\Gamma_{A}$, the rank-2 elements have opposite index ordering
compared to rank-2 $\Gamma^{A}$ elements, otherwise the elements will not be
normalized as in (50). For instance,

$\left.\text{tr}(\gamma^{ij}\gamma_{k\ell})\right|_{\begin{subarray}{c}j>i\\\
k>\ell\end{subarray}}=4\left(\eta^{ij}\eta_{k\ell}-\delta^{i}_{k}\delta^{j}_{\ell}+\delta^{i}_{\ell}\delta^{j}_{k}\right)_{\begin{subarray}{c}j>i\\\
k>\ell\end{subarray}}=4\delta^{i}_{\ell}\delta^{j}_{k}$ (61)

would have an opposite sign if $\ell>k$. With that in mind, we collect the
results for $D^{00}_{AB}$:

$\displaystyle D^{00}_{\mathbb{1}\mathbb{1}}$
$\displaystyle=t_{0}C^{00}_{\mathbb{1}\mathbb{1}}=-\frac{1}{4},$ (62)
$\displaystyle D^{00}_{00}$ $\displaystyle=t_{1}C^{00}_{00}=-\frac{1}{4},$
(63) $\displaystyle D^{00}_{ij}$
$\displaystyle=t_{1}C^{00}_{ij}=-\frac{1}{4}\eta_{ij},$ (64) $\displaystyle
D^{00}_{i0\;j0}$ $\displaystyle=t_{2}C^{00}_{i0\;j0}=-\frac{1}{4}\eta_{ij},$
(65) $\displaystyle D^{00}_{ij\;k\ell}$
$\displaystyle=t_{2}C^{00}_{ij\;kl}=-\frac{1}{4}\eta_{ik}\eta_{j\ell},$ (66)
$\displaystyle D^{00}_{05\;05}$
$\displaystyle=t_{3}C^{00}_{05\;05}=\frac{1}{4},$ (67) $\displaystyle
D^{00}_{i5\;j5}$ $\displaystyle=t_{3}C^{00}_{i5\;j5}=\frac{1}{4}\eta_{ij},$
(68) $\displaystyle D^{00}_{55}$ $\displaystyle=t_{4}C^{00}_{55}=\frac{1}{4}.$
(69)

Using these results into (58), we find

$\displaystyle\left(\gamma^{0}\right)_{ij}\left(\gamma^{0}\right)_{k\ell}$
$\displaystyle=-\frac{1}{4}\left(C\right)_{ik}\left(-C^{\dagger}\right)_{\ell
j}-\frac{1}{4}\left(\gamma^{0}C\right)_{ik}\left(C^{\dagger}\gamma^{0}\right)_{\ell
j}$
$\displaystyle-\frac{1}{4}\eta_{mn}\left(\gamma^{m}C\right)_{ik}\left(C^{\dagger}\gamma^{n}\right)_{\ell
j}$
$\displaystyle-\frac{1}{4}\eta_{mm}\left(\gamma^{0m}C\right)_{ik}\left(C^{\dagger}\gamma^{0n}\right)_{\ell
j}$
$\displaystyle-\frac{1}{4}\eta_{mn}\eta_{pq}\left(\gamma^{mq}C\right)_{ik}\left(C^{\dagger}\gamma^{np}\right)_{\ell
j}+$
$\displaystyle+\frac{1}{4}\left(\gamma^{0}\gamma_{5}C\right)_{ik}\left(-C^{\dagger}\gamma^{0}\gamma_{5}\right)_{\ell
j}$
$\displaystyle+\frac{1}{4}\eta_{mn}\left(\gamma^{m}\gamma_{5}C\right)_{ik}\left(-C^{\dagger}\gamma^{n}\gamma_{5}\right)_{\ell
j}+$
$\displaystyle+\frac{1}{4}\left(\gamma_{5}C\right)_{ik}\left(-C^{\dagger}\gamma_{5}\right)_{\ell
j}.$ (70)

When we multiply this expansion by
$\psi^{\dagger}_{i}\psi_{k}^{\dagger}\psi_{\ell}\psi_{j}$ to get (48), only
the terms with antisymmetric $\Theta^{A}$ will contribute. The symmetry
property of $\Theta^{A}$ depends on its rank:

${\Theta^{A}}^{\text{T}}=\left(\Gamma^{A}C\right)^{\text{T}}=t_{r}C^{2}\Gamma^{A}C=t_{r}C^{2}\Theta^{A}.$
(71)

