# Exclusion bounds for neutral gauge bosons

Zoltán Péli<EMAIL_ADDRESS>Institute for Theoretical Physics, ELTE
Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
Zoltán Trócsányi<EMAIL_ADDRESS>Institute for Theoretical Physics,
ELTE Eötvös Loránd University and HUN-REN ELTE Theoretical Physics Research
Group, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary, also at
University of Debrecen, Bem tér 18/A, 4026 Debrecen, Hungary

###### Abstract

We study how the recent experimental results constrain the gauge sectors of
U(1) extensions of the standard model using a novel representation of the
parameter space. We determine the bounds on the mixing angle between the
massive gauge bosons, or equivalently, the new gauge coupling as a function of
the mass $M_{Z^{\prime}}$ of the new neutral gauge boson $Z^{\prime}$ in the
approximate range $(10^{-2},10^{4})$ GeV/$c^{2}$. We consider the most
stringent bounds obtained from direct searches for the $Z^{\prime}$. We also
exhibit the allowed parameter space by comparing the predicted and measured
values of the $\rho$ parameter and those of the mass of the $W$ boson.
Finally, we discuss the prospects of $Z^{\prime}$ searches at future
colliders.

##  I Introduction

The standard model of particle interactions (SM) has been tested to high
precision both in low energy experiments as well as at high energy colliders
[1]. Most recently the large LHC experiments find spectacular agreement
between their experimental results and the SM predictions [2, 3], which leaves
very little room for new physics. Nevertheless, we do not have doubt that the
SM cannot describe all observations in the microworld. Most notably, the
masses of neutrinos, the baryon asymmetry and the origin of dark matter in the
Universe are clear indications of the need for physics beyond the standard
model (BSM). The nature of this new physics however, remains elusive.

There is also a 5$\sigma$ tension between the measured value of the muon
anomalous magnetic moment $a_{\mu}$ [4] and the SM prediction when the
hadronic vacuum polarization to the photon is extracted from the measured
total hadronic cross section in electron-positron annihilation at low energies
[5]. A natural explanation for such a difference is the contribution of a new
heavy neutral gauge boson $Z^{\prime}$ [6, 7, 8]. Presently however, the size
of the deviation between the SM prediction and the measurement is heavily
debated. The Budapest-Marseille-Wuppertal lattice collaboration computed the
hadronic vacuum polarization of the photon from first principles [9], and
found a much less significant ($<2\sigma$) tension for $a_{\mu}$ between
theory and experiment. Nevertheless, the exploration of the effects of a
$Z^{\prime}$ on measurements is interesting because $U(1)_{z}$ extensions
provide the simplest possible way to explain a potential fifth fundamental
force.

Due to their simplicity, $U(1)_{z}$ extensions have a more then forty year old
history [10] and remain popular [1] at present. They have been investigated
since the operation of the experiments of the Large Electron Positron collider
in various forms, like gauged extra U(1) symmetry [11], which can also be
broken by a new scalar [12], giving rise to a new massive gauge boson, often
call dark photon 111After diagonalization of the mass matrix of the neutral
gauge bosons, $A^{\prime}$. Lately, more complete U(1) extensions of the SM
have been studied with the goal of explaining several beyond the standard
model phenomena simultaneously, such as the superweak extension of the
standard model (SWSM) [14]. Even the complete one-loop renormalization of the
Dark Abelian Sector Model with identical gauge and scalar sectors as in the
SWSM, but with somewhat different fermion content has been carried out in Ref.
[15].

The continuing theoretical interest is met with similarly ubiquitous
experimental searches for dark photons or more generally, new neutral gauge
bosons. The experimental limits are typically presented in the parameter space
of the dark photons, which provide serendipitous discovery potential for other
types of vector particles [16]. Constraints have been placed on visible
$A^{\prime}$ decays by beam-dump, collider, fixed-target, and rare-meson-decay
experiments, as well as on invisible $A^{\prime}$ decays. New experiments have
also been proposed to explore further the parameter space in the future [17,
18].

In this work we discuss the presently most constraining experimental limits in
some part of the parameter space of general U(1) extensions that contain
right-handed neutrinos, not charged under the SM interactions, in the particle
spectrum. We focus on two regions: the case of light and heavy $Z^{\prime}$,
$\xi=M_{Z^{\prime}}/M_{Z}\ll 1$ and $\xi\gg 1$, or quantitatively
$M_{Z^{\prime}}\in[0.02,10]$ GeV and $M_{Z^{\prime}}\in[0.2,5]$ TeV 222The
cases of an intermediate mass $Z^{\prime}$, $M_{Z^{\prime}}\approx M_{Z}$, and
a very light $Z^{\prime}$, $M_{Z^{\prime}}\lesssim 20$ MeV have been discussed
elsewhere.. The most stringent limits for a heavy $Z^{\prime}$ are provided by
direct searches at the LHC in Drell-Yan pair production $pp\to
Z^{\prime}+X\to\ell^{+}\ell^{-}+X$ [20, 21]. For a light $Z^{\prime}$ those
have been obtained in direct searches for an invisibly decaying dark photon in
the NA64 [22] and BaBar [23] experiments, as well as for a dark photon
decaying into an electron-positron pair in the FASER detector [24]. Similar
studies have already been published. For instance, Refs. [25, 26] focus on
$U(1)_{z}$ extensions with a selection of benchmark $z$ charges in the light
$M_{Z^{\prime}}$ region. In the present work we use a parametrization of the
$z$ charges valid for any charge assignments that satisfy the conditions of
anomaly cancellations and gauge invariance. The exclusion limits depend on a
specific single combination of the free $z$ charges such that we also take
into account the uncertainty due to the choice of the renormalization scale
where the $z$ charges are set, neglected in Refs. [25, 26]

As one might expect, in the regions far away from the mass of the $Z$ boson
the mixing angle $\theta_{Z}$ between the massive neutral gauge bosons is
small experimentally (below $10^{-3}$), so that one can use expansions around
$\theta_{Z}=0$. In this limit the couplings of the $Z^{\prime}$ to chiral
fermions are approximately vector like and universal in the sense that they
all depend on a unique combination of the $z$ charge of the right-handed
neutrinos and Brout-Englert-Higgs (BEH) field. We also discuss how results of
electroweak precision measurements constrain the parameter space.

##  II Model definition

We consider the extensions of the standard model by a $U(1)_{z}$ gauge group
with a complex scalar field $\chi$ and three generations of right handed
neutrinos. The new fields are neutral under the standard model gauge
interactions. An example for such a model is the superweak extension of the
standard model (SWSM) [14]. However, in the present work we require only gauge
and gravity anomaly cancellation, otherwise leave the $z$ charges arbitrary.
In this section we collect the details of the model only to the extent used in
the present analyses.

###  II.1 Scalar sector

In the scalar sector, in addition to the $SU(2)_{\mathrm{L}}$-doublet Brout-
Englert-Higgs field

$\phi=\left(\\!\\!\begin{array}[]{c}\phi^{+}\\\
\phi^{0}\end{array}\\!\\!\right)=\frac{1}{\sqrt{2}}\left(\\!\\!\begin{array}[]{c}\phi_{1}+\mathrm{i}\phi_{2}\\\
\phi_{3}+\mathrm{i}\phi_{4}\end{array}\\!\\!\right)\,,$ (1)

the model contains a complex scalar SM singlet $\chi$. The Lagrangian of the
scalar fields contains the potential energy

$\begin{split}V(\phi,\chi)&=-\mu_{\phi}^{2}|\phi|^{2}-\mu_{\chi}^{2}|\chi|^{2}+\left(|\phi|^{2},|\chi|^{2}\right)\left(\\!\\!\begin{array}[]{cc}\lambda_{\phi}&\frac{\lambda}{2}\\\
\frac{\lambda}{2}&\lambda_{\chi}\end{array}\\!\\!\right)\left(\\!\\!\begin{array}[]{c}|\phi|^{2}\\\
|\chi|^{2}\end{array}\\!\\!\right)\\\ &\subset-\mathcal{L}\end{split}$ (2)

where $|\phi|^{2}=|\phi^{+}|^{2}+|\phi^{0}|^{2}$. After spontaneous symmetry
breaking, we parametrize the scalar fields as

$\begin{split}\phi=\frac{1}{\sqrt{2}}\binom{-\mathrm{i}\sqrt{2}\sigma^{+}}{v+h^{\prime}+\mathrm{i}\sigma_{\phi}}\,,\quad\chi=\frac{1}{\sqrt{2}}(w+s^{\prime}+\mathrm{i}\sigma_{\chi})\end{split}$
(3)

where $v$ and $w$ are the vacuum expectation values (VEVs) of $\phi$ and
$\chi$. The fields $h^{\prime}$ and $s^{\prime}$ are real scalars,
$\sigma^{+}$ is a charged, while $\sigma_{\phi}$ and $\sigma_{\chi}$ are
neutral Goldstone bosons that are gauge eigenstates.

