# Shedding Flavor on Dark via Freeze-in:
$U(1)_{B-3L_{i}}$ Gauged Extensions

Basabendu Barman  Purusottam Ghosh
Anish Ghoshal  Lopamudra Mukherjee

###### Abstract

We consider a singlet fermionic dark matter (DM) $\chi$ in a gauged
$U(1)_{B-3L_{i}}$ extension of the Standard Model (SM), with $i\in
e\,,\mu\,,\tau$, and derive bounds on the allowed parameter space, considering
its production via freeze-in mechanism. The DM communicates with the SM only
through flavorful vector-portal $Z_{\text{B3L}}$ due to its non-trivial charge
$x$ under $U(1)_{B-3L_{i}}$, which also guarantees the stability of the DM
over the age of the Universe for $x\neq\\{\pm 3/2,\pm 3\\}$. Considering
$Z_{\text{B3L}}$ to lie within the mass range of a few MeV up to a few GeV, we
obtain constraints on the gauge coupling $g_{\text{B3L}}$ from the requirement
of producing right relic abundance. Taking limits from various (present and
future) experimental facilities, e.g., NuCal, NA64, FASER, SHiP into account,
we show that the relic density allowed parameter space for the frozen in DM
can be probed with $g_{\text{B3L}}\gtrsim 10^{-8}$ for both
$m_{\chi}<m_{\text{ZB3L}}/2$ and $m_{\chi}\gtrsim m_{\text{ZB3L}}$, while
$g_{\text{B3L}}\lesssim 10^{-8}$ remains mostly unconstrained. We also briefly
comment on the implications of neutrino mass generation via Type-I seesaw and
anomalous $(g-2)_{\mu}$ in context with $B-3L_{\mu}$ gauged symmetry.

PI/UAN-2022-709FT

HRI-RECAPP-2021-013

##  1 Introduction

The minimal Standard Model (SM) of eletroweak and strong interactions has
accidental global symmetries, namely the baryon number $B$ and the lepton
number $L_{i}$, with $i\in e\,,\mu\,,\tau$. These symmetries are anomalous,
and a common strategy for venturing to gauge theories beyond the SM is to
gauge some linear combination of these symmetries which is anomaly-free.
Almost four decades ago, it was observed that $U(1)_{B-L}$ [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]
symmetry is an anomaly free gauge extension of SM that treats all fermion
generations on the same footing111Models where the generational universality
is violated by gauging quantities like $L_{\mu}-L_{\tau}$ has received lots of
interest in recent years [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]..
However, it is important to note that the new abelian gauge groups under which
the SM fields carry non-trivial charges, are strongly constrained by the
requirement of anomaly cancellation and the structure of the Cabibbo-
Kobayashi-Maskawa (CKM) and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing
matrices. These taken together into account, leave a limited number of global
symmetries of the SM that can be gauged with the addition of only three right-
handed neutrinos: $U(1)_{B-L}$, $U(1)_{B-3L_{i}}$ and combinations of these
groups. Observationally, as pointed out in [39], the flavour changing coupling
of the new gauge boson to active neutrinos in the mass basis can distinguish
between $U(1)_{B-L}$ and $U(1)_{B-3L_{i}}$, albeit quark and charged lepton
flavour transitions are strongly suppressed irrespective of the gauge group.
Gauged $U(1)_{B-3L_{i}}$ symmetry (where $i\in e\,,\mu\,,\tau$), has also been
discussed in the context of neutrino mass generation, low-energy unification
and flavour anomalies [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

Moreover, the failure of the SM to provide a viable dark matter (DM)
candidate, whose existence in the Universe has already been proven
unequivocally from several astrophysical [51, 52, 53, 54] and cosmological
[55, 56] evidences (for a review, see, e.g. Refs. [57, 58, 59]), compels one
to explore beyond the realms of the SM. The weakly interacting massive
particle (WIMP) paradigm is the most elusive one in explaining the observed
relic abundance of the DM in the Universe, as the observed DM abundance is set
typically by the weak scale interaction strength. However, the absence of
significant excess, in typically nuclear recoil experiments [60, 61, 62], is
constantly cornering the WIMP paradigm. Out of several alternate
prescriptions, the observed DM abundance can be generated by what is known as
the freeze-in [63, 64, 65, 66, 67, 68, 69, 70, 71, 72]. The DM particles, in
this case, are produced by the decay or annihilation of the bath particles.
The production ceases due to cooling of the bath temperature below the
relevant mass scale connecting the DM to the visible sector. As a result, in
contrast to the WIMP scenario, in freeze-in, the DM abundance increases with
the increase in interaction strength between the DM with the visible sector.
Even though freeze-in is perfectly capable of explaining the whole DM
abundance, it is challenging to probe because of its feeble coupling with the
visible sector. In context of $U(1)_{B-L}$, DM phenomenology has been studied
considering one of the right handed neutrinos, required for anomaly
cancellation, to be a potential DM candidate in, for example, Refs. [73, 5, 8,
74, 26] including particularly that of an ad-hoc singlet fermion as DM [20,
75, 76]. In all of these scenarios, the interaction between DM and the SM
particles is primarily mediated by the $B-L$ gauge boson. Hence, these
scenarios predict a one-to-one correspondence between the experimental
searches for the new gauge boson and DM physics.

Motivated from these, in this work we have explored the possibility of probing
freeze-in DM scenario in gauged $B-3L_{i}$ with a singlet vector-like
fermionic DM charged under the new gauge symmetry. As the DM is vector-like,
it does not contribute to the gauge anomaly that leaves us with the
opportunity to consider its charge as a free parameter. The charge assignment
allows the DM to talk to the SM only via the neutral gauge boson, also
ensuring its stability over the lifetime of the Universe. As a result, the DM
parameter space can be constrained from experimental bounds on the mass and
coupling of the new gauge boson, establishing a direct connection between DM
and gauge boson search. Also note the absence of an ad-hoc stabilizing $Z_{2}$
symmetry, which is very common in $B-L$ gauge extensions with one of the RHNs
as a DM candidate (for example, in [8]). Moreover, considering one of the RHNs
to be DM, the model albeit becomes minimal, but one has to take into account
the effect of the Higgs portal, which we do not intend to, as we aim to
investigate the status of freeze-in parameter space in the light of massive
gauge boson searches. The requirement of gauge anomaly cancellation also
introduces a heavy singlet right handed neutrino (RHN) which, along with one
or two more right handed neutrinos having vanishing $B-3L_{i}$ charges, can
take part in generating light neutrino masses through usual Type-I seesaw
mechanism. We typically focus on the regime where the massive gauge boson of
the local $B-3L_{i}$ symmetry is sub-GeV222Inflationary dynamics of such light
mediators were studied in Ref. [77] with constraints from $N_{\text{eff}}$
observables in [78]. and can be as heavy as $m_{\text{ZB3L}}\lesssim 10$ GeV.
We show, for suitable choices of the DM charge $x$ under $B-3L_{i}$ symmetry,
together with the gauge coupling $g_{\text{B3L}}$, it is possible to produce
observed relic abundance for the DM via freeze-in mechanism in the early
Universe. The central idea of this study is to establish the fact that
although the size of the gauge coupling is small enough, preventing the DM to
achieve thermal-equilibrium with the SM bath, such small coupling still
remains within the reach of various current and future planned/proposed
experiments that typically search for very weakly coupled light states.

The paper is organized as follows. The model set-up is discussed in Sec. 2,
where we tabulate the particle content mentioning their charges, together with
the relevant Lagrangian in subsection. 2.1; the light neutrino mass generation
via Type-I seesaw is discussed in subsection. 2.2; we then move on to the
calculation of DM abundance via freeze-in in Sec. 3, where we elaborate the
implementation of coupled Boltzmann equation in subsection 3.1 following the
discussion about the DM parameter space in subsection. 3.2, mentioning all the
relevant constraints. Finally, we conclude in Sec. 4.

##  2 Model Framework

Fields | $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U\left(1\right)_{B-3L_{i}}$
---|---|---|---|---
$Q_{L}^{i}$ | 3 | 2 | 1/6 | 1/3
$u_{R}^{i}$ | 3 | 1 | 2/3 | 1/3
$d_{R}^{i}$ | 3 | 1 | -1/3 | 1/3
$L^{e}$ | 1 | 2 | -1/2 | -3 $\delta_{ei}$
$L^{\mu}$ | 1 | 2 | -1/2 | -3 $\delta_{\mu i}$
$L^{\tau}$ | 1 | 2 | -1/2 | -3 $\delta_{\tau i}$
$e_{R}^{e}$ | 1 | 1 | -1 | -3 $\delta_{ei}$
$e_{R}^{\mu}$ | 1 | 1 | -1 | -3 $\delta_{\mu i}$
$e_{R}^{\tau}$ | 1 | 1 | -1 | -3 $\delta_{\tau i}$
$N_{R}^{e}$ | 1 | 1 | 0 | -3 $\delta_{ei}$
$N_{R}^{\mu}$ | 1 | 1 | 0 | -3 $\delta_{\mu i}$
$N_{R}^{\tau}$ | 1 | 1 | 0 | -3 $\delta_{\tau i}$
$H$ | 1 | 2 | 1/2 | 0
$S$ | 1 | 1 | 0 | 3
$\chi$ | 1 | 1 | 0 | $x$
Table 1: Charges of the new particles under $\text{SM}\times(B-3L_{i})$
symmetry, where $i\in e\,,\mu\,,\tau$ and $\delta_{ij}=1$ for $i=j$, else
zero.

We consider three beyond the SM (BSM) scenarios: the SM augmented via a new
$U(1)_{X}$ gauge group, with $X\equiv B-3L_{i}\,;\,i\in e\,,\mu\,,\tau$. The
generation dependent gauge charge assignment of the SM fields lead to triangle
anomalies, that require extra fields to cancel them which carry charge under
$U(1)_{B-3L_{i}}$. The condition for anomaly cancellations come from the
triangle diagrams with gauge bosons at the vertices:

$\sum_{\text{LH}}{\rm
Tr}\left[T^{a}\\{T^{b},T^{c}\\}\right]-\sum_{\text{RH}}{\rm
Tr}\left[T^{a}\\{T^{b},T^{c}\\}\right]\,,$ (2.1)

with “Tr” standing for trace and $T^{a,b,c}$ denoting the corresponding
generators. This vanishes exactly for the vector-like states as they
contribute identically to the left and right-handed representation. However,
the contributions from other fermions do not vanish in general due to their
chiral nature. The $B-3L_{i}$ charge assignment in Tab. 1 satisfies
cancellation of the gauge and the gravitational anomalies that can be
trivially checked.

###  2.1 Particle content

As mentioned above, in order to cancel the gauge anomalies we introduce three
right handed neutrions (RHN), singlet under the SM gauge symmetry, but
carrying non-trivial charges depending on the nature of $U(1)_{B-3L_{i}}$.
These RHNs take part in the anomaly cancellation as mentioned before. We
further introduce a vector-like singlet fermion, carrying some charge $x$
under the new gauge symmetry. This can serve as a potential DM candidate in
our model. The charge $x$ assigned to the DM is arbitrary, however
$x\neq\\{\pm\frac{3}{2}\,,\pm 3\\}$ such that the DM communicates to the SM
only through the new gauge boson. The same charge assignments also render the
DM stable by forbidding the Yukawa interactions, e.g.,
$\overline{\chi}\,e_{R}\,S\,,\overline{\chi}\,N_{R}\,S\,,\overline{L}\,H\,\chi$
etc. Thus, the DM in the present set-up can be made absolutely stable without
introducing any ad-hoc discrete symmetry. The full particle spectrum, together
with the charges under different gauge groups are summarized in Tab. 1. With
this particle content at our disposal, we can write the following interaction
Lagrangian

$\mathcal{L}^{{B-3L_{i}}}\supset\mathcal{L}_{\text{gauge}}+\mathcal{L}_{\text{scalar}}+\mathcal{L}_{\text{fermion}}^{{B-3L_{i}}}+\mathcal{L}_{\text{yuk}}^{{B-3L_{i}}}\,,$
(2.2)

with

$\mathcal{L}_{\text{gauge}}\supset-\frac{1}{4}\,Z_{\text{B3L}_{\mu\nu}}\,Z^{\text{B3L}{\mu\nu}}-\frac{\epsilon}{2}\,Z_{\text{B3L}\mu\nu}\,B^{\mu\nu}\,,$
(2.3)

where we consider $\epsilon=0$ at the scale of $B-3L_{i}$ breaking [79]. The
Lagrangian corresponding to the scalar sector is given by

$\mathcal{L}_{\text{scalar}}=\left|D_{\mu}\,S\right|^{2}-V(H,S)\,,$ (2.4)

where the scalar potential reads

$V(H,S)=-\mu_{H}^{2}\,\left|H\right|^{2}+\lambda_{H}\,\left|H\right|^{4}-\mu_{S}^{2}\,\left|S\right|^{2}+\lambda_{S}\,\left|S\right|^{4}+\lambda_{\text{HS}}\,\left|H\right|^{2}\,\left|S\right|^{2}\,,$
(2.5)

and the covariant derivative reads

$D_{\mu}\equiv\partial_{\mu}-i\,g_{2}\,\tau^{a}\,W_{\mu}^{a}-i\,g_{1}\,Y\,B_{\mu}-i\,g_{\text{B3L}}\,Y_{\text{B3L}}\,Z_{\text{B3L}\mu}\,,$
(2.6)

where $g_{1,2}$ corresponds to the $U(1)_{Y}$ and $SU(2)_{L}$ gauge couplings
respectively and $g_{\text{B3L}}$ is the new gauge coupling and
$Y_{\text{B3L}}$ is the corresponding charge as mentioned in Tab. 1. The
$SU(2)_{L}$ doublet $H$ mixes with the singlet $S$ once the fields acquire
non-zero vacuum expectation values (VEV). The two scalars in the unitary gauge
can be parametrized as

