# Light and Feebly Interacting Non-Abelian Vector Dark Matter Produced Through
Vector Misalignment

Fatemeh Elahi<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence & Mainz
Institute for Theoretical Physics,
Johannes Gutenberg-Universit t Mainz, 55099 Mainz, Germany  Sara Khatibi
<EMAIL_ADDRESS>Department of Physics, University of Tehran, North
Karegar Ave., Tehran 14395-547, Iran

###### Abstract

In this paper, we examine the evolution of light and feebly interacting non-
abelian dark gauge bosons dark matter in the early universe. We specifically
work on a multi-component dark matter scenario with a $SU(2)_{{}_{R}}$ gauge
symmetry, spontaneously broken by a scalar $\phi$. For sufficiently light
$\lesssim O(\rm{MeV})$ and feebly interacting $(g_{{}_{R}}\lesssim 10^{-10})$
gauge bosons, $W^{3}_{{}_{R}}$ and $W^{\pm}_{{}_{R}}\equiv W_{{}_{R}}^{1}\pm
iW_{{}_{R}}^{2}$ become the dark matter candidates. In this study, the portal
between the dark sector and the standard model sector is provided by the
right-handed electron charged under $SU(2)_{{}_{R}}$. We show that in the
region of the parameter space where $g_{{}_{R}}\sim 10^{-12}$, the dark gauge
boson $W^{3}_{{}_{R}}$ can be produced efficiently via the freeze-in
mechanism, however, this mechanism fails to produce $W^{\pm}_{{}_{R}}$
sufficiently. For smaller dark gauge coupling, the relic density of three dark
matter components can be obtained via the vector misalignment mechanism.
Vector misalignment has already been discussed in the case of dark photon dark
matter. In this study, we extend the arguments to non-abelian gauge bosons.
After discussing the evolution of $W_{{}_{R}}$s in the early universe, we
study the model’s constraints and show the parameter space’s allowed region.

††preprint: MITP-22-023

##  I Introduction

Many observations from inter-galactic to cosmic scales indicate the existence
of Dark Matter (DM) that corresponds to approximately $25\%$ of the energy
budget of the Universe. However, the nature and origin of DM are still
enigmatic. The long sought-after weakly interacting massive particle with
$O(\text{GeV})$ mass scale has been under extreme experimental scrutiny, and
yet, no promising sign has been observed Cui:2017nnn ; Aprile:2018dbl ;
Akerib:2016vxi . Thus, there has been a slow shift of interest to other DM
candidates, especially in the sub-GeV mass regime.

One of the simplest and well-studied models in this regime is the dark photon
dark matter Pospelov:2008jk ; Redondo:2008ec . These models come from
extending the Standard Model (SM) by a gauge $U(1)_{\text{D}}$ that is broken
either through spontaneous symmetry breaking (SSB) or via Stuckleberg
mechanism Kors:2005uz , resulting in a massive dark photon. The main portal to
SM in these studies is the kinetic mixing (e.g, $\epsilon
F^{\prime}_{\mu\nu}F^{\mu\nu}$). In the mass basis, the kinetic mixing term
induces a coupling between dark photon and the electromagnetic current
proportional to $\epsilon$. However, for sufficiently light dark photon
$(m_{A^{\prime}}<2m_{e})$ and small coupling, the dark photon may be long-
lived and can account for the relic abundance of DM.

The story of SM $+\leavevmode\nobreak\ U(1)_{\text{D}}$ is particularly simple
because it does not carry any adjoint index, and thus, the kinetic mixing term
is relevant even at low scales. Extending the SM with a non-abelian symmetry,
on the other hand, has extra complications because the associated gauge bosons
carry an adjoint index. In this case, the kinetic mixing operator between
$SU(N)_{\text{D}}$ and $U(1)_{Y}$ is non-renormalizable and usually ignorable.
The common practice, in such models, is using the Higgs portal or introducing
a vector-like fermion which is a doublet of the new gauge symmetry
Belyaev:2022zjx ; Belyaev:2022shr  or charging some SM particles under
$SU(N)_{\text{D}}$. In this paper, we will use the latter, and charge (a
subset of) right-handed particles under an extra $SU(2)$ (refer to as
$SU(2)_{{}_{R}}$). Since the phenomenology of electrons is relatively the
richest, we consider the right-handed electron to be charged under the
$SU(2)_{{}_{R}}$ as a portal between the dark and the SM sectors.

In this paper, we consider a multi-component DM scenario with a
$SU(2)_{{}_{R}}$ gauge symmetry, spontaneously broken by a scalar $\phi$ which
is a doublet under the $SU(2)_{{}_{R}}$. In contrast to the $U(1)$, in a non-
abelian gauge symmetry, we cannot use the Stuckelberg mechanism for having the
massive gauge bosons and so we should employ the Higgs mechanism. Thus after
the SSB, we have three massive dark gauge bosons, $W_{{}_{R}}^{3}$ and
$W_{{}_{R}}^{\pm}\equiv W_{{}_{R}}^{1}\pm iW_{{}_{R}}^{2}$. We show that in
the region of parameter space in which we are interested (light mass and small
coupling), these three gauge bosons are good DM candidates. Due to the
presence of $\phi$ in the scalar potential, the Higgs portal is inevitable. To
restrain the influence of the Higgs portal, we study the limit where
$v_{\phi}\gg v_{h}$. Since $m_{{}_{W_{{}_{R}}}}\propto g_{{}_{R}}v_{\phi}$ and
we are interested in $m_{{}_{W_{{}_{R}}}}$ in the sub-GeV scale, we are led to
consider very small gauge couplings.

The production of dark gauge bosons with such feeble couplings in the early
universe is non-trivial. The small coupling prevents the dark gauge bosons
from staying in thermal equilibrium. The canonical solutions are either
freeze-in mechanism Hall:2009bx ; Elahi:2014fsa ; Okada:2020evk ;
Delaunay:2020vdb ; Okada:2020cue , inflationary fluctuations Graham:2015rva ;
Alonso-Alvarez:2018tus ; Ema:2019yrd ; Ahmed:2020fhc ; Arvanitaki:2021qlj ,
vector misalignment Nelson:2011sf ; Arias:2012az ; Alonso-Alvarez:2019ixv ;
Nakayama:2019rhg ; Nakayama:2020rka , or the efficient transfer of energy from
an axion (or axion-like particle) to dark gauge boson Agrawal:2018vin ;
Co:2018lka ; Dror:2018pdh . Most of these mechanisms have been studied in
great detail for the case of a dark photon, but have not been employed for the
case of non-abelian dark gauge bosons, as far as we know. In this paper, we
will discuss the region of the parameter space ($g_{{}_{R}}\sim 10^{-12}$)
that freeze-in can explain the relic abundance for $W_{{}_{R}}^{3}$ DM.
However, the freeze-in mechanism cannot produce sufficient relic density for
$W_{{}_{R}}^{\pm}$. For even smaller couplings, we extend the analysis on
vector misalignment to $SU(2)_{{}_{R}}$ dark gauge boson dark matter, and we
show that all three components of dark gauge boson can be produced via vector
misalignment. Similar to dark photon dark matter production via misalignment
mechanism, the density of $W_{{}_{R}}$s depletes like $a^{-2}$ during
inflation. However, there are various solutions to avoid this depletion (e.g.,
adding a non-minimal coupling between $W_{{}_{R}}$s and the Ricci Scalar
Arias:2012az ; Alonso-Alvarez:2019ixv , or introducing coupling between the
Inflaton and $W_{{}_{R}}$s Nakayama:2019rhg ; Nakayama:2020rka ). In the
following sections, we discuss the relative advantages and disadvantages of
some of these solutions. In short, the main phenomenological difference
between these solutions is the constraint on the Hubble scale in the inflation
era ($H_{I}$), which is currently not measured.

The organization of the paper is as follows. In Section II, we explain the
model and the new degrees of freedom. Section III is dedicated to various
means of $W_{{}_{R}}$s production, including freeze-in (Subsection III.1), and
vector misalignment (Subsection III.2). In section IV, we discuss the
phenomenological implication of the model, and the concluding remarks are
presented in section V.