The gauge and mass eigenstates are related by the rotations

$\binom{h}{s}=\textbf{Z}_{S}\binom{h^{\prime}}{s^{\prime}}\,,\quad\binom{\sigma_{Z}}{\sigma_{Z^{\prime}}}=\textbf{Z}_{G}\binom{\sigma_{\phi}}{\sigma_{\chi}}\,,$
(4)

with

$\textbf{Z}_{X}=\begin{pmatrix}\cos\theta_{X}&-\sin\theta_{X}\\\
\sin\theta_{X}&~{}~{}\cos\theta_{X}\end{pmatrix}$ (5)

where we denoted the mass eigenstates with $h$, $s$ and $\sigma_{Z}$,
$\sigma_{Z^{\prime}}$. The angles $\theta_{S}$ and $\theta_{G}$ are the scalar
and Goldstone mixing angles that can be determined by the diagonalization of
the mass matrix of the real scalars and that of the neutral Goldstone bosons.
In the following, we are going to use the abbreviations ${\rm
c}_{X}=\cos\theta_{X}$ and ${\rm s}_{X}=\sin\theta_{X}$ for mixing angles.

###  II.2 Gauge sector

The field strength tensors of the U(1) gauge groups are gauge invariant,
kinetic mixing is allowed between the gauge fields belonging to the
hypercharge U(1)Y and the new U(1)z gauge symmetries. Equivalently, one can
choose a basis in which the gauge-field strengths do not mix 333This is the
convention used in SARAH for models with multiple gauged U(1) symmetries.,
such that the covariant derivative corresponding to the U(1) gauge groups can
be parametrized as

$D^{\mathrm{U}(1)}_{\mu}=-\mathrm{i}\begin{pmatrix}y&z\end{pmatrix}\begin{pmatrix}g_{y}&-g_{yz}\\\
0&g_{z}\end{pmatrix}\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&~{}~{}\cos\alpha\end{pmatrix}\begin{pmatrix}B_{\mu}\\\
B^{\prime}_{\mu}\end{pmatrix}$ (6)

where $B_{\mu}$ and $B^{\prime}_{\mu}$ are the U(1)y and U(1)z gauge fields,
while $y$ and $z$ are the corresponding charges. The rotation angle $\alpha$
is not physical as it can be absorbed into the definition of the gauge fields
[28]. The $y$ charges are the eigenvalues of one half times the hypercharge
operator $Y$. The $z$ charges are assigned such that Yukawa terms including
the neutrinos and the scalar fields exist, and the gauge and gravity anomalies
cancel in each family. Such a charge assignment can be paramterized with two
numbers, usually chosen to be the charge of the left-handed quark doublet
$z_{q}$ and that of the right-handed u-type quarks $z_{u}$ [29]. The $z$
charge of the field $\chi$ can be fixed without the loss of generality as its
normalization can be absorbed into the rescaling of $g_{z}$. In this work we
use $z_{\chi}=-1$.

In general, the $z_{i}$ charges of the right handed neutrinos have to satisfy

$\frac{1}{3}\sum_{i=1}^{n}z_{i}=z_{u}-4z_{q}\equiv
z_{N},\quad\text{and}\quad\biggl{(}\sum_{i=1}^{n}z_{i}\biggr{)}^{3}=9\sum_{i=1}^{n}z_{i}^{3}.$
(7)

A simple and natural choice is to have $n=3$ generations of sterile neutrinos
and generation independent $z$ charges, i.e. $z_{i}=z_{N}$ for any $i=1,2,3$.
We find that for phenomenology it is more convenient to choose $z_{N}$ and the
$z$ charge of the SM scalar field $z_{\phi}$ as independent charges. We
exhibit the corresponding $z$ charge assignment in Table 1.

field | $SU(3)_{\mathrm{c}}$ | $SU(2)_{\mathrm{L}}$ | $y$ | $z$
---|---|---|---|---
$Q_{\mathrm{L}}$ | 3 | 2 | $\frac{1}{6}$ | $z_{q}=\frac{1}{3}(z_{\phi}-z_{N})$
$U_{\mathrm{R}}$ | 3 | 1 | $\frac{2}{3}$ | $z_{u}=\frac{1}{3}(4z_{\phi}-z_{N})$
$D_{\mathrm{R}}$ | 3 | 1 | $-\frac{1}{3}$ | $z_{d}=-\frac{1}{3}(2z_{\phi}+z_{N})$
$\ell_{\mathrm{L}}$ | 1 | 2 | $-\frac{1}{2}$ | $z_{\ell}=z_{N}-z_{\phi}$
$N_{\mathrm{R}}$ | 1 | 1 | 0 | $z_{N}$
$e_{\mathrm{R}}$ | 1 | 1 | $-1$ | $z_{e}=z_{N}-2z_{\phi}$
$\phi$ | 1 | 2 | $\frac{1}{2}$ | $z_{\phi}$
$\chi$ | 1 | 1 | 0 | $z_{\chi}=-1$
Table 1:  Field content and charge assignment of a generic U(1)z extension of
the SM. The field $\phi$ is the Higgs doublet and $\chi$ is a complex scalar
field, the rest are Weyl fermions. We show the representations for
SU(3)${}_{\mathrm{c}}\otimes$SU(2)L and the charges $y$ and $z$ for
U(1)${}_{Y}\otimes$U(1)z.

A $D=4$ operator corresponding to a Majorana mass term for the sterile
neutrinos is allowed only for $z_{\chi}+2z_{N}=0$, which implies $z_{N}=1/2$
with our normalization $z_{\chi}=-1$. For example, in the $B-L$ U(1)
extension, $z_{N}=1/2$ and $z_{\phi}=0$, while in the SWSM $z_{N}=1/2$ and
$z_{\phi}=1$.

The neutral gauge fields are related to their mass eigenstates $A_{\mu}$,
$Z_{\mu}$ and $Z^{\prime}_{\mu}$ via two rotations 444Our sign convention for
$\theta_{Z}$ agrees with the convention of Ref. [14], which differs by a
factor of $(-1)$ from that of Ref. [28].

$\left(\begin{array}[]{c}B_{\mu}\\\ W^{3}_{\mu}\\\
B^{\prime}_{\mu}\end{array}\right)=\left(\begin{array}[]{ccc}\mathrm{c}_{W}&-\mathrm{s}_{W}&0\\\
\mathrm{s}_{W}&\mathrm{c}_{W}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\mathrm{c}_{Z}&-\mathrm{s}_{Z}\\\
0&\mathrm{s}_{Z}&\mathrm{c}_{Z}\end{array}\right)\left(\begin{array}[]{c}A_{\mu}\\\
Z_{\mu}\\\ Z^{\prime}_{\mu}\end{array}\right).$ (8)

The two mixing angles are (i) the weak mixing angle $\theta_{\rm W}$ and (ii)
the $Z-Z^{\prime}$ mixing angle $\theta_{Z}\in[-\pi/4,\pi/4]$. The former is
defined as $\mathrm{s}_{W}=\frac{g_{y}}{g_{Z^{0}}}$ , with
$g_{Z^{0}}^{2}=g_{y}^{2}+g_{\mathrm{L}}^{2}$ , so
$e=g_{\mathrm{L}}\mathrm{s}_{W}$ where $g_{\mathrm{L}}$ is the SU(2) gauge
coupling and $e$ is the elementary charge. The new mixing angle is defined as

$\tan(2\theta_{Z})=-\frac{2\kappa}{1-\kappa^{2}-\tau^{2}}$ (9)

in terms of effective couplings

$\kappa=2\frac{g_{z}}{g_{Z^{0}}}z_{\phi}(\mu)\text{~{}~{}and~{}~{}}\tau=2\frac{g_{z}}{g_{Z^{0}}}\tan\beta$
(10)

where $\tan\beta=\frac{w}{v}$, and we introduced the effective charge

$z_{\phi}(\mu)=z_{\phi}-\frac{g_{yz}}{2g_{z}}$ (11)

as the charge $z_{\phi}$ appears always together with this ratio of the new
couplings throughout our computations. In Eq. (11) we indicated the dependence
of the couplings on the renormalization scale $\mu$ to emphasize the scale
dependence of the effective charge defined as abbreviation. It is possible to
choose a basis of the fundamental gauge fields such that $g_{yz}(\mu_{0})=0$
at a fixed, but arbitrary renormalization scale $\mu_{0}$. Clearly,
$z_{\phi}(\mu_{0})=z_{\phi}$ at this scale, i.e. $\mu_{0}$ is the scale where
all $z$ charges are set. The scale $\mu_{0}$ can be chosen at will, but the
running of $z_{\phi}(\mu)$ introduces some theoretical uncertainty to our
predictions, whose size depends on the actual choice of $\mu_{0}$. In order to
assess this uncertainty, we discuss the one-loop running of the ratio
$\eta=g_{yz}/g_{z}$ in App. A.

