where $\rho^{\rm core}\simeq\frac{m_{\rm DM}}{\langle\sigma v\rangle_{\rm
eff}t_{\rm BH}}$. For the NFW halo profile, we define $\rho_{\rm
NFW}(r)=\rho_{s}\left(\frac{r_{\rm sc}}{r}\right)\left(1+\frac{r}{r_{\rm
sc}}\right)^{-2}$, where $\rho_{\rm sc}=0.35$ $\rm GeV/{cm}^{3}$, the scale
parameter for the density profile is taken as $r_{\rm sc}=13$ kpc [103, 223],
having consistency with the data of the supermassive BH in galaxy Milky Way
[224] and having mass alike one in NGC 1068. In this context we mention that
normalization density parameter
$\rho_{N}^{\prime}\simeq\rho_{N}(r_{I}/R_{s})^{7/3}=\mathcal{N}r_{I}^{5/6}R_{s}^{-7/3}$
requiring that at $r=r_{I}$ then $\rho_{N}^{\prime}\to\rho_{3/2}^{\prime}$.
Matching the conditions of spike and outer part of the halo as
$\rho_{N}^{\prime},\rho_{N}\simeq\rho_{\rm sc}(R_{s}/r_{\rm sc})^{-\gamma}$
for $\alpha=3/2$ and $7/3$ where $R_{\rm Sc}<<r_{I}<R_{s}<<r_{\rm sc}$, we
obtain

$\displaystyle R_{s}=\begin{cases}\left(\frac{\mathcal{N}}{\rho_{\rm sc}r_{\rm
sc}}\right)^{3/4}r_{I}^{5/8},&\alpha=3/2\\\ \left(\frac{\mathcal{N}}{\rho_{\rm
sc}r_{\rm sc}}\right)^{3/4},&\alpha=7/3\end{cases}$ (106)

For NGC 1068 we consider $R_{\rm s}\simeq 0.7$ kpc which is greater than
$r_{I}$ under the parameters used throughout this paper. Finally we define the
accumulated column density of DM as

$\Sigma(r)=\int_{R_{\rm EM}}^{r}dx\;\rho_{\rm DM}(x)$ (107)

where $R_{\rm EM}$ ($\approx 30R_{\rm Sc}=2.8\times 10^{-5}$ pc) is the
position at which neutrinos are expected to be produced with respect to the
BH. The DM density $\rho_{\rm DM}$ is given by the expression in Eq. 105 and
integrating over the radius for different benchmark cases from Tab. 7 we show
the column density in the left panel of Fig. 8 starting from $r>R_{\rm EM}$.
The accumulated column density for different DM models saturate to a constant
value around $r\simeq 10$ pc and $100$ pc depending on $\alpha_{\rm sp}$ and
$\langle\sigma v\rangle_{\rm eff}$. Hence we come to a conclusion that at the
Earth, where $r\simeq 14.4$ Mpc, the accumulated column density is considered
to be a constant. On the other hand we show the accumulated column density per
unit DM mass for different cases given in Tab. 7 which will have implications
to solve the cascade equation for AGN.

##  V Results and discussions

We study different scattering and beam-dump experiments for different $U(1)$
extensions of the SM to estimate bounds on $g_{X}-M_{Z^{\prime}}$ plane.
Taking into account different cosmic bursts like GRB involving
$\nu\bar{\nu}\to e^{+}e^{-}$ scattering, blazars and AGN involving $\nu-$DM
scattering we estimate bounds on the $g_{X}-M_{Z^{\prime}}$ plane for
different $U(1)$ extensions of the SM. Finally we compare our results.

###  V.1 Limit from LEP experiment

We estimate bounds on $\\{M_{Z^{\prime}},g_{X}\\}$ from the results of LEP-II
measuring $e^{+}e^{-}\to\bar{f}f$ scattering cross section at the $Z$ peak
with $f$ being the SM fermions [225]. The scattering cross sections are
calculated including $Z^{\prime}$ exchanging diagram and it is compared with
the observed value. We find that the strongest constraint can be obtained from
the cross-section of $e^{+}e^{-}\to q\bar{q}$ process giving a hadronic final
state whose value is $\sigma=45.44\pm 0.037$ nb. Then we obtain the upper
limit of $g_{X}$ as a function of $M_{Z^{\prime}}$ requiring the estimated
cross section is within the 90% C.L. of the observed one. Limits are given in
Fig: 9-12 for the $U(1)_{X}$ and $U(1)_{q+xu}$ in the upper and lower panels.
In case of $U(1)_{q+xu}$ scenario, these limits are shown by gray shaded area.
We find similar limits for the $L_{e}-L_{\mu,\tau}$ scenarios in the upper and
lower limits of Fig. 13 and those for $B-3L_{e}$ scenario are given in Fig.
15.

###  V.2 Limit from dark photon search at LHC experiments

Also we estimate bounds on $\\{M_{Z^{\prime}},g_{X}\\}$ from the results of
CMS [148] and LHCb [145] experiments that search for dark photon $A^{\prime}$
decaying into $\mu^{+}\mu^{-}$ pair. These experiments provide us with the
constraints on dark photon mass $m_{A^{\prime}}$ and kinetic mixing parameter
$\epsilon$. We estimate the upper limit of our gauge coupling $g_{X}$ as a
function of $M_{Z^{\prime}}$ in use of the following rescaling

$g_{X}^{\rm max}(M_{Z^{\prime}}=m_{A^{\prime}})=\epsilon^{\rm
max}(m_{A^{\prime}})e\sqrt{\frac{\sigma(pp\to
A^{\prime})BR(A^{\prime}\to\mu^{+}\mu^{-})}{\sigma(pp\to
Z^{\prime})BR(Z^{\prime}\to\mu^{+}\mu^{-})}},$ (108)

where $\sigma(pp\to A^{\prime}(Z^{\prime}))$ is dark photon$(Z^{\prime})$
production cross section and $\epsilon^{\rm max}(m_{A^{\prime}})$ is the
experimental upper limit of kinetic mixing parameter as a function of dark
photon mass. We estimated the $A^{\prime}(Z^{\prime})$ production cross
sections using CalcHEP3.5 implementing relevant interactions. Limits are given
in Fig: 9-12 for the $U(1)_{X}$ and $U(1)_{q+xu}$ in the upper and lower
panels. We find similar limits for the $B-3L_{\mu}$ scenario which are given
in the upper panel of Fig. 16.

###  V.3 Limit from BaBar experiment

At the BaBar experiment, $e^{+}e^{-}$ collision produces dark photon via the
process $e^{+}e^{-}\to A^{\prime}\gamma$. They search for visible $A^{\prime}$
decaying into $\\{e^{+}e^{-},\mu^{+}\mu^{-},\text{light mesons}\\}$ final
states and invisible $A^{\prime}$ decaying into invisible final sate such as
neutrinos. We rescale the upper limit of gauge coupling in the $U(1)_{B-L}$
case given in [146, 147] as a function of $Z^{\prime}$ mass to obtain that of
$g_{X}$ in our model. For $Z^{\prime}$ decaying into visible modes, the
rescaling formula is

$g_{X}^{\rm max}(M_{Z^{\prime}})=g_{B-L}^{\rm
max}(M_{Z^{\prime}})\sqrt{\frac{\sigma(e^{+}e^{-}\to\gamma
Z^{\prime}_{B-L})BR(Z^{\prime}_{B-L}\to\text{visible
states})}{\sigma(e^{+}e^{-}\to\gamma Z^{\prime})BR(Z^{\prime}\to\text{visible
states})}},$ (109)

where $Z^{\prime}_{B-L}$ indicate $Z^{\prime}$ boson in $U(1)_{B-L}$ case. For
$Z^{\prime}$ decaying into invisible modes, the rescaling formula is

$g_{X}^{\rm max}(M_{Z^{\prime}})=g_{B-L}^{\rm
max}(M_{Z^{\prime}})\sqrt{\frac{\sigma(e^{+}e^{-}\to\gamma
Z^{\prime}_{B-L})BR(Z^{\prime}_{B-L}\to\bar{\nu}\nu)}{\sigma(e^{+}e^{-}\to\gamma
Z^{\prime})BR(Z^{\prime}\to\bar{\nu}\nu)}},$ (110)

where all neutrino modes are summed up. Corresponding bounds are shown in
Figs. 9-12 for $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios and in Fig. 15 for
$B-3L_{e}$ scenario, respectively. In case of $U(1)_{q+xu}$ scenario these
bounds belong to the gray shaded region. Bounds for $L_{e}-L_{\mu,\tau}$
scenarios are shown in the upper and lower panels of Fig. 13.

###  V.4 Limits from dilepton searches from CMS and ATLAS

Studying dilepton production at $\sqrt{s}=13$ TeV proton-proton collider, we
compare the production cross section $(\sigma^{\rm{model}})$ of dilepton mode
(electron and muon) in our model having a trial value of $U(1)$ coupling
$g^{\rm{model}}$ with the dilepton production cross section
$(\sigma^{\rm{CMS/ATLAS}})$ given by the CMS[226] and ATLAS[165]
collaborations at the LHC using sequential standard model(SSM) [227]. We used
the CMS and ATLAS data with $\frac{\Gamma}{M_{Z^{\prime}}}=3\%$ where $\Gamma$
stands for the total decay width of $Z^{\prime}$. The CMS(ATLAS) collaboration
produced the result with $140(139)$ fb-1 luminosity. Hence we estimate bounds
on $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios following

$g_{X}=g^{\rm{model}}\sqrt{\frac{\sigma^{\rm{CMS/ATLAS}}}{\sigma^{\rm{model}}}}$
(111)

The model cross sections were estimated using the narrow width approximation
as

$\sigma^{\rm model}=\frac{8\pi^{2}}{3}~{}\sum_{q,\bar{q}}\int dx\int
dy~{}q(x,Q),\bar{q}(x,Q)\times\frac{\Gamma(Z^{\prime}\to
q\bar{q})}{M_{Z^{\prime}}}\delta(\hat{s}-M_{Z^{\prime}})\times\rm{BR}(Z^{\prime}\to
2\ell)$ (112)

where $q(x,Q)$ and $\bar{q}(x,Q)$ are the parton distribution functions of the
quark and antiquark respectively and $\hat{s}=xys$ is the invariant mass
squared of the colliding quark at $\sqrt{s}$, the center of mass energy. We
set the factorization scale at $Q=M_{Z^{\prime}}$ in the parton distribution
function CTEQ6L[228]. These bounds are shown by CMS2$\ell$ and ATLAS2$\ell$ in
Figs. 9-12. We find that around $150$ GeV $\leq M_{Z^{\prime}}\leq 6$ TeV CMS
and ATLAS limits are almost comparable. In this mass range so-far LHC produces
strongest upper limits depending on $M_{Z^{\prime}}$. We scale the
corresponding results from CMS(ATLAS) to a projected 3 ab-1 luminosity using

$g_{X}^{\rm projected}\simeq g_{X}^{\rm{current}}\sqrt{\frac{139(140)~{}\rm
fb^{-1}}{\mathcal{L}_{\rm projected}}}$ (113)

shown by CMS$2\ell$-3 and ATLAS$2\ell$-3 respectively in the same figures.
Using the same strategy we estimate bounds at current and projected
luminosities of the LHC using only the two electron [226, 165], two muon [226,
165] and two tau [229, 230] final state from the CMS and ATLAS. Then we show
bounds for the $B-3L_{e}$ scenario in Fig. 15 and for $B-3L_{\mu}$ and
$B-3L_{\tau}$ scenarios in upper and lower panels of Fig. 16, respectively. In
case of $B-3L_{\tau}$ we used the CMS(ATLAS) result at $2.2(36)$ fb-1
luminosity.

###  V.5 Limits from LEP-II and prospective bounds from ILC

We have estimated limits on $M_{Z^{\prime}}/g_{X}$ for different chiral (for
different charges) and flavored scenarios from LEP-II and prospective ILC at
$\sqrt{s}=250$ GeV, 500 GeV and 1 TeV, respectively in Sec. II. Solving them
for a range of $M_{Z^{\prime}}$ considering $M_{Z^{\prime}}>>\sqrt{s}$, we
find different diagonal lines which are shown in Figs. 9-12 for different
charges of the $U(1)_{X}$ (upper panel) and $U(1)_{q+xu}$ (lower panel)
scenarios. Similar results have been derived for $L_{e}-L_{\mu}$ and
$L_{e}-L_{\tau}$ scenarios and shown in Fig. 13. Due to electronic charge
assignment, these bounds are same. Under the $B-3L_{e}$ scenario the first
generation of lepton doublets is charged under the $U(1)$ gauge group, the
corresponding constrains are shown in Fig. 15. In case of $U(1)_{X}$,
$U(1)_{q+xu}$ and $B-3L_{e}$ scenarios we find that bounds obtained from LEP-
II bounds are comparatively week compared to the LHC bounds, however,
prospective bounds obtained from ILC at $\sqrt{s}=$ 250 GeV, 500 GeV and 1 TeV
(ILC250, ILC500 and ILC1000) could provide stronger bounds than LHC. However,
In case of $L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$ scenarios LEP-II provides
strong bound for heavy $Z^{\prime}$ and prospective bounds from ILC could be
stronger than LEP-II bounds. In future, such constrains could be improved
involving initial and final state radiations, vector boson fusion process
considering neutral and charged current interactions at electron-positron and
muon colliders.

###  V.6 Constraints from proton/electron beam dump experiments

Here we summarize our estimation of constraints from beam dump experiments.
For proton beam dump experiments LSND, PS191, NOMAD and CHARM, we estimate
bound curves in our models by rescaling those of U(1)B-L case [231].
Approximately, the curves for the upper bound on $\\{m_{Z^{\prime}},g_{X}\\}$
plane are derived applying the scaling [232, 17]

$\tau_{Z^{\prime}}(g_{B-L}^{\rm max})\sim\tau_{Z^{\prime}}(g_{X}^{\rm
max},x_{H},x_{\Phi})~{},$ (114)

where $g_{B-L}$ indicates the gauge coupling in the U(1)B-L model, and
$\tau_{Z^{\prime}}$ is the lifetime of the $Z^{\prime}$. The curves for the
lower bound are also scaled by using a formula

$g_{X}^{\rm low}\sim g_{B-L}^{\rm low}\sqrt{\frac{{\rm BR}(M\to
Z^{\prime}_{B-L}\gamma)~{}{\rm BR}(Z^{\prime}_{B-L}\to
e^{+}e^{-})\tilde{\tau}_{Z^{\prime}}}{{\rm BR}(M\to Z^{\prime}\gamma)~{}{\rm
BR}(Z^{\prime}\to e^{+}e^{-})\tilde{\tau}_{Z^{\prime}_{B-L}}}}~{},$ (115)

where $\tilde{\tau}$ is lifetime with gauge coupling taken to be unity, and
$Z^{\prime}$ is produced through meson decay processes with $M=\pi^{0}$ for
LSND, PS191 and NOMAD, and $M=\eta$ for CHARM. In our estimation, meson decay
branching ratio is obtained using the method given in Ref. [232].

For proton beam dump experiment $\nu$-cal, the $Z^{\prime}$ is dominantly
produced via bremsstrahlung process. Here we adopt the results in ref. [19]
for the constraint on $\\{M_{Z^{\prime}},g_{X}\\}$ where chiral structure of
the $Z^{\prime}$ interactions is taken into account in estimating the
$Z^{\prime}$ production cross section.

For electron beam dump experiments NA64, E774, Orsay and KEK, the bound curves
are derived by rescaling the bounds of U(1)B-L case as we did for the proton
beam dump case. We approximately estimate the constraint for the upper region
on $\\{M_{Z^{\prime}},g_{X}\\}$ plane by rescaling in use of Eq. 114, which is
the same as the proton beam dump case. The constraint for the lower region is
calculated by [17]

$g_{X}^{\rm low}\sim g_{B-L}^{\rm low}\sqrt{\frac{2{\rm
BR}(Z^{\prime}_{B-L}\to
e^{+}e^{-})\tilde{\tau}_{Z^{\prime}}}{(x_{\ell}^{2}+x_{e}^{2}){\rm
BR}(Z^{\prime}\to e^{+}e^{-})\tilde{\tau}_{Z^{\prime}_{B-L}}}}~{},$ (116)

where $Z^{\prime}$ is considered to be produced through the bremsstrahlung
process.

For electron beam dump experiments E137 and E141, we adopt the results in ref.
[19] for the constraint on $\\{M_{Z^{\prime}},g_{X}\\}$ where chiral structure
of the $Z^{\prime}$ interactions is taken into account in estimating the
$Z^{\prime}$ production cross section. Following this line we estimate
prospective bounds for DUNE-beam dump (DUNE-BD), FASER and FASER2 for
$U(1)_{q+xu}$ case utilizing the B$-$L scenario given in [19]. The same
methodology was followed to estimate constraints for the electron beam-dump
scenario in ILC-beam dump (ILC-BD). These limits are give in Fig: 9-12 for the
$U(1)_{X}$ and $U(1)_{q+xu}$ in the upper and lower panels. For $U(1)_{q+xu}$
scenario we show only the prospective DUNE-BD, FASER, FASER2 and ILC-BD
bounds. Remaining bounds belong to the gray shaded region. We find similar
limits for the $B-3L_{e}$ scenario which are given in Fig. 15 whereas for
$B-3L_{\mu}$ and $B-3L_{\tau}$ the limits belong to the gray shaded area. The
electron/positron beam-dump scanerions except for the above cases will appear
in the $L_{i}-L_{j}$ scenarios where $L_{e}-L_{\mu,\tau}$ scenarios are shown
in the upper and lower panels of Fig. 13.