##  II DM Model

We consider a multi-component DM scenario, where the SM symmetry group is
enlarged by a gauge $SU(2)_{{}_{R}}$ symmetry, that is spontaneously broken by
$\phi$ which is a doublet of $SU(2)_{{}_{R}}$, and singlet of SM gauge groups.
The subscript $R$ indicates that (a subset of) right-handed particles are
charged under $SU(2)_{{}_{R}}$. Since the phenomenology of electrons is
relatively the richest, we narrow our attention to the case where only the
right-handed electron is a doublet of $SU(2)_{{}_{R}}$:
$E_{{}_{R}}=\left(e^{\prime}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ e\right)^{T}_{{}_{R}}$. The SM electron is shown by $e$
and the $e^{\prime}$ is a new fermion with the same quantum numbers as the
right-handed electron. We charge the right-handed electron under
$SU(2)_{{}_{R}}$ to introduce a portal between SM and dark sectors. Since
$E_{{}_{R}}$ has $U(1)_{Y}$ charge as well, our model is anomalous. Ref.
Elahi:2019jeo  discusses the solution to the anomaly problem in detail.
Excluding the particles needed in the UV to make the model anomaly free, the
Lagrangian of the model, governing the phenomenology of the model, becomes the
following:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{{}_{\text{SM}}}+\frac{1}{4}W^{a}_{{}_{R}\mu\nu}W^{a\mu\nu}_{{}_{R}}+(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)-\mu_{{}_{\phi}}^{2}\phi^{\dagger}\phi+\lambda_{{}_{\phi}}(\phi^{\dagger}\phi)^{2}$
(1) $\displaystyle+\bar{E_{{}_{R}}}(\imath
D\\!\\!\\!/)E_{{}_{R}}+\left(\frac{y_{e}}{\Lambda_{1}}(\bar{L}H)(\phi^{\dagger}E_{{{}_{R}}})+h.c.\right)$
$\displaystyle+\xi_{H\phi}(H^{\dagger}H)(\phi^{\dagger}\phi)+\epsilon_{{}_{W}}\frac{(\phi^{\dagger}\tau^{a}\phi)}{\Lambda_{2}^{2}}W^{a}_{{{}_{R}}\mu\nu}B^{\mu\nu},$

where $W^{a}_{{{}_{R}}\mu\nu}$ is the field tensor of $SU(2)_{{}_{R}}$ and
$B_{\mu\nu}$ is that of hypercharge. The covariant derivative is defined as
$D_{\mu}=\partial_{\mu}+ig_{{{}_{R}}}\frac{\tau^{a}}{2}W_{{}_{R}\mu}^{a}+iQ_{Y}g_{{}_{Y}}B_{{}_{\mu}}$,
where $g_{{{}_{R}}}$ is the new gauge coupling and $g_{{}_{Y}}$ is the
hypercharge. Note that $\phi$ is not charged under $U(1)_{Y}$, and thus
$Q_{Y}^{\phi}=0$, while $Q_{Y}^{E_{{}_{R}}}=-1$. After $SU(2)_{{}_{R}}$ SSB,
the three dark gauge bosons acquire the same mass,

$m_{{}_{W_{{}_{R}}}}=\frac{g_{{}_{R}}v_{\phi}}{2},$

where the $v_{\phi}$ is the vacuum expectation value (vev) of the $\phi$.
Similarly, the mass of $\phi$ becomes
$m_{\phi}=\sqrt{2\lambda_{\phi}}v_{\phi}$.

|  | $\hskip 18.06749ptSU(2)_{{}_{R}}\hskip 18.06749pt$ |  | $\hskip 18.06749ptU(1)_{X}\hskip 18.06749pt$
---|---|---|---|---
$W_{{}_{R}}$ |  | 3 |  | $0$
$\phi$ |  | 2 |  | $1/2$
$E_{{}_{R}}$ |  | 2 |  | $1/2$

$\xrightarrow{\text{SSB}}$ $\hskip 32.52127ptU(1)_{\text{D}}\hskip 32.52127pt$
$e^{\prime}$ $1$ $W_{{}_{R}}^{\pm}$ $\pm 1$ $W_{{}_{R}}^{3}$ 0

Table 1: The charges of the new particles under the $SU(2)_{{}_{R}}\times
U(1)_{X}\xrightarrow{\text{SSB}}U(1)_{D}$ symmetries. Due to the residual
$U(1)_{D}$, in the limit where $m_{{}_{W_{{}_{R}}}}\ll m_{e^{\prime}}$,
$W^{\pm}_{{}_{R}}$ are stable DM candidates.

In this setup, because $e_{{}_{R}}$ is charged under $SU(2)_{{}_{R}}$, its
Yukawa coupling is modified as the second term in the second line of Eq. 1.
The SM Yukawa coupling is induced, after $\phi$ gains a non-zero vev:
$y_{e}^{\text{SM}}=y_{e}v_{\phi}/\Lambda_{1}$. The mass of $e^{\prime}$, on
the other hand, comes from the fermions in the UV that are added to cancel the
gauge anomaly (for more details, see Ref. Elahi:2019jeo ); thus,
$m_{e^{\prime}}\sim O\left(v_{\phi}\right)\gg m_{e}$. The third line of Eq. 1
shows the Higgs portal, and the non-renormalizable kinetic mixing term between
$SU(2)_{{}_{R}}$ and $U(1)_{Y}$. We are interested in the limit where
$v_{\phi}\gg v_{h}$, so that the mixing angle between the two scalars is close
to 0:

$\alpha\equiv\frac{1}{2}\text{tan}^{-1}\frac{\xi_{H\phi}v_{h}v_{\phi}}{v_{h}^{2}\lambda_{H}-v_{\phi}^{2}\lambda_{\phi}}\simeq
0,$

closing the Higgs portal. One may wonder that in the limit of large
$v_{\phi}$, the kinetic mixing term becomes important. The main consequence of
this term is the mixing between $W_{{}_{R}}^{3}$ and $B$, which induces a new
contribution to $W_{{}_{R}}^{3}ee$ coupling:
$g_{{}_{R}}+\epsilon_{{}_{W}}g_{{}_{Y}}\frac{v_{\phi}^{2}}{\Lambda_{2}^{2}}.$
Given that the “natural” value of $\epsilon_{W}$ is about
$g_{{}_{R}}g_{Y}/(16\pi^{2})<g_{{}_{R}}$, and $v_{\phi}/\Lambda_{2}<1$, we
expect $\epsilon_{{}_{W}}g_{{}_{Y}}\frac{v_{\phi}^{2}}{\Lambda_{2}^{2}}\ll
g_{{}_{R}}$. Hence, for the rest of the paper, we will neglect this term,
unless stated otherwise.

Studying Eq. 1, we notice that there is an accidental global $U(1)_{{}_{X}}$
symmetry with charges shown in Table 1. Since $\phi$ has a non-zero charge
under $U(1)_{{}_{X}}$, once $\phi$ gets a vev, $SU(2)_{{}_{R}}\times
U(1)_{{}_{X}}$ is spontaneously broken to a residual global $U(1)_{D}$. The
charges of particles after SSB follows the relation $(I_{3})_{{}_{R}}+X$,
where the $X$ is $U(1)_{X}$ charge. The existence of this symmetry at the low
energy guarantees the stability of $W^{\pm}_{{}_{R}}$ in the limit where
$m_{{}_{W_{{}_{R}}}}<m_{e^{\prime}}$ (since in this limit the
$W^{\pm}_{{}_{R}}\rightarrow e^{\prime}e$ decay channel is kinematically
closed). Furthermore, for the light mass limit $m_{{}_{W_{{}_{R}}}}<2m_{e}$,
the $W_{{}_{R}}^{3}$ is not stable and can decay to $3\gamma$ or
$\nu\bar{\nu}$ (through a loop of electrons) 111The two-photon decay channel
is forbidden by the Yang theorem Yang:1950rg , however, for sufficiently small
coupling, it can be a long-lived DM candidate. As a result, in this model, we
have two stable DM particles $(W^{\pm}_{{}_{R}})$ and a long-lived DM particle
$(W_{{}_{R}}^{3})$. In the following, we will discuss the production of dark
gauge bosons DM as well as the phenomenological constraints on this model.

##  III DM Production

In this section, we discuss the production of the non-abelian dark gauge
bosons in the early universe and their relic density at present. We are
interested in the regime where dark gauge bosons are light (e.g,
$m_{{}_{W_{{}_{R}}}}\sim(\text{eV- MeV})$) and feebly interacting. The gauge
coupling, in this regime, is small enough that prevents dark gauge bosons from
reaching thermal equilibrium in the early universe. So they cannot be produced
through thermal processes. Instead, different non-thermal production
mechanisms can produce light DM abundance. The freeze-in mechanism is one of
the non-thermal procedures that can produce the dark gauge bosons
Okada:2020evk ; Delaunay:2020vdb ; Okada:2020cue . Moreover, the light dark
gauge bosons can be produced during inflation by the vector misalignment
mechanism Nelson:2011sf ; Arias:2012az ; Alonso-Alvarez:2019ixv ;
Nakayama:2019rhg ; Nakayama:2020rka .