In terms of the mixing angles and effective couplings, the masses of the gauge
bosons are $M_{W}=\frac{1}{2}g_{\rm L}v$,

$M_{Z}=\frac{M_{W}}{\mathrm{c}_{W}}\sqrt{R(\mathrm{c}_{Z},\mathrm{s}_{Z})}\,,\quad
M_{Z^{\prime}}=\frac{M_{W}}{\mathrm{c}_{W}}\sqrt{R(\mathrm{s}_{Z},-\mathrm{c}_{Z})}\,,$
(12)

with $R(x,y)=\bigl{(}x-\kappa y\bigr{)}^{2}+\bigl{(}\tau y\bigr{)}^{2}.$ The
coupling parameters $\kappa$ and $\tau$ can be expressed in terms of the
experimentally more accessible parameters $M_{Z^{\prime}}$ and $\theta_{Z}$ as

$\kappa=-\mathrm{c}_{Z}\mathrm{s}_{Z}~{}\frac{M_{Z}^{2}-M_{Z^{\prime}}^{2}}{\mathrm{c}_{Z}^{2}M_{Z}^{2}+\mathrm{s}_{Z}^{2}M_{Z^{\prime}}^{2}}~{}~{}~{}~{}\text{and}~{}~{}~{}\tau=\frac{M_{Z}M_{Z^{\prime}}}{\mathrm{c}_{Z}^{2}M_{Z}^{2}+\mathrm{s}_{Z}^{2}M_{Z^{\prime}}^{2}}\,.$
(13)

Taking the ratio $\kappa/\tau$, Eqs. (10) and (13) imply

$-\mathrm{s}_{Z}\mathrm{c}_{Z}(1-\xi^{2})=\frac{\xi z_{\phi}(\mu)}{\tan\beta}$
(14)

where we remind the reader that $\xi=M_{Z^{\prime}}/M_{Z}$.

###  II.3 Modified $\rho$ parameter

The well known SM tree-level relationship between the masses of the $W$ and
$Z$ bosons is usually expressed as $\rho=1$ where

$\rho=\frac{M_{W}^{2}}{\mathrm{c}_{W}^{2}M_{Z}^{2}}\,.$ (15)

In the extended model it is no longer equal to one at the tree level as it is
modified to

$\rho=1-\mathrm{s}_{Z}^{2}\,(1-\xi^{2})\,.$ (16)

Experimentally, from global fits [1] one has

$\rho=1.00038\pm 0.00020\,,$ (17)

which implies that $\mathrm{s}_{Z}^{2}\ll 1$ for either a light or a heavy
$Z^{\prime}$ boson. Utilizing the smallness of $\mathrm{s}_{Z}$, we can also
express the $\rho$ parameter in terms of the effective couplings,
$\rho=1-\kappa^{2}/(1-\tau^{2})+\mathcal{O}\bigl{(}\mathrm{s}_{Z}^{4}\bigr{)}$,
or using the Lagrangian couplings and $M_{Z^{\prime}}$ as

$\rho=1-\frac{v^{2}}{M_{Z}^{2}-M_{Z^{\prime}}^{2}}\big{(}z_{\phi}(\mu)\,g_{z}\big{)}^{2}+\mathcal{O}\bigl{(}\mathrm{s}_{Z}^{4}\bigr{)}\,,$
(18)

which we use below. Equivalently, we can express $\rho$ using $\tan\beta$. In
the limit of a heavy $Z^{\prime}$ we have $\rho\simeq
1+\big{(}z_{\phi}(\mu)/\tan\beta\big{)}^{2}$, whereas a light $Z^{\prime}$
implies $\rho\simeq 1-\big{(}z_{\phi}(\mu)\,\xi/\tan\beta\big{)}^{2}$.

###  II.4 Vector-axial vector couplings of the Z-prime boson

Direct $Z^{\prime}$ searches at colliders are most often based on the Drell-
Yan process, hence on decays of the $Z^{\prime}$ into fermion pairs. The
relevant theoretical predictions, discussed in detail in the following
sections, rely on the interaction of the $Z^{\prime}$ boson and the fermions,
which in the Dirac basis reads as (neglecting the mixing among the neutrinos)

$\mathcal{L}_{\text{NC}}^{(Z^{\prime})}=-\frac{e}{2\mathrm{s}_{W}\mathrm{c}_{W}}Z^{\prime}_{\mu}\sum_{f}\bar{f}\gamma^{\mu}\big{(}v_{Z^{\prime},f}-a_{Z^{\prime},f}\gamma_{5}\big{)}f\,.$
(19)

We recall the vector and axial vector couplings $v_{Z^{\prime},f}$ and
$a_{Z^{\prime},f}$ using a parametrization convenient to our analysis in Table
2, obtained using the chiral couplings presented in App. B.

$f$ | $v_{Z^{\prime},f}$ | $a_{Z^{\prime},f}$
---|---|---
$\nu$ | $-\frac{1}{2}\mathrm{s}_{Z}+\frac{1}{2}\Big{(}-\kappa+2\frac{\tau}{\tan\beta}z_{N}\Big{)}\mathrm{c}_{Z}$ | $-\frac{1}{2}\bigl{(}\mathrm{s}_{Z}+\kappa\mathrm{c}_{Z}\bigr{)}$
$\ell$ | $-\Big{(}-\frac{1}{2}+2\mathrm{s}_{W}^{2}\Big{)}\mathrm{s}_{Z}+\frac{1}{2}\Big{(}-3\kappa+2\frac{\tau}{\tan\beta}z_{N}\Big{)}\mathrm{c}_{Z}$ | $\frac{1}{2}\bigl{(}\mathrm{s}_{Z}+\kappa\mathrm{c}_{Z}\bigr{)}$
$u$ | $-\Big{(}\frac{1}{2}-\frac{4}{3}\mathrm{s}_{W}^{2}\Big{)}\mathrm{s}_{Z}+\frac{1}{6}\Big{(}5\kappa-2\frac{\tau}{\tan\beta}z_{N}\Big{)}\mathrm{c}_{Z}$ | $-\frac{1}{2}\bigl{(}\mathrm{s}_{Z}+\kappa\mathrm{c}_{Z}\bigr{)}$
$d$ | $-\Big{(}-\frac{1}{2}+\frac{2}{3}\mathrm{s}_{W}^{2}\Big{)}\mathrm{s}_{Z}-\frac{1}{6}\Big{(}\kappa+2\frac{\tau}{\tan\beta}z_{N}\Big{)}\mathrm{c}_{Z}$ | $\frac{1}{2}\bigl{(}\mathrm{s}_{Z}+\kappa\mathrm{c}_{Z}\bigr{)}$
Table 2: The vector and axial-vector couplings of the $Z^{\prime}$ boson to
fermions in U(1)z extensions of the SM. The corresponding couplings of the $Z$
boson $v_{Z,f}$ and $a_{Z,f}$ are obtained by the replacement
$(\mathrm{c}_{Z},\>\mathrm{s}_{Z})\to(\mathrm{s}_{Z},\>-\mathrm{c}_{Z})$ in
$v_{Z^{\prime},f}$ and $a_{Z^{\prime},f}$.