$H=\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}(h+v_{d})\end{pmatrix}\,;~{}S=\frac{1}{\sqrt{2}}\,\left(s+v_{s}\right)\,,$
(2.7)

with $v_{d,s}$ being the corresponding VEVs. On minimizing $V(H,S)$ we obtain

$\displaystyle\mu_{H}^{2}=v_{d}^{2}\,\lambda_{H}+\frac{1}{2}\,\lambda_{HS}\,v_{s}^{2}\,$
$\displaystyle\mu_{S}^{2}=v_{s}^{2}2\,\lambda_{S}+\frac{1}{2}\,\lambda_{HS}\,v_{d}^{2}\,.$
(2.8)

The CP even mass matrix reads

$\mathcal{M}_{\text{CPE}}^{2}=\left(\begin{array}[]{cc}2v_{d}^{2}\,\lambda_{H}&v_{d}\,v_{s}\,\lambda_{HS}\\\
v_{d}\,v_{s}\,\lambda_{HS}&2\,v_{s}^{2}\,\lambda_{S}\\\ \end{array}\right)\,,$
(2.9)

with eigenvalues

$m_{1,2}=\frac{1}{2}\,\Bigl{(}A+C\mp\sqrt{A^{2}+4\,B^{2}-2\,A\,C+C^{2}}\Bigr{)}\,,$
(2.10)

with $A=2\,v_{d}^{2}\,\lambda_{H}\,,B=v_{d}\,v_{s}\,\lambda_{HS}$ and
$C=2\,v_{s}^{2}\,\lambda_{S}$. We can diagonalize this mass matrix using a
$2\times 2$ rotation matrix that leaves two physical eigenstates in the mass
basis

$\begin{pmatrix}h\\\ s\end{pmatrix}=\begin{pmatrix}c_{\theta}&-s_{\theta}\\\
s_{\theta}&c_{\theta}\end{pmatrix}\,\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}\,,$ (2.11)

where $h_{1}$ is SM-like Higgs with mass $\sim 125$ GeV and the mixing angle
is given by

$\sin 2\theta=\frac{2\,v_{d}\,v_{s}}{m_{2}^{2}-m_{1}^{2}}\,\lambda_{HS}\,.$
(2.12)

On spontaneous breaking of $U(1)_{B-3L_{i}}$, we get a massive neutral gauge
boson with mass

$m_{\text{ZB3L}}=3\,g_{\text{B3L}}\,v_{s}\,.$ (2.13)

One may then trade-off the renormalizable couplings appearing in $V(H,S)$ with
the physical masses and mixings

$\displaystyle\lambda_{S}=\frac{1}{2\,v_{s}^{2}}\,\left(m_{1}^{2}\,\sin^{2}\theta+m_{2}^{2}\,\cos^{2}\theta\right)$
$\displaystyle\lambda_{H}=\frac{1}{2\,v_{d}^{2}}\,\left(m_{1}^{2}\,\cos^{2}\theta+m_{2}^{2}\,\sin^{2}\theta\right)$
$\displaystyle\lambda_{\text{HS}}=\frac{\sin
2\theta}{2\,v_{d}\,v_{s}}\,\left(m_{1}^{2}-m_{2}^{2}\right)\,.$ (2.14)

In presence of the mixing between the two scalars, the tree level interactions
of the SM Higgs with the SM fermion and gauge bosons get modified. In such a
scenario, the LHC sets a bound on the mixing $|\sin\theta|\leq 0.36$, from the
measurement of signal strength in the di-photon channel [80, 81, 82]. The
strongest bound on singlet scalar-SM Higgs mixing appears from $W$-boson mass
correction at NLO [83] for the mass range $250\lesssim m_{2}\lesssim 850$ GeV
for $0.2\lesssim\sin\theta\lesssim 0.3$. For $m_{2}>$ 850 GeV, the bounds from
the requirement of perturbativity and unitarity turn out to be dominant,
requiring $\sin\theta\lesssim 0.2$. For $m_{2}<$ 250 GeV, the LHC and LEP
direct search and measured Higgs signal strength restrict the mixing angle
$\sin\theta\lesssim 0.25$ [84, 85]. Also, the SM Higgs in the present model
can decay into a pair of $Z_{\text{B3L}}$, which is tightly constrained from
the measurement of Higgs invisible decay branching that demands
$\text{Br}(h\to\text{invisible})<0.11$ [86]. This bound, however, is trivially
satisfied since $\Gamma(h_{1}\to
Z_{\text{B3L}}Z_{\text{B3L}})\propto\sin^{2}\theta g_{\text{B3L}}^{2}$ and
$g_{\text{B3L}}\sim\mathcal{O}(10^{-8})$, required for freeze-in as we shall
see in Sec. 3. In the entire analysis we shall fix $\sin\theta=10^{-2}$ and
the mass of the non-standard Higgs to $m_{2}=500$ GeV. Note that, due to large
$v_{s}$, both the scalar couplings $\lambda_{\text{HS}}$ and $\lambda_{S}$ are
of the same order as that of $g_{\text{B3L}}$.

The Lagrangian for the fermion sector under the $U(1)_{B-3L_{i}}$ turns out to
be

$\mathcal{L}_{\text{fermion}}^{{B-3L_{i}}}\supset\overline{N_{R}}\,i\gamma^{\mu}\,D_{\mu}\,N_{R}+\overline{\chi}\,i\gamma^{\mu}\,D_{\mu}\,\chi-\sum_{i\neq
j,k}\frac{1}{2}\,(M_{N})_{jk}\,\overline{N_{R}^{c}}^{j}\,N_{R}^{k}-m_{\chi}\,\overline{\chi}\,\chi,$
(2.15)

where we can write a bare mass term $m_{\chi}$ for the DM, without violating
the gauge symmetry, thanks to its vector-like transformation.

Finally, the Yukawa interactions are given by

$-\mathcal{L}_{\text{yuk}}^{{B-3L_{i}}}\supset\sum_{j\neq
i}\frac{1}{2}\,y_{ij}\,\overline{N_{R}^{c}}^{i}\,N_{R}^{j}\,S+\sum_{\alpha\neq
i}\,\sum_{j\neq i}\,y_{\alpha
j}\,\overline{L_{\alpha}}\,\widetilde{H}\,N_{R_{j}}+y_{L_{i}}\,\overline{L^{i}}\,\widetilde{H}\,N_{R}^{i}+\text{H.c}\,,$
(2.16)

where $\alpha\in e\,,\mu\,,\tau$, $j\in e\,,\mu\,,\tau$ . Before moving on, it
is important to note that the phenomenology of the present set-up is governed
by the following set of independent parameters
$\\{x\,,g_{\text{B3L}}\,,m_{\text{ZB3L}}\,,m_{\chi}\\}\,$. As we shall see in
Sec. 3, the choice of the charge $x$ and the gauge coupling $g_{\text{B3L}}$
crucially determine the detection prospects for freeze-in.

###  2.2 Generation of the SM neutrino mass

In the present set-up the light neutrino mass $m_{\nu}$ can be generated via
standard Type-I seesaw [87, 88] mechanism as

$m_{\nu}=-M_{D}\,(M_{R})^{-1}\,M_{D}^{T}\,,$ (2.17)

where $M_{D}$ and $M_{R}$ can be derived from the Yukawa interaction of SM
leptons in Eq. (2.15) and singlet fermions in Eq. (2.16). Depending on the
charge of the $B-3L_{i}$ gauge symmetry, these matrices can be expressed as
follows

$\displaystyle M_{D}^{e}=\begin{pmatrix}y_{ee}\,\frac{v_{d}}{\sqrt{2}}&0&0\\\
0&y_{\mu\mu}\,\frac{v_{d}}{\sqrt{2}}&y_{\mu 3}\,\frac{v_{d}}{\sqrt{2}}\\\
0&y_{\tau 2}\,\frac{v_{d}}{\sqrt{2}}&y_{\tau\tau}\,\frac{v_{d}}{\sqrt{2}}\
\end{pmatrix}\,;M_{R}^{e}=\begin{pmatrix}0&y_{12}\,\frac{v_{s}}{\sqrt{2}}&y_{13}\,\frac{v_{s}}{\sqrt{2}}\\\
y_{21}\,\frac{v_{s}}{\sqrt{2}}&M_{22}&M_{23}\\\
y_{31}\,\frac{v_{s}}{\sqrt{2}}&M_{32}&M_{33}\end{pmatrix}\,,$ (2.18)

$\displaystyle
M_{D}^{\mu}=\begin{pmatrix}y_{ee}\,\frac{v_{d}}{\sqrt{2}}&0&y_{e\tau}\,\frac{v_{d}}{\sqrt{2}}\\\
0&y_{\mu\mu}\,\frac{v_{d}}{\sqrt{2}}&0\\\ y_{\tau
e}\,\frac{v_{d}}{\sqrt{2}}&0&y_{\tau\tau}\,\frac{v_{d}}{\sqrt{2}}\
\end{pmatrix}\,;M_{R}^{\mu}=\begin{pmatrix}M_{11}&y_{12}\,\frac{v_{s}}{\sqrt{2}}&M_{13}\\\
y_{21}\,\frac{v_{s}}{\sqrt{2}}&0&y_{23}\,\frac{v_{s}}{\sqrt{2}}\\\
M_{31}&y_{32}\,\frac{v_{s}}{\sqrt{2}}&M_{33}\end{pmatrix}\,,$ (2.19)

$\displaystyle
M_{D}^{\tau}=\begin{pmatrix}y_{ee}\,\frac{v_{d}}{\sqrt{2}}&y_{e\mu}\,\frac{v_{d}}{\sqrt{2}}&0\\\
y_{\mu e}\,\frac{v_{d}}{\sqrt{2}}&y_{\mu\mu}\,\frac{v_{d}}{\sqrt{2}}&0\\\
0&0&y_{\tau\tau}\,\frac{v_{d}}{\sqrt{2}}\
\end{pmatrix}\,;M_{R}^{\tau}=\begin{pmatrix}M_{11}&M_{12}&y_{13}\,\frac{v_{s}}{\sqrt{2}}\\\
M_{21}&M_{22}&y_{23}\,\frac{v_{s}}{\sqrt{2}}\\\
y_{13}\,\frac{v_{s}}{\sqrt{2}}&y_{23}\,\frac{v_{s}}{\sqrt{2}}&0\end{pmatrix}\,.$
(2.20)

The light neutrino mass matrix is diagonalized utilizing the PMNS matrix
$U_{\text{PMNS}}$ as

$m_{\nu}=U_{\text{PMNS}}\,.m_{\nu}^{\text{diag}}\,.U_{\text{PMNS}}^{T}\,,$
(2.21)

where
$m_{\nu}^{\text{diag}}\equiv\text{diag}\left(m_{1}\,,m_{2}\,,m_{3}\right)$ and
the PMNS matrix is parametrized as

$U_{\text{PMNS}}=\left(\begin{array}[]{ccc}c_{12}\,c_{13}&s_{12}\,c_{13}&s_{13}\\\
-s_{12}\,c_{23}-c_{12}\,s_{23}\,s_{13}&c_{12}\,c_{23}-s_{12}\,s_{23}\,s_{13}&s_{23}\,c_{13}\\\
s_{12}\,s_{23}-c_{12}\,c_{23}\,s_{13}&-s_{12}\,c_{23}s_{13}-c_{12}\,s_{23}&c_{23}\,c_{13}\\\
\end{array}\right)\,,$ (2.22)

where $c_{ij}\equiv\cos\theta_{ij}$ is the cosine of the mixing angles. We
also assume the charge lepton mass matrix to be diagonal. A detailed study of
neutrino mass and mixing, appropriately fitting the observed data, has been
discussed in [89] in the context of $U(1)_{B-3L_{i}}$. Here we do not go into
such details, we only perform a simple estimation to obtain the possible mass
range of the RHNs that can provide the light neutrino mass in the correct
ballpark. The estimation of possible RHN mass range is necessary in order to
calculate the DM relic abundance in the next section. Neutrino oscillation
experiments are sensitive to the mass-squared differences and the mixing
angles, and the value of these parameters used in the analysis in the
$3\sigma$ range are as follows [90]

$\displaystyle|\Delta
m^{2}_{\text{sol}}|\equiv\left|m^{2}_{2}-m^{2}_{1}\right|\in[6.79-8.01]\times
10^{-5}{\text{eV}^{2}}\,,\left|\Delta
m^{2}_{\text{atm}}\right|\equiv|m^{2}_{3}-m^{2}_{1}|\in[2.35-2.54]\times
10^{-3}{\text{eV}^{2}}\,,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sin^{2}\theta_{12}\in[0.27,0.35]\,,\sin^{2}\theta_{23}\in[0.43,0.60]\,,\sin^{2}\theta_{13}\in[0.019,0.024]\,.$
(2.23)

Since the sign of $\Delta m_{31}^{2}$ is undetermined, distinct neutrino mass
hierarchies are possible. This, in turn, puts bound on model parameters,
particularly, on the VEV $v_{s}$ and relevant Yukawa couplings. We consider
the lightest active neutrino to be massless such that $\Delta
m_{\text{sol}}^{2}\equiv m_{2}^{2}$ and $\Delta m_{\text{atm}}^{2}\equiv
m_{3}^{2}$. We choose the case of $U(1)_{B-3L_{e}}$ for illustration, while
for the other two gauge groups similar argument follows.