In the following, we discuss how and under what conditions the freeze-in and
vector misalignment mechanisms can produce the correct relic density for the
dark gauge bosons in our non-abelian DM scenario.

###  III.1 Freeze-in

Assuming dark gauge bosons start with negligible abundance at the reheat
Temperature ($T_{\rm{RH}}$), they can slowly get produced through their feeble
interactions with right-handed $e$ and $e^{\prime}$. This mechanism is known
as freeze-in Hall:2009bx .

To produce the $W_{{}_{R}}^{3}$, among the possible processes, the
contribution from $ee\rightarrow\gamma W_{{}_{R}}^{3}$ is dominant
222Interested readers are encouraged to see Appendix A for a detailed
discussion on the contribution of other potential processes.. The
$W^{3}_{{}_{R}}$ density within the freeze-in framework can be obtained from
the Boltzmann equation Hall:2009bx ; Okada:2020evk ; Kolb:1990vq :

$\dot{n}_{W_{{}_{R}}^{3}}+3Hn_{W_{{}_{R}}^{3}}=\frac{T}{256\pi^{6}}\int_{m_{{}_{W_{{}_{R}}}}^{2}}^{\infty}ds\
d\Omega\ p_{ee}\
p_{\gamma_{W_{{}_{R}}^{3}}}|\mathcal{M}|^{2}_{ee\rightarrow\gamma_{W_{{}_{R}}^{3}}}K_{1}(\sqrt{s}/T)/\sqrt{s},$
(2)

where
$p_{ij}\equiv\sqrt{\frac{(s-(m_{i}+m_{j})^{2})(s-(m_{i}-m_{j})^{2})}{4s}}$,
and

$|\mathcal{M}|^{2}_{ee\rightarrow\gamma_{W_{{}_{R}}^{3}}}=\frac{8\pi\alpha
g_{{}_{R}}^{2}(m_{e}^{4}+3m_{{}_{W_{{}_{R}}}}^{2}m_{e}^{2}-2um_{e}^{2}+tu)}{(t-m_{e}^{2})^{2}},$
(3)

with $s,t,u$ being the ordinary Mandelstam variables. After doing the
integrals in Eq. 2 and changing the variable to $Y_{{}_{W_{{}_{R}}^{3}}}\equiv
n_{W_{{}_{R}}^{3}}/s$, we get

$\frac{dY_{{}_{W_{{}_{R}}^{3}}}}{dT}\simeq-\frac{32\pi\alpha
g_{{}_{R}}^{2}}{s(T)H(T)}\left[1+\text{log}\left(\frac{m_{e}}{T}\right)\right]T^{2}m_{{}_{W_{{}_{R}}}}K_{1}\left(\frac{m_{{}_{W_{{}_{R}}}}}{T}\right),$
(4)

in the limit $m_{{}_{W_{{}_{R}}}}<m_{e}\ll E_{\text{cm}}$. The Hubble
parameter and the entropy density of the universe are shown by $H(T)$ and
$s(T)$, respectively, and they are given by:

$H(T)=1.66\sqrt{g_{\star}}\frac{T^{2}}{M_{\mathrm{pl}}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
s(T)=\frac{2\pi^{2}}{45}g_{\star}T^{3}.$

where $g_{\star}=106.75$ is the effective relativistic degrees of freedom and
$M_{\mathrm{pl}}=1.2\times 10^{19}$ GeV is the non-reduced Planck mass. By
integrating Eq. 4, with respect to temperature
$(T\in[m_{{}_{W_{{}_{R}}}},\infty])$, the yield of the $W_{{}_{R}}^{3}$ is
obtained Hall:2009bx ; Kolb:1990vq :

$Y^{0}_{{}_{W^{3}_{{}_{R}}}}\simeq\frac{360\ \alpha\
g_{{}_{R}}^{2}M_{\mathrm{pl}}}{1.66\times
g_{\star}^{3/2}m_{{}_{W_{{}_{R}}}}}\left[0.73+\text{log}\left(\frac{m_{e}}{m_{{}_{W_{{}_{R}}}}}\right)\right].$
(5)

The relic abundance of $W_{{}_{R}}^{3}$ produced from $ee\rightarrow
W_{{}_{R}}^{3}\gamma$ is

$\Omega_{{}_{W_{{}_{R}}^{3}}}h^{2}=\frac{m_{{}_{W_{{}_{R}}}}Y^{0}_{{}_{W^{3}_{{}_{R}}}}s_{0}}{\rho_{c}/h^{2}}\simeq
10^{24}g_{{}_{R}}^{2}\left[0.73+\text{log}\left(\frac{m_{e}}{m_{{}_{W_{{}_{R}}}}}\right)\right],$
(6)

where $s_{0}=2890/\rm{cm}^{3}$ is the entropy density of the present time and
$\rho_{c}/h^{2}=1.05\times 10^{-5}\ \rm{GeV}/\rm{cm}^{3}$ is the critical
density over the square of the reduced Hubble constant. The Planck
collaboration reported $0.119$ value for DM relic density Ade:2015xua . As Eq.
6 shows, the freeze-in mechanism can explain the relic abundance for the
${W_{{}_{R}}^{3}}$, if $g_{{}_{R}}\simeq 10^{-12}$. However, as we have shown
in Appendix A, the freeze-in scenario cannot produce sufficient
$W^{\pm}_{{}_{R}}$ DM in the early universe.

For smaller gauge coupling, we need another non-thermal DM production
mechanism. In the next subsection, we explain how the vector misalignment
mechanism can produce all three components of the dark gauge bosons in our
non-abelian scenario.

###  III.2 Misalignment

The misalignment is known in the context of scalar/pseudo-scalar (like axion
and axion-like particles) Preskill:1982cy ; Abbott:1982af ; Dine:1982ah . In
the early universe when the Hubble parameter, $H$, is much larger than the
mass of a spatially homogeneous scalar, the initial value of the scalar takes
a random non-zero value. At the transition point, when the $H$ is in order of
the mass of the scalar, the scalar field starts to oscillate. In the late
universe when the $H$ is much smaller than the mass of the scalar, the field
oscillates around the minimum of the potential and its energy density is
proportional to $a^{-3}$ ($a$ is the scale factor). Thus, it redshifts like
non-relativistic matter.

Originally, Ref. Nelson:2011sf  showed that the same production mechanism can
apply for an abelian light vector DM in a Friedmann-Robertson-Walker (FRW)
background. However, at early times (during inflation), the energy density of
the vector field is proportional to the $a^{-2}$ and the vector field dilutes
with expansion and ends up with negligible abundance. Therefore, multiple
solutions were proposed to solve this problem, one of them being a new direct
coupling between the gauge boson and the curvature scalar Arias:2012az ;
Alonso-Alvarez:2019ixv .

The works of Ref. Arias:2012az ; Alonso-Alvarez:2019ixv  are in the context of
a dark photon. In this paper, we want to extend this mechanism to non-abelian
gauge bosons. The action of the non-abelian vector field, with non-minimal
coupling to the Ricci scalar, is given by:

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{1}{2}\left(M_{\mathrm{pl}}^{2}+\frac{\kappa}{6}W_{{}_{R}\mu}^{b}W_{{}_{R}}^{b\mu}\right)R-\frac{1}{4}W^{b}_{{}_{R}\mu\nu}W^{b\mu\nu}_{{}_{R}}-\frac{1}{2}m_{{}_{W_{{}_{R}}}}^{2}W_{{}_{R}\mu}^{b}W_{{}_{R}}^{b\mu}\right),$
(7)

where $R$ is the Ricci scalar, and $\kappa$ is the coupling between the gauge
bosons and the Ricci scalar. The adjoint index is shown by $b=1,2,3$, and it
means that the action can be written for all three components of dark gauge
bosons. The last term in the above action is the mass term for the non-abelian
dark gauge bosons that, as explained before, comes from the Higgs mechanism.
The FRW metric is as follows:

$g=\operatorname{diag}\left(-1,a^{2}(t),a^{2}(t),a^{2}(t)\right),$ (8)

where $a(t)$ is the scale factor. As shown in the appendix B, the equations of
motion (EOM) for the homogeneous physical field ($\mathcal{W}^{b}_{i}\equiv
W_{{}_{R}i}^{b}/a$) from the action (Eq.7) can be obtained as follows,

$\ddot{\mathcal{W}^{b}_{i}}+3H\dot{\mathcal{W}^{b}_{i}}+\left(m_{{}_{W_{{}_{R}}}}^{2}+\frac{1-\kappa}{6}R\right)\mathcal{W}^{b}_{i}=0,$
(9)

and the time component $W_{{}_{R}0}^{b}=0$. Therefore, the EOM for the spatial
component of a non-abelian vector field is the same as the harmonic oscillator
with Hubble damping term. In other words, for $\kappa=1$ the EOM of the vector
is the same as the EOM of the scalar with minimal coupling to gravity. This
way, the problem of scaling the dark vector is solved. However, it suffers
from some other issues (e.g., isocurvature bounds and ghost instability in the
theory), which we will discuss in more detail later in this section. Leaving
these issues to a more thorough model building, the usual procedure of the
misalignment mechanism can be applied for the light non-abelian vector case.