Expanding these couplings in terms of the small parameter $\mathrm{s}_{Z}$ one
obtains the following expressions (recall that $\xi=M_{Z^{\prime}}/M_{Z}$):

$\begin{split}v_{Z^{\prime},\nu}\simeq\frac{z_{N}\,\xi}{\tan\beta}-\frac{1}{2}\mathrm{s}_{Z}\xi^{2},&\quad
a_{Z^{\prime},\nu}\simeq-\frac{1}{2}\mathrm{s}_{Z}\xi^{2},\\\
v_{Z^{\prime},\ell}\simeq\frac{z_{N}\,\xi}{\tan\beta}+\frac{1}{2}\mathrm{s}_{Z}\bigl{(}4\mathrm{c}_{W}^{2}-3\xi^{2}\bigr{)},&\quad
a_{Z^{\prime},\ell}\simeq\frac{1}{2}\mathrm{s}_{Z}\xi^{2},\\\
v_{Z^{\prime},u}\simeq-\frac{z_{N}\,\xi}{3\tan\beta}+\frac{1}{6}\mathrm{s}_{Z}\bigl{(}-8\mathrm{c}_{W}^{2}+5\xi^{2}\bigr{)},&\quad
a_{Z^{\prime},u}\simeq-\frac{1}{2}\mathrm{s}_{Z}\xi^{2},\\\
v_{Z^{\prime},d}\simeq-\frac{z_{N}\,\xi}{3\tan\beta}+\frac{1}{6}\mathrm{s}_{Z}\bigl{(}4\mathrm{c}_{W}^{2}-\xi^{2}\bigr{)},&\quad
a_{Z^{\prime},d}\simeq\frac{1}{2}\mathrm{s}_{Z}\xi^{2}\,.\end{split}$ (20)

It is useful to distinguish the cases of $\xi\to\infty$ (heavy $Z^{\prime}$)
and $\xi\to 0$ (light $Z^{\prime}$). In the case of a heavy $Z^{\prime}$ boson
Eq. (14) implies that

$\frac{\xi}{\tan\beta}\simeq\frac{\mathrm{s}_{Z}\xi^{2}}{z_{N}\mathscr{Z}}$
(21)

for small values of $|\mathrm{s}_{Z}|$. In Eq. (21) we introduced the
effective charge ratio

$\mathscr{Z}(\mu)=\frac{z_{\phi}(\mu)}{z_{N}}$ (22)

that contains all dependence on the specific U(1) extension. As the effective
charge of the BEH field depends on the renormalization scale, so does
$\mathscr{Z}$, which we suppress in the following, but take into account as
theoretical uncertainty of our predictions as discussed in App. A.

Then in the limit of small neutral gauge mixing and heavy $Z^{\prime}$, the
V-A couplings simplify to

$\begin{split}v_{Z^{\prime},\nu}\simeq\mathrm{s}_{Z}\xi^{2}\biggl{(}\frac{1}{\mathscr{Z}}-\frac{1}{2}\biggr{)},&\quad
a_{Z^{\prime},\nu}\simeq-\frac{1}{2}\mathrm{s}_{Z}\xi^{2},\\\
v_{Z^{\prime},\ell}\simeq\mathrm{s}_{Z}\xi^{2}\biggl{(}\frac{1}{\mathscr{Z}}-\frac{3}{2}\biggr{)},&\quad
a_{Z^{\prime},\ell}\simeq\frac{1}{2}\mathrm{s}_{Z}\xi^{2},\\\
v_{Z^{\prime},u}\simeq-\frac{\mathrm{s}_{Z}\xi^{2}}{3}\biggl{(}\frac{1}{\mathscr{Z}}-\frac{5}{2}\biggr{)},&\quad
a_{Z^{\prime},u}\simeq-\frac{1}{2}\mathrm{s}_{Z}\xi^{2},\\\
v_{Z^{\prime},d}\simeq-\frac{\mathrm{s}_{Z}\xi^{2}}{3}\biggl{(}\frac{1}{\mathscr{Z}}+\frac{1}{2}\biggr{)},&\quad
a_{Z^{\prime},d}\simeq\frac{1}{2}\mathrm{s}_{Z}\xi^{2}\,.\end{split}$ (23)

As for a light $Z^{\prime}$, Eq. (14) implies

$\frac{\xi}{\tan\beta}\simeq-\frac{\mathrm{s}_{Z}}{z_{N}\mathscr{Z}}\,,$ (24)

and for the V-A couplings one has negligible $a_{Z^{\prime},f}$, and

$\begin{split}v_{Z^{\prime},\nu}\simeq-\frac{\mathrm{s}_{Z}}{\mathscr{Z}},&\quad
v_{Z^{\prime},\ell}\simeq\mathrm{s}_{Z}\biggl{(}-\frac{1}{\mathscr{Z}}+2\mathrm{c}_{W}^{2}\biggr{)},\\\
v_{Z^{\prime},u}\simeq\frac{\mathrm{s}_{Z}}{3}\biggl{(}\frac{1}{\mathscr{Z}}-4\mathrm{c}_{W}^{2}\biggr{)},&\quad
v_{Z^{\prime},d}\simeq\frac{\mathrm{s}_{Z}}{3}\biggl{(}\frac{1}{\mathscr{Z}}+2\mathrm{c}_{W}^{2}\biggr{)}\,.\end{split}$
(25)

We may also use the new gauge couplings as input parameters. To write the V-A
couplings as functions of $g_{z}$, we first observe that Eq. (13) implies

$\begin{split}\kappa&\simeq-\mathrm{s}_{Z}\quad\text{for}\quad\xi\to 0,\\\
\kappa&\simeq\mathrm{s}_{Z}\xi^{2}\quad\text{for}\quad\xi\to\infty.\end{split}$
(26)

Then, for a heavy $Z^{\prime}$ the axial couplings are $a_{Z^{\prime},f}\simeq
z_{N}\frac{2g_{z}}{g_{Z^{0}}}a_{f}^{(h)}$ and $v_{Z^{\prime},f}\simeq
z_{N}\frac{2g_{z}}{g_{Z^{0}}}v_{f}^{(h)}$. Using the definition of $\kappa$ in
Eq. (10), we obtain

$a_{\nu}^{(h)}=a_{u}^{(h)}=-\frac{1}{4}\,,\quad
a_{\ell}^{(h)}=a_{d}^{(h)}=+\frac{1}{4}\,,$ (27)

while from (23)

$\begin{split}v_{\nu}^{(h)}&=1-\frac{1}{2}\mathscr{Z}\,,\quad\>v_{\ell}^{(h)}=1-\frac{3}{2}\mathscr{Z}\,,\\\
v_{u}^{(h)}&=-\frac{1}{3}+\frac{5}{6}\mathscr{Z}\,,\;v_{d}^{(h)}=-\frac{1}{3}\
-\frac{1}{6}\mathscr{Z}\,.\end{split}$ (28)

A light $Z^{\prime}$ boson implies $a_{Z^{\prime},f}\simeq 0$ and
$v_{Z^{\prime},f}\simeq z_{N}\frac{2g_{z}}{g_{Z^{0}}}v_{f}^{(\ell)}$ where

$\begin{split}v_{\nu}^{(\ell)}&=1\,\,,\qquad\qquad\quad\;\;\;v_{\ell}^{(\ell)}=1-2\mathrm{c}_{W}^{2}\mathscr{Z}\,,\\\
v_{u}^{(\ell)}&=-\frac{1}{3}\
+\frac{4}{3}\mathrm{c}_{W}^{2}\mathscr{Z}\,,\;v_{d}^{(\ell)}=-\frac{1}{3}-\frac{2}{3}\mathrm{c}_{W}^{2}\mathscr{Z}\,.\end{split}$
(29)

##  III Direct Z-prime boson searches

Collider experiments such as LEP, Tevatron and the LHC performed direct
searches for a $Z^{\prime}$ boson. The nonobservation of such a particle can
be and was translated into exclusion bands for the parameters of certain
models predicting a $Z^{\prime}$ boson.

According to Ref. [29], the results of the LEPII experiment show that
$Z^{\prime}$ is either heavier than the largest center-of-mass energy
($209~{}\mathrm{GeV}$) or $|z_{\ell}g_{z}|\lesssim 10^{-3}$. Tevatron searched
for a $Z^{\prime}$ in the range
$200~{}\mathrm{GeV}<M_{Z^{\prime}}<800\mathrm{GeV}$ [31]. Finally, ATLAS [20]
and CMS [21] at the LHC performed the most recent searches up to
$M_{Z^{\prime}}<5500\mathrm{GeV}$. Below $M_{Z}$, the NA64 and BaBar
experiments together with FASER below the mass of the pion provide strong
bounds for a light $Z^{\prime}$ boson [22, 23]. In this study we focus on the
exclusion bounds obtained from ATLAS and CMS for a heavy, and NA64, BaBar and
FASER for a light $Z^{\prime}$.