###  V.7 Constraints from electron-(anti)neutrino scattering in neutrino
experiments

We can also obtain constraints on $\\{M_{Z^{\prime}},g_{X}\\}$ from
electron-(anti)neutrino scattering processes that are tested by neutrino
experiments such as BOREXINO and TEXONO. The electron-(anti)neutrino
scattering cross sections under the existence of $Z^{\prime}$ interactions are
estimated to obtain the constraints on $Z^{\prime}$ mass and coupling. The
differential cross section can be written as

$\frac{d\sigma(\nu e)}{dT}=\left.\frac{d\sigma(\nu e)}{dT}\right|_{\rm
SM}+\left.\frac{d\sigma(\nu
e)}{dT}\right|_{Z^{\prime}}+\left.\frac{d\sigma(\nu e)}{dT}\right|_{\rm Int}$
(117)

where $T$ is the electron recoil energy and the first, second and third terms
in the RHS denote the contributions from the pure SM interactions, the pure
$Z^{\prime}$ interactions and interference between the SM and $Z^{\prime}$
interactions. The terms in RHS are written by [17]. The purely SM contribution
is

$\displaystyle\left.\frac{d\sigma(\nu e)}{dT}\right|_{\rm SM}$
$\displaystyle=\frac{2G_{F}^{2}m_{e}}{\pi
E_{\nu}^{2}}\left(a^{2}_{1}E^{2}_{\nu}+a_{2}^{2}(E_{\nu}-T)^{2}-a_{1}a_{2}m_{e}T\right),$
(118)

where $E_{\nu}$ is the initial neutrino energy. Here the coefficients $a_{1}$
and $a_{2}$ are written as

$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\left\\{\sin^{2}\theta_{W}+\frac{1}{2},\ \sin^{2}\theta_{W},\
\sin^{2}\theta_{W}-\frac{1}{2},\ \sin^{2}\theta_{W}\right\\}\ {\rm for}\
\\{\nu_{e}e,\bar{\nu}_{e}e,\nu_{\beta}e,\bar{\nu}_{\beta}e\\},$ $\displaystyle
a_{2}$ $\displaystyle=$ $\displaystyle\left\\{\sin^{2}\theta_{W},\
\sin^{2}\theta_{W}+\frac{1}{2},\ \sin^{2}\theta_{W},\
\sin^{2}\theta_{W}-\frac{1}{2}\right\\}\ {\rm for}\
\\{\nu_{e}e,\bar{\nu}_{e}e,\nu_{\beta}e,\bar{\nu}_{\beta}e\\},$ (119)

where $\beta=\\{\mu,\tau\\}$. The contribution from $Z^{\prime}$ exchanging
diagram can be derived as

$\displaystyle\left.\frac{d\sigma(\overset{(-)}{\nu}_{\alpha}e)}{dT}\right|_{Z^{\prime}}$
$\displaystyle=\frac{g_{X}^{4}(x_{\ell})^{2}m_{e}}{4\pi
E_{\nu}^{2}(2m_{e}T+M^{2}_{Z^{\prime}})}[(2E^{2}_{\nu}-2E_{\nu}T+T^{2})(b^{2}_{1}+b^{2}_{2})\pm
2b_{1}b_{2}(2E_{\nu}-T)T-m_{e}T(b^{2}_{1}-b_{2}^{2})],$ (120)

where $b_{1}=\frac{x_{\ell}+x_{e}}{2}$ and $b_{2}=\frac{x_{\ell}-x_{e}}{2}$
with $x_{\ell,e}$, and the negative sign of $\pm$ is for anti neutrino
processes. The contribution from interference between the SM and $Z^{\prime}$
are also obtained, depending on the processes, such that

$\displaystyle\left.\frac{d\sigma(\nu_{e}e)}{dT}\right|_{\rm int}$
$\displaystyle=$ $\displaystyle\frac{G_{F}g_{X}^{2}x_{\ell}m_{e}}{\sqrt{2}\pi
E^{2}_{\nu}(2m_{e}T+M^{2}_{Z^{\prime}})}[2E_{\nu}^{2}(b_{1}+b_{2})+(2E^{2}_{\nu}-2E_{\nu}T+T^{2})(b_{1}c_{1}+b_{2}c_{2})]$
$\displaystyle+T(2E_{\nu}-T)(b_{1}c_{2}+b_{2}c_{1})-m_{e}T(b_{1}-b_{2}+b_{1}c_{1}-b_{2}c_{2})],$
$\displaystyle\left.\frac{d\sigma(\bar{\nu}_{e}e)}{dT}\right|_{\rm int}$
$\displaystyle=$ $\displaystyle\frac{G_{F}g_{X}^{2}x_{\ell}m_{e}}{\sqrt{2}\pi
E^{2}_{\nu}(2m_{e}T+M^{2}_{Z^{\prime}})}[2(E_{\nu}-T)^{2}(b_{1}+b_{2})+(2E^{2}_{\nu}-2E_{\nu}T+T^{2})(b_{1}c_{1}+b_{2}c_{2})]$
$\displaystyle-T(2E_{\nu}-T)(b_{1}c_{2}+b_{2}c_{1})-m_{e}T(b_{1}-b_{2}+b_{1}c_{1}-b_{2}c_{2})],$
$\displaystyle\left.\frac{d\sigma(\overset{(-)}{\nu}_{\beta}e)}{dT}\right|_{\rm
int}$ $\displaystyle=$
$\displaystyle\frac{G_{F}g_{X}^{2}x_{\ell}m_{e}}{\sqrt{2}\pi
E^{2}_{\nu}(2m_{e}T+M^{2}_{Z^{\prime}})}[(2E_{\nu}^{2}-2E_{\nu}T-T^{2})2(b_{1}c_{1}+b_{2}c_{1})\pm
T(2E_{\nu}-T)(b_{1}c_{2}+b_{2}c_{1})]$ (121) $\displaystyle-
m_{e}T(b_{1}c_{1}-b_{2}c_{2})],$

where $c_{1}=-1/2+2\sin^{2}\theta_{W}$ and $c_{2}=-1/2$. Then we calculate the
differential cross sections and estimate the upper limit on
$\\{M_{Z^{\prime}},g_{X}\\}$ for each experiment in the following way:

BOREXINO: The $\nu_{e}$-$e$ scattering is measured by the experiment with
$\langle E_{\nu}\rangle=862$ keV and $T\simeq[270,665]$ keV for ${}^{7}B_{e}$
solar neutrino. To obtain the constraint on $\\{m_{Z^{\prime}},g_{X}\\}$, we
require the total cross section should not be more than $8\%$ above that of
the SM prediction [233].

TEXONO: The experiment measures $\bar{\nu}_{e}$-$e$ scatterings using 187 kg
of CsI(Tl) scintillating crystal array with 29882/7369 kg-day of reactor
ON/OFF data with electron recoil energy of $T\simeq[3,8]$ MeV. We estimate the
$\chi^{2}$ value as

$\chi^{2}=\sum_{\rm bin}\frac{(R_{\rm data}-R_{\rm th})^{2}}{\Delta R^{2}},$
(122)

where $R_{\rm data}$ and $R_{\rm th}$ are the event ratios observed by the
experiment and predicted by the cross-section from Eq. 117, $\Delta R$ is the
experimental uncertainty, for each recoil energy bin taken from data in ref.
[117], and anti-neutrino flux in the same reference is applied. The upper
limit curve on $\\{M_{Z^{\prime}},g_{X}\\}$ plane is then obtained for $90\%$
C.L from the $\chi^{2}$ fit.

GEMMA: The experiment measures $\bar{\nu}_{e}$-$e$ scattering with 1.5 kg HPGe
detector where energy of neutrino is $\langle E_{\nu}\rangle\sim 1$-$2$ MeV
and flux is $2.7\times 10^{13}$ cm-2s-1. We estimated the $\chi^{2}$ value
using the formula Eq. 122 for the data from ref. [234] with 13000 ON-hours and
3000 OFF-hours. Then we obtain the upper curve on $\\{M_{Z^{\prime}},g_{X}\\}$
with $90\%$ C.L.

CHARM-II: The experiment measures $\nu_{\mu}(\bar{\nu}_{\mu})$-electron
scattering where $2677\pm 82$ and $2752\pm 88$ events are obtained for
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ cases. The mean neutrino energy is $\langle
E_{\nu_{\mu}}\rangle=23.7$ GeV and $\langle E_{\bar{\nu}_{\mu}}\rangle=19.1$
GeV, and the range of measured recoil energy is 3-24 GeV. We estimated the
$\chi^{2}$ value using the formula Eq. 122 for the data from ref. [120, 121],
and obtain the upper curve on $\\{M_{Z^{\prime}},g_{X}\\}$ with $90\%$ C.L.

These limits are given in Fig: 9-12 for the $U(1)_{X}$ and $U(1)_{q+xu}$ in
the upper and lower panels. In case of $U(1)_{q+xu}$ scenario these bounds
belong to the gray shaded region. Bounds for $L_{e}-L_{\mu,\tau}$ scenarios
are shown in the upper and lower panels of Fig. 13 whereas for
$L_{\mu}-L_{\tau}$ scenario these limits belong to the gray shaded region. We
find similar limits for the $B-3L_{e}$ scenario which are given in Fig. 15
whereas for $B-3L_{\mu}$ and $B-3L_{\tau}$ the limits belong to the gray
shaded area.

Figure 9: Limits on $g_{X}-M_{Z^{\prime}}$ plane for the $U(1)_{X}$ (upper
panel) and $U(1)_{q+xu}$ (lower) scenarios considering $x_{H}=-1$ and $x=-1$,
respectively. Figure 10: Limits on $g_{X}-M_{Z^{\prime}}$ plane for B$-$L
scenario for $U(1)_{X}$ with $x_{H}=0$ and $U(1)_{q+xu}$ with $x=1$
respectively.

Figure 11: Limits on $g_{X}-M_{Z^{\prime}}$ plane for the $U(1)_{X}$ (upper
panel) and $U(1)_{q+xu}$ (lower) scenarios considering $x_{H}=1$ and $x=0$,
respectively.

Figure 12: Limits on $g_{X}-M_{Z^{\prime}}$ plane for the $U(1)_{X}$ (upper
panel) and $U(1)_{q+xu}$ (lower) scenarios considering $x_{H}=2$ and $x=2$,
respectively.

###  V.8 Constraints from coherent neutrino-nucleus scattering

We also consider constraint on $\\{M_{Z^{\prime}},g_{X}\\}$ from coherent
elastic neutrino-nucleus scattering (CE$\nu$NS) measured by COHERENT
experiment with CsI and Ar targets [235]. In this work, we rescale the
constraint on $U(1)_{B-L}$ case in Ref. [236, 237] by comparing number of
events in $U(1)_{B-L}$ and other cases. The number of events at COHERENT
experiment is estimated, adopting formulas in the reference, as follows. The
differential cross section for CE$\nu$NS process is given by [238, 239]

$\frac{d\sigma_{\nu-N}}{dT}(E,T)=\frac{G_{F}^{2}M}{\pi}\left(1-\frac{MT}{2E^{2}}\right)Q_{{\rm
SM}+Z^{\prime}}^{2},$ (123)

where $T$ is the recoil energy, $E$ is the neutrino energy, $M$ is the mass of
target nucleus and $Q_{{\rm SM}+Z^{\prime}}$ is the factor coming from
SM+$Z^{\prime}$ interactions. In the models $Q_{{\rm SM}+Z^{\prime}}$ is
written such that

$Q_{{\rm
SM}+Z^{\prime}}=\left(g^{p}_{V}(\nu_{\ell})+2\epsilon^{uV}_{\ell\ell}+\epsilon^{dV}_{\ell\ell}\right)ZF_{Z}(|{\bf
q}^{2}|)+\left(g^{n}_{V}(\nu_{\ell})+\epsilon^{uV}_{\ell\ell}+2\epsilon^{dV}_{\ell\ell}\right)NF_{N}(|{\bf
q}^{2}|),$ (124)

where $Z(N)$ is the number of proton(neutron) in the target nucleus,
$g^{p(n)}_{V}$ is the neutrino-proton(neutron) coupling in the SM and
$F_{Z(N)}(|{\bf q}^{2}|)$ is the from factors of the proton(neutron) for the
target nucleus. The effective coupling $\epsilon^{qV}_{\ell\ell}$ is
explicitly given by

$\epsilon^{qV}_{\ell\ell}=\frac{g_{X}^{2}x_{\ell}x_{q}}{\sqrt{2}G_{F}({\bf
q}^{2}+m^{2}_{Z^{\prime}})}.$ (125)

For the neutrino-proton(neutron) coupling, we adopt the values of
$g_{V}^{p}(\nu_{e})=0.0401$, $g_{V}^{p}=0.0318$ and $g_{V}^{n}=-0.5094$ for
the SM [240, 241]. For the form factors $F_{Z(N)}(|{\bf q}^{2}|)$, we apply
Helm parametrization [242] using proton rms radii $\\{R_{p}({\rm
Cs}),R_{p}({\rm I}),R_{p}({\rm Ar})\\}=\\{4.804,4.749,3.448\\}$ [fm] and
neutron rms radii $\\{R_{n}({\rm Cs}),R_{n}({\rm I}),R_{n}({\rm
Ar})\\}=\\{5.01,4.94,3.55\\}$ [fm] [243, 244, 245].

For the CE$\nu$NS event rate in the COHERENT experiment, we use the neutrino
fluxes that depend on the neutrino fluxes produced from the Spallation Neutron
Source (SNS) at the Oak Ridge National Laboratories. They are written as

$\displaystyle\frac{dN_{\nu_{\mu}}}{dE}$ $\displaystyle=$
$\displaystyle\eta\delta\left(E-\frac{m_{\pi}^{2}-m_{\mu}^{2}}{2m_{\pi}}\right),$
$\displaystyle\frac{dN_{\nu_{\bar{\mu}}}}{dE}$ $\displaystyle=$
$\displaystyle\eta\frac{64E^{2}}{m_{\mu}^{3}}\left(\frac{3}{4}-\frac{E}{m_{\mu}}\right),$
$\displaystyle\frac{dN_{\nu_{e}}}{dE}$ $\displaystyle=$
$\displaystyle\eta\frac{192E^{2}}{m^{3}_{\mu}}\left(\frac{1}{2}-\frac{E}{m_{\mu}}\right),$
(126)

where $\eta=rN_{\rm POT}/(4\pi L^{2})$ with $r$, $N_{\rm POT}$ and $L$ being
respectively the number of neutrinos per flavor that are produced for each
proton-on-target (POT), the number of POT, and the distance between the source
and the detector. For these values, we adopt $r=9\times 10^{-2}$, $N_{\rm
POT}=13.7\times 10^{22}$ and $L=27.5$ m for Ar detector, and $r=0.08$, $N_{\rm
POT}=17.6\times 10^{22}$ and $L=19.5$ m for CsI detector. The theoretical
number of events for each energy bin in the COHERENT experiment is estimated
by

$N_{i}=N(\mathcal{N})\int^{T_{i+1}}_{T_{i}}dTA(T)\int_{E_{\rm min}}^{E_{\rm
max}}dE\sum_{\nu=\nu_{e},\nu_{\mu},\nu_{\bar{\mu}}}\frac{dN_{\nu}}{dE}\frac{d\sigma_{\nu-N}}{dT}(E,T),$
(127)

where $i$ distinguishes recoil energy bin, $E_{\rm
min(max)}=\sqrt{MT/2}(m_{\mu}/2)$ and $A(T)$ is the energy-dependent
reconstruction efficiency. We estimate the upper bound of coupling $g_{X}$ for
each mass in our models by rescaling that of $U(1)_{B-L}$ case in ref. [237]
where we compare the number of events for the upper bound on $U(1)_{B-L}$ and
compare it with the number of events in each model to find the upper bound on
$g_{X}$. Limits obtained from the coherent scenarios are shown in the upper
panels of Fig. 9-12 for $U(1)_{X}$ case. These limits for the $U(1)_{q+xu}$
are not separately shown as they belong to the gray shaded region. Similar
bounds for $B-3L_{e}$ are shown in Fig. 15.

###  V.9 $g-2$ of muon and electron

In the context of the general $U(1)_{X}$ extension of the SM, the left and
right-handed fermions have different charges under the general $U(1)_{X}$
gauge group. These charges are family universal. However, in the case of
flavoured scenarios the general $U(1)$ charge could depend on the generation
of the fermions. Hence we write the general interaction Lagrangian as

$\displaystyle\mathcal{L}_{\rm
int}=g_{X}\overline{f}(q_{L}^{f}P_{L}+q_{R}^{f}P_{R})\gamma_{\mu}fZ^{\mu}$
(128)

which can be reduced to leptonic interactions with $Z^{\prime}$. The vector
and axial-vector couplings in general $U(1)_{X}$ extension can contribute to
the magnetic moment of the muon and electron involving $Z^{\prime}$ in the
loop satisfying $(g-2)_{\mu}$ results from [246]. Hence we can estimate bounds
on the $g_{X}-M_{Z^{\prime}}$ plane for different charges in general
$U(1)_{X}$ scenarios and flavored $U(1)$ scenarios. From [247, 248, 249] we
write down the magnetic moment of muon (electron) using one loop $Z^{\prime}$
mediated processes as

$\displaystyle\Delta{a_{\ell}}=\frac{g_{X}^{2}}{32\pi^{2}}\int_{0}^{1}dy\frac{\Big{\\{}2y(1-y)(y-4)-4y^{3}\frac{m_{\ell}^{2}}{M_{Z^{\prime}}^{2}}\Big{\\}}x_{a}^{2}+2(1-y)y^{2}x_{b}^{2}}{(1-y)(1-y\frac{m_{\ell}^{2}}{M_{{Z^{\prime}}^{2}}})+y\frac{m_{\ell}^{2}}{M_{Z^{\prime}}^{2}}}$
(129)

where $m_{\ell}=m_{\mu(e)}$, $x_{a}=-\frac{3}{2}x_{H}-2x_{\Phi}$ and
$x_{b}=-\frac{1}{2}x_{H}$ for the general $U(1)_{X}$ scenario. In the case of
the $U(1)_{q+xu}$ scenario, we have $x_{a}=-\frac{5+x}{3}$ and
$x_{b}=-\frac{x-1}{3}$ respectively. Similarly, we can calculate these
coefficients for the flavored cases using the corresponding charges of the
left and right-handed charged leptons. The current experimental ranges of muon
$(g-2)$ is given by [250]

$\displaystyle\Delta a_{\mu}=(24.9\pm 4.98)\times 10^{-10},$ (130)

where the deviation from the world average of SM prediction is 5.0 $\sigma$
level. In addition to that, the current experimental ranges of electron
$(g-2)$ are

$\displaystyle\Delta a_{e}(^{133}\text{Cs})=-(8.8\pm 3.6)\times
10^{-13}~{}~{}\text{\cite[cite]{[\@@bibref{Number}{Parker:2018vye}{}{}]}},\,\,\,\,\,\Delta
a_{e}(^{87}\text{Rb})=(4.8\pm 3.0)\times
10^{-13}~{}~{}\text{\cite[cite]{[\@@bibref{Number}{Morel:2020dww}{}{}]}}\,,$
(131)

where the deviations from the SM prediction are, respectively, $2.4\sigma$ and
$1.6\sigma$. Since positive $\Delta a_{e}$ is obtained from $Z^{\prime}$
interaction we only show the parameter region that satisfies the second range
of $\Delta a_{e}$. Limits obtained from $(g-2)_{e,\mu}$ are given in Figs.
10-12 for different charges of $U(1)_{X}$ and $U(1)_{q+xu}$ respectively. We
find $(g-2)_{e}$ limits for $L_{e}-L_{\mu,\tau}$ scenarios in the upper and
lower panels of Fig. 13. We show $(g-2)_{\mu}$ limits for $L_{\mu}-L_{\tau}$
scenario in Fig. 14. We show the limits for $B-3L_{e}$ and $B-3L_{\mu}$ in
Figs.15 and upper panel of 16 for $(g-2)_{e}$ and $(g-2)_{\mu}$ respectively.