We choose $y_{\mu\mu}=y_{\mu 3}=y_{\tau 2}=y_{\tau\tau}\equiv y_{\ell}$,
$y_{12}=y_{13}\equiv y$ and $M_{22}=M_{33}\equiv M$. Now, for
$v_{s}=m_{\text{ZB3L}}/g_{\text{B3L}}=10^{10}$ GeV, light neutrino mass can be
generated for $y_{\ell}\sim 10^{-7}$, $y_{ee}\sim 10^{-2}$, $y\sim 10^{-2}$
with $M=1$ TeV. Even if we consider $M=10$ TeV, we can still obtain the light
neutrino mass, by properly tuning the couplings. However, in that case, the
RHNs are beyond the present collider reach. Since the Yukawas do not determine
the outcome of the DM phenomenology, we can safely fix them to some benchmark
values such that light neutrino mass is satisfied. For the rest of the
analysis we will consider the RHN mass to be 1 TeV. We find, with
$y_{\ell}\sim 10^{-7}$, the lifetime of 1 TeV RHN turns out to be
$\tau_{N}\sim 10^{-12}$ sec, which is much shorter than the typical BBN
timescale333For certain choices of lightest neutrino mass, RHN having such
gauge interactions can be long lived enough to give interesting collider
signatures like displaced vertices, as studied in [91]..

##  3 Dark matter freeze-in

The interaction strength that controls DM freeze-in production also controls
the reaction rate of the new gauge mediator with the SM bath. Therefore, for
very small $g_{\text{B3L}}$, typically required for freeze-in, the production
of $Z_{\text{B3L}}$ is also heavily suppressed and can be assumed to have a
zero initial abundance. It is thus necessary to to consider a two-step
production, i.e., to first produce $Z_{\text{B3L}}$ from the thermal bath and
subsequently the DM from the $Z_{\text{B3L}}$ produced [92, 93]. Whether the
produced $Z_{\text{B3L}}$ shall be in thermal equilibrium or not, will depend
on the choice of the gauge coupling $g_{\text{B3L}}$ that determines the
interaction rate between the SM bath and $Z_{\text{B3L}}$. Thus, there exists
a bound on the size of $g_{\text{B3L}}$ from these requirements. In order to
determine that we need to compare the reaction rate of $Z_{\text{B3L}}$
production with the Hubble rate $\mathcal{H}$ given by
$\mathcal{H}=\frac{\pi}{3}\,\sqrt{\frac{g_{\star\rho}}{10}}\,\frac{T^{2}}{M_{\text{pl}}}\,,$
for a standard radiation dominated Universe, where $g_{\star\rho}$ is the
total number of relativistic degrees of freedom contributing to the energy
density. Next we note that there exist several channels for the production of
$Z_{\text{B3L}}$ from the thermal bath. All these channels can be broadly
classified into three categories based on their time of occurrence as

  * •

$v_{s}=v_{d}=0$: At this stage both $B-3L_{i}$ and EW symmetry is exact as
none of the scalars in the theory gets a non-zero VEV. As one can see from Eq.
(2.4), that a non-zero number density of $Z_{\text{B3L}}$ can still build up
from $S\to Z_{\text{B3L}}Z_{\text{B3L}}$ decay, however at this stage
$Z_{\text{B3L}}$’s are massless.

  * •

$v_{s}\neq 0\,,v_{d}=0$: Once $B-3L_{i}$ is broken (but electroweak symmetry
is exact), $Z_{\text{B3L}}$ is massive and can be pair-produced produced both
from on-shell decay of the scalar $S\to Z_{\text{B3L}}Z_{\text{B3L}}$, along
with the $2\to 2$ scattering channels: $SS\to
Z_{\text{B3L}}\,Z_{\text{B3L}}\,,N_{R}\,N_{R}\to
Z_{\text{B3L}}\,Z_{\text{B3L}}\,,f\,f\to
Z_{\text{B3L}}\,Z_{\text{B3L}}\,,f\,f\to\,W^{3}(B)\,Z_{\text{B3L}}$, where
$f\in$ SM fermions (chiral) and $V\in\\{W_{\mu}^{3}\,,B_{\mu}\\}$ are the SM
gauge bosons without the longitudinal degrees of freedom. The SM Higgs doublet
is parametrized as $H^{T}=\bigl{\\{}H^{+}\,,H^{0}\bigr{\\}}$, with four
propagating massless Goldstone degrees of freedom. Note that, the scalar $S$
can remain in thermal equilibrium with the SM bath through the portal coupling
$\lambda_{\text{HS}}$, which can be $\sim\mathcal{O}(1)$ before EWSB. On the
other hand, after EWSB, $\lambda_{\text{HS}}$ becomes of the order of
$g_{\text{B3L}}$ (cf. Eq. (2.1)), however the physical scalar $h_{2}$ can
still maintain thermal equilibrium due to the four point gauge interactions,
e.g., $h_{2}\,h_{2}\to W^{+}W^{-},\,ZZ$.

  * •

$v_{s,d}\neq 0$: Once EW symmetry is also broken, then all states are massive
and the SM gauge bosons obtain the longitudinal degrees of freedom. We then
obtain two CP-even physical scalars $h_{1,2}$ as elaborated before. Along with
this, now the SM fermions are massive and the physical gauge bosons are
$W^{\pm}\,,Z\,,\gamma$.

In Fig. 1 we have illustrated the scheme of two-step freeze-in by a cartoon,
where we show $Z_{\text{B3L}}$’s are first thermalized (yellow blob) after
being produced from the bath (green blob) and finally the DM yield builds up
from the thermalized $Z_{\text{B3L}}$ density (gray blob).

###  3.1 Freeze-in yield

As advocated earlier, the computation of DM yield depends on whether
$Z_{\text{B3L}}$ can be considered to be a part of the SM bath. Here we derive
the condition under which this is true. We first note before EWSB the
production rate of $Z_{\text{B3L}}$ is given by

$\Gamma_{\text{SM}\to Z_{\text{B3L}}}^{v_{d}=0}=\begin{cases}\Gamma_{S\to
Z_{\text{B3L}}Z_{\text{B3L}}}\,\frac{K_{1}(m_{S}/T)}{K_{2}(m_{S}/T)}\,&\text{for
decay}\,,\\\\[8.0pt] n_{Z_{\text{B3L}}}\,\langle\sigma
v\rangle_{Z_{\text{B3L}}Z_{\text{B3L}}}&\text{for scattering}\,,\end{cases}$
(3.1)

Figure 1: DM production via two-step freeze-in.

Figure 2: Top Left: Decay rates of $S\to Z_{\text{B3L}}Z_{\text{B3L}}$ for
different choices of $m_{S}$ shown via different colors, where the shaded
regions are discarded from BBN, requiring $\tau_{S}\gtrsim 1$ sec. Top Right:
The shaded region is where the decay process $S\to
Z_{\text{B3L}}Z_{\text{B3L}}$ falls out of equilibrium before EWSB. Here
different colours correspond to different choices of $m_{\text{ZB3L}}$. Bottom
Left: Same as top right, but comparing DM production rate via decay with the
Hubble rate. Bottom Right: The shaded region depicts the region where
$\Gamma_{Z_{\text{B3L}}}^{-1}>1$ sec (see text).

where $n_{Z_{\text{B3L}}}\sim T^{3}$ and

$\Gamma_{S\to
Z_{\text{B3L}}Z_{\text{B3L}}}=\frac{9\,g_{\text{B3L}}^{2}}{32\,\pi\,r^{2}}\,m_{S}\,\sqrt{1-4\,r^{2}}\,(1-4\,r^{2}+12\,r^{4})\,,$
(3.2)

with $r=m_{\text{ZB3L}}/m_{S}$. Here $K_{i}$’s denote the modified Bessel
functions. Now, for the $2\to 2$ scattering the dominant contribution comes
from $f\,f\to W^{3}(B)\,Z_{\text{B3L}}$, which has the form
$\langle\sigma\,v\rangle\sim g_{\text{B3L}}^{2}\,g_{2(1)}^{2}/T^{2}$, while
for all other channels we have $g_{\text{B3L}}^{4}$ dependence, and hence
negligible given the size of $g_{\text{B3L}}$ required for freeze-in. It is
important to note here that, there will be an upper bound on the $S$-lifetime
from big bang nucleosynthesis (BBN). If the decay occurs after BBN with
$\tau_{S}>1$ sec, entropy production has to be less than $\sim 10\%$ (assuming
no change to light element abundances due to it) because the precision
measurements of the baryon density through BBN and CMB observations match very
well [94, 95]. If $S$ decays before BBN, entropy production constraints are
non-existent. In the top left panel of Fig. 2 we show the BBN bound arising
from the lifetime of $S$. Here all the coloured regions are discarded from the
requirement of $\tau_{S}\lesssim 1$ sec. As one can notice, the bound gets
weaker for heavier $m_{S}$ since $\tau_{S}\propto 1/m_{S}$. Thus, $m_{S}=500$
GeV allows $g_{\text{B3L}}\gtrsim 10^{-15}$ for $m_{\text{ZB3L}}\simeq 1$ GeV,
as shown by the green region. For $m_{\text{ZB3L}}\gtrsim 5$ GeV, this turns
out to be a bit stronger, requiring $g_{\text{B3L}}\gtrsim 10^{-14}$ due to
suppression from phase space in the decay width. For the same choice of
$m_{S}$, it is possible to keep $Z_{\text{B3L}}$ in equilibrium till $T\sim
m_{S}$ for $m_{\text{ZB3L}}\simeq 1$ GeV with $g_{\text{B3L}}\gtrsim 10^{-8}$
as illustrated in the top right panel (blue contour). For heavier
$Z_{\text{B3L}}$, e.g., $m_{\text{ZB3L}}=10$ GeV (orange contour) this bound
slightly changes. Since we are interested in $m_{\text{ZB3L}}\lesssim 10$ GeV,
therefore it is possible to keep $Z_{\text{B3L}}$ in thermal equilibrium with
$g_{\text{B3L}}\gtrsim 10^{-8}$ that also satisfies the BBN bound for
$m_{S}=500$ GeV. Once the electroweak symmetry is also broken $(v_{d}\neq 0)$,
$Z_{\text{B3L}}$ can be produced via the decay $h_{1(2)}\to
Z_{\text{B3L}}Z_{\text{B3L}}$ (if kinematically allowed) and also via the
dominant scattering channel $f\,f\to Z_{\text{B3L}}\,\gamma$. The full
expressions of all the decay and scattering processes contributing to
$Z_{\text{B3L}}$ production in the post-EWSB scenario are gathered in
Appendix. A. In this regime we obtain similar bounds on $g_{\text{B3L}}$ for
$Z_{\text{B3L}}\lesssim 10$ GeV, with the masses and mixing of our choice as
mentioned earlier.

Figure 3: DM production channels from on-shell decay of $Z_{\text{B3L}}$
(left) and scattering mediated by $Z_{\text{B3L}}$ (right).

It is necessary to ensure that the DM is produced out of equilibrium with the
choices of $g_{\text{B3L}}$ obtained above. This is depicted in the bottom
left panel of Fig. 2. Here we show the bound on $g_{\text{B3L}}$ and $x$
requiring $\langle\Gamma_{Z_{\text{B3L}}\to\chi\chi}\rangle<\mathcal{H}$,
where we consider the decay to be dominant DM production channel. We find,
with $x=10^{-3}$, it is possible to achieve the condition of non-
thermalization till $T\sim 1$ MeV. Larger values of $x$ certainly makes the
the DM thermal at a comparatively larger temperature simply because
$\Gamma_{Z_{\text{B3L}}\to\chi\chi}\propto x^{2}\,g_{\text{B3L}}^{2}$.
However, the decay reaction rate can not compete with the Hubble rate at a
very high temperature as it is evident from the bottom left panel of Fig. 2.
For DM production via annihilation, this condition however is trivially
satisfied since $\langle\sigma v\rangle\propto x^{2}\,g_{\text{B3L}}^{4}$.
Finally, we would like to mention that the lifetime of $Z_{\text{B3L}}$ turns
out to be $\tau_{Z_{\text{B3L}}}\ll\tau_{\text{BBN}}$ for
$g_{\text{B3L}}\gtrsim 10^{-12}$ as shown in the bottom right panel, taking
all decay channels of $Z_{\text{B3L}}$ into account.

#### Boltzmann equation for dark matter yield

As it is usually done, while considering the production of DM from any SM/BSM
particle, the latter is implicitly assumed to be in thermal equilibrium. Hence
we usually do not need to solve a system of coupled Boltzmann equations, since
the equilibrium number density is assumed for the decaying mother particle.
But, here due to very low interaction strength of $Z_{\text{B3L}}$ (due to
small $g_{\text{B3L}}$ which also a universal coupling), it will not be in
thermal equilibrium with the rest of the particles. So, first, we find the
comoving number density of $Z_{\text{B3L}}$ by solving its Boltzmann equation.
Then we use this to find the relic density of the DM. Thus the DM production
in the present scenario is a two-step process (cf. Fig. 1). Once a non-zero
number density of $Z_{\text{B3L}}$ is formed from the SM bath, it is then
possible to produce DM from the on-shell decay of $Z_{\text{B3L}}$ and/or
scattering of the bath particles mediated by $Z_{\text{B3L}}$, depending on
the mass of the DM. The corresponding Feynman graphs are shown in Fig. 3. Note
that, the decay channel is always present irrespective of the EWSB (as long as
kinematically accessible), while all the SM particles are assumed to be
massless above the EWSB scale and massive below. Note that, in the case
$v_{s}=v_{d}=0$, since the $Z_{\text{B3L}}$ is massless, hence the DM can be
produced only through $s$-channel $Z^{\prime}$ mediation (cf. Fig. 3).
However, because of $1/s\sim 1/T^{2}$ dependence of the cross-section, the
contribution to DM yield from this channel will be heavily suppressed since
$T\gg v_{s}$. Since it is evident that the evolution of DM number density is
related to the primordial number density of $Z_{\text{B3L}}$, hence one has to
solve a set of coupled Boltzmann equations (cBEQ) to capture the evolution of
DM number density properly. In Eq. (3.3) we have written down the cBEQ in
terms of the dimensionless quantity $z=m_{\chi}/T$ that acts as a proxy for
the temperature of the thermal bath. The first equation takes care of the
evolution of $Z_{\text{B3L}}$ yield, where we define the yield as the ratio of
number density to entropy density of the visible sector as $Y_{i}=n_{i}/s$ for
the $i^{\text{th}}$ particle concerned. The equilibrium yield is given by

$Y_{\text{eq}}=\frac{45}{4\,\pi^{4}}\,\frac{g_{i}}{g_{\star
s}}\,(m_{i}/T)^{2}\,K_{2}(m_{i}/T)\,,$

where $g_{i}$ is the intrinsic number of degrees of freedom of the species
concerned (which is unity for $S$) and $g_{\star s}$ is the total number of
relativistic degrees of freedom present in the thermal bath. Note that,
depending on the heaviside $\Theta$ function, $Z_{\text{B3L}}$ production
channels before and after EWSB shall appear. The second equation takes care of
the evolution of DM yield, where we consider contribution to the DM number
density both from $Z_{\text{B3L}}$ decay (first term) and scattering mediated
by $Z_{\text{B3L}}$ (second term). We solve this set of equations numerically
to obtain the final DM relic abundance, utilizing the DM production cross-
sections and decay reported in Appendix. B.