In order to compare model predictions to experimental results at colliders one
has to compute the cross section for the process $pp\to
Z^{\prime}+X\to\ell^{+}\ell^{-}+X$, which is usually performed in the narrow
width approximation

$\begin{split}\sigma\bigl{(}pp&\to
Z^{\prime}+X\to\ell^{+}\ell^{-}+X\bigr{)}=\\\ &=\sigma\bigl{(}pp\to
Z^{\prime}X\bigr{)}\mathrm{Br}\bigl{(}Z^{\prime}\to\ell^{+}\ell^{-}\bigr{)},\end{split}$
(30)

assuming that the total width of the $Z^{\prime}$ boson $\Gamma_{Z^{\prime}}$
is much smaller than its mass,
$\gamma_{Z^{\prime}}=\Gamma_{Z^{\prime}}/M_{Z^{\prime}}\ll 1$. Eq. (30) is
usually presented as

$\begin{split}\sigma\bigl{(}pp&\to
Z^{\prime}+X\to\ell^{+}\ell^{-}+X\bigr{)}=\\\
&=\frac{\pi}{6s}\biggl{(}c_{U}w_{U}\bigl{(}s,M_{Z^{\prime}}\bigr{)}+c_{D}w_{D}\bigl{(}s,M_{Z^{\prime}}\bigr{)}\biggr{)},\end{split}$
(31)

where and $U\in\\{u,c,t\\}$, $D\in\\{d,s,b\\}$ and the coefficients $c_{q}$
collect model dependent contributions to the cross section

$c_{q}=\bigl{(}2\sqrt{2}G_{F}M_{Z}^{2}\rho\bigr{)}\mathrm{Br}\bigl{(}Z^{\prime}\to\ell^{+}\ell^{-}\bigr{)}\bigl{(}a_{Z^{\prime},q}^{2}+v_{Z^{\prime},q}^{2}\bigr{)}\,.$
(32)

The hadronic structure functions $w_{U/D}$ (cf. Ref. [31]) collect the QCD
corrections. For the production of a heavy neutral gauge boson they depend
only on the $M$ of the gauge boson and the center of mass energy squared $s$,

$\begin{split}w_{U/D}&=\sum_{q\in
U/D}\int_{0}^{1}\\!\mathrm{d}x_{1}\int_{0}^{1}\\!\mathrm{d}x_{2}\int_{0}^{1}\\!\mathrm{d}z~{}\delta\biggl{(}\frac{M^{2}}{s}-z~{}x_{1}x_{2}\biggr{)}\\\
&\times\biggl{[}f_{g}\bigl{(}x_{1},M\bigr{)}\biggl{(}f_{q}\bigl{(}x_{2},M\bigr{)}+f_{\overline{q}}\bigl{(}x_{2},M\bigr{)}\biggr{)}\Delta_{gq}\bigl{(}z,M^{2}\bigr{)}\\\
&+\biggl{(}f_{q}\bigl{(}x_{1},M\bigr{)}f_{\overline{q}}\bigl{(}x_{2},M\bigr{)}\biggr{)}\Delta_{qq}\bigl{(}z,M^{2}\bigr{)}+\big{(}x_{1}\leftrightarrow
x_{2}\bigr{)}\biggr{]},\end{split}$ (33)

where the functions $f_{i}(x,\mu_{F})$ are the parton distribution functions
inside the proton for parton $i$ at factorization scale $\mu_{F}$. We use the
NNPDF3.0 NLO PDF set in our numerical computations. At the next-to-leading
order (NLO) accuracy the coefficient functions $\Delta_{ab}$ for vector boson
production are known [32]. One also needs to compute the total decay width of
the $Z^{\prime}$ to obtain the cross section (30). We collect the coefficient
functions and the decay width formulae of the $Z^{\prime}$ boson, needed to
compute the cross section in Eq. (31) in App. C.

##  IV Numerical analysis

###  IV.1 Parameter scanning

The model predictions can be expressed as functions of the free parameters of
the theory. At the most fundamental level, these are the free $z$ charges, new
couplings and VEV ratio,

$z_{\phi}\,,\>z_{N}\,,\>g_{z}\,,\>g_{yz}\,,\>\tan\beta\,,$ (34)

which are not independent and a certain combination of them appears in the
model predictions. For instance, the tree level $\rho$ parameter estimates the
constraints from the electroweak precision observables and it depends only on

$\bigl{(}\mathrm{s}_{Z}\,,\>M_{Z^{\prime}}\bigr{)}\quad\text{or}\quad\bigl{(}z_{N}\,g_{z}\,,\>M_{Z^{\prime}}\,,\>\mathscr{Z}\bigr{)}\,.$
(35)

The NA64 experiment presents exclusion bounds for a dark photon in the
$(\epsilon,M_{A^{\prime}})$ plane. Those constraints can be translated to our
model parameters using the relation derived in App. D. This shows one that
exclusion bounds depend on either

$\bigl{(}\mathrm{s}_{Z}\,,\>M_{Z^{\prime}}\,,\>\mathscr{Z}\bigr{)}\quad\text{or}\quad\bigl{(}z_{N}\,g_{z}\,,\>M_{Z^{\prime}}\,,\>\mathscr{Z}\bigr{)},$
(36)

Presently, the most stringent bounds on the parameter space for heavy
$Z^{\prime}$ bosons can be obtained from direct searches using the Drell-Yan
pair production process

$p+p\to Z^{\prime}+X\to\ell^{+}+\ell^{-}+X,$ (37)

described in Sect. III. The corresponding cross section (31) can be rewritten
as

$\begin{split}\sigma&=\frac{4\pi^{2}}{3s}\frac{\Gamma_{Z^{\prime}}}{M_{Z^{\prime}}}\mathrm{Br}\bigl{(}Z^{\prime}\to\ell^{+}\ell^{-}\bigr{)}\\\
&\times\biggl{(}\mathrm{Br}\bigl{(}Z^{\prime}\to
U\overline{U}\bigr{)}w_{U}\bigl{(}s,M_{Z^{\prime}}\bigr{)}\\\
&\quad+\mathrm{Br}\bigl{(}Z^{\prime}\to
D\overline{D}\bigr{)}w_{D}\bigl{(}s,M_{Z^{\prime}}\bigr{)}\biggr{)},\end{split}$
(38)

where the branching fractions are listed in App. C, while $w_{U/D}$ are given
by Eq. (33). In this case as well, the predictions depend on the parameter set
(36) or equivalently on

$\bigl{(}\gamma_{Z^{\prime}},M_{Z^{\prime}},\mathscr{Z}\bigr{)}$ (39)

where $\gamma_{Z^{\prime}}=\Gamma_{Z^{\prime}}/M_{Z^{\prime}}$.

###  IV.2 Constraints on a light neutral gauge boson

Light vector-type particles, usually called dark photons $(A^{\prime})$, are
often considered as a portal to a secluded sector in particle physics or
downright as dark matter candidates. Presently, the most stringent, $90\%$ CL
exclusion bound in the dark photon mass range
$M_{A^{\prime}}\in(1~{}\mathrm{MeV},8~{}\mathrm{GeV})$ comes from the combined
results of the NA64 [22], BaBar [23] and more recently the FASER [24]
experiments. The dark photon model probed in these experiments have a single
vector type coupling $\epsilon e$ to the electromagnetic current. The
parameters $\bigl{(}\epsilon,M_{A^{\prime}}\bigr{)}$ can be matched to a
generic U(1)z as detailed in App. D. We scanned the parameter planes
$\bigl{(}\mathrm{s}_{Z},M_{Z^{\prime}}\bigr{)}$ and
$\bigl{(}|z_{N}\,g_{z}|,M_{Z^{\prime}}\bigr{)}$ for several benchmark values
of $\mathscr{Z}$. Using the value for the $\rho$ parameter given in Eq. (17)
we set upper bounds

$|\mathrm{s}_{Z}|\lesssim 4.5\cdot
10^{-3}\quad\text{and}\quad|z_{N}\,g_{z}|\lesssim\frac{1.7\cdot
10^{-3}}{|\mathscr{Z}|}$ (40)

for $M_{Z^{\prime}}\ll M_{Z}$.

The experimental bounds obtained from NA64, BaBar and FASER all depend on
$v_{\ell}^{\ell}$ given in Eq. (29). The former two experiments searched for
invisible decay products of dark photons, whereas the latter one searches for
decays $A^{\prime}\to e^{+}e^{-}$ which introduces further dependence on the
corresponding branching fractions. The mapping of $\epsilon$ onto
$\mathrm{s}_{Z}$ or $|z_{N}\,g_{z}|$ and $M_{Z^{\prime}}$ thus leads to a
dependence on $\mathscr{Z}$ in the exclusion bands. For instance, as
$\mathscr{Z}$ approaches $1/(2\mathrm{c}_{W}^{2})$, the reduced vector
coupling tends to zero $v_{\ell}^{\ell}\to 0$, which renders
$|\mathrm{s}_{Z}|$ and $|z_{N}\,g_{z}|$ unconstrained. The exclusion band
obtained from the FASER experiment is even more sensitive as the branching
fraction $\mathrm{Br}(Z^{\prime}\to e^{+}e^{-})$ also depends on
$v_{\ell}^{\ell}$.

Figure 1:  $90\%$ CL exclusion bounds for light $Z^{\prime}$ bosons obtained
from the NA64, BaBar and FASER experiments. The width of the band corresponds
to the uncertainty in the number of sterile neutrino families where the decay
$Z\to N+N$ is kinematically allowed. The region above the dashed line is
excluded due to the $\rho$ parameter and the area above the gray bands is also
excluded for a selected value of $\mathscr{Z}$.