Figure 13: Limits on $g_{X}-M_{Z^{\prime}}$ plane for the $L_{e}-L_{\mu}$
(upper panel) and $L_{e}-L_{\tau}$ (lower) scenarios. Figure 14: Limits on
$g_{X}-M_{Z^{\prime}}$ plane for the $L_{\mu}-L_{\tau}$ scenario.

###  V.10 Limits from GRB

From GRB we have estimated the enhancement in energy deposition due to the
influence of $Z^{\prime}$ using Schwarzchild, HT and modified gravity
scenarios. The corresponding limits are given in Tab. 5 for different charges
for the chiral scenarios like $U(1)_{X}$, $U(1)_{q+xu}$ and flavored scenarios
like $B-3L_{i}$ and $L_{i}-L_{j}$ respectively. Hence we estimate constraints
with $10\%$ and a prospective $1\%$ precision respectively. Following the
constraints on the $U(1)$ from large VEV scenarios ($v_{\Phi}>v$), we estimate
the bounds on $g_{X}-M_{Z^{\prime}}$ plane for different $U(1)$ scenarios. The
bounds $M_{Z^{\prime}}/g_{X}$ will also satisfy the condition $v_{\Phi}>v$,
therefore, bounds lower than that will not put any constraints on
$g_{X}-M_{Z^{\prime}}$ plane. Applying this fact in Tab. 5 we find that in the
case of general $U(1)_{X}$ scenario, GRB can not grant any limit from the
Schwarzchild metric for $x_{H}=-1$ for $R^{\rm Sc}/M=3.67$, $x_{H}=0$ and $-1$
for $R^{\rm Sc}/M=5$ respectively. A similar scenario happens in the case of
HT metric for $x_{H}=-1$ using $R^{\rm HT}/M=3$ and $J/M^{2}=0.37$ where as
bounds obtained on $M_{Z^{\prime}}/g_{X}$ for different $x_{H}$ with $R^{\rm
HT}/M=5$ and $J/M^{2}=0.37$ will not survive because all these values reside
below the electroweak scale VEV. In the case of $U(1)_{q+xu}$ and
$L_{e}-L_{\mu}$ scenarios using the same fact we notice that bounds for the
Schwarzchild case with $R^{\rm Sc}/M=5$ and HT case with $R^{\rm HT}/M=5$,
$J/M^{2}=0.37$ will not survive. Both the cases of $R/M$ values will bestow
constraints in the $B-3L_{e}$ scenario, however, the bounds on
$M_{Z^{\prime}}/g_{X}$ is stronger in HT case compared to the Schwarzchild
case. Finally, we observe that for fixed charges bounds obtained from HT is
slightly stronger than the Schwarzchild case, however, the limits will be
close depending on $R/M$ and $J/M^{2}$ respectively. Therefore for simplicity,
we show only bounds obtained using the HT case solving
$`M_{Z^{\prime}}/g_{X}=\rm bounds$’ for $10^{-3}\rm{GeV}\leq
M_{Z^{\prime}}\leq 25$ TeV where ‘bounds’ stands for the surviving values of
this quantity given in Tab. 5 for $U(1)_{X}$ case, however, for the remaining
cases we consider $10^{-3}\rm{GeV}\leq M_{Z^{\prime}}\leq 10$ TeV.

In the case of modified gravity, we consider CG and BIRN cases. As we
explained, in the Schwarzchild and HT cases, in the same line we find that GRB
can not constrain $x_{H}=-1$ in CG and BIRN cases for $R^{\rm CG}/M=3.5$,
$q=0.3$, $I=0.0$ and $R^{\rm BIRN}/M=3$, $b/M=0.5$ in $U(1)_{X}$ scenario.
Whereas in CG and BIRN cases GRB affects only $x_{H}=2$ in $U(1)_{X}$ scenario
for $R^{\rm CG}/M=5$, $q=0.3$, $I=0.0$ and $R^{\rm CG}/M=5$, $q=0.5$, whereas
rest of the cases show bounds on $M_{Z^{\prime}}/g_{X}$ which estimate VEV of
the general $U(1)_{X}$ scenario lower than $v$. Due to this fact bounds
obtained on $M_{Z^{\prime}}/g_{X}$ from CG and BIRN cases with $R^{\rm
CG}/M=5$, $q=0.3$, $I=0.0$ and $R^{\rm CG}/M=5$, $q=0.5$ do not survive.
Following the same approach for the $U(1)_{q+xu}$ scenario we find that
estimated bounds on $M_{Z^{\prime}}/g_{X}$ using $R^{\rm CG}/M=3.5$, $q=0.3$,
$I=0.0$ and $R^{\rm BIRN}/M=3$, $b/M=0.5$ for CG and BIRN cases respectively
survive for all $x$, however, those estimated from the remaining choices do
not survive due to the fact that they can not satisfy the condition
$v_{\Phi}>v$. Among the flavor scenarios $B-3L_{e}$ scenario can be
constrained from CG and BIRN cases for the required choices of parameters like
$R/M$, $q$, $I$ and $b/M$ whereas in $L_{i}-L_{j}$ scenario constraints on
$M_{Z^{\prime}}/g_{X}$ can be obtained from CG and BIRN cases using $R^{\rm
CG}/M=3.5$, $q=0.3$, $I=0.0$ and $R^{\rm BIRN}/M=3$, $b/M=0.5$ respectively.
The other choice will not be able to constrain $g_{X}-M_{Z^{\prime}}$
parameters because it does not satisfy $v_{\Phi}>v$. We find that constraints
on $M_{Z^{\prime}}/g_{X}$ obtained from the CG scenario is slightly greater
than those obtained using BIRN scenario. Therefore constraints on
$g_{X}-M_{Z^{\prime}}$ obtained from CG scenario is slightly stronger than the
BIRN case. As a result we show only the constraints in the final limit plots
obtained from the CG case due to simplicity solving $`M_{Z^{\prime}}/g_{X}=\rm
bounds$’ for $10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 10$ TeV where ‘bounds’
stands for the surviving values of this quantity given in Tab. 5 for different
$U(1)$ extended scenarios.

Final limits are given in Figs. 9-12 for $U(1)_{X}$ and $U(1)_{q+Xu}$ cases.
We show different existing bounds in $U(1)_{X}$ cases with colored lines and
only relevant ones in case of $U(1)_{q+xu}$ with colored lines and remaining
ones with gray shaded regions. We find that the strongest GRB contours
obtained from the modified gravity model CG (blue dot and dot-dashed lines)
are almost matching with the HT (black dot and dot-dashed lines) case. There
is no limits for $x_{H}=-1$ in $U(1)_{X}$ case. The bounds are shown in the
upper panel of Fig. 9. In case of $x=-1$ in $U(1)_{q+xu}$ scenario, given in
the lower panel of the same figure, we find the bounds from the GRB scenarios
belong to the gray shaded region. In Fig. 10 we show $x_{H}=0$ and $x=1$ from
both $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios representing the bounds on
$g_{X}-M_{Z^{\prime}}$ plane for B$-$L scenario. We find that GRB contours
with $10\%$ precision are comparable with the bounds obtained by BOREXINO,
TEXONO and CHARM-II depending on $M_{Z^{\prime}}$. However, the bounds with
prospective $1\%$ precision could provide a strong limit on the gauge coupling
$6\times 10^{-6}\leq g_{X}\leq 7\times 10^{-5}$ for $0.025$ GeV $\leq
M_{Z^{\prime}}\leq 0.25$ GeV. The prospective bounds with $1\%$ precision
crosses the bounds obtained from the beam-dump scenarios at FASER, FASER2 and
ILC-BD which could be tested in future. Limits on the $U(1)$ gauge coupling
for $x_{H}=1$ and $x=0$ are shown in the upper and lower panels of Fig. 11 for
$U(1)_{X}$ and $U(1)_{q+xu}$ respectively. We find that in $U(1)_{X}$ case
prospective searches from FASER, FASER2 and ILC-BD could provide complementary
search reaches to probe $Z^{\prime}$ around $0.06$ GeV. On the other hand
limits from existing bounds, specially from $\nu-$cal covers the area where
prospective searches could cross over in case of $U(1)_{q+xu}$ scenario.
Bounds on the $U(1)$ gauge coupling for different $M_{Z^{\prime}}$ for
$x_{H}(x)=2$ are shown in Fig. 12 in the lower and upper panels for $U(1)_{X}$
and $U(1)_{q+xu}$ respectively providing same experience like the previous
charge assignment. We find that limits from dark photon searches at LHCb
($2.2$ GeV $\leq M_{Z^{\prime}}\leq 70$ GeV) and CMS ($1$ GeV $\leq
M_{Z^{\prime}}\leq 9$ GeV) provide stronger limit than the other existing
searches which occupy the parameter regions predicted by GRB. Finally we find
that GRB contours could provide strong limits on the $g_{X}$ for a narrow
range $100$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV for any charge of $U(1)_{X}$
and $U(1)_{q+xu}$ scenarios which could be probed by future collider
experiments. We find that among the existing beam-dump scenarios $\nu-$cal
provides the strongest bounds in case of $U(1)_{X}$ and $U(1)_{q+xu}$
scenarios. Other existing bounds from different beam-dump scenarios are shown
by different colored lines for $U(1)_{X}$ case whereas by a collective gray
shaded regions in case $U(1)_{q+xu}$ involving E774, NA64, E141, E137, Orsay,
NoMad, PS191 and CHARM etc. BaBar(vis and invis) provides stronger bounds on
$g_{X}-M_{Z^{\prime}}$ plane within $0.5$ GeV $\leq M_{Z^{\prime}}\leq 10$
GeV. We find that limits estimated from $(g-2)_{e}$ (Rb and CS) and
$(g-2)_{\mu}$ are weak compared to the GRB bounds for different charges
(expect $x_{H}(x)=-1$ where $g-2$ limits are negative electron and muon
respectively) of the $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios respectively. We
also find that limits estimated from GEMMA and COHERENT experiments are weak
compared to the GRB limits in case of $U(1)_{X}$ scenario, however, they
belong to the gray shaded area in case of $U(1)_{q+xu}$ scenario,
respectively. Limits from BaBaR(vis and invis) are weaker than scattering
experiments like TEXONO, BOREXINO (belong to the gray shaded region in
$U(1)_{q+xu}$) and dark photon searches from LHCb and CMS limits, respectively
apart from a narrow window of $Z^{\prime}$ mass between $9$ GeV $\leq
M_{Z^{\prime}}\leq 10$ GeV. Limits obtained from GRB with $10\%$ and $1\%$
precisions are weaker than the current LEP-II bounds for $M_{Z^{\prime}}>209$
GeV in general $U(1)$ scenarios for different charges. Prospective ILC bounds
are stronger than the GRB bounds for $M_{Z^{\prime}}>\sqrt{s}_{\rm ILC}$.

Figure 15: Limits on $g_{X}-M_{Z^{\prime}}$ plane for $B-3L_{e}$ scenario.

Figure 16: Limits on $g_{X}-M_{Z^{\prime}}$ plane for $B-3L_{\mu}$ (upper
panel) and $B-3L_{\tau}$ (lower panel) scenarios.

Estimating bounds from GRB we find limits on the flavored scenarios like
$L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$ scenarios in the upper and lower panels
of Fig. 13. We find that limits from $10\%$ precision is comparable with
TEXONO bounds for $L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$ scenarios. Whereas
prospective bounds with $1\%$ precision could be stronger. The GRB contours
with $10\%$ and $1\%$ precision cross the prospective ILC-BD line at
$\\{M_{Z^{\prime}},g_{X}\\}=\\{0.02\rm{GeV},2.5\times 10^{-5}\\}$ and
$\\{0.05\rm{GeV},1.5\times 10^{-5}\\}$, respectively. Like $U(1)_{X}$ and
$U(1)_{q+xu}$ cases GRB contours could provide strong limits on the $g_{X}$
for a narrow range $100$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV. The
$Z^{\prime}-$electron coupling is same in $L_{e}-L_{\mu}$ and $L_{e}-L_{\tau}$
cases causing same bounds where $Z^{\prime}-$electron interaction is involved.
We find that in cases of $L_{i}-L_{j}$, Orsay and E137 provide stronger bound
up to $\\{M_{Z^{\prime}},g_{X}\\}=\\{0.08~{}\rm{GeV},10^{-6}\\}$ and
$\\{0.25~{}\rm{GeV},10^{-7}\\}$ respectively whereas E774 and E141 provide
comparatively weaker bounds for $M_{Z^{\prime}}\leq 0.02$ GeV. Limits
estimated from CHARM-II, GEMMA and BaBar(vis and invis for $M_{Z^{\prime}}\leq
0.02$ GeV) are weaker than the GRB bounds. Bounds obtained on $g_{X}$ for
$Z\to 4\mu$ search in case of $L_{e}-L_{\mu}$ scenario provide stronger bound
for $10$ GeV $\leq M_{Z^{\prime}}\leq 50$ GeV where $U(1)$ coupling varies
between $0.003\leq g_{X}\leq 0.04$. There is no bound in this range for
$L_{e}-L_{\tau}$ scenario because muon has no direct coupling with
$Z^{\prime}$ in this case. Following the same line we provide bounds from
CHARM-II in case of $L_{e}-L_{\mu}$ scenario. In addition to that we find
constrains from [232] where $e^{-}e^{+}$ collision at KLOE experiment [253,
254] provide a stronger bound on $g_{X}$ for a narrow range $0.6$ GeV $\leq
M_{Z^{\prime}}\leq 1$ GeV which can reach up to $4\times 10^{-4}\leq g_{X}\leq
3\times 10^{-4}$ touching the prospective bounds from GRB with $1\%$ precision
at $\\{M_{Z^{\prime}},g_{X}\\}=\\{1~{}\rm{GeV},2\times 10^{-4}\\}$. The bounds
obtained from KLOE are weaker compared to the other bounds in case of
$L_{e}-L_{\tau}$ scenario. We show the limits [232] obtained from the APEX
[155] and A1 [153, 152] experiments in case $L_{e}-L_{(\mu,\tau)}$ scenarios
to be weak compared to the limits estimated for GRB. The $(g-2)_{\mu(e)}$
bounds for $L_{e}-L_{\mu(\tau)}$ scenario is weak compared to the GRB
constraints. We find that GRB bounds with $10\%$ and prospective $1\%$
precision are stronger than the Super-K bounds obtained from [231]. Limits
obtained from GRB with $10\%$ and $1\%$ precisions are weaker than the current
LEP-II bounds for $M_{Z^{\prime}}>209$ GeV in case of $L_{e}-L_{\mu}$ and
$L_{e}-L_{\tau}$ scenarios. Prospective ILC bounds are stronger than the GRB
bounds for $M_{Z^{\prime}}>\sqrt{s}_{\rm ILC}$.

Bounds estimated from the GRB scenario in $B-3L_{e}$ case can be found in Fig.
15. The limit obtained with $10\%$ precision is comparable with the results
obtained from TEXONO experiment. We find that for the $B-3L_{e}$ case, GEMMA
provides limits around $6\times 10^{-6}\leq g_{X}\leq 10^{-4}$ for $0.03$ GeV
$\leq M_{Z^{\prime}}\leq 0.3$ GeV which is comparable with the limits obtained
from TEXONO and GRB with $10\%$ precision respectively, however, bounds
obtained from the GRB analysis with a prospective $1\%$ precision could be a
factor of five stronger than these bounds. Considering the limits from KOLE
[253, 254], APEX [155], A1 [153, 152] and NA48/2 [154] experiments from [232,
231] which are weaker than the estimated GRB bounds. Limits obtained from GRB
with $10\%$ precision are comparable with the current LEP-II bounds for
$M_{Z^{\prime}}>209$ GeV whereas bounds with $1\%$ precision are stronger than
the current LEP-II limits. Prospective ILC bounds are stronger than the GRB
bounds for $M_{Z^{\prime}}>\sqrt{s}_{\rm ILC}$.

In flavored scenarios, we find that strongest bounds from GRB could also
appear within $100$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV where corresponding
$U(1)$ coupling varies between $0.02\leq g_{X}\leq 0.03$. When $Z^{\prime}$ is
light such that $M_{Z^{\prime}}\simeq\mathcal{O}(0.01)$ GeV, the contours from
GRB could intersect with the prospective limits from FASER, FASER2 (in
$B-3L_{e}$ case only) and ILC-BD respectively which could be probed in future.
We find that BaBar (vis and invis) provide stronger bounds for
$M_{Z^{\prime}}\geq 0.3(0.11)$ GeV in case of $L_{e}-L_{\mu}$ scenario and
$M_{Z^{\prime}}\geq 0.9(0.2)$ GeV in case of $L_{e}-L_{\tau}$ scenario,
respectively compared to the bounds estimated from GRB using $10\%(1\%)$
precision, respectively. We find that in case of $B-3L_{e}$ scenario $\nu-$cal
provides strongest bound among the existing beam-dump scenarios up to
$M_{Z^{\prime}}\leq 0.3$ GeV. Prospective bounds from the beam-dump scenario
from DUNE (DUNE-BD) are weaker than $\nu-$cal scenario for $M_{Z^{\prime}}\leq
0.27$ GeV.

###  V.11 Limits from blazars and AGN

The total cross section for $\nu$-DM scattering process depends on the
mediator mass, here $M_{Z^{\prime}}$ and general $U(1)$ coupling, $g_{X}$ in
addition to the neutrinos energies. In this analysis we will estimate
constraints on these parameters. Therefore we write
$\sigma=\sigma(g_{X},M_{Z^{\prime}})$ and the cascade equation given in Eq. 95
gets modified as

$\frac{d\Phi}{dx}=-\sigma(g_{X},M_{Z^{\prime}})\frac{\Sigma(r)}{m_{\rm
DM}}\;\Phi+\frac{\Sigma(r)}{m_{\rm
DM}}\int_{E^{\prime}}^{\infty}dE\;\frac{d\sigma(g_{X},M_{Z^{\prime}})}{dE^{\prime}}\Phi(E)$
(132)

using the fact $E_{\nu}^{2}+{E_{\nu}^{\prime}}^{2}>>m_{\rm
DM}(E_{\nu}^{\prime}-E_{\nu})$ we use the following parametrization to solve
the cascade equation

$\alpha_{1}=\frac{g_{X}^{4}\;\Sigma}{8\pi
M_{Z^{\prime}}^{4}}(1\;\text{TeV}),\;\;\alpha_{2}=\frac{m_{DM}}{M_{Z^{\prime}}^{2}}(1\;\text{TeV})$
(133)

where $\alpha_{1,2}$ are dimensionless quantities. This parametrization will
be used for the differential scattering cross-sections involving three types
of DM candidates. The second term in Eq. 132 is proportional to the
differential scattering cross section and it can be parameterized in terms of
$\alpha_{1,2}$; hence, it can be written in terms of $\alpha_{1,2}$ and
$m_{\rm DM}$. In our analysis, we fix $m_{\rm DM}$. Therefore this term
depends on $\alpha_{1,2}$ only. To constrain the parameter space of $g_{X}$
and $M_{Z^{\prime}}$ using blazar and AGN event, we apply the following steps:

  1. (i)

Using the initial flux of blazar and AGN given in Eqs. 96 and 100
respectively, we compute the final attenuated flux $\Phi_{\tilde{a}}$ by
solving the cascade equation for different values of $\alpha_{1,2}$ which is
equivalent to different values of $g_{X}$ and $M_{Z^{\prime}}$ related by Eq.
133.