From the above equation, we can see the non-minimal coupling to gravity in the
action causes the effective mass in the EOM for the non-abelian vector fields.
According to the time evolution of the Ricci scalar, the effective mass
changes, particularly manifests itself in the inflationary era. Since our non-
abelian gauge bosons acquire the mass from the Higgs mechanism, during
inflation, the bare mass of the vector fields is zero. In the middle of the
radiation-dominant era, the SSB happens and the non-abelian vector fields get
mass. In the appendix C, the EOM for the homogeneous non-abelian vector field
is solved for the inflation and post-inflationary era with details. We find
that the field value at the end of inflation is as follows:

$\mathcal{W}^{b}_{i,e}\simeq\mathcal{W}^{b}_{i,s}\mathrm{e}^{-\frac{1}{2}\beta_{-}N_{tot}},$
(10)

where the $\mathcal{W}^{b}_{i,s}$ is the initial value of the field at the
start of inflation, the $N_{tot}$ indicates the total number of e-folds of
inflation, and $\beta_{-}=3-\sqrt{1+8\kappa}$. After inflation and before the
SSB happens, the solution of the EOM is
$\mathcal{W}^{b}_{i}(t)\simeq\mathcal{W}^{b}_{i,e}$. After the SSB and at the
point where $m_{{}_{W_{{}_{R}}}}\sim\frac{3}{2}H_{\star}$ (where $H_{\star}$
is the Hubble parameter at the DM production time), the evolution of the field
transits to an oscillation mode. Finally at late times when
$m_{{}_{W_{{}_{R}}}}\gg H$, the vector field oscillates around the minimum of
the potential and the solution of the EOM is as follows:

$\mathcal{W}^{b}_{i}(t)=\mathcal{W}^{b}_{i,e}(\frac{a_{\star}}{a})^{\frac{3}{2}}\cos(m_{{}_{W_{{}_{R}}}}(t-t_{\star})),$
(11)

where $a_{\star}$ is the scalar factor at the DM production time
($t_{\star}$). The energy-momentum tensor can be found from the following:

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L})}{\delta
g^{\mu\nu}}.$ (12)

The energy density of the homogeneous non-abelian vector field is as follows
Arias:2012az ; Alonso-Alvarez:2019ixv ; Golovnev:2008cf :

$\rho_{{}_{\mathcal{W}^{b}}}=T_{00}=\frac{1}{2}\left(\dot{\mathcal{W}^{b}_{i}}\dot{\mathcal{W}^{b}_{i}}+m_{{}_{W_{{}_{R}}}}^{2}\mathcal{W}^{b}_{i}\mathcal{W}^{b}_{i}+(1-\kappa)H^{2}\mathcal{W}^{b}_{i}\mathcal{W}^{b}_{i}+2(1-\kappa)H\dot{\mathcal{W}^{b}_{i}}\mathcal{W}^{b}_{i}\right).$
(13)

Since in the late time $H\ll m_{{}_{W_{{}_{R}}}}$, the energy density
approximately becomes,

$\rho_{{}_{\mathcal{W}^{b}}}\simeq\frac{1}{2}m_{{}_{W_{{}_{R}}}}^{2}(\mathcal{W}^{b}_{i,e})^{2}\left(\frac{a_{\star}}{a}\right)^{3}.$
(14)

The energy density of the vector field in this region is inversely
proportional to the $a^{3}$ (space volume element) and thus redshifts like
cold matter. From the conservation of comoving entropy
($S=sa^{3}=\frac{2\pi}{45}g_{\star S}(T)T^{3}a^{3}$) we have,

$\left(\frac{a_{\star}}{a}\right)^{3}=\frac{g_{\star S}(T_{0})}{g_{\star
S}(T_{\star})}\left(\frac{T_{0}}{T_{\star}}\right)^{3}.$ (15)

As a result, the current energy density of the non-abelian vector field
becomes,

$\rho^{0}_{{}_{\mathcal{W}^{b}}}\simeq 6.5\leavevmode\nobreak\
\left(\frac{\mathrm{keV}}{\mathrm{cm}^{3}}\right)\mathcal{F}\left(T_{\star}\right)\sqrt{\frac{m_{{}_{W_{{}_{R}}}}}{\mathrm{eV}}}\left(\frac{\mathcal{W}^{b}_{i,e}}{10^{12}\mathrm{GeV}}\right)^{2},$
(16)

where
$\mathcal{F}\left(T_{\star}\right)\equiv\left(g_{\star}\left(T_{\star}\right)/3.36\right)^{3/4}(3.91/g_{\star
S}(T_{\star}))$.

The current energy density is proportional to the square of the field value at
the end of inflation $(\mathcal{W}^{b}_{i,e})^{2}$ which is obtained from Eq.
10. Since the $\rho_{\mathrm{DM}}=1.25\frac{\mathrm{keV}}{\mathrm{cm}^{3}}$
Planck:2018jri , the relic density of the non-abelian vector can be found as
follows,

$\frac{\Omega_{\mathcal{W}^{b}}}{\Omega_{\mathrm{DM}}}\simeq
5.2\mathcal{F}\left(T_{\star}\right)\sqrt{\frac{m_{{}_{W_{{}_{R}}}}}{\mathrm{eV}}}\left(\frac{\mathcal{W}^{b}_{i,e}}{10^{12}\mathrm{GeV}}\right)^{2}.$
(17)

After considering the homogeneous field, we should care about the fluctuation
analysis. One important constraint in the above scenario comes from the cosmic
microwave background (CMB) limits on the isocurvature fluctuations. During
inflation, adiabatic perturbations are generated. Since the inflaton decays to
other components of the universe during the reheating, these adiabatic
fluctuations are transmitted to these product particles. However in our
scenario, in addition to the inflaton during inflation, the dark gauge bosons
exist. Since inflaton is not responsible for the production of the dark gauge
bosons during inflation, another source of perturbation known as isocurvature
fluctuations are generated. Isocurvature fluctuations may not match other
particle fluctuations and will leave footprints on the CMB. Since no such
fluctuations have been observed yet Planck:2018jri , our scenario is
constrained. However, this constraint is mostly sensitive to $H_{I}$ and not
model parameters. The isocurvature constraint in our model should be very
similar to that of the dark photon model Alonso-Alvarez:2019ixv .

It should be mentioned that the non-minimal coupling scenario suffers from
ghost instability. In the short wavelength, the kinetic term of the
longitudinal mode of the vector field has a wrong sign. Thus, some alternative
approaches have been introduced Nakayama:2019rhg ; Nakayama:2020rka ;
Cline:2003gs ; Sbisa:2014pzo ; Karciauskas:2010as ; Himmetoglu:2008zp ;
Himmetoglu:2009qi . For instance, in Ref. Nakayama:2019rhg , the author
suggests modifying the kinetic term of the gauge boson instead of adding a
non-minimal coupling to the Ricci scalar in the action. Ref. Nakayama:2019rhg
introduces a coupling between the inflaton and gauge bosons in the form of
$f^{2}(\phi)F_{\mu\nu}F^{\mu\nu}$, where $f(\phi)$ is a function of the
inflaton field $\phi$. If $f^{2}(\phi)$ redshifts as $a^{2}$ or $a^{-4}$
during inflation while approaching $f(\phi)\rightarrow 1$ by the end of
inflation, one can get a consistent model for a vector coherent oscillation
where during inflation the vector field has homogeneous condensate and at the
late time, it has a coherent oscillation without any instability problem in
the theory. The same model with the non-abelian vector field coupling to a
scalar field via the gauge kinetic function is also feasible. The important
phenomenological difference between the scenario with non-minimal coupling to
gravity and the one with the modified kinetic term is the constraint coming
from isocurvature perturbation. Future measurements of $H_{I}$ would favor one
of these scenarios. In the following, however, we will be oblivious to the
cosmological UV completion of the model and only study the terrestrial
constraints on the model.

##  IV DM Phenomenology

Light dark non-abelian gauge boson $(W_{{}_{R}}^{3})$ that couples to
electrons has a similar phenomenology to the dark photon, which has been
discussed extensively in the literature. In particular, the following
constraints apply to $W_{{}_{R}}^{3}$.