Our findings for selected benchmark values of $\mathscr{Z}$ are summarized in
Fig. 1. The regions in the parameter planes above the dashed line and gray
bands are excluded at $90\%$ CL. The dashed lines correspond to the
experimental value of the $\rho$ parameter in Eq. (17), whereas the regions
above the gray bands correspond to the exclusions by direct searches at fixed
values of the effective charge ratio $\mathscr{Z}$. The width of the gray
bands is the uncertainty due to the number of right handed neutrinos lighter
than $M_{Z^{\prime}}/2$. A light $Z^{\prime}$ boson may always decay into the
three families of active neutrinos, but decays into right handed neutrinos may
be kinematically forbidden depending on the specific values of
$M_{Z^{\prime}}$.

###  IV.3 Constraints on a heavy neutral gauge boson

Direct searches for heavy $Z^{\prime}$ bosons were performed at the LEPII,
Tevatron and LHC as well and are of continued interest for future colliders
[17, 33]. We perform the scan in the parameter sets given in Eq. (36) using
the 95 $\%$ CL exclusion bands presented by the ATLAS [20] and CMS [21]
experiments. Our findings are summarized in Figs. 2. The exclusion limit by
the $\rho$ parameter is represented again with a dashed line: the region above
it is excluded. Analytically this correspond to

$|\mathrm{s}_{Z}|\lesssim
0.0025\biggl{[}\frac{1~{}\mathrm{TeV}}{M_{Z^{\prime}}}\biggr{]}\quad\text{and}\quad|g_{z}\,z_{N}|\lesssim\frac{0.11}{\mathscr{Z}}\biggl{[}\frac{M_{Z^{\prime}}}{1~{}\mathrm{TeV}}\biggr{]}.$
(41)

Figure 2:  $95\%$ CL exclusion bounds for heavy $Z^{\prime}$ bosons obtained
from the CMS and ATLAS experiments at the LHC for fixed ratios
$\gamma_{Z^{\prime}}$. The region above the dashed line is excluded due to the
$\rho$ parameter and the area above the gray line is also excluded for a
select value of $\mathscr{Z}$.

The collider searches by ATLAS and CMS are performed for fixed values of the
ratio $\gamma_{Z^{\prime}}$. We chose the datasets corresponding to the
largest $(\gamma_{Z^{\prime}}=10\%)$ and smallest
$(\gamma_{Z^{\prime}}=0.5\%)$ presented values. It is possible that the cross
section (38) is large enough so that the process is excluded experimentally
for given values of the input parameters (36), but the corresponding ratio
$\gamma_{Z^{\prime}}$ is larger than that searched for in the experiment in a
region whose lower boundary is denoted by a solid curve in Figs. 2. The
hatched region corresponds to this exact case, i.e. where no strict exclusion
applies. The region shaded in red in the parameter plane presented in Figs. 2
is excluded at 95 % CL. The branching fractions and the ratio
$\gamma_{Z^{\prime}}$ in Eq. (38) depend on the charge ratio $\mathscr{Z}$,
hence do so the exclusion bounds. We find that there exist a value of
$\mathscr{Z}$ both for $\mathrm{s}_{Z}$ and $z_{N}\,g_{z}$ which corresponds
to a loosest, i.e. the most conservative bound on these parameters. For the
mixing $\mathrm{s}_{Z}$ this value is $\mathscr{Z}\simeq 0.6$. Any other fixed
$\mathscr{Z}$ value presents a more severe bound than that shown in Fig. 2. It
is interesting that the cross section in Eq. (38) diverges as
$|\mathscr{Z}|\to 0$ (vanishing $z_{N}$ charge), which means that only zero
mixing $\mathrm{s}_{Z}=0$ is allowed for small $\mathscr{Z}$. Conversely, Eq.
(38) saturates at a finite value for $|\mathscr{Z}|\to\infty$, the
corresponding exclusion bands are shown in the left hand side plots of Fig. 2.

As for $z_{N}\,g_{z}$ the most conservative bound corresponds to
$\mathscr{Z}\simeq 0.54$ as can be seen in the plots on the right hand side of
Fig. 2. As opposed to the exclusion bound on $\mathrm{s}_{Z}$, in this case
the cross section is finite at $\mathscr{Z}=0$, hence one has a well defined
exclusion bound on $z_{N}\,g_{z}$ for $\mathscr{Z}=0$.

###  IV.4 Constraints on the parameter space of specific U(1) extensions

We showcase the exclusion bounds on two specific U(1)z extensions, one with a
light $Z^{\prime}$ boson and one with a heavy $Z^{\prime}$ boson. We consider
here the uncertainty due to the RGE running of $\eta$ and we also use the mass
of the $W$ boson as a constraint.

The tree level $\rho$ parameter discussed in Sect. II.3 is a useful quantity
to gauge the exclusion from electroweak precision observables in a model
independent manner. The effect of one-loop BSM corrections might become
important for a given region in the parameter space and thus the use of a
precise prediction is warranted. The drawback is that the radiative BSM
corrections are in general complicated functions of the free parameters and
the $z$ charges. Once the $z$ charges are set and $\eta$ is considered as an
uncertainty, there are two free parameters from the gauge sector
($M_{Z^{\prime}}$ and either $\mathrm{s}_{Z}$ or $g_{z}$) and two from the
scalar sector ($M_{S}$ and $\mathrm{s}_{S}$). Using these four parameters we
compute the complete one-loop corrections to $M_{W}$ in U(1)z extensions
presented in Ref. [34] based on the computational method of Ref. [35], consult
also Ref. [36] for the renormalization of $\mathrm{s}_{Z}$. Our input
parameters are

$M_{W}^{\text{SM}}=80.353~{}\mathrm{GeV},\quad
M_{W}^{\text{exp.}}=80.377~{}\mathrm{GeV},$ (42)

with a combined experimental and theoretical uncertainty of
$\sigma=15\,\mathrm{MeV}$.

The BSM corrections can either amount to a positive or negative contribution
to $M_{W}$. A heavy $Z^{\prime}$ boson and a light $S$ scalar ($M_{S}<M_{H}$)
increase, while a light $Z^{\prime}$ boson and a heavy $S$ scalar
($M_{S}>M_{H}$) decrease the predicted value of $M_{W}$. In this work we focus
on the effect of the gauge sector and thus we present exclusion bounds
obtained from $M_{W}$ at $\mathrm{s}_{S}=0$, i.e. when the extended scalar
sector does not affect the mass of the $W$ boson.

Our case study for a U(1)z extension with a light $Z^{\prime}$ boson is the
SWSM (recall that $z_{N}=1/2$ and $z_{\phi}=1$). This model can explain the
observed dark matter abundance in the Universe with freeze-out scenario if
$10\,\mathrm{MeV}\lesssim M_{Z^{\prime}}\lesssim m_{\pi}\ll M_{Z}$ and the
dark matter candidate is the lightest sterile neutrino, which is considered to
be lighter than $M_{Z^{\prime}}/2$, while the other sterile neutrinos are much
heavier [28].

Figure 3:  Exclusion bounds for models with a light $M_{Z^{\prime}}$ and
$z_{N}=1/2$ and $z_{\phi}=1$. The red region is excluded at $90~{}\%$ CL. The
green region is the preferred parameter space of the SWSM. The width of the
lines take into account the uncertainty in the $\eta$ parameter. The gray line
corresponds to the NA64, BaBar and FASER experiments whereas the dashed ones
correspond to the bounds from $M_{W}$ and $\rho$.

Our findings are summarized in Fig. 3. The region in red is excluded at
$90~{}\%$ CL. The gray band is the lower boundary of the exclusion region from
the NA64, BaBar and FASER experiments and the width of the gray band
corresponds to the combined uncertainty from decays of the $Z^{\prime}$ boson
and the running of $\eta$. Solving the RGEs of App. A, we find that the
largest possible value of $\eta$ is $0.4$, hence $\mathscr{Z}$ can take values
in the range $(1.6,2.0)$. The dashed lines correspond to the bound from
$M_{W}$ computed with $\mathrm{s}_{S}=0$ and to the bound from the tree level
$\rho$ parameter as a reference. The scalar sector has the potential to
significantly affect the bound obtained from $M_{W}$. In fact, for a heavy
scalar ($M_{S}\gg M_{h}$) and a light $Z^{\prime}$ boson one may write the BSM
correction $\delta M_{W}^{\text{BSM}}$ to the mass of the $W$ boson as

$\delta
M_{W}^{\text{BSM}}\simeq-\left[5.6\big{(}100\mathrm{s}_{Z}\,\big{)}^{2}+1.5\big{(}10\,\mathrm{s}_{S}\big{)}^{2}\biggl{(}1+0.57\log\biggl{(}\frac{M_{S}}{1~{}\mathrm{TeV}}\biggr{)}\biggr{)}\right]\mathrm{MeV},$
(43)

which is independent of the $z$ charges. For instance, a scalar with a mass
$M_{S}\simeq 1~{}\mathrm{TeV}$ and a mixing of $\mathrm{s}_{S}\simeq 0.2$
would increase the difference $|M_{W}-M_{W}^{\text{exp.}}|$ above $2\sigma$,
excluding any nonzero value of $\mathrm{s}_{Z}$.