  2. (ii)

Using Eq. 97 and $\Phi_{\tilde{a}}$ we calculate the expected number of events
at IceCube for the attenuated flux corresponding to blazar and AGN.

  3. (iii)

Finally we require that the number of events be at least $10\%$ of the total
observed events with no $\nu$-DM scattering (i.e 0.1 events in case of TXS
0506+056 and 3 events in case of AGN) for deriving a limit at $90\%$ C.L. on
$g_{X}-M_{Z^{\prime}}$ plane.

Now we estimate constrains on $g_{X}-M_{Z^{\prime}}$ plane for chiral and
flavored scenarios from the blazar and AGN cases considering $m_{\rm
DM}=3M_{Z^{\prime}}$.

Constraints from blazar and AGN cases considering $x_{H}(x)=-1$, $0$, $1$ and
$2$ for $U(1)_{X}$ (upper panel) and $U(1)_{q+xu}$ (lower panel) scenarios in
Figs. 9\- 12, respectively. We mention that $x_{H}=0$ and $x=1$ cases shown in
Fig. 10 under $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios resemble the B$-$L
scenario. We denote the Dirac(D1) and complex scalar(CS) type DM candidates
with $U(1)$ charge $Q_{\chi}=1000$ (under the perturbative limit
$g|Q_{\chi}|\leq\sqrt{4\pi}$) and Majorana RHN DM (M2) with $U(1)$ charge
$-5$. The blazar (B) scenarios for the corresponding D1, CS and M2 cases are
represented by red solid (B-D1), dashed (B-CS) and dot-dashed (B-M2) lines
whereas the AGN (A) scenarios are represented by corresponding green lines.
The B-D1 (A-D1) and B-CS (A-CS) derive stronger limits which almost overlap
with each other due to the fact that
$E_{\nu}^{2}+{E_{\nu}^{\prime}}^{2}>>m_{\rm DM}(E_{\nu}^{\prime}-E_{\nu})$ and
same $U(1)$ charges whereas the limits obtained from the Majorana DM are
weaker compared to the Dirac and complex scalar cases due its different $U(1)$
charge. Among the blazar and AGN cases we obtain that AGN cases are stronger
than the respective blazar cases. Comparing with existing bounds we find that
limits obtained from the blazar and AGN scenarios are weak compared to the
limits obtained from the scattering experiments like COHERENT, CHARM-II,
GEMMA, BOREXINO, TEXONO and dark photon searches at BaBar(vis and invis), LHCb
and CMS, respectively. The existing bounds from different beam-dump scenarios
like NA64, E774 and E141 covers the region for $M_{Z^{\prime}}\leq 0.02$ GeV
for the Dirac and CS cases with $Q_{\chi}=1000$, however, bounds from KEK,
Orsay, NoMad, CHARM and E137 provide strong bound for $M_{Z^{\prime}}$ range
under consideration. Among the beam-dump scenarios strongest existing bound
comes from $\nu-$cal which is stronger than the prospective search reach for
the beam-dump scenario at DUNE (DUNE-BD) for $M_{Z^{\prime}}\leq 0.4$ GeV.
$\nu-$cal provides stronger bound than AGN (D1 and CS) cases for
$M_{Z^{\prime}}\leq 0.002$ GeV. The crossovers of the AGN (D1 and CS) bounds
with the prospective FASER, FASER2 and ILC-BD scenarios also appear in the
sahded regions for both $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios. We find that
bounds obtained from GRB are roughly $\mathcal{O}(2)-\mathcal{O}(3)$ of
magnitude stronger than the blazar and AGN bounds depending on
$M_{Z^{\prime}}$.

We estimate bounds on the $g_{X}-M_{Z^{\prime}}$ plane for $L_{e}-L_{\mu}$ and
$L_{\mu}-L_{\tau}$ scenarios constraining the $\nu-$DM scattering from blazar
and AGN data observed by IceCube. The limits are shown in the upper panel of
Fig. 13 for $L_{e}-L_{\mu}$ and in Fig. 14 for $L_{\mu}-L_{\tau}$ scenarios,
respectively. We find that in $L_{e}-L_{\mu}$ scenario, limits from blazar and
AGN are weaker than the estimated limits obtained from BaBar(vis and invis),
CHARM-II, TEXONO, BOREXINO and GRB scenarios. On the other hand we find that
limits obtained after studying Majorana DM from blazar (B-M2) and AGN (A-M2)
are the weakest in these two scenarios. Limits estimated from B-D1, B-CS are
also weak in the flavored scenarios. However, in case of $L_{\mu}-L_{\tau}$
scenario, A-D1 and A-CS are stronger than the $(g-2)_{\mu}$ limits for
$M_{Z^{\prime}}\leq 0.02$ GeV. AGN scenario also provide a strong bound for
the narrow range $0.0175$ GeV$\leq M_{Z^{\prime}}\leq 0.03$ GeV, however,
limits for $M_{Z^{\prime}}\geq 0.03$ GeV belong to gray shaded region whose
boundary consists of the strongest limits from BOREXINO, CCFR [255], BaBar
4$\mu$ [231] and $4\mu$ search from CMS [256] and ATLAS [257], respectively.
It is interesting to notice that in case of $L_{\mu}-L_{\tau}$, limits
obtained from white dwarf starts [258, 231] are stronger than BOREXINO.
Remaining part of the boundary of the gray shaded region consists of CCFR and
4$\mu$ search from BaBar respectively. We find that $4\mu$ search from the CMS
can constrain the parameter regions from $L_{e}-L_{\mu}$ and
$L_{\mu}-L_{\tau}$ scenarios [256] providing bounds for $4$ GeV $\leq
M_{Z^{\prime}}\leq 70$ GeV where limits on $g_{X}$ could reach up to
$1.5\times 10^{-3}$ [256]. In case of $L_{\mu}-L_{\tau}$ we find that limits
from new $(g-2)_{\mu}$ data provide strong bound on the gauge coupling within
$4\times 10^{-4}\leq g_{X}\leq 10^{-3}$ for $0.01$ GeV $\leq
M_{Z^{\prime}}\leq 0.2$ GeV where AGN could provide comparable limits from
Dirac (D2) and complex scalar (CS) DM candidates, respectively. In addition to
that bounds can be obtained on the gauge coupling from NA62 experiment using
$K\to\mu+E_{T}^{\rm miss}$ mode following [259, 260, 261] which is stronger
than the bounds obtained from $(g-2)_{\mu}$ experiment. Whereas bounds from
NA62 using $K\to 3\mu+E_{T}^{\rm miss}$ is comparable to the bounds obtained
from the $4\mu$ search from the BaBar experiment belonging to the gray shaded
area. We find that a prospective limit from the Belle-II experiment at 50
$\rm{ab}^{-1}$ luminosity can vary within $3.2\times 10^{-4}\leq g_{X}\leq
1.6\times 10^{-3}$ for $0.007$ GeV $\leq M_{Z^{\prime}}\leq 5$ GeV [262].
Within this range, we find that limits obtained analysing AGN (D1 and CS) data
comparing $\nu-$DM scattering could provide comparable bound for
$M_{Z^{\prime}}=0.01$ GeV with prospective reach from Belle-II at 50 ab-1
luminosity. Prospective sensitivities from NA64$\mu$ [263] and COHERENT [231]
experiment could be stronger than the limits obtained analysing the AGN data
for Dirac (D1) and complex scalar (CS) DM candidates.

We estimate limits on the $g_{X}-M_{Z^{\prime}}$ plane constraining $\nu-$DM
scattering from blazar and AGN data observed by IceCube experiment in the
context of $B-3L_{\mu}$ scenario. The limit plots are shown in the upper panel
of Fig. 16. The gray-shaded region represents the strongest limits obtained by
the neutrino Non-Standard Interaction (NSI) and it is taken from [30]. We show
the estimated limits from the dark photon searches from LHCb, CMS experiments
respectively. In addition to that we add limits obtained from the $4\mu$
search from CMS[256]. We find that limits obtained by constraining $\nu-$DM
scattering using blazar and AGN results from IceCube are weaker than existing
limits. We also estimate limits on the gauge coupling with respect to the
$Z^{\prime}$ mass using latest $(g-2)_{\mu}$ data which belong to the grey-
shaded region.

###  V.12 Theoretical limits from neutrino-DM scattering

We consider the theoretical limits on $\nu$-DM interaction. The interaction
affects the Cosmic Microwave Background (CMB) spectrum. The effect
parametrizes a dimensionless parameter

$\displaystyle u_{\nu\rm DM}\equiv\frac{\sigma_{\nu\rm DM}}{\sigma_{\rm
Th}}\left(\frac{m_{\rm DM}}{100{\rm GeV}}\right)^{-1}$ (134)

where $\sigma_{\rm Th}$ is the Thomson scattering cross section and
$\sigma_{\nu\rm DM}$ is the $\nu-$DM scattering cross section. We can obtain
the upper bounds of the total cross-section for neutrino DM scattering from
555Planck CMB and Baryon Acoustic Oscillation (BAO) data could be found inn
[89] and in this case $\sigma_{\rm Th}u_{\nu\rm DM}<2.56\times 10^{-28}{\rm
cm^{2}}({\rm CMB+BAO})$ which is weaker than the Lyman-$\alpha$. the
Lyman-$\alpha$ data [90] as follows

$\displaystyle\sigma_{\nu\rm DM}\left(\frac{m_{\rm DM}}{100{\rm
GeV}}\right)^{-1}$ $\displaystyle=$ $\displaystyle\sigma_{\rm Th}u_{\nu\rm
DM}<3.66\times 10^{-30}~{}{\rm cm^{2}}$ (135)

Comparing the theoretically estimated cross sections with respect to the
observed upper limit on the $\nu-$DM scattering cross section from the
Lyman-$\alpha$ line, we estimate limits on the $U(1)$ gauge coupling with
respect to $M_{Z^{\prime}}$ for the chiral and flavored scenarios. We find
that in the chiral cases for different $x_{H}$ or $x$, Lyman-$\alpha$ line
with Dirac and complex scalar DM candidates (Lyman-D and Lyman-S,
respectively) match with each other due to large value of $Q_{\chi}$ and
$E_{\nu}^{2}+{E_{\nu}^{\prime}}^{2}>>m_{\rm{DM}}(E_{\nu}^{\prime}-E_{\nu})$
under the perturbative limit $g|Q_{\chi}|<\sqrt{4\pi}$. The Majorana DM
(Lyman-M) case is weak compared to the Dirac (D) and complex scalar (S) cases.
Through out the chiral cases Lyman-$\alpha$ (Lyman-M, Lyman-D and Lyman-S)
lines are weak compared to the limits obtained from the scattering experiment,
beam-dump experiment and limits obtained from GRB scenarios (current precesion
with $10\%$ and prospective precision with $1\%$) shown in Figs. 9-12.

We find the above behavior of the Lyman lines in the case of $L_{e}-L_{\mu}$
and $L_{e}-L_{\tau}$ scenarios shown in the upper and lower panels of Fig. 13.
Significant difference occurs in the $L_{\mu}-L_{\tau}$ scenario as shown in
Fig. 14. Under the perturbative constraints we find that limits obtained from
the Lyman-D and Lyman-S scenarios could provide the strongest bound on the
$U(1)$ gauge coupling for $0.004$ GeV $\leq M_{Z^{\prime}}\leq 0.06$ GeV.
Bounds estimated from considering Majorana DM candidate represented by Lyman-M
are lying in the grsay shaded region. Limits from Lyman-D and Lyman-S could be
comparable with the bounds obtained from NA62 experiment considering
$K\to\mu+E_{T}^{\rm miss}$, $(g-2)_{\mu}$ and prospective bounds from
COHERENT, NA64$\mu$ and Belle-II. These bounds could be probed by different
experiments in near future. In the context of $L_{\mu}-L_{\tau}$ scenario, the
shaded region in green is disfavored if we want to explain $b\to
s\mu^{+}\mu^{-}$ anomalies [264].

We show that estimated bounds obtained from the case of $B-3L_{i}$ scenario
are shown in Figs. 15 and Fig. 16 respectively. Like the chiral cases, in this
flavored scenario, limits obtained from Lyman-M are weaker than Lyman-D and
Lyman-S, however, all the lines belong to the gray-shaded region.

##  VI Conclusions

In this paper, we consider chiral and flavored scenarios where interactions of
$Z^{\prime}$ depend on the corresponding $U(1)$ charges of the fermions. We
incorporate cosmic bursts involving GRB 221009A which is the brightest GRB of
all time, cosmic blazar (TXS 0506+056) and AGN (NGC 1068) to probe neutrino
interactions with $Z^{\prime}$ from different models. In the context of GRB,
we study electron-positron pair production from neutrino anti-neutrino
interaction involving SM and $Z^{\prime}$ gauge boson. Estimating energy
deposition of SM and BSM process involving Sc, HT and modified gravity
scenarios (BRIN and CG) we constrain $U(1)$ gauge coupling with respect to
$Z^{\prime}$ mass on $g_{X}-M_{Z^{\prime}}$ plane. As a part of chiral
aspects, we used $U(1)_{X}$ and $U(1)_{q+xu}$ scenarios where we changed the
corresponding charges to estimate bounds. We found that for $x_{H}=-1$, no
bounds from GRB could be obtained from the fact that the estimated VEV of
$U(1)_{X}$ scenario will be lower than electroweak VEV which is not possible
in our model set-up. In addition to that we find estimated bounds with $10\%$
precision are comparable with TEXONO results. Prospective bounds with $1\%$
precision are stronger than the previous cases in the case of $U(1)_{X}$ and
$U(1)_{q+xu}$ scenarios which cross over prospective bounds from FASER, FASER2
and ILC-BD which could be verified in future. A similar inference could be
drawn for the $L_{e}-L_{\mu,\tau}$, $B-3L_{e}$ scenarios where FASER, FASER2,
beam-dump search at ILC could be interesting to probe a parameter space
favoured by GRB depending on the interaction between $Z^{\prime}$ and the SM
fermions. To study neutrino interactions from blazars and AGN we use IceCube
data in the context of $\nu-$DM scattering which could probe light
$Z^{\prime}$ providing bounds on $g_{X}-M_{Z^{\prime}}$ plane for different
models. We introduced complex scalar, Dirac and Majorana DM candidates with
respective $U(1)$ charges under the perturbative limit. We find that in case
of $L_{\mu}-L_{\tau}$ scenario $\nu-DM$ scattering compared with IceCube data
could provide strong bounds for $0.006$ GeV $\leq M_{Z^{\prime}}\leq 0.02$ GeV
where bounds on $U(1)$ coupling could vary between $3\times 10^{-4}\leq
g_{X}\leq 7\times 10^{-4}$. These bounds are comparable with recent
$(g-2)_{\mu}$ observation and bounds obtained by COHERENT $\mu-$nucleon
scattering. Belle-II, NA62 and NA64μ experiments could probe this parameter
region in the future. We also find the theoretical limits on $\nu-$DM
scattering comparing with Lyman-$\alpha$ line for Dirac and complex scalar DM
candidates in the context of $L_{\mu}-L_{\tau}$ scenario which could provide
strong limits for $0.004$ GeV $\leq M_{Z^{\prime}}\leq 0.08$ GeV where the
corresponding gauge coupling will vary between $1.75\times 10^{-4}\leq
g_{X}\leq 8\times 10^{-4}$ under perturbative limit which could be probed by
NA62, Belle-II and NA64$\mu$ experiments in future.

###### Acknowledgements.

SKA thanks the Department of Ministry of Education, Culture, Sports, Science
and Technology of Japan for MEXT fellowship to study in Japan. This work is
supported by INFN and MIUR [GL] and the Fundamental Research Funds for the
Central Universities [TN].

##  Appendix A Cross sections of $\nu\bar{\nu}\to e^{-}e^{+}$ process in
center of mass frame in general $U(1)$ scenarios:

###  A.1 Kinematics

We define the quantities $p_{1}$ as initial neutrino momentum, $p_{2}$ as
initial anti-neutrino momentum, $k_{1}$ as final charged lepton momentum,
$k_{2}$ as final charged lepton momentum. The $4-$vector representation of
these momenta are $p_{1}=(E,0,0,E)$, $p_{2}=(E,0,0,-E)$,
$k_{1}=(E,0,E\sin\theta,E\cos\theta)$ and
$k_{2}=(E,0,-E\sin\theta,-E\cos\theta)$. The scalar products and neutrino
energy in laboratory frame are given by $k_{1}\cdot p_{1}=k_{2}\cdot
p_{2}=E^{2}(1-cos\theta)$ and $k_{1}\cdot p_{2}=k_{2}\cdot
p_{1}=E^{2}(1+cos\theta)$.