  1. I.

Light $W_{{}_{R}}^{3}$ can be produced inside hot ($\sim O(20$ MeV)) and dense
stars. If $W_{{}_{R}}^{3}$ is sufficiently weakly coupled, it can escape the
star without re-scattering. Since it carries some energy, this process
contributes to the cooling rate of the star. The bounds from the Sun,
Horizontal Branch (HB), and Red Giants (RG) are shown in green, orange, and
red shaded regions, respectively, in Fig. 1 An:2014twa ; An:2013yfc ;
Redondo:2013lna . If the dark sector’s particle mass is close to the plasma
frequency, the dark sector particle can be produced resonantly and thus leave
a more significant impact on the cooling rate of the star. Thus, these cooling
rates are more sensitive at some particular mass of $W_{{}_{R}}^{3}$.

Figure 1: The current bounds on $W_{{}_{R}}^{3}$ are demonstrated. The shaded
regions are the constraints from stellar cooling: Sun (green), Horizontal
Branch (orange), and Red Giants (red) An:2014twa ; An:2013yfc ;
Redondo:2013lna . The bound from the decay of $W_{{}_{R}}^{3}\rightarrow
3\gamma$ is shown by the blue line An:2014twa . Various XENON1T analyses probe
different regions of parameter: the low electron recoil region is shown by the
dark-cyan line XENON:2020rca , the analysis on S2 signal (as explained in the
text) gives the purple line XENON:2019gfn , and the extension of this analysis
in lighter mass regime provides a constraint shown by the magenta line
XENON:2021myl . The projected bound coming from the absorption of DM in the
super-CDMS experiment is shown by the dashed gray-line Bloch:2016sjj . The
LAMPOST constraint is shown by the dark-yellow line Chiles:2021gxk .

  2. II.

Due to the coupling of dark gauge bosons with electrons, $W_{{}_{R}}^{3}$ can
decay to three photons through a loop of electrons. This process has the
following decay rate Redondo:2008ec ; Pospelov:2008jk ; McDermott:2017qcg ;
Bhoonah:2018gjb :

$\Gamma_{W_{{}_{R}}^{3}\rightarrow\gamma\gamma\gamma}=\frac{17\left(g_{{}_{R}}\right)^{2}\alpha^{3}}{2^{5}3^{6}5^{3}\pi^{2}}\frac{m_{{}_{W_{{}_{R}}}}^{9}}{m_{e}^{8}}.$
(18)

The decay of $W^{3}_{R}$ can alter the ionization history, and thus leave a
footprint in the cosmic microwave background (CMB) data. However, the
constraint becomes faint if $\tau_{W^{3}_{R}\rightarrow 3\gamma}\gtrsim
10^{25}\ s$ Fradette:2014sza ; Poulin:2016anj . Furthermore, one can find the
constraint on the energy injection of $\gamma$-rays coming from the decay of
$W^{3}_{R}\rightarrow 3\gamma$ in the present time Pospelov:2008jk ;
An:2014twa :

$E_{\gamma}\frac{d\phi_{gal}}{dE_{\gamma}}=\frac{\Gamma_{W^{3}_{R}\rightarrow
3\gamma}}{4\pi
m_{{}_{W_{{}_{R}}}}}E_{\gamma}\frac{dN}{dE_{\gamma}}\rho_{sol}R_{sol},$ (19)

where $\rho_{sol}$ is the energy density of $W^{3}_{R}$ the sun’s position,
and $R_{sol}\simeq 8.3\ \text{kpc}$ is the distance between the sun and the
center of the galaxy. A limit from the observation of the diffuse $\gamma$-ray
was estimated in Redondo:2008ec  using the monochromatic photon injection
found in Yuksel:2007dr . Interestingly, the constraints from CMB and the
galactic diffuse $\gamma$-ray are overlapping and thus are shown with a single
blue line in Fig. 1.

  3. III.

When DM enters the XENON1T experiment, it can scatter off of xenon nuclei or
an electron. For the case of nuclei recoil, a scintillation is generated as a
result of the de-excitation of the excited xenon to its ground state. This
scintillation signal is called S1. In the case of electron recoil, one or a
few electrons may be freed via atomic ionization. A series of electric fields
are used to drift the freed electrons from its interaction point into gaseous
xenon, which produces a secondary scintillation signal called S2 (the purple
line in Fig. 1) XENON:2019gfn . The ratio $S1/S2$ is used to distinguish
between low electron recoil (ER) (dark-cyan line in Fig. 1) XENON:2020rca
from nuclear recoil 333The analysis on nuclear recoil is not sensitive to the
region of the parameter of our interest.. The Magenta line in Fig. 1 is a
development on XENON1T-S2 analysis, written as XENON1T-SE on the plot, to
increase the sensitivity to smaller mass region XENON:2021myl . 444It is worth
mentioning that the XENON1T experiment reported an excess of the ER events
below 7 keV with a 3-4 $\sigma$ local significance XENON:2020rca . Similar to
Ref. Alonso-Alvarez:2020cdv ; Okada:2020evk , our model could also explain
this excess at the benchmark of  $m_{{}_{W_{{}_{R}}}}=2.8\leavevmode\nobreak\
\mathrm{keV},\hskip 14.45377ptg_{{{}_{R}}}=2.26\times 10^{-16}.$ However,
further measurements with XENONnT ruled this excess out XENON:2022ltv .

  4. IV.

The absorption of light DM by cold atoms in direct detection can also provide
strong constraints on DM interactions. The absorption cross section is similar
to that of the photon which carries the momentum of $m_{{}_{W_{{}_{R}}}}$, but
with a different coupling to electrons An:2014twa

$\text{Rate per
atom}\simeq\frac{\rho_{DM}}{m_{{}_{W_{{}_{R}}}}}\left(\frac{g_{{}_{R}}}{e}\right)^{2}\sigma_{\gamma}(|\vec{q}|=m_{{}_{W_{{}_{R}}}}),$
(20)

where $\rho_{DM}$ is the local energy density of $W^{3}_{R}$ DM. The dashed-
gray line shows the projected 90% C.L. sensitivities for Super-CDMS using Ge
(20 kg-years) Bloch:2016sjj .

  5. V.

Recently, the LAMPOST ( Light $A^{\prime}$ Multilayer Periodic Optical SNSPD
Target) experiment has searched for a DM (e.g, dark photon, axion, or in our
case dark non-abelian gauge boson) that converts to a photon. This experiment
uses a superconducting nanowire single-photon detector (SNSPD) to detect the
incoming photon. The LAMPOST experiment is sensitive to $O(\rm{eV})$ DM
Chiles:2021gxk . The region of the parameter space that the LAMPOST experiment
can probe is shown by the dark-yellow line in Fig. 1. This bound is sensitive
to the kinetic mixing coupling, where we have assumed
$\epsilon_{W}(v_{\phi}/\Lambda_{2})^{2}\simeq g_{{}_{R}}e/(16\pi^{2})$.

  6. VI.

Since $W_{R}^{3}$ couples to $e_{R}$, it will contribute to electron $g-2$,
through diagram 2. The contribution of $W_{R}^{3}$ is the following
Queiroz:2014zfa ; Leveille:1977rc ; Ayazi:2021dca :

$\Delta a_{e}^{NP}\simeq-\frac{1}{3}\left(\frac{g_{{}_{R}}m_{e}}{\pi
m_{{}_{W_{{}_{R}}}}}\right)^{2}\left(1+O\left(\frac{m_{e}}{m_{e^{\prime}}}\right)\right),$
(21)

Where the contribution of $W_{{}_{R}}^{\pm}$ is $O(m_{e}/m_{e^{\prime}})$
suppressed. The new physics contribution to $(g-2)_{e}$ has to $\Delta
a_{e}^{NP}\lesssim 10^{-13}$ Parker:2018vye ; Morel:2020dww . Since we are
considering very weakly coupled (e.g, $g_{{}_{R}}<10^{-12}$) and $O(\rm{eV-
MeV})$ DM, this constraint does not appear on the constraint plot.

Figure 2: The Feynman diagram contributes to $(g-2)_{e}$ that involves
$W_{{}_{R}}$.