$\displaystyle\frac{dY_{Z_{\text{B3L}}}}{dz}$ $\displaystyle=$
$\displaystyle\frac{s(m_{\chi})}{\mathcal{H}(m_{\chi})}\frac{1}{z^{2}}\langle\sigma
v_{{}_{ff\to Z_{\text{B3L}}\gamma}}\rangle\big{(}{Y_{\rm
eq}}^{2}-Y_{Z_{\text{B3L}}}Y_{\rm
eq}\big{)}-\frac{z}{\mathcal{H}(m_{\chi})}\langle\Gamma_{Z_{\text{B3L}}\to\chi\,\chi}\rangle
Y_{Z_{\text{B3L}}}$
$\displaystyle+\frac{z}{\mathcal{H}(m_{\chi})}\Big{(}\langle\Gamma_{S\to
Z_{\text{B3L}}Z_{\text{B3L}}}\rangle\Theta[z_{\rm
ew}-z]\big{(}Y_{S}^{\text{eq}}-Y_{Z_{\text{B3L}}}\big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\langle\Gamma_{h_{1,2}\to
Z_{\text{B3L}}Z_{\text{B3L}}}\rangle\Theta[z-z_{\rm
ew}]\big{(}Y_{h_{1,2}}^{\text{eq}}-Y_{Z_{\text{B3L}}}\big{)}\Big{)}$
$\displaystyle\frac{dY_{\chi}}{dz}$ $\displaystyle=$
$\displaystyle\frac{z}{\mathcal{H}(m_{\chi})}\langle\Gamma_{Z_{\text{B3L}}\to\chi~{}\chi}\rangle
Y_{Z_{\text{B3L}}}+\frac{s(m_{\chi})}{\mathcal{H}(m_{\chi})}\,\frac{1}{z^{2}}\,\langle\sigma
v_{\text{SM}\,\text{SM}\to\chi\chi}\rangle\,Y_{\text{eq}}^{2}\,,$ (3.3)

Figure 4: Evolution of yields $Y_{i}$ with inverse temperature $z$
($=\frac{m_{\chi}}{T}$). Here we have fixed $x=10^{-3}$ such that right relic
abundance is satisfied by the asymptotic DM yield shown via the thick black
curve.

where $s(m_{\chi})=\frac{2\pi^{2}}{45}\,g_{\star s}\,m_{\chi}^{3}$ and
$\mathcal{H}(T=m_{\chi})=\sqrt{\frac{\pi^{2}g_{\star\rho}}{90}}\,\frac{m_{\chi}^{2}}{M_{\rm
pl}}$. In order to visualize the evolution of the yields with the
dimensionless parameter $z$, we plot $Y_{\chi\,,Z_{\text{B3L}}}$, together
with $Y_{\text{eq}}$ in Fig. 4. Here we see, as expected, $Z_{\text{B3L}}$ is
first produced from the thermal bath via decay and scattering of the bath
particles, as denoted by the black dot-dashed curve. Once it is produced, it
starts following the equilibrium distribution from $z\sim 10^{-3}$ onward. As
the $Z_{\text{B3L}}\to\chi\chi$ decay is completed, the yield $Y_{\chi}$
saturates around $z\sim 10^{-1}$ and remains fixed till today, while the
$Z_{\text{B3L}}$ abundance keeps tracking the equilibrium. The final DM relic
density is determined using

$\Omega_{\chi}\,h^{2}=\left(2.75\times
10^{8}\right)\left(\frac{m_{\chi}}{\text{GeV}}\right)Y_{\chi}(T_{0})$ (3.4)

where $T_{0}$ is the temperature of the Universe at the present epoch. We must
also remember that the PLANCK [96] allowed relic density allows:
$\Omega_{\text{DM}}h^{2}=0.11933\pm 0.00091\,,$ which we will use to constrain
the relic density allowed parameter space.

###  3.2 Viable parameter space

Here we illustrate the relic density allowed parameter space in
$g_{\text{B3L}}-m_{\text{ZB3L}}$ plane for some benchmark choices of the DM
mass, and for different choices of DM charge $x$. This is also the
experimental plane which is of our present interest. Thus, we are going to
project both already existing and future bounds from different experiments on
this plane, together with contours satisfying observed DM abundance. We fix
two benchmark values of the DM mass: 10 MeV and 6 GeV. In the former case the
dominant contribution to the DM relic abundance comes from on-shell decay of
$Z_{\text{B3L}}$, together with sub-dominant contribution from SM scattering.
In the later case, on the other hand, the entire contribution arises from
scattering since the decay contribution is completely switched off as
$m_{\text{ZB3L}}<2\,m_{\chi}$. The black tilted straight line contours in Fig.
5 correspond to different choices of $x$ satisfying the observed relic
abundance. Here we see, larger $x$ is capable of producing the correct relic
for smaller $g_{\text{B3L}}$, while the bound on $g_{\text{B3L}}$ is
sufficiently relaxed for smaller $x$. Since decay plays the dominant role in
setting the right relic abundance in this case, the DM yield can be
analytically expressed as

$Y_{\chi}\approx\frac{135}{g_{\star
s}\,\sqrt{g_{\star\rho}}}\,\frac{x^{2}\,g_{\text{B3L}}^{2}}{4\,\pi^{5}}\,\frac{M_{\text{pl}}\,m_{\text{ZB3L}}^{3}}{m_{\chi}^{4}}\,\frac{\sqrt{1-4\,r^{2}}\,(1+2\,r^{2})}{r^{4}}\,,$
(3.5)

with $r=m_{\text{ZB3L}}/m_{\chi}$. This shows, for a fixed DM mass $m_{\chi}$
and fixed $x$, in order to satisfy the PLANCK data, one has to increase
$g_{\text{B3L}}$ with the increase in $m_{\text{ZB3L}}$ since $Y_{\chi}\propto
x^{2}\,g_{\text{B3L}}^{2}/m_{\text{ZB3L}}$. This pattern follows in Fig. 5. On
the other hand, for $m_{\chi}>m_{\text{ZB3L}}/2$, the only contribution to DM
yield arises from 2-to-2 scattering processes mediated by $Z_{\text{B3L}}$.
The DM yield in that case can be approximated as (in the limit of massless SM
states)

$\frac{dY_{\chi}}{dz}\approx\frac{g_{\text{B3L}}^{4}\,x^{2}}{\pi^{8}\,\sqrt{g_{\star\rho}}\,g_{\star
s}}\,\frac{M_{\text{pl}}}{m_{\chi}}\Biggl{[}4\,K_{1}(z)^{2}+z\,\sqrt{\pi}\,G_{1,3}^{3,0}\left(z^{2}\Bigg{|}\begin{array}[]{c}1\\\
-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\\\ \end{array}\right)\Biggr{]}\,,$ (3.6)

where $G^{m,n}_{p,q}$ are the Meijer’s $G$ functions, and in the case of
freeze-in with $m_{\chi}/T_{\text{FI}}\simeq\mathcal{O}(1)$ at the time of
freeze-in, the numerical value turns out to be $\mathcal{O}(10^{-1})$. Due to
$g_{\text{B3L}}^{4}$ dependence, in order to satisfy the relic abundance, one
has to choose a larger $x$ for a given $g_{\text{B3L}}$ compared to the case
where DM production takes place via decay. Note that, the expression in Eq.
(3.6) is independent of the mediator mass. This is exactly what is reflected
in Fig. 6, where we see the straight line contours of right relic abundance,
shown via different colours, do not depend on $m_{\text{ZB3L}}$.

#### Constraints on the dark matter parameter space

Since the SM fermions and RHN channels are always present irrespective of the
$B-3L_{i}$ gauge choices, the relic density allowed parameter space remains
almost unchanged for different choices of the $U(1)_{B-3L_{i}}$ gauge
symmetry. The main constraint on $g_{\text{B3L}}$ and $m_{\text{ZB3L}}$ arise
from the search of $U(1)_{B-3L_{i}}$ gauge boson in different collider and
beam dump experimental facilities, together with constraints from BBN on the
lifetime of $Z_{\text{B3L}}$ and also from non-standard neutrino interactions
(NSI) between neutrinos and matter (for recent review, see, for example [97,
98]). Apart from BBN, at low masses and small couplings, there is also a very
strong bound from Supernova 1987A (SN1987A) [99, 100, 101, 102], which occurs
due to the presence of the weakly coupled vector boson that can significantly
contribute to the energy loss during supernova explosion, thereby shortening
the duration of the observable neutrino pulse emitted during core collapse. We
mostly follow the constraints derived in Ref. [39], where the authors have
performed an updated analysis, and contrast current constraints and the
sensitivity reach of proposed experiments for $U(1)_{B-3L_{i}}$ gauge boson
for neutrino observables and quark flavour transitions with beam dump,
collider and precision experiments. As we will see, depending on the choice of
the gauge symmetry, the relic density allowed parameter space for the DM can
become more constrained or comparatively relaxed.

As it is evident from Fig. 5 and Fig. 6, larger values of $x$, i.e., smaller
$g_{\text{B3L}}\lesssim 10^{-9}$ are typically unconstrained from present as
well as future experimental sensitivities. For DM mass of 10 MeV, this
corresponds to $x\gtrsim 10^{-2}$, while for a DM of mass 6 GeV this is true
for $x\gtrsim 10^{5}$, due to the absence of decay. However, all such
$g_{\text{B3L}}$ values can still produce the observed relic abundance. In
case of $U(1)_{B-3L_{e}}$, with $x\gtrsim 10^{-3}$, corresponding to
$g_{\text{B3L}}\gtrsim 10^{-8}$, searches from beam dump and fixed target
experiments like E137, E141, E774 [103], NuCal/U70 [104, 105] , LSND [106]
already discard $m_{\text{ZB3L}}\lesssim 200$ MeV. Future novel fixed target
experiments with high intensity beams, e.g., SHiP [107] or the LHC forward
detector FASER [108, 109, 110, 111], can provide a much better sensitivity in
probing the relic density allowed parameter space for $m_{\text{ZB3L}}\sim
1.8$ GeV. Increasing $g_{\text{B3L}}$ further makes the viable parameter space
available to $e^{+}e^{-}$ colliders like Belle-II [112], which is found to be
more sensitive than the existing limits from BaBar [113, 114]. For
$m_{\text{ZB3L}}\lesssim 0.1$ GeV, there arise bounds due to elastic neutrino-
nucleus scattering (CE$\nu$NS) on liquid argon, measured at the COHERENT [115]
facility. Bounds from the neutrino non-standard interactions (NSI) [116, 117]
and Borexino [118, 119, 120] can constraint the parameter space for
$g_{\text{B3L}}\gtrsim 10^{-5}$. We have adopted the NSI limits derived in
[117], where the authors have considered lepton flavor dependent
$U(1)^{\prime}$ interactions444For similar limits from NSI see Ref. [121]..
For $m_{\text{ZB3L}}\lesssim 10$ MeV, the $\Delta N_{\text{eff}}$ bound due to
heating of the neutrino bath during neutrino decoupling in the early Universe
[122, 123], becomes more stringent, which is substantially weaker for
$m_{\text{ZB3L}}>10$ MeV (cf.Fig. 6).

Figure 5: The black tilted straight contours having different patterns satisfy
PLANCK observed relic density for different choices of the DM charge $x$, in
increasing order from top to bottom. The DM mass is fixed at 10 MeV. Bounds
from different experiments are shown in different colours.

Figure 6: Same as Fig. 5, but with DM mass $m_{\chi}=6$ GeV. Here the straight
lines with different colors denote right relic abundance for different choices
of $x$ in decreasing order from bottom to top (see text).