###  A.2 Amplitudes and Cross sections of $\nu_{e}\bar{\nu}_{e}\to e^{-}e^{+}$

We use the following gauge interactions,

$\displaystyle{\cal
L}_{gauge}=Z^{\mu}\left(\bar{\nu}_{L}c_{L}^{\nu}\gamma_{\mu}\nu_{L}+\sum_{i=L,R}\bar{e}_{i}c_{i}^{e}\gamma_{\mu}e_{i}\right)+\left(W^{+\mu}\bar{\nu}_{L}c_{+}\gamma_{\mu}e_{L}+h.c.\right)+Z^{\prime\mu}\left(\bar{\nu}_{L}{\tilde{c}}_{L}^{\nu}\gamma_{\mu}\nu_{L}+\sum_{i=L,R}\bar{e}_{i}{\tilde{c}}_{i}^{e}\gamma_{\mu}e_{i}\right)$
(136)

where $\\{L,R\\}$ are the projections and $\\{c_{i}^{e},\tilde{c}_{i}^{e}\\}$
are the corresponding couplings. The amplitudes for $\nu\bar{\nu}\rightarrow
e^{+}e^{-}$ are given by

$\displaystyle i{\cal M}_{1}$ $\displaystyle=$
$\displaystyle\frac{-ic_{\pm}^{2}}{M_{W}^{2}}\bar{v}(k_{1})\gamma_{\mu}P_{L}u(p_{1})\bar{u}(p_{2})\gamma^{\mu}P_{L}v(k_{2}),$
$\displaystyle i{\cal M}_{2}$ $\displaystyle=$
$\displaystyle\frac{ic_{L}^{\nu}}{M_{Z}^{2}}\bar{v}(p_{2})\gamma_{\mu}P_{L}u(p_{1})\bar{u}(k_{1})\gamma^{\mu}\left(c_{L}^{e}P_{L}+c_{R}^{e}P_{R}\right)v(k_{2}),$
$\displaystyle i{\cal M}_{3}$ $\displaystyle=$
$\displaystyle\frac{ic_{L}^{\nu}}{M_{Z^{\prime}}^{2}}\bar{v}(p_{2})\gamma_{\mu}P_{L}u(p_{1})\bar{u}(k_{1})\gamma^{\mu}\left(\tilde{c}_{L}^{e}P_{L}+\tilde{c}_{R}^{e}P_{R}\right)v(k_{2}),$
(137)

where ${\cal M}_{1}$, ${\cal M}_{2}$ and ${\cal M}_{3}$ are amplitudes for the
$W$, $Z$ and $Z^{\prime}$ mediated processes in Fig.2. The square of the
amplitudes are given by

$\displaystyle\frac{1}{4}\sum_{spin}\left|{\cal M}_{1}\right|^{2}$
$\displaystyle=$ $\displaystyle\frac{4c_{\pm}^{4}}{M_{W}^{4}}(k_{1}\cdot
p_{2})(p_{1}\cdot k_{2}),$ $\displaystyle\frac{1}{4}\sum_{spin}\left|{\cal
M}_{2}\right|^{2}$ $\displaystyle=$ $\displaystyle\frac{4c_{L}^{\nu
2}(c_{L}^{e2}+c_{R}^{e2})}{M_{Z}^{4}}(k_{1}\cdot p_{2})(p_{1}\cdot k_{2}),$
$\displaystyle\frac{1}{4}\sum_{spin}\left|{\cal M}_{3}\right|^{2}$
$\displaystyle=$ $\displaystyle\frac{4\tilde{c}_{L}^{\nu
2}(\tilde{c}_{L}^{e2}+\tilde{c}_{R}^{e2})}{M_{Z^{\prime}}^{4}}(k_{1}\cdot
p_{2})(p_{1}\cdot k_{2}),$ (138)

and the cross terms are given by

$\displaystyle\frac{1}{4}\sum_{spin}\left({\cal M}_{1}{\cal
M}_{2}^{\ast}+{\cal M}_{1}^{\ast}{\cal M}_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{4c_{\pm}^{2}c_{L}^{\nu}c_{L}^{e}}{M_{W}^{2}M_{Z}^{2}}(k_{1}\cdot
p_{2})(p_{1}\cdot k_{2}),$ $\displaystyle\frac{1}{4}\sum_{spin}\left({\cal
M}_{1}{\cal M}_{3}^{\ast}+{\cal M}_{1}^{\ast}{\cal M}_{3}\right)$
$\displaystyle=$
$\displaystyle\frac{4c_{\pm}^{2}\tilde{c}_{L}^{\nu}\tilde{c}_{L}^{e}}{M_{W}^{2}M_{Z^{\prime}}^{2}}(k_{1}\cdot
p_{2})(p_{1}\cdot k_{2}),$ $\displaystyle\frac{1}{4}\sum_{spin}\left({\cal
M}_{2}{\cal M}_{3}^{\ast}+{\cal M}_{2}^{\ast}{\cal M}_{3}\right)$
$\displaystyle=$
$\displaystyle\frac{4c_{L}^{\nu}\tilde{c}_{L}^{\nu}(c_{L}^{e}\tilde{c}_{L}^{2}+c_{R}^{e}\tilde{c}_{R}^{2})}{M_{Z}^{2}M_{Z^{\prime}}^{2}}(k_{1}\cdot
p_{2})(p_{1}\cdot k_{2}).$ (139)

The total cross section is given by $\sigma v_{rel}=\frac{1}{64\pi E^{2}}\int
d\cos\theta\overline{\sum(|{\cal M}_{i}|^{2}+|{\cal M}_{j}|^{2}+{\cal
M}_{i}{\cal M}_{j}^{\ast}+{\cal M}_{i}^{\ast}{\cal M}_{j})}$ which reduces to

$\displaystyle\sigma
v_{rel}=\frac{E^{2}}{6\pi}\left(\frac{c_{\pm}^{4}}{M_{W}^{4}}+\frac{c_{L}^{\nu
2}(c_{L}^{e2}+c_{R}^{e2})}{M_{Z}^{4}}+\frac{\tilde{c}_{L}^{\nu
2}(\tilde{c}_{L}^{e2}+\tilde{c}_{R}^{e2})}{M_{Z^{\prime}}^{4}}+\frac{c_{\pm}^{2}c_{L}^{\nu}c_{L}^{e}}{M_{W}^{2}M_{Z}^{2}}+\frac{c_{\pm}^{2}\tilde{c}_{L}^{\nu}\tilde{c}_{L}^{e}}{M_{W}^{2}M_{Z^{\prime}}^{2}}+\frac{c_{L}^{\nu}\tilde{c}_{L}^{\nu}(c_{L}^{e}\tilde{c}_{L}^{\nu}+c_{R}^{e}\tilde{c}_{R}^{\nu})}{M_{Z}^{2}M_{Z^{\prime}}^{2}}\right).$
(140)

###  A.3 Cross sections of $\nu_{\mu,\tau}\bar{\nu}_{\mu,\tau}\to e^{-}e^{+}$

If neutrino is not $\nu_{e}$, ${\cal M}_{1}=0$. Therefore, the total cross
section is given by

$\displaystyle\sigma v_{rel}=\frac{E^{2}}{6\pi}\left(\frac{c_{L}^{\nu
2}(c_{L}^{e2}+c_{R}^{e2})}{M_{Z}^{4}}+\frac{\tilde{c}_{L}^{\nu
2}(\tilde{c}_{L}^{e2}+\tilde{c}_{R}^{e2})}{M_{Z^{\prime}}^{4}}+\frac{c_{L}^{\nu}\tilde{c}_{L}^{\nu}(c_{L}^{e}\tilde{c}_{L}^{\nu}+c_{R}^{e}\tilde{c}_{R}^{\nu})}{M_{Z}^{2}M_{Z^{\prime}}^{2}}\right).$
(141)

##  Appendix B $\nu-$DM scattering cross sections in laboratory frame in
general $U(1)$ scenarios:

###  B.1 Kinematics

We define the quantities $p_{1}$ as initial neutrino momentum, $p_{2}$ as
initial DM momentum, $k_{1}$ as final neutrino momentum, $k_{2}$ as final DM
momentum. The $4-$vector representation of these momenta are
$p_{1}=(E_{\nu}^{\prime},0,0,E_{\nu}^{\prime})$, $p_{2}=(m_{\chi},0,0,0)$,
$k_{1}=(E_{\nu},0,E_{\nu}\sin\theta,E_{\nu}\cos\theta)$ and
$k_{2}=(E_{\chi},0,-E_{\nu}\sin\theta,E_{\nu}^{\prime}-E_{\nu}\cos\theta)$
where $E_{\chi}=E_{\nu}^{\prime}+m_{\chi}-E_{\nu}$. The scalar products and
neutrino energy in laboratory frame are given by $k_{1}\cdot
p_{1}=E_{\nu}^{\prime}E_{\nu}(1-\cos\theta)$, $k_{1}\cdot
p_{2}=m_{\chi}E_{\nu}$, $k_{2}\cdot
p_{1}=m_{\chi}E_{\nu}^{\prime}-E_{\nu}^{\prime}E_{\nu}(1-\cos\theta)$,
$E_{\nu}=\frac{m\chi
E_{\nu}^{\prime}}{m_{\chi}+E_{\nu}^{\prime}(1-\cos\theta)}$.

###  B.2 Dirac fermion DM

We use the following gauge interactions,

$\displaystyle{\cal
L}_{z^{\prime}}=Z^{\prime}_{\mu}\left(\bar{\nu}_{L}g_{\nu}\gamma_{\mu}\nu_{L}+\tilde{g}_{\chi}\bar{\chi}\gamma_{\mu}\chi\right)$
(142)

where $g_{\nu}=g_{X}Q_{\ell}$ and $\tilde{g}_{\chi}=g_{X}Q_{\chi}$. The
amplitude for $\nu DM\rightarrow\nu DM$ are given by

$\displaystyle i{\cal
M}=\frac{i\tilde{g}_{\chi}g_{\nu}}{(p_{1}-k_{1})^{2}-M_{Z^{\prime}}^{2}+i\Gamma_{Z^{\prime}}M_{Z^{\prime}}}\bar{u}(k_{1})\gamma_{\mu}u(p_{1})\bar{u}(k_{2})\gamma^{\mu}u(p_{2}).$
(143)

The square of the amplitude is given by

$\displaystyle\frac{1}{4}\sum_{spins}\left|{\cal
M}\right|^{2}=\frac{4g_{\nu}^{2}\tilde{g}_{\chi}^{2}}{((p_{1}-k_{1})^{2}-M_{Z^{\prime}}^{2})^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}}\left((k_{1}\cdot
k_{2})(p_{1}\cdot p_{2})+(k_{1}\cdot p_{2})(p_{1}\cdot
k_{2})-m_{\chi}^{2}(k_{1}\cdot p_{1})\right).$ (144)

The differential cross section is given by

$\displaystyle\frac{d\sigma}{dE_{\nu}}=\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}m_{\chi}}{8\pi
E_{\nu}^{\prime
2}\\{\left(2m_{\chi}(E_{\nu}^{\prime}-E_{\nu})+M_{Z^{\prime}}^{2}\right)^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}\\}}\left(E_{\nu}^{\prime
2}+E_{\nu}^{2}-m_{\chi}(E_{\nu}^{\prime}-E_{\nu})\right).$ (145)

The total cross section is given by

$\displaystyle\sigma=\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}m_{\chi}}{8\pi
E_{\nu}^{\prime 2}}\int_{-1}^{1}d\cos\theta\frac{\left(E_{\nu}^{\prime
2}+E_{\nu}^{2}-m_{\chi}(E_{\nu}^{\prime}-E_{\nu})\right)}{\left(2m_{\chi}(E_{\nu}^{\prime}-E_{\nu})+M_{Z^{\prime}}^{2}\right)^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}}.$
(146)

###  B.3 Majorana fermion DM

We use the following gauge interactions,

$\displaystyle{\cal
L}_{Z^{\prime}}=Z^{\prime}_{\mu}\left(\bar{\nu}_{L}g_{\nu}\gamma_{\mu}\nu_{L}+\tilde{g}_{\chi}\bar{\chi}\gamma_{\mu}\gamma_{5}\chi\right),$
(147)

The amplitude for $\nu DM\rightarrow\nu DM$ are given by

$\displaystyle i{\cal
M}=\frac{ig_{\nu}\tilde{g}_{\chi}}{(p_{1}-k_{1})^{2}-M_{Z^{\prime}}^{2}+i\Gamma_{Z^{\prime}}M_{Z^{\prime}}}\bar{u}(k_{1})\gamma_{\mu}u(p_{1})\bar{u}(k_{2})\gamma^{\mu}\gamma_{5}u(p_{2}).$
(148)

The square of the amplitude is given by

$\displaystyle\frac{1}{4}\sum_{spin}\left|{\cal
M}\right|^{2}=\frac{4g_{\nu}^{2}\tilde{g}_{\chi}^{2}}{((p_{1}-k_{1})^{2}-M_{Z^{\prime}}^{2})^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}}\left((k_{1}\cdot
k_{2})(p_{1}\cdot p_{2})+(k_{1}\cdot p_{2})(p_{1}\cdot
k_{2})-m_{\chi}^{2}(k_{1}\cdot p_{1})\right).$ (149)

The differential cross section is given by

$\displaystyle\frac{d\sigma}{dE_{\nu}}=\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}m_{\chi}}{8\pi
E_{\nu}^{\prime
2}\\{\left(2m_{\chi}(E_{\nu}^{\prime}-E_{\nu})+M_{Z^{\prime}}^{2}\right)^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}\\}}\left(E_{\nu}^{\prime
2}+E_{\nu}^{2}-m_{\chi}(E_{\nu}^{\prime}-E_{\nu})\right).$ (150)

The total cross section is given by

$\displaystyle\sigma=\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}m_{\chi}}{8\pi
E_{\nu}^{\prime 2}}\int_{-1}^{1}d\cos\theta\frac{\left(E_{\nu}^{\prime
2}+E_{\nu}^{2}-m_{\chi}(E_{\nu}^{\prime}-E_{\nu})\right)}{\left(2m_{\chi}(E_{\nu}^{\prime}-E_{\nu})+M_{Z^{\prime}}^{2}\right)^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}}.$
(151)

###  B.4 Complex scalar DM

We use the following gauge interactions,

$\displaystyle{\cal
L}_{Z^{\prime}}=Z^{\prime}_{\mu}\left(\bar{\nu}_{L}g_{\nu}\gamma_{\mu}\nu_{L}+\tilde{g}_{\chi}(\chi^{*}\partial_{\mu}\chi-\partial_{\mu}\chi^{*}\chi\right),$
(152)

The amplitude for $\nu DM\rightarrow\nu DM$ are given by

$\displaystyle i{\cal
M}=\frac{ig_{\nu}\tilde{g}_{\chi}}{(p_{1}-k_{1})^{2}-M_{Z^{\prime}}^{2}+i\Gamma_{Z^{\prime}}M_{Z^{\prime}}}\bar{u}(k_{1})\gamma_{\mu}P_{L}u(p_{1})(k_{2}+p_{2})^{\mu}.$
(153)

The square of the amplitude is given by

$\displaystyle\frac{1}{2}\sum_{spin}\left|{\cal
M}\right|^{2}=\frac{1}{2}\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}}{((p_{1}-k_{1})^{2}-M_{Z^{\prime}}^{2})^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}}\left(2(k_{1}\cdot
k_{2}+k_{1}\cdot p_{2})(p_{1}\cdot k_{2}+p_{1}\cdot
p_{2})-(k_{2}+p_{2})^{2}(k_{1}\cdot p_{1})\right).$ (154)

The differential cross section is given by

$\displaystyle\frac{d\sigma}{dE_{\nu}}=\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}m_{\chi}}{8\pi
E_{\nu}^{\prime
2}\\{\left(2m_{\chi}(E_{\nu}^{\prime}-E_{\nu})+M_{Z^{\prime}}^{2}\right)^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}\\}}\left(2E_{\nu}^{\prime}E_{\nu}-m_{\chi}(E_{\nu}^{\prime}-E_{\nu})\right).$
(155)

The total cross section is given by

$\displaystyle\sigma=\frac{g_{\nu}^{2}\tilde{g}_{\chi}^{2}}{8\pi
E_{\nu}^{\prime
2}}\int_{-1}^{1}d\cos\theta\frac{E_{\nu}^{2}\left(2E_{\nu}^{\prime}E_{\nu}-m_{\chi}(E_{\nu}^{\prime}-E_{\nu})\right)}{\left(2m_{\chi}(E_{\nu}^{\prime}-E_{\nu})+M_{Z^{\prime}}^{2}\right)^{2}+\Gamma_{Z^{\prime}}^{2}M_{Z^{\prime}}^{2}}.$
(156)

For light $Z^{\prime}$, $\Gamma_{Z}^{\prime}M_{Z^{\prime}}$ term in the
denominators could be suppressed.