The aforementioned constraints are due to the similarities between
$W_{{}_{R}}^{3}$ and dark photon. However, due to the non-abelian nature of
$SU(2)_{{}_{R}}$, we need to consider the constraint coming from the self-
interaction (SI) between dark gauge bosons. The SI between DM is under great
debate. On the one hand, large self-interaction can explain the small scale
structure problems that are associated with CCDM555collision-less cold dark
matter(Ref. Tulin:2017ara , and references therein) . On the other hand, large
self interaction of DM (e.g, $\sigma/m_{\text{DM}}>0.47\
\text{cm}^{2}/\text{g}=4.59\times 10^{3}\,{\mathrm{GeV}}^{-3}$ at
$v_{\text{DM}}\simeq 10^{-3}$) is excluded by the observation of merging
clusters Harvey:2015hha ; Kaplinghat:2015aga . The elastic self-interaction
between dark gauge bosons is the following:

$\sigma_{(W_{R}W_{R}\rightarrow W_{R}W_{R})}=\frac{1}{2}\int
dt\frac{t}{2\left(\frac{s}{2}-m_{{}_{W_{{}_{R}}}}^{2}\right)^{2}}\frac{1}{64\pi
s}|\mathcal{M}|^{2},$ (22)

where $|\mathcal{M}|^{2}$ is the squared matrix element of the $W_{R}$ self-
interaction, and $s,t$ are the the ordinary Mandelstam variables. In the non-
relativistic limit, this cross section is

$\displaystyle\sigma_{(W_{R}W_{R}\rightarrow W_{R}W_{R})}$
$\displaystyle=\frac{g_{{}_{R}}^{4}(320m_{{}_{W_{{}_{R}}}}^{8}-1200m_{{}_{W_{{}_{R}}}}^{6}m_{\phi}^{2}+8801m_{{}_{W_{{}_{R}}}}^{4}m_{\phi}^{4}-4208m_{{}_{W_{{}_{R}}}}^{2}m_{\phi}^{6}+520m_{\phi}^{8})}{1152\pi
m_{{}_{W_{{}_{R}}}}^{2}m_{\phi}^{4}(m_{\phi}^{2}-4m_{{}_{W_{{}_{R}}}}^{2})^{2}}$
$\displaystyle\simeq\frac{65}{144\pi}\frac{g_{{}_{R}}^{4}}{m_{{}_{W_{{}_{R}}}}^{2}}\hskip
36.135pt\text{for}\hskip 36.135ptm_{\phi}\gg m_{{}_{W_{{}_{R}}}}.$ (23)

Therefore, for $m_{{}_{W_{{}_{R}}}}\lesssim 10^{7}\text{eV}\times
g_{{}_{R}}^{4/3}$, merging cluster does not impose any constraint.

One may also wonder about the phenomenology of $W_{{}_{R}}$s at colliders.
With high luminosity LHC (HL-LHC), the minimum cross section that can be
probed is roughly $\sigma^{\min}_{NP}\simeq O(1)\ \text{ab}$, since the
integrated luminosity is expected to be about $3\ \text{ab}^{-1}$ CMS:2013xfa
. The cross section for producing $W_{{}_{R}}$ at colliders is proportional to
$\left(g_{{}_{R}}\right)^{2}$. For $g_{{}_{R}}<10^{-12}$, the production cross
section will be orders of magnitude smaller than $\sigma^{\min}_{NP}$. Thus,
$W_{{}_{R}}$ is unlikely to be produced at the upgraded LHC.

###  IV.1 $e^{\prime}$ Phenomenology

The production of the $e^{\prime}$ at colliders is proportional to
$g_{Y}^{2}\simeq O(1)$. Therefore, depending on the mass of $e^{\prime}$, its
production at LHC or FCC is plausible FCC:2018evy . The pair production of
$e^{\prime}$ can happen through the following s-channel process at the LHC
where both of $e^{\prime}$s decay to $eW_{{}_{R}}^{\pm}$:

$pp\rightarrow\gamma^{*}/Z^{*}\rightarrow
e^{\prime}e^{\prime}\rightarrow(eW_{{}_{R}}^{\pm})(eW_{{}_{R}}^{\pm}).$

Since we assume small gauge coupling, the $e^{\prime}$ should be a long-lived
particle. However, because $e^{\prime}$s have electromagnetic charge, they
will leave two tracks in the tracker, where the curve of the tracks is
dependent on the mass of $e^{\prime}$. As a result, the detector signature is
two displaced vertices from the primary $pp$ interaction point. In the final
state, we have two displaced, isolated leptons (two electrons) and missing
energy due to the $W_{{}_{R}}^{\pm}$s. The related Feynman diagram is shown in
Fig. 3. Recently, the CMS collaboration has published their results on such
searches for the center of mass-energy $\sqrt{s}=13$ TeV with an integrated
luminosity of $118$ fb-1. The current bound corresponds to
$m_{e^{\prime}}>610\ \text{GeV}$ CMS:2021kdm  for $e^{\prime}$ with a life
time $c\tau_{e^{\prime}}<1$ cm. This bound on the $e^{\prime}$ mass, ensures
us the stability of the $W_{{}_{R}}^{\pm}$ since the only decay channel for
dark gauge boson $(W_{{}_{R}}^{\pm}\rightarrow ee^{\prime})$ is kinematically
forbidden.

Figure 3:  The Feynman diagram for $e^{\prime}$ pair production at the LHC.

##  V Summary

In this paper, we examined a multi-component dark matter (DM) scenario with
light-feebly interacting non-abelian dark gauge bosons. In this region of the
parameter space, $W_{{}_{R}}^{3}$ and $W_{{}_{R}}^{\pm}\equiv
W_{{}_{R}}^{1}\pm iW_{{}_{R}}^{2}$ associated with the gauge
$SU(2)_{{{}_{R}}}$ are the DM candidates. Due to their small gauge coupling,
the evolution of $W_{{}_{R}}$s in the early universe is non-trivial.

We charged right-handed electrons under the $SU(2)_{{}_{R}}$ symmetry to
provide a portal between the dark and the standard model sectors. We
investigated the production of $W_{{}_{R}}$ dark gauge bosons via freeze-in
and vector misalignment mechanisms. We found that for $g_{{}_{R}}\simeq
10^{-12}$, the relic density of $W^{3}_{{}_{R}}$ freezing in from
$ee\rightarrow\gamma W_{{}_{R}}^{3}$ can account for the total DM abundance in
the universe. However, the obtained relic abundance for $W_{{}_{R}}^{\pm}$ via
the freeze-in mechanism is negligible.

For smaller gauge couplings, the vector misalignment mechanism can be used to
produce the non-abelian dark gauge bosons. We extended the action of the non-
abelian vector field with non-minimal coupling to the Ricci scalar term and
showed the obtained equation of motion (EOM) for the spatial component of a
homogeneous non-abelian vector field is the same as the harmonic oscillator
with Hubble damping term. The non-minimal coupling to Ricci scalar leads to an
effective mass in the EOM, which has the most important effect on the
inflationary era. In contrast to the $U(1)$ Stuckelberg mechanism, we must
consider massless gauge bosons before $SU(2)_{{}_{R}}$ is spontaneously broken
by a scalar $\phi$. However, as long as the $SU(2)_{{}_{R}}$ breaking scale is
before the Hubble rate matches the $W_{{}_{R}}$ mass, the story of non-abelian
gauge symmetries becomes similar to dark photons. In the late epoch when the
Hubble parameter is much smaller than the mass of the non-abelian vector, the
field oscillates around the minimum of the potential. This coherent
oscillation acts as a non-relativistic matter and its abundance can explain
the observed relic density of DM in the present time. We should mention that
the vector misalignment mechanism can produce all components of the light dark
gauge boson.

Due to the interaction between $W_{{}_{R}}$ and right-handed electrons, this
model can be probed in various astrophysical observations and terrestrial
experiments. Particularly, we demonstrated the parameter space excluded by the
stellar cooling, the CMB and the galactic diffuse $\gamma$-ray observations,
and different direct detection experiments (like, XENON1T, Super-CDMS, and
LAMPOST). Other bounds such as DM self-interaction or electron $g-2$ provide
mild constraints on this model.

Therefore, the phenomenology of this model is very similar to an ordinary dark
photon dark matter model. The existence of a heavy electron-like particle and
$W^{\pm}_{{}_{R}}$ distinguishes this model from dark photon models.

## Acknowledgments

We would like to thank N. Khosravi, H. Mehrabpour and P. Schwaller for useful
discussions. The work of FE was supported by the Cluster of Excellence
Precision Physics, Fundamental Interactions, and Structure of Matter (PRISMA+
EXC 2118/1) funded by the German Research Foundation(DFG) within the German
Excellence Strategy (Project ID 39083149), and by grant 05H18UMCA1 of the
German Federal Ministry for Education and Research (BMBF).