Measurements of muonic neutrino tridents:
$\nu_{\mu}\to\nu_{\mu}\,\mu^{+}\,\mu^{-}$ performed at the CCFR [124, 125]
facility has found no excess over the SM expectations, thus posing a bound on
$g_{\text{B3L}}-m_{\text{ZB3L}}$ plane in the case of $U(1)_{B-3L_{\mu}}$, as
shown by the solid white curve in the top right panel of Fig. 5 and Fig. 6.
Similar to the $B-3L_{e}$ scenario, in $B-3L_{\mu}$ case, future sensitivity
of FASER2 and SHiP can provide extremely good probe to the relic density
allowed parameter space for $g_{\text{B3L}}\lesssim 10^{-6}$. Importantly, DM
parameter space with $g_{\text{B3L}}\gtrsim 10^{-5}$ can be probed with the
planned upgraded run of NA64 with a dedicated muon beam (shown by the blue
curve). Again, bounds from NSI and COHERENT becomes important for
$m_{\text{ZB3L}}\lesssim 0.1$ GeV. Here we also show the 3$\sigma$ preferred
region of the observed anomalous $(g-2)_{\mu}$ excess in red (See Appendix C
for the computation) [126, 127, 128, 129, 130]. However, a common explanation
of both $(g-2)_{\mu}$ anomaly and DM abundance is possible with
$g_{\text{B3L}}\sim 10^{-3}$, for which one has to tune the $x$-charge
accordingly to ensure freeze-in production remains non-thermal.

Due to hadronic final state, the parameter space for $U(1)_{B-3L_{\tau}}$
scenario is comparatively relaxed than the other two cases. The most stringent
bound comes from NSI-induced neutrino oscillations that typically constraints
$g_{\text{B3L}}\gtrsim 10^{-5}$. Large $g_{\text{B3L}}$ values are constrained
from the measurement of scattering of tau neutrinos produced in the decay of
$Z_{\text{B3L}}$ with DONuT detector [131]555$\nu_{\tau}$ related NSI and
decays can also become important if DUNE experiment involves modified beam
flux as shown in Ref. [132, 133].. LEP measurement of partial width of $Z$
into taus can also constraint a part of the parameter space for large
$g_{\text{B3L}}$ [134] (shown in orange in the bottom panel of Fig. 5 and Fig.
6). We also project bounds from the proposed scattering and neutrino detector
by SHiP, known as SND@SHiP (white dashed curve) [107] and also a rather weaker
bound from SND@LHC (brown dot-dashed curve) [135]. Although the SND@SHiP
proposal benefits from a much larger neutrino flux, its sensitivity is limited
by systematic uncertainties, resulting in roughly the same reach as FASER$\nu$
(blue solid curve) [136]. The reach of SND@SHiP can be further improved by
reducing the flux uncertainty as proposed by the DsTau experiment [137, 138]
at the CERN-SPS, that plans to study tau-neutrino production by directly
measuring $D_{s}\to\nu_{\tau}\,X$ decay. The search for a light $Z^{\prime}$
boson at high-intensity facilities such as the near detector complex of the
Deep Underground Neutrino Experiment (DUNE) [139, 140] has been studied in
[141, 102]. Here we see, DUNE can probe heavier $Z_{\text{B3L}}$ and smaller
$g_{\text{B3L}}$, covering most of the region of existing beam-dump searches.
The DUNE loses its sensitivity in the visible channel for
$m_{\text{ZB3L}}<2\,m_{e}$, where the dominant decay turns out to be into
light neutrino pairs [142]. The relic density allowed parameter space for
$g_{\text{B3L}}\lesssim 10^{-6}$ is totally unconstrained from experimental
bounds for $m_{\text{ZB3L}}\gtrsim 10^{-2}$ GeV for the $U(1)_{B-3L_{\tau}}$
scenario. However, future projections from FASER2 and SHiP can potentially
probe a part of the parameter space for $m_{\text{ZB3L}}$ around 1 GeV. It is
interesting to note that experiments like SHiP and FASER can probe different
mass regime of $Z_{\text{B3L}}$, hence a different DM parameter space,
depending on the flavour structure. For $B-3L_{\mu}$ and $B-3L_{\tau}$ future
projections from SHiP and FASER cover mostly higher $m_{\text{ZB3L}}$ than in
$B-3L_{e}$. Finally, we note that the $Z_{\text{B3L}}$ mass also lies within
the window of the new “X17” boson of mass $17$ MeV, observed by the ATOMKI
collaboration [143, 144] in the decay of excited states of Beryllium yielding
$e^{+}\,e^{-}$ pairs 666Such scenarios are also being searched for in positron
beam-dump experiment like PADME [145]..

In the absence of kinetic mixing between $U(1)_{Y}$ and $U(1)_{X}$ gauge
groups, the models under consideration will contribute to flavour changing
quark transitions at one loop level. As a result, the light $Z_{\text{B3L}}$
can be produced in meson decays and subsequently decay to leptons to which it
couples to. Hence constraints from FCNC decays such as $B\to
K^{(*)}\ell^{+}\ell^{-}$, $B^{+}\to K^{+}\nu\bar{\nu}$,
$B\to\pi+\text{invisible}$, $K\to\pi+\text{invisible}$ etc. can, in general,
put strong bounds on the $\\{g_{\text{B3L}},m_{\text{ZB3L}}\\}$ parameter
space. For example, LHCb has put upper limits on the branching fractions of
$\mathcal{B}(B^{0}\to
K^{0*}Z^{\prime})\times\mathcal{B}(Z^{\prime}\to\mu^{+}\mu^{-})$ [146] and
$\mathcal{B}(B^{+}\to
K^{+}Z^{\prime})\times\mathcal{B}(Z^{\prime}\to\mu^{+}\mu^{-})$ [147] where
$Z^{\prime}$ is a light ($0.2\,\lesssim m_{Z^{\prime}}\,\lesssim 4$ GeV)
hidden sector boson produced as a resonance. Such experimental constraints are
also relevant for our models of interest. The $U(1)_{B-3L_{\mu}}$ model, is
especially compelling since it not only allows us to explain the observed
anomalous magnetic moment of the muon777The same $Z_{\text{B3L}}$-mediated
diagram (Fig. 7) will also contribute to $\Delta a_{e}$ in case of
$U(1)_{B-3L_{e}}$ scenario, however, this contribution can not explain the
relative sign in $\Delta a_{e}$., but could also have the potential to explain
the anomalies in $b\to s\ell\ell$ decays888For studies of light vector boson
models in the context of rare FCNC decays and $(g-2)_{\mu}$ see [148, 149,
150, 151, 152, 153, 154].. However, the realm of the sub-GeV vector boson with
$g_{\text{B3L}}\lesssim 10^{-5}$, as required for the dark matter freeze-in,
is beyond the reach of current experimental sensitivity. As already pointed
out in [39], future experimental searches would require a sensitivity of
$\mathcal{B}(B\to K(e^{+}e^{-}/\mu^{+}\mu^{-}))\leq 10^{-15}$ for
$m_{\text{ZB3L}}\sim 1$ GeV in order to probe the relic favoured parameter
space shown in Figs. 5 and 6.

##  4 Conclusion

In this work we have explored the viable parameter space for freeze-in
production of dark matter (DM) in flavoured gauge extensions of the SM, where
the SM gauge symmetry is augmented by an abelian $U(1)_{B-3L_{i}}$ local gauge
group, with $i\in e\,,\mu\,,\tau$. Apart from anomaly cancellation, three
generations of right handed neutrinos (RHNs) take part in explaining the
origin of small neutrino masses via Type-I seesaw. The DM is assumed to be a
singlet vector-like neutral lepton that carries a non-trivial charge under the
new gauge symmetry which in turn also protects the DM from decaying. The
charges of the new particles are assigned in a way that apart from the heavy
neutral gauge boson $Z_{\text{B3L}}$, the DM does not have any other portal to
the visible sector. Thus, the DM phenomenology is decided by the new gauge
coupling $g_{\text{B3L}}$ and mass of the new gauge boson $m_{\text{ZB3L}}$,
together with the DM charge $x$ and DM mass $m_{\chi}$.

Due to the universal coupling of $Z_{\text{B3L}}$, we employ the two-step
freeze-in mechanism, where $Z_{\text{B3L}}$ is first produced from the thermal
bath and then its further decay leads to the production of the DM $\chi$.
Besides the decay, the particles in the bath can also undergo $2\to 2$
annihilation mediated by $Z_{\text{B3L}}$, pair-producing the DM particles. We
find, it is possible to satisfy the PLANCK observed relic density for
$m_{\text{ZB3L}}\lesssim m_{\chi}$ and $m_{\text{ZB3L}}\gtrsim 2\,m_{\chi}$.
Particularly, in the former scenario the DM abundance becomes independent of
the mediator mass $m_{Z_{\text{B3L}}}$.

For gauge coupling values so tiny as mandated from the requirement of the
freeze-in criterion in the early Universe, existing experimental searches are
not sensitive to flavor observables like, e.g., flavour changing transitions
in the mass basis. However some constraints arise on flavour non-universal
couplings through non-standard interactions (NSI) due to its couplings to
baryons or electrons are present, as evident, e.g., from $U(1)_{B-3L_{\mu}}$
gauge boson from recent resonance searches for $B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}$ at the LHCb. We typically focus on the region of the
parameter space where the mass $m_{\text{ZB3L}}\lesssim 10$ GeV, which is of
interest in several energy and intensity frontier experiments. We find the DM
parameter space becomes very tightly constrained for the case of
$U(1)_{B-3L_{e}}$ and $U(1)_{B-3L_{\mu}}$ scenarios, while for
$U(1)_{B-3L_{\tau}}$ the constraints are much relaxed as one can see from the
bottom panel of Fig. 5 and Fig. 6, depending on the size of $g_{\text{B3L}}$.
We show, experiments like FASER, FASER2, FASER$\nu$ and SHiP etc. will be able
to probe a substantial part of the freeze-in parameter space for size of the
BSM gauge coupling $g_{\text{B3L}}\simeq\mathcal{O}(10^{-8})$ and for
$m_{\text{ZB3L}}\sim 1$ GeV, while larger coupling values can be constrained
from experiments like Belle-II, BaBar and from non-standard neutrino
interactions (NSI).

We thus conclude, in a well-motivated UV-complete model with possible
explanation of the hierarchy of the flavor structure in the leptonic sector,
one may envisage a flavorful vector-portal non-thermal DM candidate via
freeze-in. Additionally, one will be able to probe the predicted tiny freeze-
in couplings in several current and upcoming light dark world of experimental
facilities, and see imprints of non-thermal DM which eluded detection
prospects in the past with traditional DM direct searches.

## Acknowledgements

BB received funding from the Patrimonio Autónomo - Fondo Nacional de
Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José
de Caldas (MinCiencias - Colombia) grant 80740-465-2020. This project has
received funding /support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
860881-HIDDeN. LM received funding from NSF Grant No. PHY1915142. PG would
like to acknowledge the support from DAE, India for the Regional Centre for
Accelerator based Particle Physics (RECAPP), Harish Chandra Research
Institute. The authors would like to thank Sudip Jana, Soumitra Nandi and
Nobuchika Okada for useful discussions, and Arindam Das and P S Bhupal Dev for
providing valuable feedback on the manuscript.

##  Appendix A $Z_{\text{B3L}}$ production from SM bath

Here we have gathered the expressions for all decay and annihilation cross-
sections contributing to $Z_{\text{B3L}}$ production, specifically after EWSB.
The $h_{1,2}\to Z_{\text{B3L}}Z_{\text{B3L}}$ decay width reads

$\Gamma_{h_{1(2)}\to
Z_{\text{B3L}}Z_{\text{B3L}}}=\frac{9\,g_{\text{B3L}}^{2}\,\sin^{2}(\cos^{2})\theta}{32\,\pi\,r^{2}}\,\frac{m_{\text{ZB3L}}^{2}}{m_{1(2)}}\,\sqrt{1-4\,r^{2}}\,(1-4\,r^{2}+12\,r^{4})\,;\,r=m_{\text{ZB3L}}/m_{1(2)}\,,$
(A.1)

Next we have the most dominating annihilation channel $ff\to
Z_{\text{B3L}}\,,\gamma$, $(f\in q,\ell)$, which give rise to a cross-section

$\displaystyle\sigma(s)_{qq\to
Z_{\text{B3L}}\gamma}=\frac{g_{\text{B3L}}^{2}\,m_{W}^{4}}{\mathcal{A}}\,s^{5/2}\,\sqrt{1-\frac{m_{\text{ZB3L}}^{2}}{s}}\,\Biggl{(}1-\frac{4\,m_{q}^{2}}{s}\Biggr{)}^{-1/4}\,\mathcal{B}\,,$
(A.2)

where

$\displaystyle\mathcal{A}=3888\,\pi^{2}\,m_{Z}^{2}\,\pi\,v_{d}^{2}\,\left(s-4m_{q}^{2}\right)\,\left(s-m_{\text{ZB3L}}^{2}\right)^{3}$
$\displaystyle\left[m_{\text{ZB3L}}^{2}s-4m_{q}^{2}\left(m_{\text{ZB3L}}^{2}-s\right)\right]\,\left[4m_{q}^{2}\,\left(m_{\text{ZB3L}}^{2}-s\right)+s\left(2s-m_{\text{ZB3L}}^{2}\right)\right]\,,$
(A.3)

and

$\displaystyle\mathcal{B}=\frac{\mathcal{C}}{s^{2}}+\frac{\mathcal{D}}{s^{4}}\,\log\Biggl{[}\frac{s\left(m_{\text{ZB3L}}^{2}-2s\right)-4m_{q}^{2}\left(m_{\text{ZB3L}}^{2}-s\right)}{4m_{q}^{2}\left(m_{\text{ZB3L}}^{2}-s\right)-m_{\text{ZB3L}}^{2}s}\Biggr{]}\,,$
(A.4)

with

$\displaystyle\mathcal{C}=\left(4m_{q}^{2}-s\right)\left(s-m_{\text{ZB3L}}^{2}\right)^{3}$
$\displaystyle\left[16m_{q}^{4}-s^{2}\left(\frac{\left(s-4m_{q}^{2}\right)^{2}\left(m_{\text{ZB3L}}^{2}-s\right)^{4}}{s^{6}}-1\right)+8m_{q}^{2}m_{\text{ZB3L}}^{2}+m_{\text{ZB3L}}^{4}-2m_{\text{ZB3L}}^{2}s\right]\,,$
(A.5)

and

$\displaystyle\mathcal{D}=\left(-8m_{q}^{4}-4m_{q}^{2}\left(m_{\text{ZB3L}}^{2}-s\right)+m_{\text{ZB3L}}^{4}+s^{2}\right)\Biggl{[}-16m_{q}^{4}\left(m_{\text{ZB3L}}^{2}-s\right)^{4}+8m_{q}^{2}s\left(m_{\text{ZB3L}}^{2}-s\right)^{4}$
$\displaystyle-
m_{\text{ZB3L}}^{2}s^{2}\left(m_{\text{ZB3L}}^{2}-2s\right)\left(m_{\text{ZB3L}}^{2}-s\right)^{2}\Biggr{]}\,.$
(A.6)