## References

  * [1] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * [2] G. Bertone, D. Hooper, and J. Silk, “Particle dark matter: Evidence, candidates and constraints,” Phys. Rept. 405 (2005) 279–390, arXiv:hep-ph/0404175.
  * [3] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [4] Fermi-LAT Collaboration, M. Ackermann et al., “Constraining Dark Matter Models from a Combined Analysis of Milky Way Satellites with the Fermi Large Area Telescope,” Phys. Rev. Lett. 107 (2011) 241302, arXiv:1108.3546 [astro-ph.HE].
  * [5] Fermi-LAT Collaboration, A. A. Abdo et al., “Constraints on Cosmological Dark Matter Annihilation from the Fermi-LAT Isotropic Diffuse Gamma-Ray Measurement,” JCAP 04 (2010) 014, arXiv:1002.4415 [astro-ph.CO].
  * [6] P. Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” Phys. Lett. B 67 (1977) 421–428.
  * [7] T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131 (1979) 95–99.
  * [8] M. Gell-Mann, P. Ramond, and R. Slansky, “Complex Spinors and Unified Theories,” Conf. Proc. C 790927 (1979) 315–321, arXiv:1306.4669 [hep-th].
  * [9] R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys. Rev. Lett. 44 (1980) 912.
  * [10] J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) x U(1) Theories,” Phys. Rev. D22 (1980) 2227.
  * [11] S. Weinberg, “Baryon and Lepton Nonconserving Processes,” Phys. Rev. Lett. 43 (1979) 1566–1570.
  * [12] A. Davidson, “$B-L$ as the fourth color within an $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{R}\times\mathrm{U}(1)$ model,” Phys. Rev. D 20 (1979) 776.
  * [13] A. Davidson, M. Koca, and K. C. Wali, “U(1) as the Minimal Horizontal Gauge Symmetry,” Phys. Rev. Lett. 43 (1979) 92.
  * [14] R. E. Marshak and R. N. Mohapatra, “Quark - Lepton Symmetry and B-L as the U(1) Generator of the Electroweak Symmetry Group,” Phys. Lett. 91B (1980) 222–224.
  * [15] R. N. Mohapatra and R. E. Marshak, “Local B-L Symmetry of Electroweak Interactions, Majorana Neutrinos and Neutron Oscillations,” Phys. Rev. Lett. 44 (1980) 1316–1319. [Erratum: Phys. Rev. Lett.44,1643(1980)].
  * [16] A. Das, N. Okada, and D. Raut, “Enhanced pair production of heavy Majorana neutrinos at the LHC,” Phys. Rev. D97 no. 11, (2018) 115023, arXiv:1710.03377 [hep-ph].
  * [17] K. Chakraborty, A. Das, S. Goswami, and S. Roy, “Constraining general U(1) interactions from neutrino-electron scattering measurements at DUNE near detector,” JHEP 04 (2022) 008, arXiv:2111.08767 [hep-ph].
  * [18] K. Asai, A. Das, J. Li, T. Nomura, and O. Seto, “Probing for chiral $Z^{\prime}$ gauge boson through scattering measurement experiments,” arXiv:2307.09737 [hep-ph].
  * [19] K. Asai, A. Das, J. Li, T. Nomura, and O. Seto, “Chiral Z’ in FASER, FASER2, DUNE, and ILC beam dump experiments,” Phys. Rev. D 106 no. 9, (2022) 095033, arXiv:2206.12676 [hep-ph].
  * [20] M. Hashimoto, S. Iso, and Y. Orikasa, “Radiative symmetry breaking from flat potential in various U(1)’ models,” Phys. Rev. D 89 no. 5, (2014) 056010, arXiv:1401.5944 [hep-ph].
  * [21] T. Appelquist, B. A. Dobrescu, and A. R. Hopper, “Nonexotic Neutral Gauge Bosons,” Phys. Rev. D 68 (2003) 035012, arXiv:hep-ph/0212073.
  * [22] M. Carena, A. Daleo, B. A. Dobrescu, and T. M. P. Tait, “$Z^{\prime}$ gauge bosons at the Tevatron,” Phys. Rev. D70 (2004) 093009, arXiv:hep-ph/0408098 [hep-ph].
  * [23] R. Foot, “New Physics From Electric Charge Quantization?,” Mod. Phys. Lett. A 6 (1991) 527–530.
  * [24] X. G. He, G. C. Joshi, H. Lew, and R. R. Volkas, “NEW Z-prime PHENOMENOLOGY,” Phys. Rev. D 43 (1991) 22–24.
  * [25] X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, “Simplest Z-prime model,” Phys. Rev. D 44 (1991) 2118–2132.
  * [26] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, “Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams,” Phys. Rev. Lett. 113 (2014) 091801, arXiv:1406.2332 [hep-ph].
  * [27] A. Dasgupta, P. S. B. Dev, T. Han, R. Padhan, S. Wang, and K. Xie, “Searching for Heavy Leptophilic $Z^{\prime}$: from Lepton Colliders to Gravitational Waves,” arXiv:2308.12804 [hep-ph].
  * [28] H.-S. Lee and E. Ma, “Gauged $B-x_{i}L$ origin of $R$ Parity and its implications,” Phys. Lett. B 688 (2010) 319–322, arXiv:1001.0768 [hep-ph].
  * [29] 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, arXiv:hep-ph/0010118.
  * [30] M. Bauer, P. Foldenauer, and M. Mosny, “Flavor structure of anomaly-free hidden photon models,” Phys. Rev. D 103 no. 7, (2021) 075024, arXiv:2011.12973 [hep-ph].
  * [31] D. Eichler, M. Livio, T. Piran, and D. N. Schramm, “Nucleosynthesis, Neutrino Bursts and Gamma-Rays from Coalescing Neutron Stars,” Nature 340 (1989) 126–128.
  * [32] H. A. Bethe and J. R. Wilson, “Revival of a stalled supernova shock by neutrino heating,” Astrophys. J. 295 (1985) 14–23.
  * [33] J. Cooperstein, L. van den Horn, and E. A. Baron, “Neutrino flows in collapsing stars: a two-fluid model,” ApJ 309 (1986) 653.
  * [34] J. Cooperstein, L. J. van den Horn, and E. A. Baron, “Neutrino Pair Energy Deposition in Supernovae,” ApJ 321 (1987) L129.
  * [35] J. Goodman, A. Dar, and S. Nussinov, “Neutrino Annihilation in type II Supernovae,” Astrophys. J. Lett. 314 (1987) L7–L10.
  * [36] D. Eichler, M. Livio, T. Piran, and D. N. Schramm, “Nucleosynthesis, neutrino bursts and $\gamma$-rays from coalescing neutron stars,” Nature (London) 340 no. 6229, (July, 1989) 126–128.
  * [37] M. Jaroszynski, “Neutrino Emission and Annihilation Near Tori around Black Holes,” Acta Astron. 43 (July, 1993) 183–191.
  * [38] A. R. Prasanna and S. Goswami, “Energy deposition due to neutrino pair annihilation near rotating neutron stars,” Phys. Lett. B 526 (2002) 27–33, arXiv:astro-ph/0109058.
  * [39] J. D. Salmonson and J. R. Wilson, “General relativistic augmentation of neutrino pair annihilation energy deposition near neutron stars,” Astrophys. J. 517 (1999) 859–865, arXiv:astro-ph/9908017.
  * [40] J. D. Salmonson and J. R. Wilson, “Neutrino annihilation between binary neutron stars,” Astrophys. J. 561 (2001) 950–956, arXiv:astro-ph/0108196.
  * [41] K. Asano and T. Fukuyama, “Neutrino pair annihilation in the gravitation of gamma-ray burst sources,” Astrophys. J. 531 (2000) 949–955, arXiv:astro-ph/0002196.
  * [42] K. Asano and T. Fukuyama, “Relativistic effects on neutrino pair annihilation above a Kerr black hole with the accretion disk,” Astrophys. J. 546 (2001) 1019–1026, arXiv:astro-ph/0009453.
  * [43] G. J. Mathews and J. R. Wilson, “Binary induced neutron star compression, heating, and collapse,” Astrophys. J. 482 (1997) 929–941, arXiv:astro-ph/9701142.
  * [44] M. Ruffert and H. T. Janka, “Gamma-ray bursts from accreting black holes in neutron star mergers,” Astron. Astrophys. 344 (Apr., 1999) 573–606, arXiv:astro-ph/9809280 [astro-ph].
  * [45] R. Popham, S. E. Woosley, and C. Fryer, “Hyperaccreting black holes and gamma-ray bursts,” The Astrophysical Journal 518 no. 1, (Jun, 1999) 356–374. https://doi.org/10.1086/307259.
  * [46] S. Harikae, K. Kotake, T. Takiwaki, and Y. ichiro Sekiguchi, “A GENERAL RELATIVISTIC RAY-TRACING METHOD FOR ESTIMATING THE ENERGY AND MOMENTUM DEPOSITION BY NEUTRINO PAIR ANNIHILATION IN COLLAPSARS,” The Astrophysical Journal 720 no. 1, (Aug, 2010) 614–625. https://doi.org/10.1088/0004-637x/720/1/614.
  * [47] T. D. Matteo, R. Perna, and R. Narayan, “Neutrino trapping and accretion models for gamma-ray bursts,” The Astrophysical Journal 579 no. 2, (Nov, 2002) 706–715. https://doi.org/10.1086/342832.
  * [48] S. Fujibayashi, Y. Sekiguchi, K. Kiuchi, and M. Shibata, “Properties of neutrino-driven ejecta from the remnant of a binary neutron star merger: Pure radiation hydrodynamics case,” The Astrophysical Journal 846 no. 2, (Sep, 2017) 114. https://doi.org/10.3847/1538-4357/aa8039.
  * [49] O. Just, M. Obergaulinger, H. T. Janka, A. Bauswein, and N. Schwarz, “Neutron-star merger ejecta as obstacles to neutrino-powered jets of gamma-ray bursts,” Astrophys. J. Lett. 816 no. 2, (2016) L30, arXiv:1510.04288 [astro-ph.HE].
  * [50] F. Foucart, M. D. Duez, L. E. Kidder, R. Nguyen, H. P. Pfeiffer, and M. A. Scheel, “Evaluating radiation transport errors in merger simulations using a monte carlo algorithm,” Phys. Rev. D 98 (Sep, 2018) 063007. https://link.aps.org/doi/10.1103/PhysRevD.98.063007.
  * [51] F. Foucart, M. D. Duez, F. Hebert, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel, “Monte-Carlo Neutrino Transport in Neutron Star Merger Simulations,” ApJ 902 no. 1, (Oct., 2020) L27, arXiv:2008.08089 [astro-ph.HE].
  * [52] R. D. Blandford and R. L. Znajek, “Electromagnetic extraction of energy from Kerr black holes,” Monthly Notices of the Royal Astronomical Society 179 no. 3, (07, 1977) 433–456.
  * [53] G. Lambiase and L. Mastrototaro, “Effects of modified theories of gravity on neutrino pair annihilation energy deposition near neutron stars,” ApJ. 904 (2020) 1–8.
  * [54] G. Lambiase and L. Mastrototaro, “Neutrino pair annihilation ($\nu{\bar{\nu}}\rightarrow e^{-}e^{+}$) in the presence of quintessence surrounding a black hole,” Eur. Phys. J. C 81 no. 10, (2021) 932, arXiv:2012.09100 [astro-ph.HE].
  * [55] G. G. Raffelt, Stars as laboratories for fundamental physics: The astrophysics of neutrinos, axions, and other weakly interacting particles.  5, 1996.
  * [56] G. G. Raffelt, “Muon-neutrino and tau-neutrino spectra formation in supernovae,” Astrophys. J. 561 (2001) 890–914, arXiv:astro-ph/0105250.
  * [57] H. T. Janka, “Neutrino Emission from Supernovae,” arXiv:1702.08713 [astro-ph.HE].
  * [58] M. Hellgren and J. Suhonen, “Neutral-current supernova neutrino-nucleus scattering off 127I and 133Cs,” Phys. Rev. C 106 no. 2, (2022) 025808.
  * [59] E. Burns et al., “GRB 221009A: The BOAT,” Astrophys. J. Lett. 946 no. 1, (2023) L31, arXiv:2302.14037 [astro-ph.HE].
  * [60] e. a. R. Battiston, “Observation of anomalous electron fluxes induced by grb221009a on cses-01 low-energy charged particle detector,” The Astrophysical Journal Letters 946 no. 1, (Mar, 2023) L29. https://dx.doi.org/10.3847/2041-8213/acc247.
  * [61] R. Alves Batista, “GRB 221009A: a potential source of ultra-high-energy cosmic rays,” arXiv:2210.12855 [astro-ph.HE].
  * [62] B. P. Abbott et al. Astrophys. J. Lett. 848 no. 2, (2017) L12, arXiv:1710.05833 [astro-ph.HE].
  * [63] K. Murase, M. Mukhopadhyay, A. Kheirandish, S. S. Kimura, and K. Fang, “Neutrinos from the Brightest Gamma-Ray Burst?,” Astrophys. J. Lett. 941 no. 1, (2022) L10, arXiv:2210.15625 [astro-ph.HE].
  * [64] M. D. Fulton et al., “The optical light curve of grb 221009a: The afterglow and the emerging supernova,” The Astrophysical Journal Letters 946 no. 1, (Mar, 2023) L22. https://dx.doi.org/10.3847/2041-8213/acc101.
  * [65] E. Guarini, I. Tamborra, D. Bégué, and A. Rudolph, “Probing gamma-ray bursts observed at very high energies through their afterglow,” Mon. Not. Roy. Astron. Soc. 253 (2023) 149–162, arXiv:2301.10256 [astro-ph.HE].
  * [66] N. Breton, “Geodesic structure of the Born-Infeld black hole,” Class. Quant. Grav. 19 (2002) 601–612.
  * [67] E. Babichev, C. Charmousis, and M. Hassaine, “Charged Galileon black holes,” JCAP 05 (2015) 031, arXiv:1503.02545 [gr-qc].
  * [68] IceCube, Fermi-LAT, MAGIC, AGILE, ASAS-SN, HAWC, H.E.S.S., INTEGRAL, Kanata, Kiso, Kapteyn, Liverpool Telescope, Subaru, Swift NuSTAR, VERITAS, VLA/17B-403 Collaboration, M. G. Aartsen et al., “Multimessenger observations of a flaring blazar coincident with high-energy neutrino IceCube-170922A,” Science 361 no. 6398, (2018) eaat1378, arXiv:1807.08816 [astro-ph.HE].
  * [69] P. Gondolo and J. Silk, “Dark matter annihilation at the galactic center,” Phys. Rev. Lett. 83 (1999) 1719–1722, arXiv:astro-ph/9906391.
  * [70] S. L. Shapiro and D. C. Heggie, “Effect of stars on the dark matter spike around a black hole: A tale of two treatments,” Phys. Rev. D 106 no. 4, (2022) 043018, arXiv:2209.08105 [astro-ph.GA].
  * [71] C. Boehm and P. Fayet, “Scalar dark matter candidates,” Nucl. Phys. B 683 (2004) 219–263, arXiv:hep-ph/0305261.
  * [72] C. A. Argüelles, A. Kheirandish, and A. C. Vincent, “Imaging Galactic Dark Matter with High-Energy Cosmic Neutrinos,” Phys. Rev. Lett. 119 no. 20, (2017) 201801, arXiv:1703.00451 [hep-ph].
  * [73] A. Olivares-Del Campo, C. Bœhm, S. Palomares-Ruiz, and S. Pascoli, “Dark matter-neutrino interactions through the lens of their cosmological implications,” Phys. Rev. D 97 no. 7, (2018) 075039, arXiv:1711.05283 [hep-ph].
  * [74] J.-W. Wang, A. Granelli, and P. Ullio, “Direct Detection Constraints on Blazar-Boosted Dark Matter,” Phys. Rev. Lett. 128 no. 22, (2022) 221104, arXiv:2111.13644 [astro-ph.HE].
  * [75] A. Granelli, P. Ullio, and J.-W. Wang, “Blazar-boosted dark matter at Super-Kamiokande,” JCAP 07 no. 07, (2022) 013, arXiv:2202.07598 [astro-ph.HE].
  * [76] A. Ambrosone, M. Chianese, D. F. G. Fiorillo, A. Marinelli, and G. Miele, “Starburst Nuclei as Light Dark Matter Laboratories,” arXiv:2210.05685 [astro-ph.HE].
  * [77] J. B. Dent, B. Dutta, J. L. Newstead, and I. M. Shoemaker, “Bounds on Cosmic Ray-Boosted Dark Matter in Simplified Models and its Corresponding Neutrino-Floor,” Phys. Rev. D 101 no. 11, (2020) 116007, arXiv:1907.03782 [hep-ph].
  * [78] K.-Y. Choi, J. Kim, and C. Rott, “Constraining dark matter-neutrino interactions with IceCube-170922A,” Phys. Rev. D 99 no. 8, (2019) 083018, arXiv:1903.03302 [astro-ph.CO].
  * [79] M. Blennow, E. Fernandez-Martinez, A. Olivares-Del Campo, S. Pascoli, S. Rosauro-Alcaraz, and A. V. Titov, “Neutrino Portals to Dark Matter,” Eur. Phys. J. C 79 no. 7, (2019) 555, arXiv:1903.00006 [hep-ph].
  * [80] J. M. Cline, S. Gao, F. Guo, Z. Lin, S. Liu, M. Puel, P. Todd, and T. Xiao, “Blazar Constraints on Neutrino-Dark Matter Scattering,” Phys. Rev. Lett. 130 no. 9, (2023) 091402, arXiv:2209.02713 [hep-ph].
  * [81] F. Ferrer, G. Herrera, and A. Ibarra, “New constraints on the dark matter-neutrino and dark matter-photon scattering cross sections from TXS 0506+056,” arXiv:2209.06339 [hep-ph].
  * [82] IceCube Collaboration, R. Abbasi et al., “Evidence for neutrino emission from the nearby active galaxy NGC 1068,” Science 378 no. 6619, (2022) 538–543, arXiv:2211.09972 [astro-ph.HE].
  * [83] D. M. Crenshaw and S. B. Kraemer, “Resolved spectroscopy of the narrow-line region in ngc 1068: kinematics of the ionized gas,” Astrophys. J. Lett. 532 (2000) L101, arXiv:astro-ph/0002438.
  * [84] W. Jaffe et. al., “The central dusty torus in the active nucleus of NGC 1068,” Nature 429 (2004) 47–49.
  * [85] G. Mangano, A. Melchiorri, P. Serra, A. Cooray, and M. Kamionkowski, “Cosmological bounds on dark matter-neutrino interactions,” Phys. Rev. D 74 (2006) 043517, arXiv:astro-ph/0606190.
  * [86] C. Boehm, M. J. Dolan, and C. McCabe, “A Lower Bound on the Mass of Cold Thermal Dark Matter from Planck,” JCAP 08 (2013) 041, arXiv:1303.6270 [hep-ph].
  * [87] B. Bertoni, S. Ipek, D. McKeen, and A. E. Nelson, “Constraints and consequences of reducing small scale structure via large dark matter-neutrino interactions,” JHEP 04 (2015) 170, arXiv:1412.3113 [hep-ph].
  * [88] R. J. Wilkinson, C. Boehm, and J. Lesgourgues, “Constraining Dark Matter-Neutrino Interactions using the CMB and Large-Scale Structure,” JCAP 05 (2014) 011, arXiv:1401.7597 [astro-ph.CO].
  * [89] M. R. Mosbech, C. Boehm, S. Hannestad, O. Mena, J. Stadler, and Y. Y. Y. Wong, “The full Boltzmann hierarchy for dark matter-massive neutrino interactions,” JCAP 03 (2021) 066, arXiv:2011.04206 [astro-ph.CO].
  * [90] D. C. Hooper and M. Lucca, “Hints of dark matter-neutrino interactions in Lyman-$\alpha$ data,” Phys. Rev. D 105 no. 10, (2022) 103504, arXiv:2110.04024 [astro-ph.CO].
  * [91] P. Fayet, D. Hooper, and G. Sigl, “Constraints on light dark matter from core-collapse supernovae,” Phys. Rev. Lett. 96 (2006) 211302, arXiv:hep-ph/0602169.
  * [92] IceCube Collaboration, A. McMullen, A. Vincent, C. Arguelles, and A. Schneider, “Dark matter neutrino scattering in the galactic centre with IceCube,” JINST 16 no. 08, (2021) C08001, arXiv:2107.11491 [astro-ph.HE].