##  Appendix A Sub-dominant Freeze-In Processes

If the operator responsible for the production of $W_{{}_{R}}$ is
renormalizable, Ref. Hall:2009bx  shows that the production of $W_{{}_{R}}$ is
efficient at lower temperatures when $T\sim m_{{}_{W_{{}_{R}}}}$. If the
operator is non-renormalizable, however, the production of $W_{{}_{R}}$ is
dominated at high temperatures $T\simeq T_{\rm{Max}}$ Elahi:2014fsa ;
Garcia:2017tuj ; Bernal:2019mhf , where $T_{\rm{Max}}=T_{\rm{RH}}$ assuming
the inflaton decays instantly. Let us denote $T_{\phi}$ to indicate the
temperature at which $\phi$ acquires a vev. In the following, we compute the
relic abundance of $W_{{}_{R}}$ in the two regimes of $T>T_{\phi}$ and
$T<T_{\phi}$. We will see that the processes discussed in this appendix lead
to a negligible abundance of $W_{{}_{R}}$.

###  A.1 $T_{\phi}<T<T_{\rm{RH}}<\Lambda_{2}$

For $T>T_{\phi}$, the non-renormalizable kinetic mixing term can contribute to
the production of $W_{{}_{R}}$ at high temperatures via $f\bar{f}\rightarrow
W_{{}_{R}}\leavevmode\nobreak\ \phi\phi$ (Fig. 4), where $f$ is any fermion
that has a non-zero charge under the hypercharge symmetry666SM fermions plus
$e^{\prime}$ and other fermions in the UV needed to cancel the anomaly of the
theory.. The Boltzmann equation of $W_{{}_{R}}$ in this regime is
Elahi:2014fsa :

$\dot{n}_{W_{{}_{R}}}+3Hn_{W_{{}_{R}}}=\frac{T}{(4\pi)^{7}}\int_{0}^{\infty}ds\
s^{3/2}|\mathcal{M}|^{2}K_{1}\left(\frac{\sqrt{s}}{T}\right)\simeq\frac{3N_{f}}{16\pi^{5}}\frac{\epsilon_{W}^{2}g_{{}_{R}}^{2}\alpha_{Y}^{2}T^{8}}{\Lambda_{2}^{4}},$
(A.1)

where $|\mathcal{M}|^{2}\simeq
16\pi^{2}\epsilon_{W}^{2}g_{{}_{R}}^{2}\alpha_{Y}^{2}s/\Lambda_{2}^{4}$, with
$\alpha_{Y}=g_{Y}^{2}/(4\pi)$, and $N_{f}$ is the number of fermions with a
hypercharge charge. Changing the variable to $Y_{{}_{W_{{}_{R}}}}$, we get

$\frac{dY_{{}_{W_{{}_{R}}}}}{dT}\simeq-\frac{1}{s(T)H(T)}\frac{3N_{f}}{16\pi^{5}}\frac{\epsilon_{W}^{2}g_{{}_{R}}^{2}\alpha_{Y}^{2}T^{7}}{\Lambda_{2}^{4}},$
(A.2)

integrating Eq. A.2, in the limit of $([T_{\phi},T_{\rm{RH}}])$, the yields of
the $W_{{}_{R}}$ is obtained:

$Y_{{}_{W_{{}_{R}}}}\simeq\frac{45N_{f}\epsilon_{W}^{2}\alpha^{2}\
g_{{}_{R}}^{2}}{32\pi^{7}\times 1.66\times
g_{\star}^{3/2}}\frac{M_{\text{pl}}(T^{3}_{\text{RH}}-T^{3}_{\phi})}{\Lambda_{2}^{4}}.$
(A.3)

In this temperature range, other contributions to $W_{{}_{R}}$ relic abundance
will be negligible as these processes are dominant at $T\sim
m_{{}_{W_{{}_{R}}}}$.

Figure 4: The freeze-in production of $W_{{}_{R}}$ in the early universe
before $\phi$ acquires a vev via the non-renormalizable operator
$\frac{\phi^{\dagger}\tau^{a}\phi}{\Lambda_{2}^{2}}W_{{{}_{R}},\mu\nu}^{a}B^{\mu\nu}$.
The relic abundance of $W_{{}_{R}}$ coming from this process depends on the UV
parameters. However, a realistic estimate reveals that the amount of
$W_{{}_{R}}$ produced through this process is orders of magnitude smaller than
the observed relic density of dark matter.

As Eq. A.3 shows, the relic abundance of $W_{{}_{R}}$ before $\phi$ acquires a
vev is sensitive to the UV scale parameters, which are irrelevant at low
energy phenomenology. Nonetheless, for illustrative purposes, let us choose
the following benchmark to gain an intuition about the contribution of UV-FI
to $W_{{}_{R}}$ production:

$\displaystyle\Lambda_{2}=10^{7}\ \text{GeV},\hskip 21.68121pt$ $\displaystyle
T_{\text{RH}}=10^{6}\ \text{GeV},\hskip 21.68121ptT_{\phi}=10^{5}\
\text{GeV},$ $\displaystyle g_{{}_{R}}=10^{-6},\hskip 21.68121pt\epsilon_{W}=$
$\displaystyle g_{{}_{R}}g_{Y}/(16\pi^{2})\simeq 2\times 10^{-8},\hskip
21.68121ptN_{f}=30.$

For this benchmark, the relic abundance is
$\Omega_{{}_{W_{{}_{R}}}}h^{2}\simeq
10^{-18}\left(\frac{m_{{}_{W_{{}_{R}}}}}{\text{GeV}}\right)\ll\Omega_{\text{DM}}h^{2}$.
Hence, we can safely neglect the contribution from UV-FI to the $W_{{}_{R}}$
relic abundance.

One may also wonder about the production of $W^{\pm}_{{}_{R}}$ through the
plasmon effect Dvorkin:2019zdi : $\gamma^{\star}\rightarrow
W_{{}_{R}}^{3}\rightarrow W_{{}_{R}}^{+}W_{{}_{R}}^{-}$. However, this process
is proportional to $\epsilon_{W}^{2}g_{{}_{R}}^{4}$. Given that
$\epsilon_{W}<g_{{}_{R}}$, this process is highly suppressed and thus will not
leave any impact on CMB.

###  A.2 $m_{{}_{W_{{}_{R}}}}<T<T_{\phi}$

At low temperatures $T<T_{\phi},m_{\phi}$, the dominant diagrams leading to
production of $W_{{}_{R}}$ are shown in Fig 5. After $\phi$ acquires a vev,
$e^{\prime}$ gains a mass $m_{e^{\prime}}\sim v_{\phi}$, and thus immediately
becomes non-relativistic. In this regime, $e^{\prime}$ number density
decreases because of (a) annihilation to a pair of photon777 For simplicity,
in this subsection we use $\gamma$ instead of B., (b) annihilation to
$W_{{}_{R}}^{3}\gamma$, or (c) decay to $eW^{\pm}_{{}_{R}}$:

$\dot{n}_{e^{\prime}}+3Hn_{e^{\prime}}=-\langle\sigma
v\rangle_{{}_{e^{\prime}e^{\prime}\rightarrow\gamma\gamma}}(n_{e^{\prime}}^{2}-n_{e^{\prime},\text{eq}}^{2})-\langle\sigma
v\rangle_{{}_{e^{\prime}e^{\prime}\rightarrow
W^{3}_{{}_{R}}\gamma}}n_{e^{\prime}}^{2}-n_{e^{\prime}}\Gamma_{e^{\prime}\rightarrow
eW^{\pm}_{{}_{R}}}.$ (A.4)

In the regime where $g_{{}_{R}}\ll\alpha$, the production of $W_{{}_{R}}$ from
$e^{\prime}$ is overwhelmed by $e^{\prime}e^{\prime}\rightarrow\gamma\gamma$.
For $g_{{}_{R}}\lesssim 10^{-10}$, the relic abundance of $W_{{}_{R}}$ from
$e^{\prime}$ is roughly $\Omega_{W_{{}_{R}}}=\Omega_{W^{\pm}_{{}_{R}}}\lesssim
10^{-26}\Omega_{\text{DM}}$. This relic abundance is estimated by assuming
that $e^{\prime}$ freezes out after its annihilation to a pair of photon, and
the remaining $e^{\prime}$ decays to $eW^{\pm}_{{}_{R}}$ through the dark
gauge coupling.

Figure 5: The Feynman diagrams that influence $n_{e^{\prime}}$ in the early
universe. Once $e^{\prime}$ becomes non-relativistic, its number density
depletes rapidly because of its annihilation to a pair of photon (or B). In
the limit where $g_{{}_{R}}\ll\alpha$, the production of $W_{{}_{R}}^{3,\pm}$
is very inefficient.