We shall work concretely in the Weyl representation, where
$C=-i\gamma^{0}\gamma^{2}$. In that case, $C^{2}=-\mathbb{1}$ and so
$\Theta^{A}$ is antisymmetric for ranks such that $t_{r}=1$ which correspond
to $r=\\{0,3,4\\}$. Then we are interested in four terms in the expansion
above:

$\displaystyle\left(\gamma^{0}\right)_{ij}\left(\gamma^{0}\right)_{k\ell}$
$\displaystyle=-\frac{1}{4}\left(C\right)_{ik}\left(-C^{\dagger}\right)_{\ell
j}$
$\displaystyle-\frac{1}{4}\left(\gamma^{0}\gamma_{5}C\right)_{ik}\left(-C^{\dagger}\gamma^{0}\gamma_{5}\right)_{\ell
j}+$
$\displaystyle+\frac{1}{4}\eta_{mn}\left(\gamma^{m}\gamma_{5}C\right)_{ik}\left(-C^{\dagger}\gamma^{n}\gamma_{5}\right)_{\ell
j}$
$\displaystyle+\frac{1}{4}\left(\gamma_{5}C\right)_{ik}\left(-C^{\dagger}\gamma_{5}\right)_{\ell
j}+\cdots\;.$ (72)

Explicitly in terms of $\gamma$ matrices, we have

$\displaystyle\left(i\gamma^{0}\right)_{ij}\left(i\gamma^{0}\right)_{k\ell}$
$\displaystyle=-\frac{1}{4}\left(\gamma^{0}\gamma^{2}\right)_{ik}\left(\gamma^{0}\gamma^{2}\right)_{\ell
j}$
$\displaystyle-\frac{1}{4}\left(\gamma^{2}\gamma_{5}C\right)_{ik}\left(\gamma^{2}\gamma_{5}\right)_{\ell
j}-$
$\displaystyle-\frac{1}{4}\left(\gamma^{3}\right)_{ik}\left(\gamma^{3}\right)_{\ell
j}-\frac{1}{4}\left(\gamma^{0}\gamma_{5}\right)_{ik}\left(\gamma^{0}\gamma_{5}\right)_{\ell
j}$
$\displaystyle-\frac{1}{4}\left(\gamma^{1}\right)_{ik}\left(\gamma^{1}\right)_{\ell
j}-$
$\displaystyle-\frac{1}{4}\left(\gamma^{1}\gamma^{3}\right)_{ik}\left(\gamma^{1}\gamma^{3}\right)_{\ell
j}+\cdots\;.$ (73)

Finally, we express the particle-particle Fierz rearrangement in a manifestly
real way as

$\displaystyle(\bar{\psi}\psi)^{2}$
$\displaystyle=\frac{1}{4}\left(\psi^{\dagger}\gamma^{0}\gamma^{2}\psi^{*}\right)\left[\psi^{\text{T}}\left(\gamma^{0}\gamma^{2}\right)^{*}\psi\right]$
$\displaystyle-\frac{1}{4}\left(\psi^{\dagger}\gamma^{2}\gamma_{5}\psi^{*}\right)\left[\psi^{\text{T}}\left(\gamma^{2}\gamma_{5}\right)^{*}\psi\right]+$
$\displaystyle+\frac{1}{4}\left(\psi^{\dagger}\gamma^{3}\psi\right)\left[\psi^{\text{T}}\left(\gamma^{3}\right)^{*}\psi\right]$
$\displaystyle+\frac{1}{4}\left(\psi^{\dagger}\gamma^{0}\gamma_{5}\psi\right)\left[\psi^{\text{T}}\left(\gamma^{0}\gamma_{5}\right)^{*}\psi\right]+$
$\displaystyle+\frac{1}{4}\left(\psi^{\dagger}\gamma^{1}\psi\right)\left[\psi^{\text{T}}\left(\gamma^{1}\right)^{*}\psi\right]$
$\displaystyle-\frac{1}{4}\left(\psi^{\dagger}\gamma^{1}\gamma^{3}\psi\right)\left[\psi^{\text{T}}\left(\gamma^{1}\gamma^{3}\right)^{*}\psi\right]$
$\displaystyle=\frac{1}{4}\sum_{a}\xi_{a}\left(\psi^{\dagger}O^{a}\psi\right)\left[\psi^{\text{T}}\left(O^{a}\right)^{*}\psi\right],$
(74)

with $\xi_{a}=1$ for
$\\{\gamma^{0}\gamma^{2},\,\gamma^{3},\,\gamma^{0}\gamma^{5},\,\gamma^{1}\\}$
and $\xi_{a}=-1$ for $\gamma^{2}\gamma_{5}$ and $\gamma^{1}\gamma^{3}$.

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