  * [93] K. Murase and I. M. Shoemaker, “Neutrino Echoes from Multimessenger Transient Sources,” Phys. Rev. Lett. 123 no. 24, (2019) 241102, arXiv:1903.08607 [hep-ph].
  * [94] J. A. Carpio, A. Kheirandish, and K. Murase, “Time-delayed neutrino emission from supernovae as a probe of dark matter-neutrino interactions,” JCAP 04 (2023) 019, arXiv:2204.09650 [hep-ph].
  * [95] J. H. Davis and J. Silk, “Spectral and Spatial Distortions of PeV Neutrinos from Scattering with Dark Matter,” arXiv:1505.01843 [hep-ph].
  * [96] W. Yin, “Highly-boosted dark matter and cutoff for cosmic-ray neutrinos through neutrino portal,” EPJ Web Conf. 208 (2019) 04003, arXiv:1809.08610 [hep-ph].
  * [97] Y. Zhang, “Speeding up dark matter with solar neutrinos,” PTEP 2022 no. 1, (2022) 013B05, arXiv:2001.00948 [hep-ph].
  * [98] Y. Jho, J.-C. Park, S. C. Park, and P.-Y. Tseng, “Cosmic-Neutrino-Boosted Dark Matter ($\nu$BDM),” arXiv:2101.11262 [hep-ph].
  * [99] Y. Farzan and S. Palomares-Ruiz, “Dips in the Diffuse Supernova Neutrino Background,” JCAP 06 (2014) 014, arXiv:1401.7019 [hep-ph].
  * [100] A. Das and M. Sen, “Boosted dark matter from diffuse supernova neutrinos,” Phys. Rev. D 104 no. 7, (2021) 075029, arXiv:2104.00027 [hep-ph].
  * [101] D. Bardhan, S. Bhowmick, D. Ghosh, A. Guha, and D. Sachdeva, “Bounds on boosted dark matter from direct detection: The role of energy-dependent cross sections,” Phys. Rev. D 107 no. 1, (2023) 015010, arXiv:2208.09405 [hep-ph].
  * [102] Y.-H. Lin, W.-H. Wu, M.-R. Wu, and H. T.-K. Wong, “Searching for Afterglow: Light Dark Matter Boosted by Supernova Neutrinos,” Phys. Rev. Lett. 130 no. 11, (2023) 111002, arXiv:2206.06864 [hep-ph].
  * [103] J. M. Cline and M. Puel, “NGC 1068 constraints on neutrino-dark matter scattering,” arXiv:2301.08756 [hep-ph].
  * [104] J. McDonald, “Gauge singlet scalars as cold dark matter,” Phys. Rev. D 50 (1994) 3637–3649, arXiv:hep-ph/0702143.
  * [105] V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf, and G. Shaughnessy, “Complex Singlet Extension of the Standard Model,” Phys. Rev. D 79 (2009) 015018, arXiv:0811.0393 [hep-ph].
  * [106] M. Ibe, S. Matsumoto, and T. T. Yanagida, “The GeV-scale dark matter with B–L asymmetry,” Phys. Lett. B 708 (2012) 112–118, arXiv:1110.5452 [hep-ph].
  * [107] C.-W. Chiang, T. Nomura, and J. Tandean, “Dark Matter and Higgs Boson in a Model with Discrete Gauge Symmetry,” Phys. Rev. D 87 no. 7, (2013) 073004, arXiv:1205.6416 [hep-ph].
  * [108] N. Okada and S. Okada, “$Z^{\prime}$-portal right-handed neutrino dark matter in the minimal U(1)X extended Standard Model,” Phys. Rev. D 95 no. 3, (2017) 035025, arXiv:1611.02672 [hep-ph].
  * [109] N. Okada, S. Okada, and D. Raut, “Natural $Z^{\prime}$ -portal Majorana dark matter in alternative U(1) extended standard model,” Phys. Rev. D 100 no. 3, (2019) 035022, arXiv:1811.11927 [hep-ph].
  * [110] N. Okada, S. Okada, and Q. Shafi, “Light $Z^{\prime}$ and dark matter from U(1)X gauge symmetry,” Phys. Lett. B 810 (2020) 135845, arXiv:2003.02667 [hep-ph].
  * [111] P. Fileviez Pérez and C. Murgui, “Dark Matter and The Seesaw Scale,” Phys. Rev. D 98 no. 5, (2018) 055008, arXiv:1803.07462 [hep-ph].
  * [112] T. G. Rizzo, “The Bactrian Effect: Multiple Resonances and Light Dirac Dark Matter,” JHEP 04 (2021) 248, arXiv:2102.03647 [hep-ph].
  * [113] P. Lipari, “Lepton spectra in the earth’s atmosphere,” Astropart. Phys. 1 (1993) 195–227.
  * [114] A. Fedynitch, R. Engel, T. K. Gaisser, F. Riehn, and T. Stanev, “Calculation of conventional and prompt lepton fluxes at very high energy,” EPJ Web Conf. 99 (2015) 08001, arXiv:1503.00544 [hep-ph].
  * [115] C. Argüelles, A. Kheirandish, and A. C. Vincent, “Constraining Dark Matter Neutrino Interaction with High Energy Neutrinos,” PoS ICHEP2016 (2016) 512, arXiv:1612.08472 [hep-ph].
  * [116] A. C. Vincent, C. A. Argüelles, and A. Kheirandish, “High-energy neutrino attenuation in the Earth and its associated uncertainties,” JCAP 11 (2017) 012, arXiv:1706.09895 [hep-ph].
  * [117] TEXONO Collaboration, M. Deniz et al., “Measurement of Nu(e)-bar -Electron Scattering Cross-Section with a CsI(Tl) Scintillating Crystal Array at the Kuo-Sheng Nuclear Power Reactor,” Phys. Rev. D 81 (2010) 072001, arXiv:0911.1597 [hep-ex].
  * [118] Borexino Collaboration, C. Arpesella et al., “First real time detection of Be-7 solar neutrinos by Borexino,” Phys. Lett. B 658 (2008) 101–108, arXiv:0708.2251 [astro-ph].
  * [119] Borexino Collaboration, G. Alimonti et al., “The Borexino detector at the Laboratori Nazionali del Gran Sasso,” Nucl. Instrum. Meth. A 600 (2009) 568–593, arXiv:0806.2400 [physics.ins-det].
  * [120] CHARM-II Collaboration, P. Vilain et al., “Measurement of differential cross-sections for muon-neutrino electron scattering,” Phys. Lett. B 302 (1993) 351–355.
  * [121] CHARM-II Collaboration, P. Vilain et al., “Precision measurement of electroweak parameters from the scattering of muon-neutrinos on electrons,” Phys. Lett. B 335 (1994) 246–252.
  * [122] COHERENT Collaboration, D. Akimov et al., “COHERENT Collaboration data release from the first observation of coherent elastic neutrino-nucleus scattering,” arXiv:1804.09459 [nucl-ex].
  * [123] COHERENT Collaboration, D. Akimov et al., “First Measurement of Coherent Elastic Neutrino-Nucleus Scattering on Argon,” Phys. Rev. Lett. 126 no. 1, (2021) 012002, arXiv:2003.10630 [nucl-ex].
  * [124] COHERENT Collaboration, D. Akimov et al., “COHERENT Collaboration data release from the first detection of coherent elastic neutrino-nucleus scattering on argon,” arXiv:2006.12659 [nucl-ex].
  * [125] A. G. Beda, E. V. Demidova, A. S. Starostin, V. B. Brudanin, V. G. Egorov, D. V. Medvedev, M. V. Shirchenko, and T. Vylov, “GEMMA experiment: Three years of the search for the neutrino magnetic moment,” Phys. Part. Nucl. Lett. 7 (2010) 406–409, arXiv:0906.1926 [hep-ex].
  * [126] M. Lindner, F. S. Queiroz, W. Rodejohann, and X.-J. Xu, “Neutrino-electron scattering: general constraints on Z′ and dark photon models,” JHEP 05 (2018) 098, arXiv:1803.00060 [hep-ph].
  * [127] CHARM Collaboration, F. Bergsma et al., “Search for Axion Like Particle Production in 400-GeV Proton - Copper Interactions,” Phys. Lett. B 157 (1985) 458–462.
  * [128] NOMAD Collaboration, P. Astier et al., “Final NOMAD results on muon-neutrino —$>$ tau-neutrino and electron-neutrino —$>$ tau-neutrino oscillations including a new search for tau-neutrino appearance using hadronic tau decays,” Nucl. Phys. B 611 (2001) 3–39, arXiv:hep-ex/0106102.
  * [129] J. Blumlein and J. Brunner, “New Exclusion Limits for Dark Gauge Forces from Beam-Dump Data,” Phys. Lett. B 701 (2011) 155–159, arXiv:1104.2747 [hep-ex].
  * [130] J. Blümlein and J. Brunner, “New Exclusion Limits on Dark Gauge Forces from Proton Bremsstrahlung in Beam-Dump Data,” Phys. Lett. B 731 (2014) 320–326, arXiv:1311.3870 [hep-ph].
  * [131] M. Davier and H. Nguyen Ngoc, “An Unambiguous Search for a Light Higgs Boson,” Phys. Lett. B 229 (1989) 150–155.
  * [132] G. A. Beer et al., “Emission of Muonium Into Vacuum From a Silica Powder Layer,” Phys. Rev. Lett. 57 (1986) 671–674.
  * [133] E. M. Riordan et al., “A Search for Short Lived Axions in an Electron Beam Dump Experiment,” Phys. Rev. Lett. 59 (1987) 755.
  * [134] J. D. Bjorken, S. Ecklund, W. R. Nelson, A. Abashian, C. Church, B. Lu, L. W. Mo, T. A. Nunamaker, and P. Rassmann, “Search for Neutral Metastable Penetrating Particles Produced in the SLAC Beam Dump,” Phys. Rev. D 38 (1988) 3375.
  * [135] NA64 Collaboration, D. Banerjee et al., “Improved limits on a hypothetical X(16.7) boson and a dark photon decaying into $e^{+}e^{-}$ pairs,” Phys. Rev. D 101 no. 7, (2020) 071101, arXiv:1912.11389 [hep-ex].
  * [136] A. Bross, M. Crisler, S. H. Pordes, J. Volk, S. Errede, and J. Wrbanek, “A Search for Shortlived Particles Produced in an Electron Beam Dump,” Phys. Rev. Lett. 67 (1991) 2942–2945.
  * [137] G. Bernardi et al., “FURTHER LIMITS ON HEAVY NEUTRINO COUPLINGS,” Phys. Lett. B 203 (1988) 332–334.
  * [138] G. Bernardi et al., “Anomalous Electron Production Observed in the CERN Ps Neutrino Beam,” Phys. Lett. B 181 (1986) 173–177.
  * [139] G. Bernardi et al., “Search for Neutrino Decay,” Phys. Lett. B 166 (1986) 479–483.
  * [140] LHCb Collaboration, R. Aaij et al., “Search for hidden-sector bosons in $B^{0}\\!\to K^{*0}\mu^{+}\mu^{-}$ decays,” Phys. Rev. Lett. 115 no. 16, (2015) 161802, arXiv:1508.04094 [hep-ex].
  * [141] LHCb Collaboration, R. Aaij et al., “Search for long-lived scalar particles in $B^{+}\to K^{+}\chi(\mu^{+}\mu^{-})$ decays,” Phys. Rev. D 95 no. 7, (2017) 071101, arXiv:1612.07818 [hep-ex].
  * [142] LHCb Collaboration, R. Aaij et al., “Measurement of the $B^{\pm}$ production cross-section in pp collisions at $\sqrt{s}=$ 7 and 13 TeV,” JHEP 12 (2017) 026, arXiv:1710.04921 [hep-ex].
  * [143] LHCb Collaboration, R. Aaij et al., “Physics case for an LHCb Upgrade II - Opportunities in flavour physics, and beyond, in the HL-LHC era,” arXiv:1808.08865 [hep-ex].
  * [144] LHCb Collaboration, R. Aaij et al., “Search for Dark Photons Produced in 13 TeV $pp$ Collisions,” Phys. Rev. Lett. 120 no. 6, (2018) 061801, arXiv:1710.02867 [hep-ex].
  * [145] LHCb Collaboration, R. Aaij et al., “Search for $A^{\prime}\to\mu^{+}\mu^{-}$ Decays,” Phys. Rev. Lett. 124 no. 4, (2020) 041801, arXiv:1910.06926 [hep-ex].
  * [146] BaBar Collaboration, J. P. Lees et al., “Search for a Dark Photon in $e^{+}e^{-}$ Collisions at BaBar,” Phys. Rev. Lett. 113 no. 20, (2014) 201801, arXiv:1406.2980 [hep-ex].
  * [147] BaBar Collaboration, J. P. Lees et al., “Search for Invisible Decays of a Dark Photon Produced in ${e}^{+}{e}^{-}$ Collisions at BaBar,” Phys. Rev. Lett. 119 no. 13, (2017) 131804, arXiv:1702.03327 [hep-ex].
  * [148] CMS Collaboration, “Search for prompt production of a GeV scale resonance decaying to a pair of muons in proton-proton collisions at $\sqrts=13~\mathrmTeV,”.
  * [149] A. Anastasi et al., “Limit on the production of a low-mass vector boson in $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{U}\gamma$, $\mathrm{U}\to\mathrm{e}^{+}\mathrm{e}^{-}$ with the KLOE experiment,” Phys. Lett. B 750 (2015) 633–637, arXiv:1509.00740 [hep-ex].
  * [150] KLOE-2 Collaboration, A. Anastasi et al., “Combined limit on the production of a light gauge boson decaying into $\mu^{+}\mu^{-}$ and $\pi^{+}\pi^{-}$,” Phys. Lett. B 784 (2018) 336–341, arXiv:1807.02691 [hep-ex].
  * [151] KLOE-2 Collaboration, D. Babusci et al., “Limit on the production of a light vector gauge boson in phi meson decays with the KLOE detector,” Phys. Lett. B 720 (2013) 111–115, arXiv:1210.3927 [hep-ex].
  * [152] A1 Collaboration, H. Merkel et al., “Search for Light Gauge Bosons of the Dark Sector at the Mainz Microtron,” Phys. Rev. Lett. 106 (2011) 251802, arXiv:1101.4091 [nucl-ex].
  * [153] H. Merkel et al., “Search at the Mainz Microtron for Light Massive Gauge Bosons Relevant for the Muon g-2 Anomaly,” Phys. Rev. Lett. 112 no. 22, (2014) 221802, arXiv:1404.5502 [hep-ex].
  * [154] NA48/2 Collaboration, J. R. Batley et al., “Search for the dark photon in $\pi^{0}$ decays,” Phys. Lett. B 746 (2015) 178–185, arXiv:1504.00607 [hep-ex].
  * [155] APEX Collaboration, S. Abrahamyan et al., “Search for a New Gauge Boson in Electron-Nucleus Fixed-Target Scattering by the APEX Experiment,” Phys. Rev. Lett. 107 (2011) 191804, arXiv:1108.2750 [hep-ex].
  * [156] HADES Collaboration, G. Agakishiev et al., “Searching a Dark Photon with HADES,” Phys. Lett. B 731 (2014) 265–271, arXiv:1311.0216 [hep-ex].
  * [157] PHENIX Collaboration, A. Adare et al., “Search for dark photons from neutral meson decays in $p+p$ and $d$ \+ Au collisions at $\sqrt{s_{NN}}=$ 200 GeV,” Phys. Rev. C 91 no. 3, (2015) 031901, arXiv:1409.0851 [nucl-ex].
  * [158] WASA-at-COSY Collaboration, P. Adlarson et al., “Search for a dark photon in the $\pi^{0}\to e^{+}e^{-}\gamma$ decay,” Phys. Lett. B 726 (2013) 187–193, arXiv:1304.0671 [hep-ex].
  * [159] BESIII Collaboration, M. Ablikim et al., “Dark Photon Search in the Mass Range Between 1.5 and 3.4 GeV/$c^{2}$,” Phys. Lett. B 774 (2017) 252–257, arXiv:1705.04265 [hep-ex].
  * [160] ALEPH, DELPHI, L3, OPAL, LEP Electroweak Working Group Collaboration, J. Alcaraz et al., “A Combination of preliminary electroweak measurements and constraints on the standard model,” arXiv:hep-ex/0612034.
  * [161] ALEPH Collaboration, R. Barate et al., “Study of the muon pair production at center-of-mass energies from 20-GeV to 136-GeV with the ALEPH detector,” Phys. Lett. B 399 (1997) 329–341.
  * [162] LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3, OPAL Collaboration, R. Barate et al., “Search for the standard model Higgs boson at LEP,” Phys. Lett. B 565 (2003) 61–75, arXiv:hep-ex/0306033.
  * [163] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., “Precision electroweak measurements on the $Z$ resonance,” Phys. Rept. 427 (2006) 257–454, arXiv:hep-ex/0509008.
  * [164] CMS Collaboration, “Search for a narrow resonance in high-mass dilepton final states in proton-proton collisions using 140$~{}\mathrm{fb}^{-1}$ of data at $\sqrt{s}=13~{}\mathrm{TeV}$,”.
  * [165] ATLAS Collaboration, G. Aad et al., “Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector,” Phys. Lett. B 796 (2019) 68–87, arXiv:1903.06248 [hep-ex].
  * [166] ATLAS Collaboration Collaboration, “Technical Design Report for the Phase-II Upgrade of the ATLAS LAr Calorimeter,” Tech. Rep. CERN-LHCC-2017-018. ATLAS-TDR-027, CERN, Geneva, Sep, 2017.  https://cds.cern.ch/record/2285582.
  * [167] R. Delbourgo and A. Salam, “The gravitational correction to pcac,” Phys. Lett. B 40 (1972) 381–382.
  * [168] T. Eguchi and P. G. O. Freund, “Quantum Gravity and World Topology,” Phys. Rev. Lett. 37 (1976) 1251.
  * [169] S. Oda, N. Okada, and D.-s. Takahashi, “Classically conformal $U(1)^{\prime}$ extended standard model and Higgs vacuum stability,” Phys. Rev. D92 no. 1, (2015) 015026, arXiv:1504.06291 [hep-ph].
  * [170] E. Eichten, K. D. Lane, and M. E. Peskin, “New Tests for Quark and Lepton Substructure,” Phys. Rev. Lett. 50 (1983) 811–814.
  * [171] LEP, ALEPH, DELPHI, L3, OPAL, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavor Group Collaboration, t. S. Electroweak, “A Combination of preliminary electroweak measurements and constraints on the standard model,” arXiv:hep-ex/0312023 [hep-ex].
  * [172] ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaboration, S. Schael et al., “Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP,” Phys. Rept. 532 (2013) 119–244, arXiv:1302.3415 [hep-ex].
  * [173] H. Kroha, “Compositeness limits from E+ E- annihilation revisited,” Phys. Rev. D 46 (1992) 58–69.
  * [174] LCC Physics Working Group Collaboration, K. Fujii et al., “Tests of the Standard Model at the International Linear Collider,” arXiv:1908.11299 [hep-ex].
  * [175] T. K. Poddar, S. Goswami, and A. K. Mishra, “Energizing gamma ray bursts via $Z^{\prime}$ mediated neutrino heating,” Eur. Phys. J. C 83 no. 3, (2023) 223, arXiv:2206.03485 [hep-ph].
  * [176] K. Asano and T. Fukuyama, “Neutrino pair annihilation in the gravitation of gamma-ray burst sources,” The Astrophysical Journal 531 no. 2, (Mar, 2000) 949. https://dx.doi.org/10.1086/308513.
  * [177] S. Mukherjee and S. Chakraborty, “Horndeski theories confront the Gravity Probe B experiment,” Phys. Rev. D 97 no. 12, (2018) 124007, arXiv:1712.00562 [gr-qc].
  * [178] D. Frederiks, A. Lysenko, A. Ridnaia, D. Svinkin, A. Tsvetkova, M. Ulanov, T. Cline, and Konus-Wind Team, “Konus-Wind detection of GRB 221009A,” GRB Coordinates Network 32668 (Oct., 2022) 1\.
  * [179] P. Veres, E. Burns, E. Bissaldi, S. Lesage, O. Roberts, and Fermi GBM Team, “GRB 221009A: Fermi GBM detection of an extraordinarily bright GRB,” GRB Coordinates Network 32636 (Oct., 2022) 1.