Figure 4:  Exclusion bounds for the $B-L$ extension of the SM where one has
$z_{N}=1/2$ and $z_{\phi}=0$. The red region is excluded at $95~{}\%$ CL. The
green region is the preferred parameter space of the $B-L$ model. The width of
the lines take into account the uncertainty in the $\eta$ parameter. The gray
line corresponds to the CMS and ATLAS direct searches at
$\gamma_{Z^{\prime}}=0.1$ and at $0.005$. The dashed line in the top left
corner corresponds to the exclusion from $M_{W}$ at $\mathrm{s}_{S}=0$.

As for a U(1)z extension with a heavy $Z^{\prime}$ boson, $M_{Z}\ll
M_{Z^{\prime}}$, we choose to investigate the $B-L$ extension of the SM, which
has $z_{N}=1/2$ and $z_{\phi}=0$. It is interesting to note that in this case
$\mathscr{Z}=0$ at the default scale $\mu_{0}$, and the uncertainty from the
RG running of $\eta$ is also essentially negligible, at most about 0.1. Hence,
there is effectively no mixing between the $Z$ and $Z^{\prime}$ bosons,
$\mathrm{s}_{Z}\simeq 0$, and consequently, there is no bound from the tree
level $\rho$ parameter. Our findings are summarized in Fig. 4. The region in
red is excluded at $95~{}\%$ CL. The dashed line correspond to the exclusion
from $M_{W}$ at $\mathrm{s}_{S}=0$. Since the tree level $\rho$ parameter
equals one, this corresponds purely to one-loop BSM corrections to $M_{W}$.
The hatched region is not excluded by $M_{W}$ and the width to mass ratio
$\gamma_{Z^{\prime}}$ of the $Z^{\prime}$ boson is larger than the one
considered experimentally. The green region displays the presently allowed
parameter space of the $B-L$ model.

###  IV.5 Projections for future pp collider experiments

The High Energy LHC experiment is planned to operate at
$\sqrt{s}=27\,\mathrm{TeV}$ center of mass energy, while the Future Circular
Collider will collide particles at $\sqrt{s}=100\,\mathrm{TeV}$. The
experimental programs of both machines include direct searches for
$Z^{\prime}$ bosons. The cross section (38) for the process $p+p\to
Z^{\prime}+X\to\ell^{+}+\ell^{-}+X$, which is the main search channel for
$Z^{\prime}$ bosons, is shown on Fig. 5 for relevant values of $\sqrt{s}$. The
cross sections are inside of the gray band on Fig. 5 for any value of
$\mathscr{Z}$ for $\gamma_{Z^{\prime}}=0.1$. Note that the gray band for a
different value of $\gamma_{Z^{\prime}}$ can be obtained by the linear
rescaling of those on Fig. 5.

Figure 5:  Production cross sections $\sigma\bigl{(}pp\to Z^{\prime}\bigr{)}$
times the leptonic branching fraction
$\text{Br}\bigl{(}Z^{\prime}\to\ell^{+}+\ell^{-}\bigr{)}$ as the function of
$M_{Z}^{\prime}$ for center of mass energies $\sqrt{s}=27~{}\mathrm{TeV}$
(left, for the HE-LHC) and $100~{}\mathrm{TeV}$ (right, for FCC-pp) and fixed
ratio $\gamma_{Z^{\prime}}=0.1$. The cross section for any value of
$\mathscr{Z}$ is inside of the gray band.

Detector simulations are already available both for the HE-LHC and the FCC-hh.
We compute projected 95 $\%$ CL exclusion bands both for $|\mathrm{s}_{Z}|$
and $|g_{z}\,z_{N}|$ using the simulations for the HE-LHC at $15$ ab-1
integrated luminosity [17] and for the FCC-hh at $30$ ab-1 [37]. Our
predictions are shown in Fig. 6. The width of the gray bands correspond to the
2 $\sigma$ uncertainty of the simulation in the location of the exclusion
band.

It is noteworthy that for large $\bigl{(}\gtrsim 10~{}\mathrm{TeV}\bigr{)}$
masses of the $Z^{\prime}$ boson the cross section

$\sigma\bigl{(}p+p\to Z^{\prime}+X\to Z+W^{+}+W^{-}+X\bigr{)}$ (44)

may become comparable to or larger than the Drell-Yan pair production cross
section in Eq. (38) as the ratio of the two cross sections is

$\begin{split}\frac{\mathrm{Br}\bigl{(}Z^{\prime}\to
ZW^{+}W^{-}\bigr{)}}{\mathrm{Br}\bigl{(}Z^{\prime}\to\ell^{+}\ell^{-}\bigr{)}}&=\frac{\mathscr{Z}^{2}}{2-6\mathscr{Z}+5\mathscr{Z}^{2}}\biggl{(}C_{ff}\frac{7\mathrm{c}_{W}^{4}}{160\pi}\biggr{)}\frac{M_{Z^{\prime}}^{2}}{M_{Z}^{2}}\\\
&\simeq
0.368~{}\frac{\mathscr{Z}^{2}}{2-6\mathscr{Z}+5\mathscr{Z}^{2}}\biggl{[}\frac{M_{Z^{\prime}}}{10~{}\mathrm{TeV}}\biggr{]}^{2}.\end{split}$
(45)

Figure 6:  Projected exclusion bounds on $|\mathrm{s}_{Z}|$ at
$\mathscr{Z}=0.6$ and on $|z_{N}\,g_{z}|$ at $\mathscr{Z}=0.54$ for the HE-LHC
and FCC-pp experiments using the simulated exclusion bands obtained in Refs.
[17] and [37]. The dark gray line is the expected median exclusion limit and
the width of the gray bands correspond to the $95\%$ CL expected limit. The
dashed line represents the exclusion by the $\rho$ parameter and the red
shaded area is already excluded at $95\%$ CL.

##  V Conclusions

In this paper we studied the parameter space of U(1)z extensions of the
standard model with an additional complex scalar field and three families of
right handed neutrinos with generation independent $z$ charges and no exotic
fermions. Anomaly cancellation constrains the $z$ charges such that two $z$
charges remain arbitrary. The vector - axial vector couplings, which are
critical in the analysis presented in this work, depend on a special
combination $\mathscr{Z}$ of $z_{\phi}$ the $z$ charge of the BEH field and
$z_{N}$ as given in Eq. (22). We presented our predictions using
$\mathscr{Z}$, the mass $M_{Z^{\prime}}$ of the $Z^{\prime}$ boson and either
the $\mathrm{s}_{Z}$ mixing between $Z$ and $Z^{\prime}$, or $z_{N}\,g_{z}$
the right handed neutrino $z$ charge times the new gauge coupling. Our
exclusion bounds on these parameters used the results of the NA64, BaBar,
FASER experiments for a light and those of the ATLAS and CMS experiments for a
heavy $Z^{\prime}$ boson as constraints from direct searches. We also studied
the limits obtained from the measured values of the $\rho$ parameter and the
mass of the $W$ boson as constraints from indirect sources.

For a light $Z^{\prime}$ boson the $\rho$ parameter provides a bound on
$|\mathrm{s}_{Z}|$, independent of $\mathscr{Z}$, and a bound on
$|z_{N}\,g_{z}|$, proportional to $1/\mathscr{Z}$ as given in Eq. (40). The
bound on $|\mathrm{s}_{Z}|$ obtained from the NA64, BaBar and FASER
experiments is in general more severe, and it is proportional to
$1/\mathscr{Z}$. The constraint on $|z_{N}\,g_{z}|$ from such direct searches
depends weekly on $\mathscr{Z}$ unless $\mathscr{Z}$ is fine tuned to
$\mathscr{Z}=1/\bigl{(}2\mathrm{c}_{W}^{2}\bigr{)}$, (see Eq. (70)).

In the case of a heavy $Z^{\prime}$ boson, the value $\mathscr{Z}\simeq 0.6$
corresponds to the loosest bound on $|\mathrm{s}_{Z}|$ (or $\simeq 0.54$ on
$|z_{N}\,g_{z}|$) obtained from the ATLAS and CMS experiments, which means
that one has a model independent way to constrain the parameter space of U(1)z
extensions with different charge assignments. In this region, the $\rho$
parameter excludes a decent portion of the parameter space, but it is never as
severe as the exclusion bound obtained from direct searches.