The heavy gauge bosons of the SM can also give rise to pair production of
$Z_{\text{B3L}}$, which read

$\displaystyle\sigma(s)_{VV\to
Z_{\text{B3L}}Z_{\text{B3L}}}\simeq\frac{g_{\text{B3L}}^{2}\,m_{W}^{4}}{32\,\pi\,v_{d}^{2}\,s}\,\sqrt{\frac{s-4\,m_{\chi}^{2}}{s-4\,m_{V}^{2}}}\,\frac{(s^{2}-4\,m_{V}^{2}\,s+12\,m_{V}^{4})\,(s^{2}-4\,m_{\text{ZB3L}}^{2}\,s+12\,m_{\text{ZB3L}}^{4})}{m_{\text{ZB3L}}^{2}\,(m_{W}^{2}-m_{Z}^{2})^{2}}$
$\displaystyle\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{1}^{2})^{2}+\Gamma_{1}^{2}\,m_{1}^{2}}\Biggr{]}\,\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{2}^{2})^{2}+\Gamma_{2}^{2}\,m_{2}^{2}}\Biggr{]}\,\sin^{2}\theta\,\cos^{2}\theta+\mathcal{O}[g_{\text{B3L}}^{4}]\,,$
(A.7)

The contributions from the SM fermions give rise to

$\displaystyle\sigma(s)_{qq\to
Z_{\text{B3L}}Z_{\text{B3L}}}\simeq\frac{3\,g_{\text{B3L}}^{2}}{64\,\pi\,s}\,\cos^{2}\theta\,\sin^{2}\theta\,\frac{m_{q}^{2}\,m_{W}^{2}}{m_{\text{ZB3L}}^{2}\,v_{d}^{2}}\sqrt{(s-4\,m_{\text{ZB3L}}^{2})(s-4\,m_{q}^{2})}$
$\displaystyle\Biggl{[}\frac{s^{2}-4\,s\,m_{\text{ZB3L}}^{2}+12\,m_{\text{ZB3L}}^{4}}{m_{Z}^{2}-m_{W}^{2}}\Biggr{]}\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{1}^{2})^{2}+\Gamma_{1}^{2}\,m_{1}^{2}}\Biggr{]}\,\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{2}^{2})^{2}+\Gamma_{2}^{2}\,m_{2}^{2}}\Biggr{]}+\mathcal{O}[g_{\text{B3L}}^{4}]\,,$
(A.8)

$\displaystyle\sigma(s)_{\ell\ell\to
Z_{\text{B3L}}Z_{\text{B3L}}}\simeq\frac{9\,g_{\text{B3L}}^{2}}{64\,\pi\,s}\,\cos^{2}\theta\,\sin^{2}\theta\,\frac{m_{\ell}^{2}\,m_{W}^{2}}{m_{\text{ZB3L}}^{2}\,v_{d}^{2}}\sqrt{(s-4\,m_{\text{ZB3L}}^{2})(s-4\,m_{q}^{2})}$
$\displaystyle\Biggl{[}\frac{s^{2}-4\,s\,m_{\text{ZB3L}}^{2}+12\,m_{\text{ZB3L}}^{4}}{m_{Z}^{2}-m_{W}^{2}}\Biggr{]}\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{1}^{2})^{2}+\Gamma_{1}^{2}\,m_{1}^{2}}\Biggr{]}\,\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{2}^{2})^{2}+\Gamma_{2}^{2}\,m_{2}^{2}}\Biggr{]}+\mathcal{O}[g_{\text{B3L}}^{4}]\,,$
(A.9)

Finally, the SM scalars can also contribute the $Z_{\text{B3L}}$ number
density with cross-sections

$\displaystyle\sigma(s)_{h_{1(2)}h_{1(2)}\to
Z_{\text{B3L}}Z_{\text{B3L}}}\simeq\frac{9\,s\,g_{\text{B3L}}^{2}}{32\,\pi\,v_{d}^{2}}\,\Biggl{(}1+\frac{2\,m_{1(2)}^{2}}{s}\Biggr{)}^{2}\,\sqrt{\frac{s-4\,m_{\text{ZB3L}}^{2}}{s-4\,m_{1(2)}^{2}}}\,\sin^{2}(\cos^{2})\theta$
$\displaystyle\,\Biggl{[}\frac{s^{2}-4\,s\,m_{\text{ZB3L}}^{2}+12\,m_{\text{ZB3L}}^{4}}{m_{\text{ZB3L}}^{2}}\Biggr{]}\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{1}^{2})^{2}+\Gamma_{1}^{2}\,m_{1}^{2}}\Biggr{]}\,\Biggl{[}\frac{(m_{1}^{2}-m_{2}^{2})}{(s-m_{2}^{2})^{2}+\Gamma_{2}^{2}\,m_{2}^{2}}\Biggr{]}+\mathcal{O}[g_{\text{B3L}}^{4}]\,,$
(A.10)

##  Appendix B Decays and annhilations for freeze-in

The available decay channels for $Z_{\text{B3L}}$ are

$\displaystyle\Gamma_{Z_{\text{B3L}}\to
qq}=\frac{g_{\text{B3L}}^{2}\,m_{\text{ZB3L}}}{36\,\pi}\,\sqrt{1-4\,r^{2}}\,(1+2\,r^{2})\,;\,r=m_{q}/m_{\text{ZB3L}}\,,$
(B.1)

$\displaystyle\Gamma_{Z_{\text{B3L}}\to\ell\ell}=\frac{3\,g_{\text{B3L}}^{2}\,m_{\text{ZB3L}}}{4\,\pi}\,\sqrt{1-4\,r^{2}}\,(1+2\,r^{2})\,;\,r=m_{\ell}/m_{\text{ZB3L}}\,,$
(B.2)

$\Gamma_{Z_{\text{B3L}}\to\chi\chi}=\frac{x^{2}\,g_{\text{B3L}}^{2}\,m_{\text{ZB3L}}}{12\,\pi}\,\sqrt{1-4\,r^{2}}\,(1+2\,r^{2})\,;r=m_{\chi}/m_{\text{ZB3L}}\,.$
(B.3)

Annihilation cross-sections contributing to DM pair production read

$\displaystyle\sigma(s)_{qq\to\chi\chi}=\frac{g_{\text{B3L}}^{4}\,x^{2}\,s}{324\,\pi}\,\sqrt{\frac{s-4\,m_{\chi}^{2}}{s-4\,m_{q}^{2}}}\,\Biggl{[}\frac{1}{(s-m_{\text{ZB3L}}^{2})^{2}+\Gamma_{Z_{\text{B3L}}}^{2}\,m_{\text{ZB3L}}^{2}}\Biggr{]}\,\Biggl{(}1+\frac{2\,m_{q}^{2}}{s}\Biggr{)}\,\Biggl{(}1+\frac{2\,m_{\chi}^{2}}{s}\Biggr{)}\,.$
(B.4)

$\displaystyle\sigma(s)_{\ell\ell\to\chi\chi}=\frac{3\,x^{2}g_{\text{B3L}}^{4}\,s}{4\,\pi}\sqrt{\frac{s-4\,m_{\chi}^{2}}{s-4\,m_{\ell}^{2}}}\,\Biggl{[}\frac{1}{(s-m_{\text{ZB3L}}^{2})^{2}+\Gamma_{Z_{\text{B3L}}}^{2}\,m_{\text{ZB3L}}^{2}}\Biggr{]}\,\Biggl{(}1+\frac{2\,m_{q}^{2}}{s}\Biggr{)}\,\Biggl{(}1+\frac{2\,m_{\chi}^{2}}{s}\Biggr{)}\,.$
(B.5)

$\displaystyle\sigma(s)_{N_{R}N_{R}\to\chi\chi}=\frac{3\,x^{2}\,g_{\text{B3L}}^{4}\,s}{8\,\pi}\,\sqrt{\frac{s-4\,m_{\chi}^{2}}{s-4\,m_{N}^{2}}}\,\Biggl{[}\frac{1}{(s-m_{\text{ZB3L}}^{2})^{2}+\Gamma_{Z_{\text{B3L}}}^{2}\,m_{\text{ZB3L}}^{2}}\Biggr{]}\,\Biggl{(}1-\frac{2\,m_{N}^{2}}{s}\Biggr{)}\,\Biggl{(}1+\frac{2\,m_{\chi}^{2}}{s}\Biggr{)}\,.$
(B.6)

##  Appendix C Muon anomalous magnetic moment

Figure 7: $Z_{\text{B3L}}$ mediated one loop contribution to $\Delta a_{\mu}$.

The contribution to the anomalous magnetic moment of the muon from the one
loop diagram shown in Fig. 7 is given by

$\Delta
a_{\mu}=\frac{9\,g_{\text{B3L}}^{2}\,m_{\mu}^{2}}{8\,\pi^{2}}\,\int_{0}^{1}dx\frac{2\,x^{2}\,(1-x)}{(1-x)\,m_{\text{ZB3L}}^{2}+x^{2}\,m_{\mu}^{2}}\,.$
(C.1)