  * [180] Insight-HXMT, GECAM Collaboration, Z.-H. An et al., “Insight-HXMT and GECAM-C observations of the brightest-of-all-time GRB 221009A,” arXiv:2303.01203 [astro-ph.HE].
  * [181] D. Frederiks et al., “Properties of the extremely energetic GRB~221009A from Konus-WIND and SRG/ART-XC observations,” arXiv:2302.13383 [astro-ph.HE].
  * [182] M. A. Williams et al., “GRB 221009A: Discovery of an Exceptionally Rare Nearby and Energetic Gamma-Ray Burst,” Astrophys. J. Lett. 946 no. 1, (2023) L24, arXiv:2302.03642 [astro-ph.HE].
  * [183] S. Lesage et al., “Fermi-GBM Discovery of GRB 221009A: An Extraordinarily Bright GRB from Onset to Afterglow,” arXiv:2303.14172 [astro-ph.HE].
  * [184] W.-f. Fong, E. Berger, R. Margutti, and B. A. Zauderer, “A Decade of Short-duration Gamma-ray Burst Broadband Afterglows: Energetics, Circumburst Densities, and jet Opening Angles,” Astrophys. J. 815 no. 2, (2015) 102, arXiv:1509.02922 [astro-ph.HE].
  * [185] IceCube Collaboration, M. G. Aartsen et al., “Neutrino emission from the direction of the blazar TXS 0506+056 prior to the IceCube-170922A alert,” Science 361 no. 6398, (2018) 147–151, arXiv:1807.08794 [astro-ph.HE].
  * [186] S. Gasparyan, D. Bégué, and N. Sahakyan, “Time-dependent lepto-hadronic modelling of the emission from blazar jets with SOPRANO: the case of TXS 0506 + 056, 3HSP J095507.9 + 355101, and 3C 279,” Mon. Not. Roy. Astron. Soc. 509 no. 2, (2021) 2102–2121, arXiv:2110.01549 [astro-ph.HE].
  * [187] IceCube, “Icecube data from 2008 to 2017 related to analysis of txs 0506+056,” 2018.  https://icecube.wisc.edu/data-releases/2018/07/icecube-data-from-2008-to-2017-related-to-analysis-of-txs-0506056/,.
  * [188] J. N. Bahcall and R. A. Wolf, “Star distribution around a massive black hole in a globular cluster.,” Astrophys. J.  209 (Oct., 1976) 214–232.
  * [189] P. Ullio, H. Zhao, and M. Kamionkowski, “A Dark matter spike at the galactic center?,” Phys. Rev. D 64 (2001) 043504, arXiv:astro-ph/0101481.
  * [190] M. Gorchtein, S. Profumo, and L. Ubaldi, “Probing Dark Matter with AGN Jets,” Phys. Rev. D 82 (2010) 083514, arXiv:1008.2230 [astro-ph.HE]. [Erratum: Phys.Rev.D 84, 069903 (2011)].
  * [191] P. Padovani, F. Oikonomou, M. Petropoulou, P. Giommi, and E. Resconi, “TXS 0506+056, the first cosmic neutrino source, is not a BL Lac,” Mon. Not. Roy. Astron. Soc. 484 no. 1, (2019) L104–L108, arXiv:1901.06998 [astro-ph.HE].
  * [192] J. F. Navarro, C. S. Frenk, and S. D. M. White, “A Universal Density Profile from Hierarchical Clustering,” Astrophys. J.  490 no. 2, (Dec., 1997) 493–508, arXiv:astro-ph/9611107 [astro-ph].
  * [193] B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel, and P. Tozzi, “Dark matter substructure within galactic halos,” Astrophys. J. Lett. 524 (1999) L19–L22, arXiv:astro-ph/9907411.
  * [194] A. Klypin, A. V. Kravtsov, J. Bullock, and J. Primack, “Resolving the structure of cold dark matter halos,” Astrophys. J. 554 (2001) 903–915, arXiv:astro-ph/0006343.
  * [195] C. Power, J. F. Navarro, A. Jenkins, C. S. Frenk, S. D. M. White, V. Springel, J. Stadel, and T. R. Quinn, “The Inner structure of Lambda CDM halos. 1. A Numerical convergence study,” Mon. Not. Roy. Astron. Soc. 338 (2003) 14–34, arXiv:astro-ph/0201544.
  * [196] O. Y. Gnedin and J. R. Primack, “Dark Matter Profile in the Galactic Center,” Phys. Rev. Lett. 93 (2004) 061302, arXiv:astro-ph/0308385.
  * [197] F. Stoehr, S. D. M. White, V. Springel, G. Tormen, and N. Yoshida, “Dark matter annihilation in the halo of the Milky Way,” Mon. Not. Roy. Astron. Soc. 345 (2003) 1313, arXiv:astro-ph/0307026.
  * [198] R. Schodel et al., “A Star in a 15.2 year orbit around the supermassive black hole at the center of the Milky Way,” Nature 419 (2002) 694–696, arXiv:astro-ph/0210426.
  * [199] R. Genzel, C. Pichon, A. Eckart, O. E. Gerhard, and T. Ott, “Stellar dynamics in the Galactic centre: Proper motions and anisotropy,” Mon. Not. Roy. Astron. Soc. 317 (2000) 348, arXiv:astro-ph/0001428.
  * [200] A. M. Ghez, B. L. Klein, M. Morris, and E. E. Becklin, “High proper motion stars in the vicinity of Sgr A*: Evidence for a supermassive black hole at the center of our galaxy,” Astrophys. J. 509 (1998) 678–686, arXiv:astro-ph/9807210.
  * [201] D. Lynden-Bell and M. J. Rees, “On quasars, dust and the galactic centre,” Monthly Notices of the Royal Astronomical Society 152 no. 4, (Jul, 1971) 461–475. https://doi.org/10.1093/mnras/152.4.461.
  * [202] H. Zhao, M. G. Haehnelt, and M. J. Rees, “Feeding black holes at galactic centres by capture from isothermal cusps,” New Astron. 7 (2002) 385, arXiv:astro-ph/0112096.
  * [203] M. Milosavljevic and D. Merritt, “Formation of galactic nuclei,” Astrophys. J. 563 (2001) 34–62, arXiv:astro-ph/0103350.
  * [204] M. Volonteri, F. Haardt, and P. Madau, “The Assembly and merging history of supermassive black holes in hierarchical models of galaxy formation,” Astrophys. J. 582 (2003) 559–573, arXiv:astro-ph/0207276.
  * [205] D. Merritt, M. Milosavljevic, L. Verde, and R. Jimenez, “Dark matter spikes and annihilation radiation from the galactic center,” Phys. Rev. Lett. 88 (2002) 191301, arXiv:astro-ph/0201376.
  * [206] Y. B. Zeldovich and I. D. Novikov, Relativistic astrophysics. Vol.1: Stars and relativity.  1971\.
  * [207] J. F. Navarro, C. S. Frenk, and S. D. M. White, “The Structure of cold dark matter halos,” Astrophys. J. 462 (1996) 563–575, arXiv:astro-ph/9508025.
  * [208] J. F. Navarro, E. Hayashi, C. Power, A. Jenkins, C. S. Frenk, S. D. M. White, V. Springel, J. Stadel, and T. R. Quinn, “The Inner structure of Lambda-CDM halos 3: Universality and asymptotic slopes,” Mon. Not. Roy. Astron. Soc. 349 (2004) 1039, arXiv:astro-ph/0311231.
  * [209] D. Reed, F. Governato, L. Verde, J. Gardner, T. R. Quinn, J. Stadel, D. Merritt, and G. Lake, “Evolution of the density profiles of dark matter halos,” Mon. Not. Roy. Astron. Soc. 357 (2005) 82–96, arXiv:astro-ph/0312544.
  * [210] T. Fukushige, A. Kawai, and J. Makino, “Structure of dark matter halos from hierarchical clustering. 3. Shallowing of the Inner cusp,” Astrophys. J. 606 (2004) 625–634, arXiv:astro-ph/0306203.
  * [211] J. Diemand, B. Moore, and J. Stadel, “Convergence and scatter of cluster density profiles,” Mon. Not. Roy. Astron. Soc. 353 (2004) 624, arXiv:astro-ph/0402267.
  * [212] J. N. Bahcall and R. A. Wolf, “The star distribution around a massive black hole in a globular cluster. II. Unequal star masses.,” Astrophys. J.  216 (Sept., 1977) 883–907.
  * [213] J. L. Spudich and D. E. Koshland, “Non-genetic individuality: chance in the single cell,” Nature 262 no. 5568, (Aug, 1976) 467–471. https://doi.org/10.1038/262467a0.
  * [214] D. Merritt, “Relaxation and tidal stripping in rich clusters of galaxies. I. Evolution of the mass distribution.,” Astrophys. J.  264 (Jan., 1983) 24–48.
  * [215] IceCube, “Evidence for neutrino emission from the nearby active galaxy ngc 1068,” 2022.  https://icecube.wisc.edu/data-releases/2022/11/evidence-for-neutrino-emission-from-the-nearby-active-galaxy-ngc-1068/,.
  * [216] M. Bustamante, J. F. Beacom, and W. Winter, “Theoretically palatable flavor combinations of astrophysical neutrinos,” Phys. Rev. Lett. 115 no. 16, (2015) 161302, arXiv:1506.02645 [astro-ph.HE].
  * [217] C. A. Argüelles, T. Katori, and J. Salvado, “New Physics in Astrophysical Neutrino Flavor,” Phys. Rev. Lett. 115 (2015) 161303, arXiv:1506.02043 [hep-ph].
  * [218] G. D. Quinlan, L. Hernquist, and S. Sigurdsson, “Models of Galaxies with Central Black Holes: Adiabatic Growth in Spherical Galaxies,” Astrophys. J. 440 (1995) 554–564, arXiv:astro-ph/9407005.
  * [219] D. Merritt, “Evolution of the dark matter distribution at the galactic center,” Phys. Rev. Lett. 92 (2004) 201304, arXiv:astro-ph/0311594.
  * [220] D. Merritt, S. Harfst, and G. Bertone, “Collisionally Regenerated Dark Matter Structures in Galactic Nuclei,” Phys. Rev. D 75 (2007) 043517, arXiv:astro-ph/0610425.
  * [221] M. Baes, P. Buyle, G. K. T. Hau, and H. Dejonghe, “Observational evidence for a connection between supermassive black holes and dark matter haloes,” Mon. Not. Roy. Astron. Soc. 341 (2003) L44, arXiv:astro-ph/0303628.
  * [222] L. Ferrarese, “Beyond the bulge: a fundamental relation between supermassive black holes and dark matter halos,” Astrophys. J. 578 (2002) 90–97, arXiv:astro-ph/0203469.
  * [223] G. Gentile, C. Tonini, and P. Salucci, “Lambda CDM Halo Density Profiles: Where do actual halos converge to NFW ones?,” Astron. Astrophys. 467 (2007) 925–931, arXiv:astro-ph/0701550.
  * [224] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,” Astrophys. J. Lett. 930 no. 2, (2022) L12.
  * [225] ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaboration, S. Schael et al., “Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP,” Phys. Rept. 532 (2013) 119–244, arXiv:1302.3415 [hep-ex].
  * [226] CMS Collaboration, A. M. Sirunyan et al., “Search for resonant and nonresonant new phenomena in high-mass dilepton final states at $\sqrt{s}$ = 13 TeV,” JHEP 07 (2021) 208, arXiv:2103.02708 [hep-ex].
  * [227] P. Langacker, “The Physics of Heavy $Z^{\prime}$ Gauge Bosons,” Rev. Mod. Phys. 81 (2009) 1199–1228, arXiv:0801.1345 [hep-ph].
  * [228] J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky, and W. K. Tung, “New generation of parton distributions with uncertainties from global QCD analysis,” JHEP 07 (2002) 012, arXiv:hep-ph/0201195.
  * [229] CMS Collaboration, V. Khachatryan et al., “Search for heavy resonances decaying to tau lepton pairs in proton-proton collisions at $\sqrt{s}=13$ TeV,” JHEP 02 (2017) 048, arXiv:1611.06594 [hep-ex].
  * [230] ATLAS Collaboration, M. Aaboud et al., “Search for additional heavy neutral Higgs and gauge bosons in the ditau final state produced in 36/fb of pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” JHEP 01 (2018) 055, arXiv:1709.07242 [hep-ex].
  * [231] M. Bauer, P. Foldenauer, and J. Jaeckel, “Hunting All the Hidden Photons,” JHEP 07 (2018) 094, arXiv:1803.05466 [hep-ph].
  * [232] P. Ilten, Y. Soreq, M. Williams, and W. Xue, “Serendipity in dark photon searches,” JHEP 06 (2018) 004, arXiv:1801.04847 [hep-ph].
  * [233] G. Bellini et al., “Precision measurement of the 7Be solar neutrino interaction rate in Borexino,” Phys. Rev. Lett. 107 (2011) 141302, arXiv:1104.1816 [hep-ex].
  * [234] A. G. Beda, V. B. Brudanin, V. G. Egorov, D. V. Medvedev, V. S. Pogosov, M. V. Shirchenko, and A. S. Starostin, “Upper limit on the neutrino magnetic moment from three years of data from the GEMMA spectrometer,” arXiv:1005.2736 [hep-ex].
  * [235] COHERENT Collaboration, D. Akimov et al., “Observation of Coherent Elastic Neutrino-Nucleus Scattering,” Science 357 no. 6356, (2017) 1123–1126, arXiv:1708.01294 [nucl-ex].
  * [236] P. Melas, D. K. Papoulias, and N. Saoulidou, “Probing generalized neutrino interactions with the DUNE Near Detector,” JHEP 07 (2023) 190, arXiv:2303.07094 [hep-ph].
  * [237] M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, and Y. Y. Zhang, “Constraints on light vector mediators through coherent elastic neutrino nucleus scattering data from COHERENT,” JHEP 01 (2021) 116, arXiv:2008.05022 [hep-ph].
  * [238] J. Barranco, O. G. Miranda, and T. I. Rashba, “Probing new physics with coherent neutrino scattering off nuclei,” JHEP 12 (2005) 021, arXiv:hep-ph/0508299.
  * [239] K. Patton, J. Engel, G. C. McLaughlin, and N. Schunck, “Neutrino-nucleus coherent scattering as a probe of neutron density distributions,” Phys. Rev. C 86 (2012) 024612, arXiv:1207.0693 [nucl-th].
  * [240] M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, and Y. Y. Zhang, “Physics results from the first COHERENT observation of coherent elastic neutrino-nucleus scattering in argon and their combination with cesium-iodide data,” Phys. Rev. D 102 no. 1, (2020) 015030, arXiv:2005.01645 [hep-ph].
  * [241] J. Erler and S. Su, “The Weak Neutral Current,” Prog. Part. Nucl. Phys. 71 (2013) 119–149, arXiv:1303.5522 [hep-ph].
  * [242] R. H. Helm, “Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei,” Phys. Rev. 104 (1956) 1466–1475.
  * [243] G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera, and C. W. de Jager, “Nuclear Ground State Charge Radii from Electromagnetic Interactions,” Atom. Data Nucl. Data Tabl. 60 (1995) 177–285.
  * [244] I. Angeli and K. P. Marinova, “Table of experimental nuclear ground state charge radii: An update,” Atom. Data Nucl. Data Tabl. 99 no. 1, (2013) 69–95.
  * [245] M. Bender, K. Rutz, P. G. Reinhard, J. A. Maruhn, and W. Greiner, “Shell structure of superheavy nuclei in selfconsistent mean field models,” Phys. Rev. C 60 (1999) 034304, arXiv:nucl-th/9906030.
  * [246] Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 no. 14, (2021) 141801, arXiv:2104.03281 [hep-ex].
  * [247] J. P. Leveille, “The Second Order Weak Correction to (G-2) of the Muon in Arbitrary Gauge Models,” Nucl. Phys. B 137 (1978) 63–76.
  * [248] S. Baek, N. G. Deshpande, X. G. He, and P. Ko, “Muon anomalous g-2 and gauged L(muon) - L(tau) models,” Phys. Rev. D 64 (2001) 055006, arXiv:hep-ph/0104141.
  * [249] T. Aoyama et al., “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887 (2020) 1–166, arXiv:2006.04822 [hep-ph].
  * [250] Muon g-2 Collaboration, D. P. Aguillard et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm,” arXiv:2308.06230 [hep-ex].
  * [251] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, “Measurement of the fine-structure constant as a test of the Standard Model,” Science 360 (2018) 191, arXiv:1812.04130 [physics.atom-ph].
  * [252] L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, “Determination of the fine-structure constant with an accuracy of 81 parts per trillion,” Nature 588 no. 7836, (2020) 61–65.
  * [253] KLOE-2 Collaboration, F. Archilli et al., “Search for a vector gauge boson in $\phi$ meson decays with the KLOE detector,” Phys. Lett. B 706 (2012) 251–255, arXiv:1110.0411 [hep-ex].
  * [254] KLOE-2 Collaboration, A. Anastasi et al., “Limit on the production of a new vector boson in $\mathrm{e^{+}e^{-}}\rightarrow{\rm U}\gamma$, U$\rightarrow\pi^{+}\pi^{-}$ with the KLOE experiment,” Phys. Lett. B 757 (2016) 356–361, arXiv:1603.06086 [hep-ex].
  * [255] CCFR Collaboration, S. R. Mishra et al., “Neutrino Tridents and W Z Interference,” Phys. Rev. Lett. 66 (1991) 3117–3120.
  * [256] CMS Collaboration, A. M. Sirunyan et al., “Search for an $L_{\mu}-L_{\tau}$ gauge boson using Z$\to 4\mu$ events in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 792 (2019) 345–368, arXiv:1808.03684 [hep-ex].
  * [257] ATLAS Collaboration, G. Aad et al., “Search for a new Z’ gauge boson in $4\mu$ events with the ATLAS experiment,” JHEP 07 (2023) 090, arXiv:2301.09342 [hep-ex].
  * [258] H. K. Dreiner, J.-F. Fortin, J. Isern, and L. Ubaldi, “White Dwarfs constrain Dark Forces,” Phys. Rev. D 88 (2013) 043517, arXiv:1303.7232 [hep-ph].
  * [259] M. Ibe, W. Nakano, and M. Suzuki, “Constraints on $L_{\mu}-L_{\tau}$ gauge interactions from rare kaon decay,” Phys. Rev. D 95 no. 5, (2017) 055022, arXiv:1611.08460 [hep-ph].
  * [260] S. Martellotti, “The NA62 Experiment at CERN,” in 12th Conference on the Intersections of Particle and Nuclear Physics.  10, 2015.  arXiv:1510.00172 [physics.ins-det].
  * [261] G. Krnjaic, G. Marques-Tavares, D. Redigolo, and K. Tobioka, “Probing Muonphilic Force Carriers and Dark Matter at Kaon Factories,” Phys. Rev. Lett. 124 no. 4, (2020) 041802, arXiv:1902.07715 [hep-ph].
  * [262] Belle-II Collaboration, L. Aggarwal et al., “Snowmass White Paper: Belle II physics reach and plans for the next decade and beyond,” arXiv:2207.06307 [hep-ex].
  * [263] NA64 Collaboration, Y. M. Andreev et al., “Search for a light Z’ in the L$\mu$-L$\tau$ scenario with the NA64-e experiment at CERN,” Phys. Rev. D 106 no. 3, (2022) 032015, arXiv:2206.03101 [hep-ex].
  * [264] D. Tuckler, “$B$ Anomalies and Dark Matter in an $L_{\mu}-L_{\tau}$ Model with General Kinetic Mixing,” arXiv:2209.03397 [hep-ph].