##  Appendix B Equation of Motion for a Non-Abelian Vector Boson

The action for the non-abelian vector field with non-minimal coupling to the
Ricci scalar is given by,

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{1}{2}\left(M_{\mathrm{pl}}^{2}+\frac{\kappa}{6}W_{{}_{R}\mu}^{b}W_{{}_{R}}^{b\leavevmode\nobreak\
\mu}\right)R-\frac{1}{4}W^{b}_{{}_{R}\mu\nu}W^{b\leavevmode\nobreak\
\mu\nu}_{{}_{R}}-\frac{1}{2}m_{{}_{W_{{}_{R}}}}^{2}W_{{}_{R}\mu}^{b}W_{{}_{R}}^{b\leavevmode\nobreak\
\mu}\right).$ (B.1)

Accordingly, the equation of motion (EOM) is obtained as follows,

$\partial_{\mu}\left(\sqrt{-g}W^{b\leavevmode\nobreak\
\mu\nu}_{{}_{R}}\right)=\sqrt{-g}(\frac{\kappa}{6}R-m_{{}_{W_{{}_{R}}}}^{2})W_{{}_{R}}^{b\leavevmode\nobreak\
\nu}.$ (B.2)

If we consider a spatially homogeneous field, we can neglect the spatial
derivatives. Using the FRW metric, $\sqrt{-g}=a^{3}$ which $a$ is the scalar
factor,

$\partial_{0}[a^{3}W^{b\leavevmode\nobreak\
0\nu}_{{}_{R}}]=a^{3}(\frac{\kappa}{6}R-m_{{}_{W_{{}_{R}}}}^{2})W_{{}_{R}}^{b\leavevmode\nobreak\
\nu}.$ (B.3)

For the time component $\nu=0$,

$3HW^{b}_{{}_{R}00}+\partial_{0}W^{b}_{{}_{R}00}=-(\frac{\kappa}{6}R-m_{{}_{W_{{}_{R}}}}^{2})W_{{}_{R}0}^{b},$
(B.4)

where $H=a/\dot{a}$ is the Hubble parameter. Since

$W^{b}_{{}_{R}00}=\partial_{0}W_{{}_{R}0}^{b}-\partial_{0}W_{{}_{R}0}^{b}+g_{R}f^{b}_{cd}W_{{}_{R}0}^{c}W_{{}_{R}0}^{d}=0,$
(B.5)

where $f^{b}_{cd}$ is the antisymmetric structure constant of the
$SU(2)_{{}_{R}}$. As a result, $W_{{}_{R}0}^{b}=0$.

For the spatial component $\nu=i$,

$\partial_{0}W^{b}_{{}_{R}0i}+HW^{b}_{{}_{R}0i}+(m_{{}_{W_{{}_{R}}}}^{2}-\frac{\kappa}{6}R)W_{{}_{R}i}^{b}=0.$
(B.6)

If we neglect the spatial derivatives and consider $W_{{}_{R}0}^{b}=0$, the
EOM for the spatial component be obtained as follows,

$\ddot{W_{{}_{R}i}^{b}}+H\dot{W_{{}_{R}i}^{b}}+\left(m_{{}_{W_{{}_{R}}}}^{2}-\frac{\kappa}{6}R\right)W_{{}_{R}i}^{b}=0.$
(B.7)

By field redefinition $\mathcal{W}^{b}_{i}\equiv W_{{}_{R}i}^{b}/a$ and
considering
$R=6(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}})=6\dot{H}+12H^{2}$,

$\ddot{\mathcal{W}^{b}_{i}}+3H\dot{\mathcal{W}^{b}_{i}}+\left(m_{{}_{W_{{}_{R}}}}^{2}+\frac{1-\kappa}{6}R\right)\mathcal{W}^{b}_{i}=0,$
(B.8)

Therefore, the EOM for the spatial component of a homogeneous non-abelian
vector field is the same as the damped harmonic oscillator.

##  Appendix C Solving the EOM in the different eras

We can solve the EOM of a non-abelian vector field (B.8) for the evolving
universe,

$\mathcal{W}^{b}_{i}(t)=\alpha_{1}e^{i\omega_{-}t}+\alpha_{2}e^{i\omega_{+}t},$
(C.1)

where the frequencies
$\omega_{\pm}=\frac{3iH}{2}\pm\frac{1}{2}\sqrt{-9H^{2}+4(m_{{}_{W_{{}_{R}}}}^{2}+\frac{1-\kappa}{6}R)}$.
We can determine the above equations in different eras in the evolving
universe.

###  C.1 Inflation era

During inflation, $m_{{}_{W_{{}_{R}}}}$ is zero since the SSB does not happen
yet. The Hubble parameter is constant $H_{I}$ and $R=12H^{2}_{I}$, so we have
the damped harmonic oscillator with the constant frequency Alonso-
Alvarez:2019ixv ,

$\omega_{\pm}=\frac{iH_{I}}{2}\beta_{\pm},$ (C.2)

where $\beta_{\pm}=3\pm\sqrt{1+8\kappa}$.

$\mathcal{W}^{b}_{i}(t)=\mathcal{W}^{b}_{i,s}\left(c_{1}\mathrm{e}^{-\frac{1}{2}\beta_{-}H_{I}t}+c_{2}\mathrm{e}^{-\frac{1}{2}\beta_{+}H_{I}t}\right),$
(C.3)

where the $\mathcal{W}^{b}_{i,s}$ is the initial value of the field at the
start of inflation. For $\kappa>1$, to avoid a trans-Planckian field
excursion, $\beta_{+}$ is positive and $\beta_{-}$ is negative, so after
passing enough time the term is proportional
$\mathrm{e}^{-\frac{1}{2}\beta_{-}H_{I}t}$ will dominant. So we have,

$\mathcal{W}^{b}_{i}(t)\simeq\mathcal{W}^{b}_{i,s}c_{1}\mathrm{e}^{-\frac{1}{2}\beta_{-}H_{I}t}.$
(C.4)

Since during inflation, the scale factor has a relation with the Hubble
parameter ($a=e^{H_{I}t}$),

$\mathcal{W}^{b}_{i}(a)\simeq\mathcal{W}^{b}_{i,s}(\frac{a}{a_{s}})^{-\frac{1}{2}\beta_{-}}.$
(C.5)

We can use the number of e-folds,

$\mathcal{W}^{b}_{i}(a)\simeq\mathcal{W}^{b}_{i,s}\mathrm{e}^{-\frac{1}{2}\beta_{-}N(a)}.$
(C.6)

So the value of the field at end of inflation can express as a function of the
total number of e-folds of inflation,

$\mathcal{W}^{b}_{i,e}\simeq\mathcal{W}^{b}_{i,s}\mathrm{e}^{-\frac{1}{2}\beta_{-}N_{tot}}.$
(C.7)

###  C.2 Post-Inflation era

1\. In the first part the non-abelian gauge field has zero mass since the
spontaneous symmetry breaking (SSB) happens in the middle of the radiation
era, so $m_{{}_{W_{{}_{R}}}}=0$. Also, in the radiation era $R=0$. As result,
the frequencies become $\omega_{+}=3iH$ and $\omega_{-}=0$ and we have the
over-damped harmonic oscillator,

$\mathcal{W}^{b}_{i}(t)=\alpha_{1}+\alpha_{2}\mathrm{e}^{-3Ht}.$ (C.8)

After enough time evolution, the first term will be dominant and the value of
the field will be constant. Indeed this constant value is the value of the
field at the end of inflation,

$\mathcal{W}^{b}_{i}(t)\simeq\mathcal{W}^{b}_{i,e}.$ (C.9)

The vector field has a vanishing time derivative.
2\. After the SSB, the vector boson acquires mass. At this point
$m_{{}_{W_{{}_{R}}}}\sim H_{\star}$ and the evolution of the vector field
switch from exponential decay (constant value) to an oscillation mode. In
another word, the field rolls down the potential and starts to oscillate. We
can consider the oscillation modes of the vector field as DM degrees of
freedom. Here, $H_{\star}$ denotes the Hubble parameter at the DM production
time.
3\. For the late universe and in the limit $m_{{}_{W_{{}_{R}}}}\gg H$, the
frequencies become $\omega_{\pm}=\frac{3iH}{2}\pm m_{{}_{W_{{}_{R}}}}$,

$\mathcal{W}^{b}_{i}(t)=\gamma_{1}\mathrm{e}^{-3Ht/2}\leavevmode\nobreak\
\mathrm{e}^{-im_{{}_{W_{{}_{R}}}}t}+\gamma_{2}\mathrm{e}^{-3Ht/2}\leavevmode\nobreak\
\mathrm{e}^{im_{{}_{W_{{}_{R}}}}t}.$ (C.10)

We can find the above constants, by matching to the time of the DM production,
$t_{\star}$,

$\mathcal{W}^{b}_{i}(t)=\mathcal{W}^{b}_{i,e}(\frac{a_{\star}}{a})^{\frac{3}{2}}\cos(m_{{}_{W_{{}_{R}}}}(t-t_{\star})),$
(C.11)

where $a_{\star}$ is the scale factor at the DM production time.

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