## References

  * [1] S. Khalil and O. Seto, _Sterile neutrino dark matter in B - L extension of the standard model and galactic 511-keV line_ , _JCAP_ 10 (2008) 024 [0804.0336].
  * [2] N. Okada and O. Seto, _Higgs portal dark matter in the minimal gauged $U(1)_{B-L}$ model_, _Phys. Rev. D_ 82 (2010) 023507 [1002.2525].
  * [3] S. Kanemura, O. Seto and T. Shimomura, _Masses of dark matter and neutrino from TeV scale spontaneous $U(1)_{B-L}$ breaking_, _Phys. Rev. D_ 84 (2011) 016004 [1101.5713].
  * [4] M. Lindner, D. Schmidt and T. Schwetz, _Dark Matter and neutrino masses from global U(1) B−L symmetry breaking_, _Phys. Lett. B_ 705 (2011) 324 [1105.4626].
  * [5] N. Okada and Y. Orikasa, _Dark matter in the classically conformal B-L model_ , _Phys. Rev. D_ 85 (2012) 115006 [1202.1405].
  * [6] T. Basak and T. Mondal, _Constraining Minimal $U(1)_{B-L}$ model from Dark Matter Observations_, _Phys. Rev. D_ 89 (2014) 063527 [1308.0023].
  * [7] S. Kanemura, T. Matsui and H. Sugiyama, _Neutrino mass and dark matter from gauged $U(1)_{B-L}$ breaking_, _Phys. Rev. D_ 90 (2014) 013001 [1405.1935].
  * [8] N. Okada and S. Okada, _$Z^{\prime}_{BL}$ portal dark matter and LHC Run-2 results_, _Phys. Rev. D_ 93 (2016) 075003 [1601.07526].
  * [9] A. Biswas, S. Choubey and S. Khan, _Galactic gamma ray excess and dark matter phenomenology in a $U(1)_{B-L}$ model_, _JHEP_ 08 (2016) 114 [1604.06566].
  * [10] D. Nanda and D. Borah, _Common origin of neutrino mass and dark matter from anomaly cancellation requirements of a $U(1)_{B-L}$ model_, _Phys. Rev. D_ 96 (2017) 115014 [1709.08417].
  * [11] S. Singirala, R. Mohanta, S. Patra and S. Rao, _Majorana Dark Matter in a new $B−L$ model_, _JCAP_ 11 (2018) 026 [1710.05775].
  * [12] P. Bandyopadhyay, E.J. Chun and R. Mandal, _Implications of right-handed neutrinos in $B-L$ extended standard model with scalar dark matter_, _Phys. Rev. D_ 97 (2018) 015001 [1707.00874].
  * [13] V. De Romeri, E. Fernandez-Martinez, J. Gehrlein, P.A.N. Machado and V. Niro, _Dark Matter and the elusive $Z′$ in a dynamical Inverse Seesaw scenario_, _JHEP_ 10 (2017) 169 [1707.08606].
  * [14] D. Borah, D. Nanda, N. Narendra and N. Sahu, _Right-handed neutrino dark matter with radiative neutrino mass in gauged B $-$ L model_, _Nucl. Phys. B_ 950 (2020) 114841 [1810.12920].
  * [15] N. Okada, S. Okada and D. Raut, _Natural Z’ -portal Majorana dark matter in alternative U(1) extended standard model_ , _Phys. Rev. D_ 100 (2019) 035022 [1811.11927].
  * [16] A. Das, N. Okada, S. Okada and D. Raut, _Probing the seesaw mechanism at the 250 GeV ILC_ , _Phys. Lett. B_ 797 (2019) 134849 [1812.11931].
  * [17] A. Biswas, D. Borah and D. Nanda, _Type III seesaw for neutrino masses in U(1) B−L model with multi-component dark matter_, _JHEP_ 12 (2019) 109 [1908.04308].
  * [18] A. Das, S. Goswami, K.N. Vishnudath and T. Nomura, _Constraining a general U(1) ′ inverse seesaw model from vacuum stability, dark matter and collider_, _Phys. Rev. D_ 101 (2020) 055026 [1905.00201].
  * [19] V. Baules, N. Okada and S. Okada, _Braneworld Cosmological Effect on Freeze-in Dark Matter Density and Lifetime Frontier_ , 1911.05344.
  * [20] R.N. Mohapatra and N. Okada, _Dark Matter Constraints on Low Mass and Weakly Coupled B-L Gauge Boson_ , _Phys. Rev. D_ 102 (2020) 035028 [1908.11325].
  * [21] R.N. Mohapatra and N. Okada, _Freeze-in Dark Matter from a Minimal B-L Model and Possible Grand Unification_ , _Phys. Rev. D_ 101 (2020) 115022 [2005.00365].
  * [22] M. Dutta, S. Bhattacharya, P. Ghosh and N. Sahu, _Singlet-Doublet Majorana Dark Matter and Neutrino Mass in a minimal Type-I Seesaw Scenario_ , _JCAP_ 03 (2021) 008 [2009.00885].
  * [23] A. Biswas, D. Borah and D. Nanda, _Type III Seesaw and Two-Component Dark Matter in $U(1)_{B-L}$ Model_, _Springer Proc. Phys._ 248 (2020) 297.
  * [24] N. Okada, S. Okada, D. Raut and Q. Shafi, _Dark matter $Z^{\prime}$ and XENON1T excess from $U(1)_{X}$ extended standard model_, _Phys. Lett. B_ 810 (2020) 135785 [2007.02898].
  * [25] P. Ghosh, S. Mahapatra, N. Narendra and N. Sahu, _Experimentally Verifiable $U(1)_{B-L}$ Symmetric Model with Type-II Seesaw and Dark Matter_, 2107.11951.
  * [26] N. Nath, N. Okada, S. Okada, D. Raut and Q. Shafi, _Light $Z^{\prime}$ and Dirac fermion dark matter in the $B-L$ model_, 2112.08960.
  * [27] X.-G. He, G.C. Joshi, H. Lew and R.R. Volkas, _Simplest Z-prime model_ , _Phys. Rev. D_ 44 (1991) 2118.
  * [28] S. Baek, N.G. Deshpande, X.-G. He and P. Ko, _Muon anomalous $g-2$ and gauged ${L}_{\mu}{-l}_{\tau}$ models_, _Phys. Rev. D_ 64 (2001) 055006.
  * [29] E. Ma, D.P. Roy and S. Roy, _Gauged L(mu) - L(tau) with large muon anomalous magnetic moment and the bimaximal mixing of neutrinos_ , _Phys. Lett. B_ 525 (2002) 101 [hep-ph/0110146].
  * [30] E. Salvioni, A. Strumia, G. Villadoro and F. Zwirner, _Non-universal minimal Z’ models: present bounds and early LHC reach_ , _JHEP_ 03 (2010) 010 [0911.1450].
  * [31] J. Heeck and W. Rodejohann, _Gauged $L_{\mu}-L_{\tau}$ Symmetry at the Electroweak Scale_, _Phys. Rev. D_ 84 (2011) 075007 [1107.5238].
  * [32] K. Harigaya, T. Igari, M.M. Nojiri, M. Takeuchi and K. Tobe, _Muon g-2 and LHC phenomenology in the $L_{\mu}-L_{\tau}$ gauge symmetric model_, _JHEP_ 03 (2014) 105 [1311.0870].
  * [33] C.D. Carone, _Flavor-nonuniversal dark gauge bosons and the muon g−2_ , _Physics Letters B_ 721 (2013) 118.
  * [34] W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, _Quark flavor transitions in $L_{\mu}-L_{\tau}$ models_, _Phys. Rev. D_ 89 (2014) 095033 [1403.1269].
  * [35] Y. Farzan, _A model for large non-standard interactions of neutrinos leading to the LMA-Dark solution_ , _Phys. Lett. B_ 748 (2015) 311 [1505.06906].
  * [36] Y. Farzan and I.M. Shoemaker, _Lepton Flavor Violating Non-Standard Interactions via Light Mediators_ , _JHEP_ 07 (2016) 033 [1512.09147].
  * [37] A. Biswas, S. Choubey and S. Khan, _FIMP and Muon ( $g-2$) in a U$(1)_{L_{\mu}-L_{\tau}}$ Model_, _JHEP_ 02 (2017) 123 [1612.03067].
  * [38] A. Biswas and A. Shaw, _Reconciling dark matter, $R_{K^{(*)}}$ anomalies and $(g-2)_{\mu}$ in an ${L_{\mu}-L_{\tau}}$ scenario_, _JHEP_ 05 (2019) 165 [1903.08745].
  * [39] M. Bauer, P. Foldenauer and M. Mosny, _Flavor structure of anomaly-free hidden photon models_ , _Phys. Rev. D_ 103 (2021) 075024 [2011.12973].
  * [40] E. Ma, _Gauged B - 3L(tau) and radiative neutrino masses_ , _Phys. Lett. B_ 433 (1998) 74 [hep-ph/9709474].
  * [41] E. Ma and U. Sarkar, _Gauged B - 3L(tau) and baryogenesis_ , _Phys. Lett. B_ 439 (1998) 95 [hep-ph/9807307].
  * [42] L.N. Chang, O. Lebedev, W. Loinaz and T. Takeuchi, _Constraints on gauged B - 3 L(tau) and related theories_ , _Phys. Rev. D_ 63 (2001) 074013 [hep-ph/0010118].
  * [43] P.B. Pal and U. Sarkar, _Gauged B - 3L(tau), low-energy unification and proton decay_ , _Phys. Lett. B_ 573 (2003) 147 [hep-ph/0306088].
  * [44] Q. Chang, X.-Q. Li and Y.-D. Yang, _Family Non-universal Z-prime effects on anti-B(q) - B(q) mixing, B — $>$ X(s) mu+ mu- and B(s) —$>$ mu+ mu- Decays_, _JHEP_ 02 (2010) 082 [0907.4408].
  * [45] H.-S. Lee and E. Ma, _Gauged $B-x_{i}L$ origin of $R$ Parity and its implications_, _Phys. Lett. B_ 688 (2010) 319 [1001.0768].
  * [46] H. Okada, _Dark Matters in Gauged $B-3L_{i}$ Model_, 1212.0492.
  * [47] E.J. Chun, A. Das, J. Kim and J. Kim, _Searching for flavored gauge bosons_ , _JHEP_ 02 (2019) 093 [1811.04320].
  * [48] B.C. Allanach and J. Davighi, _Third family hypercharge model for ${R}_{K^{\left(\ast\right)}}$ and aspects of the fermion mass problem_, _JHEP_ 12 (2018) 075 [1809.01158].
  * [49] D. Borah, M. Dutta, S. Mahapatra and N. Sahu, _Muon (g-2) and XENON1T excess with boosted dark matter in $U(1)_{L_{\mu}-L_{\tau}}$ model_, _Phys. Lett. B_ 820 (2021) 136577 [2104.05656].
  * [50] D. Borah, M. Dutta, S. Mahapatra and N. Sahu, _Lepton Anomalous Magnetic Moment with Singlet-Doublet Fermion Dark Matter in Scotogenic $U(1)_{L_{\mu}-L_{\tau}}$ Model_, 2109.02699.
  * [51] F. Zwicky, _Die Rotverschiebung von extragalaktischen Nebeln_ , _Helv. Phys. Acta_ 6 (1933) 110.
  * [52] F. Zwicky, _On the Masses of Nebulae and of Clusters of Nebulae_ , _Astrophys. J._ 86 (1937) 217.
  * [53] V.C. Rubin and W.K. Ford, Jr., _Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions_ , _Astrophys. J._ 159 (1970) 379.
  * [54] D. Clowe, M. Bradač, A.H. Gonzalez, M. Markevitch, S.W. Randall, C. Jones et al., _A direct empirical proof of the existence of dark matter_ , _Astrophys. J. Lett._ 648 (2006) L109 [astro-ph/0608407].
  * [55] W. Hu and S. Dodelson, _Cosmic microwave background anisotropies_ , _Ann. Rev. Astron. Astrophys._ 40 (2002) 171 [astro-ph/0110414].
  * [56] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [57] G. Jungman, M. Kamionkowski and K. Griest, _Supersymmetric dark matter_ , _Phys. Rept._ 267 (1996) 195 [hep-ph/9506380].
  * [58] G. Bertone, D. Hooper and J. Silk, _Particle dark matter: Evidence, candidates and constraints_ , _Phys. Rept._ 405 (2005) 279 [hep-ph/0404175].
  * [59] J.L. Feng, _Dark Matter Candidates from Particle Physics and Methods of Detection_ , _Ann. Rev. Astron. Astrophys._ 48 (2010) 495 [1003.0904].
  * [60] XENON collaboration, _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302 [1805.12562].
  * [61] XENON collaboration, _Projected WIMP sensitivity of the XENONnT dark matter experiment_ , _JCAP_ 11 (2020) 031 [2007.08796].
  * [62] PandaX collaboration, _Dark matter direct search sensitivity of the PandaX-4T experiment_ , _Sci. China Phys. Mech. Astron._ 62 (2019) 31011 [1806.02229].
  * [63] J. McDonald, _Thermally generated gauge singlet scalars as selfinteracting dark matter_ , _Phys. Rev. Lett._ 88 (2002) 091304 [hep-ph/0106249].
  * [64] L.J. Hall, K. Jedamzik, J. March-Russell and S.M. West, _Freeze-In Production of FIMP Dark Matter_ , _JHEP_ 03 (2010) 080 [0911.1120].
  * [65] X. Chu, T. Hambye and M.H.G. Tytgat, _The Four Basic Ways of Creating Dark Matter Through a Portal_ , _JCAP_ 05 (2012) 034 [1112.0493].
  * [66] N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen and V. Vaskonen, _The Dawn of FIMP Dark Matter: A Review of Models and Constraints_ , _Int. J. Mod. Phys. A_ 32 (2017) 1730023 [1706.07442].
  * [67] M. Duch, B. Grzadkowski and D. Huang, _Strongly self-interacting vector dark matter via freeze-in_ , _JHEP_ 01 (2018) 020 [1710.00320].
  * [68] A. Biswas, D. Borah and A. Dasgupta, _UV complete framework of freeze-in massive particle dark matter_ , _Phys. Rev. D_ 99 (2019) 015033 [1805.06903].
  * [69] S. Heeba, F. Kahlhoefer and P. Stöcker, _Freeze-in production of decaying dark matter in five steps_ , _JCAP_ 11 (2018) 048 [1809.04849].
  * [70] B. Barman, S. Bhattacharya and M. Zakeri, _Non-Abelian Vector Boson as FIMP Dark Matter_ , _JCAP_ 02 (2020) 029 [1905.07236].
  * [71] B. Barman and A. Ghoshal, _Scale Invariant FIMP Miracle_ , 2109.03259.
  * [72] B. Barman and A. Ghoshal, _Probing pre-BBN era with Scale Invarint FIMP_ , 2203.13269.
  * [73] Z.M. Burell and N. Okada, _Supersymmetric minimal B-L model at the TeV scale with right-handed Majorana neutrino dark matter_ , _Phys. Rev. D_ 85 (2012) 055011 [1111.1789].
  * [74] N. Okada, S. Okada and Q. Shafi, _Light $Z′$ and dark matter from U(1)X gauge symmetry_, _Phys. Lett. B_ 810 (2020) 135845 [2003.02667].
  * [75] L.M.G. de la Vega, L.J. Flores, N. Nath and E. Peinado, _Complementarity between dark matter direct searches and CE $\nu$NS experiments in U(1)’ models_, _JHEP_ 09 (2021) 146 [2107.04037].
  * [76] A. Ghoshal, N. Okada and A. Paul, _eV Hubble Scale Inflation with Radiative Plateau: Very light Inflaton, Reheating & Dark Matter in $B-L$ Extensions_, 2203.03670.
  * [77] A. Paul, A. Ghoshal, A. Chatterjee and S. Pal, _Inflation, (P)reheating and Neutrino Anomalies: Production of Sterile Neutrinos with Secret Interactions_ , _Eur. Phys. J. C_ 79 (2019) 818 [1808.09706].
  * [78] A. Paul, A. Chatterjee, A. Ghoshal and S. Pal, _Shedding Light on Dark Matter and Neutrino Interactions from Cosmology_ , 2104.04760.
  * [79] S. Okada, _$Z^{\prime}$ Portal Dark Matter in the Minimal $B-L$ Model_, _Adv. High Energy Phys._ 2018 (2018) 5340935 [1803.06793].
  * [80] T. Robens and T. Stefaniak, _Status of the Higgs Singlet Extension of the Standard Model after LHC Run 1_ , _Eur. Phys. J. C_ 75 (2015) 104 [1501.02234].
  * [81] T. Robens and T. Stefaniak, _LHC Benchmark Scenarios for the Real Higgs Singlet Extension of the Standard Model_ , _Eur. Phys. J. C_ 76 (2016) 268 [1601.07880].
  * [82] G. Chalons, D. Lopez-Val, T. Robens and T. Stefaniak, _The Higgs singlet extension at LHC Run 2_ , _PoS_ ICHEP2016 (2016) 1180 [1611.03007].
  * [83] D. López-Val and T. Robens, _$\Delta$ r and the W-boson mass in the singlet extension of the standard model_, _Phys. Rev. D_ 90 (2014) 114018 [1406.1043].
  * [84] CMS collaboration, _Search for a Higgs boson in the mass range from 145 to 1000 GeV decaying to a pair of W or Z bosons_ , _JHEP_ 10 (2015) 144 [1504.00936].
  * [85] M.J. Strassler and K.M. Zurek, _Discovering the Higgs through highly-displaced vertices_ , _Phys. Lett. B_ 661 (2008) 263 [hep-ph/0605193].
  * [86] CMS collaboration, _Searches for invisible decays of the Higgs boson in pp collisions at $\sqrt{s}$ = 7, 8, and 13 TeV_, _JHEP_ 02 (2017) 135 [1610.09218].
  * [87] R.N. Mohapatra and G. Senjanović, _Neutrino mass and spontaneous parity nonconservation_ , _Phys. Rev. Lett._ 44 (1980) 912.
  * [88] M. Gell-Mann, P. Ramond and R. Slansky, _Complex Spinors and Unified Theories_ , _Conf. Proc. C_ 790927 (1979) 315 [1306.4669].
  * [89] W. Wang and Z.-L. Han, _$U(1)_{B-3L_{\alpha}}$ extended scotogenic models and single-zero textures of neutrino mass matrices_, _Phys. Rev. D_ 101 (2020) 115040 [1911.00819].
  * [90] Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * [91] A. Das, P.S.B. Dev and N. Okada, _Long-lived TeV-scale right-handed neutrino production at the LHC in gauged $U(1)_{X}$ model_, _Phys. Lett. B_ 799 (2019) 135052 [1906.04132].
  * [92] T. Hambye, M.H.G. Tytgat, J. Vandecasteele and L. Vanderheyden, _Dark matter from dark photons: A taxonomy of dark matter production_ , _Phys. Rev. D_ 100 (2019) 095018.
  * [93] R.N. Mohapatra and N. Okada, _Dark matter constraints on low mass and weakly coupled $b-l$ gauge boson_, _Phys. Rev. D_ 102 (2020) 035028.
  * [94] G. Steigman, _Neutrinos And Big Bang Nucleosynthesis_ , _Adv. High Energy Phys._ 2012 (2012) 268321 [1208.0032].
  * [95] M. Kaplinghat, S. Tulin and H.-B. Yu, _Direct Detection Portals for Self-interacting Dark Matter_ , _Phys. Rev. D_ 89 (2014) 035009 [1310.7945].
  * [96] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [97] Y. Farzan and M. Tortola, _Neutrino oscillations and Non-Standard Interactions_ , _Front. in Phys._ 6 (2018) 10 [1710.09360].
  * [98] _Neutrino Non-Standard Interactions: A Status Report_ , vol. 2, 2019.  10.21468/SciPostPhysProc.2.001.
  * [99] J.H. Chang, R. Essig and S.D. McDermott, _Revisiting Supernova 1987A Constraints on Dark Photons_ , _JHEP_ 01 (2017) 107 [1611.03864].
  * [100] S. Knapen, T. Lin and K.M. Zurek, _Light Dark Matter: Models and Constraints_ , _Phys. Rev. D_ 96 (2017) 115021 [1709.07882].
  * [101] D. Croon, G. Elor, R.K. Leane and S.D. McDermott, _Supernova Muons: New Constraints on $Z$’ Bosons, Axions and ALPs_, _JHEP_ 01 (2021) 107 [2006.13942].
  * [102] P.S.B. Dev, B. Dutta, K.J. Kelly, R.N. Mohapatra and Y. Zhang, _Light, long-lived B $-$ L gauge and Higgs bosons at the DUNE near detector_, _JHEP_ 07 (2021) 166 [2104.07681].
  * [103] J.D. Bjorken, R. Essig, P. Schuster and N. Toro, _New Fixed-Target Experiments to Search for Dark Gauge Forces_ , _Phys. Rev. D_ 80 (2009) 075018 [0906.0580].
  * [104] J. Blumlein and J. Brunner, _New Exclusion Limits for Dark Gauge Forces from Beam-Dump Data_ , _Phys. Lett. B_ 701 (2011) 155 [1104.2747].
  * [105] J. Blümlein and J. Brunner, _New Exclusion Limits on Dark Gauge Forces from Proton Bremsstrahlung in Beam-Dump Data_ , _Phys. Lett. B_ 731 (2014) 320 [1311.3870].
  * [106] R. Essig, R. Harnik, J. Kaplan and N. Toro, _Discovering New Light States at Neutrino Experiments_ , _Phys. Rev. D_ 82 (2010) 113008 [1008.0636].
  * [107] SHiP collaboration, _A facility to Search for Hidden Particles (SHiP) at the CERN SPS_ , 1504.04956.
  * [108] J.L. Feng, I. Galon, F. Kling and S. Trojanowski, _Dark Higgs bosons at the ForwArd Search ExpeRiment_ , _Phys. Rev. D_ 97 (2018) 055034 [1710.09387].
  * [109] FASER collaboration, _FASER’s physics reach for long-lived particles_ , _Phys. Rev. D_ 99 (2019) 095011 [1811.12522].
  * [110] FASER collaboration, _Technical Proposal for FASER: ForwArd Search ExpeRiment at the LHC_ , 1812.09139.
  * [111] FASER collaboration, _FASER: ForwArd Search ExpeRiment at the LHC_ , 1901.04468.
  * [112] Belle-II collaboration, _The Belle II Physics Book_ , _PTEP_ 2019 (2019) 123C01 [1808.10567].
  * [113] BaBar collaboration, _Search for a Dark Photon in $e^{+}e^{-}$ Collisions at BaBar_, _Phys. Rev. Lett._ 113 (2014) 201801 [1406.2980].
  * [114] BaBar collaboration, _Search for Invisible Decays of a Dark Photon Produced in ${e}^{+}{e}^{-}$ Collisions at BaBar_, _Phys. Rev. Lett._ 119 (2017) 131804 [1702.03327].
  * [115] COHERENT collaboration, _First Measurement of Coherent Elastic Neutrino-Nucleus Scattering on Argon_ , _Phys. Rev. Lett._ 126 (2021) 012002 [2003.10630].
  * [116] P. Coloma, B.A. Dobrescu, C. Frugiuele and R. Harnik, _Dark matter beams at LBNF_ , _JHEP_ 04 (2016) 047 [1512.03852].
  * [117] P. Coloma, M.C. Gonzalez-Garcia and M. Maltoni, _Neutrino oscillation constraints on U(1)’ models: from non-standard interactions to long-range forces_ , _JHEP_ 01 (2021) 114 [2009.14220].
  * [118] D.W.P.d. Amaral, D.G. Cerdeno, P. Foldenauer and E. Reid, _Solar neutrino probes of the muon anomalous magnetic moment in the gauged $\mathrm{U}{(1)}_{L_{\mu}-{L}_{\tau}}$_, _JHEP_ 12 (2020) 155 [2006.11225].
  * [119] G. Bellini et al., _Precision measurement of the 7Be solar neutrino interaction rate in Borexino_ , _Phys. Rev. Lett._ 107 (2011) 141302 [1104.1816].
  * [120] Borexino collaboration, _First Simultaneous Precision Spectroscopy of $pp$, 7Be, and $pep$ Solar Neutrinos with Borexino Phase-II_, _Phys. Rev. D_ 100 (2019) 082004 [1707.09279].
  * [121] J. Heeck, M. Lindner, W. Rodejohann and S. Vogl, _Non-Standard Neutrino Interactions and Neutral Gauge Bosons_ , _SciPost Phys._ 6 (2019) 038 [1812.04067].
  * [122] A. Kamada and H.-B. Yu, _Coherent Propagation of PeV Neutrinos and the Dip in the Neutrino Spectrum at IceCube_ , _Phys. Rev. D_ 92 (2015) 113004 [1504.00711].
  * [123] M. Escudero, D. Hooper, G. Krnjaic and M. Pierre, _Cosmology with A Very Light L μ $-$ Lτ Gauge Boson_, _JHEP_ 03 (2019) 071 [1901.02010].
  * [124] S.R. Mishra, S.A. Rabinowitz, C. Arroyo, K.T. Bachmann, R.E. Blair, C. Foudas et al., _Neutrino tridents and w-z interference_ , _Phys. Rev. Lett._ 66 (1991) 3117.
  * [125] W. Altmannshofer, S. Gori, J. Martín-Albo, A. Sousa and M. Wallbank, _Neutrino Tridents at DUNE_ , _Phys. Rev. D_ 100 (2019) 115029 [1902.06765].
  * [126] M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, _Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data_, _Eur. Phys. J. C_ 77 (2017) 827 [1706.09436].
  * [127] M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, _A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\boldsymbol{\alpha}(m_{Z}^{2})}$_, _Eur. Phys. J. C_ 80 (2020) 241 [1908.00921].
  * [128] Muon g-2 collaboration, _Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL_ , _Phys. Rev. D_ 73 (2006) 072003 [hep-ex/0602035].
  * [129] T. Aoyama et al., _The anomalous magnetic moment of the muon in the Standard Model_ , _Phys. Rept._ 887 (2020) 1 [2006.04822].
  * [130] Muon g-2 collaboration, _Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm_ , _Phys. Rev. Lett._ 126 (2021) 141801 [2104.03281].
  * [131] DONuT collaboration, _Final tau-neutrino results from the DONuT experiment_ , _Phys. Rev. D_ 78 (2008) 052002 [0711.0728].
  * [132] A. Ghoshal, A. Giarnetti and D. Meloni, _On the role of the $\nu_{\tau}$ appearance in DUNE in constraining standard neutrino physics and beyond_, _JHEP_ 12 (2019) 126 [1906.06212].
  * [133] A. Ghoshal, A. Giarnetti and D. Meloni, _Neutrino Invisible Decay at DUNE: a multi-channel analysis_ , _J. Phys. G_ 48 (2021) 055004 [2003.09012].
  * [134] E. Ma and D.P. Roy, _Phenomenology of the $B$ \- 3L($\tau$) gauge boson_, _Phys. Rev. D_ 58 (1998) 095005 [hep-ph/9806210].
  * [135] SHiP collaboration, _SND@LHC_ , 2002.08722.
  * [136] F. Kling, _Probing light gauge bosons in tau neutrino experiments_ , _Phys. Rev. D_ 102 (2020) 015007 [2005.03594].
  * [137] S. Aoki et al., _Study of tau-neutrino production at the CERN SPS_ , 1708.08700.
  * [138] DsTau collaboration, _DsTau: Study of tau neutrino production with 400 GeV protons from the CERN-SPS_ , _JHEP_ 01 (2020) 033 [1906.03487].
  * [139] DUNE collaboration, _Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF_ , 1512.06148.
  * [140] DUNE collaboration, _Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 1: The LBNF and DUNE Projects_ , 1601.05471.
  * [141] J.M. Berryman, A. de Gouvea, P.J. Fox, B.J. Kayser, K.J. Kelly and J.L. Raaf, _Searches for Decays of New Particles in the DUNE Multi-Purpose Near Detector_ , _JHEP_ 02 (2020) 174 [1912.07622].
  * [142] P. Bakhti, Y. Farzan and M. Rajaee, _Secret interactions of neutrinos with light gauge boson at the DUNE near detector_ , _Phys. Rev. D_ 99 (2019) 055019 [1810.04441].
  * [143] A.J. Krasznahorkay, M. Csatlós, L. Csige, Z. Gácsi, J. Gulyás, M. Hunyadi et al., _Observation of anomalous internal pair creation in ${}^{8}\mathrm{Be}$: A possible indication of a light, neutral boson_, _Phys. Rev. Lett._ 116 (2016) 042501.
  * [144] J.L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, T.M.P. Tait et al., _Protophobic Fifth-Force Interpretation of the Observed Anomaly in 8Be Nuclear Transitions_, _Phys. Rev. Lett._ 117 (2016) 071803 [1604.07411].
  * [145] E. Nardi, C.D.R. Carvajal, A. Ghoshal, D. Meloni and M. Raggi, _Resonant production of dark photons in positron beam dump experiments_ , _Phys. Rev. D_ 97 (2018) 095004 [1802.04756].
  * [146] LHCb collaboration, _Search for hidden-sector bosons in $B^{0}\\!\to K^{*0}\mu^{+}\mu^{-}$ decays_, _Phys. Rev. Lett._ 115 (2015) 161802 [1508.04094].
  * [147] LHCb collaboration, _Search for long-lived scalar particles in $B^{+}\to K^{+}\chi(\mu^{+}\mu^{-})$ decays_, _Phys. Rev. D_ 95 (2017) 071101 [1612.07818].
  * [148] D. Aristizabal Sierra, F. Staub and A. Vicente, _Shedding light on the $b\to s$ anomalies with a dark sector_, _Phys. Rev. D_ 92 (2015) 015001 [1503.06077].
  * [149] A. Datta, J. Liao and D. Marfatia, _A light $Z^{\prime}$ for the $R_{K}$ puzzle and nonstandard neutrino interactions_, _Phys. Lett. B_ 768 (2017) 265 [1702.01099].
  * [150] A. Datta, J. Kumar, J. Liao and D. Marfatia, _New light mediators for the $R_{K}$ and $R_{K^{*}}$ puzzles_, _Phys. Rev. D_ 97 (2018) 115038 [1705.08423].
  * [151] F. Sala and D.M. Straub, _A New Light Particle in B Decays?_ , _Phys. Lett. B_ 774 (2017) 205 [1704.06188].
  * [152] Z. Kang and Y. Shigekami, _$(g-2)_{\mu}$ versus flavor changing neutral current induced by the light $(B-L)_{\mu\tau}$ boson_, _JHEP_ 11 (2019) 049 [1905.11018].
  * [153] D. Borah, L. Mukherjee and S. Nandi, _Low scale U(1) X gauge symmetry as an origin of dark matter, neutrino mass and flavour anomalies_, _JHEP_ 12 (2020) 052 [2007.13778].
  * [154] J.-Y. Cen, Y. Cheng, X.-G. He and J. Sun, _Flavor Specific $U(1)_{B_{q}-L_{\mu}}$ Gauge Model for Muon $g-2$ and $b\to s\bar{\mu}\mu$ Anomalies_, 2104.05006.