We also used detector simulations for the HE-LHC and FCC experiments to
provide projected exclusion bounds on the parameter space. We found that the
minimum of the excluded $|\mathrm{s}_{Z}|$ values will improve by one order of
magnitude in HE-LHC compared to LHC and an additional one order of magnitude
in FCC-hh compared to HE-LHC. As for $|g_{z}\,z_{N}|$ we find no such
improvement. The exclusion bound obtained from the LHC and the projected ones
are based on leptonic final states. We find that for very large values of
$M_{Z^{\prime}}$ $(>10~{}\mathrm{TeV})$ the decay $Z^{\prime}\to
Z+W^{+}+W^{-}$ might become dominant over $Z^{\prime}\to\ell^{+}+\ell^{-}$
depending on $\mathscr{Z}$. This means that we propose direct searches at FCC
for the final state $Z+W^{+}+W^{-}$ as it is well motivated experimentally.

### Acknowledgment

We appreciate collaboration with Josú Hernández-García at early stages of this
project. We are grateful to members of the PPPheno Group at ELTE for useful
discussions and especially to K. Seller for his useful contributions. This
research was supported by the Excellence Programme of the Hungarian Ministry
of Culture and Innovation under contract TKP2021-NKTA-64 and by the Hungarian
Scientific Research Fund PD-146527.

##  Appendix A Running eta parameter

We defined the ratio of the mixing coupling $g_{yz}$ and the new gauge
coupling $g_{z}$ as

$\eta=\frac{g_{yz}}{g_{z}}\,.$ (46)

It always appears as a correction to the $z$ charge of the BEH field as in Eq.
(11). Like any other coupling, $\eta$ depends on the renormalization scale
$\mu$, as described by the renormalization group equations (RGEs)

$\begin{split}\dot{g}_{y}&=\frac{g_{y}}{16\pi^{2}}\biggl{(}\frac{41}{6}g_{y}^{2}+\frac{5}{3}g_{z}^{2}\,\eta\,b_{\eta}\biggr{)},\\\
\dot{g}_{z}&=\frac{g_{z}^{3}}{16\pi^{2}}\biggl{(}\frac{5}{9}+\frac{4000}{369}z_{N}^{2}+\frac{10}{41}b_{\eta}^{2}\biggr{)},\\\
\dot{\eta}&=\frac{g_{y}^{2}}{16\pi^{2}}b_{\eta}\end{split}$ (47)

where

$b_{\eta}=\frac{16}{3}z_{N}-\frac{41}{3}\biggl{(}z_{\phi}-\frac{\eta}{2}\biggr{)}=\frac{z_{N}}{3}\bigl{(}16-41\mathscr{Z}\bigr{)}$
(48)

is a linear function of $\eta$. The derivative $\dot{f}=\partial f/\partial t$
is meant with respect to $t=\ln\bigl{(}\mu/\mu_{0}\bigr{)}$. We chose the $z$
charge assignment according to Tab. 1 at an arbitrarily chosen fixed scale
$\mu_{0}$ where $\eta(\mu_{0})=0$. Then one can investigate the uncertainty
due to the choice of the unknown scale $\mu_{0}$.

The scale $\mu_{0}$ can be chosen arbitrarily. In our study the most
reasonable choices are either $\mu_{0}=m_{\rm t}$ or $\mu_{0}=M_{Z^{\prime}}$.
The values of $M_{Z^{\prime}}$ considered here are at maximum a few tens of
$\mathrm{TeV}$, then the running of $\eta$ from $\mu_{0}=M_{Z^{\prime}}$ down
to the electroweak scale does not affect the value of $\mathscr{Z}$ in any way
relevant for the phenomenology considered in this work. One may set $\mu_{0}$
as high as ${M}_{\rm Pl}$, in which case $\eta({m_{\rm t}})=\rm{O}(1)$, with
exact value depending on $z_{N}$ and $z_{\phi}$. For instance, in the SWSM,
choosing $\eta({M}_{\rm Pl})=0$ at the Planck-scale, the renormalization group
running with GUT normalization implies that $\eta\simeq 0.67$ at the
electroweak scale [28]. Here we choose $\mu_{0}=m_{\rm{t}}$. Then one initial
condition, $g_{y}(m_{\rm{t}})\simeq 0.36$ is known, while the initial
conditions for $g_{z}$, $\eta$ as well as the $z$ charges $z_{N}$ and
$z_{\phi}$ are free parameters. One can show that the coupling $g_{z}(\mu)$
has a Landau pole below the Planck scale ${M}_{\rm Pl}$ if $g_{z}(m_{\rm t})$
is larger than a critical value. Assuming a constant $\eta$, this value is

$\alpha_{z}(m_{\rm t})=\frac{g_{z}(m_{\rm
t})^{2}}{4\pi}\gtrsim\frac{11.95}{41+800z_{N}^{2}+18b_{\eta}^{2}}\,.$ (49)

Taking the running of $\eta$ into account, this formula is not exact and the
actual upper bound on $g_{z}(m_{\rm t})$ to avoid the Landau pole below the
Planck scale is about $15\%$ lower. The loosest constraint obtained from
avoiding the Landau pole corresponds to $z_{N}=z_{\phi}=0$ with $\eta(m_{\rm
t})=0$. Then one has the upper bound $g_{z}(m_{\rm t})\lesssim 1.91$. Any
different $z$ charge assignment results in a considerably more severe upper
limit. For instance, in the SWSM it is $g_{z}(m_{\rm t})\lesssim 0.22$. Then
the initial conditions $\eta({M}_{\rm Pl})=0$ and $g_{z}(m_{\rm
t})\in[0,0.22]$ in the one-loop RGEs of Eq. (47) yield $\eta$ at the
electroweak scale in the range $\eta(m_{\rm t})\in[0.4,0.375]$ (the larger
$g_{z}(m_{\rm t})$, the smaller $\eta(m_{\rm t})$).

##  Appendix B Chrial couplings of the Z and $Z^{\prime}$ bosons

We list here the chiral couplings of the $Z^{\prime}$ bosons to fermions in
terms of the neutral mixing angle and effective couplings $\kappa$ and $\tau$
in Table 3. The chiral couplings to the $Z$ boson can be obtained by the
replacement
$(\mathrm{c}_{Z},\>\mathrm{s}_{Z})\to(\mathrm{s}_{Z},\>-\mathrm{c}_{Z})$.

$f$ | $C^{R}_{Z^{\prime},f\overline{f}}$ | $C^{L}_{Z^{\prime},f\overline{f}}$
---|---|---
$\nu$ | $\frac{\tau}{\tan\beta}z_{N}\mathrm{c}_{Z}$ | $-\mathrm{s}_{Z}+\bigl{(}-\kappa+\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$
$\ell$ | $-2\mathrm{s}_{W}^{2}\mathrm{s}_{Z}+\bigl{(}-2\kappa+\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$ | $\bigl{(}1-2\mathrm{s}_{W}^{2}\bigr{)}\mathrm{s}_{Z}+\bigl{(}-\kappa+\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$
$u$ | $\frac{4}{3}\mathrm{s}_{W}^{2}\mathrm{s}_{Z}+\bigl{(}\frac{4}{3}\kappa-\frac{1}{3}\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$ | $\bigl{(}-1+\frac{4}{3}\mathrm{s}_{W}^{2}\bigr{)}\mathrm{s}_{Z}+\frac{1}{3}\bigl{(}\kappa-\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$
$d$ | $-\frac{2}{3}\mathrm{s}_{W}^{2}\mathrm{s}_{Z}-\bigl{(}\frac{2}{3}\kappa+\frac{1}{3}\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$ | $\bigl{(}1-\frac{2}{3}\mathrm{s}_{W}^{2}\bigr{)}\mathrm{s}_{Z}+\frac{1}{3}\bigl{(}\kappa-\frac{\tau}{\tan\beta}z_{N}\bigr{)}\mathrm{c}_{Z}$
Table 3: The chiral couplings of the $Z^{\prime}$ boson to fermions in U(1)z
extensions of the SM.

We recall here the chiral couplings of the neutrinos, for a detailed
discussion see Ref. [38]. As the neutral currents are written in terms of
flavor eigenstates, the interactions between the neutral gauge bosons and the
propagating mass eigenstate neutrinos include also the neutrino mixing
matrices:

${\mbox{\boldmath$\Gamma$}}^{\mu}_{V\nu_{i}\nu_{j}}=-\mathrm{i}e\gamma^{\mu}\Big{(}{\mbox{\boldmath$\Gamma$}}^{L}_{V\nu\nu}P_{L}+{\mbox{\boldmath$\Gamma$}}^{R}_{V\nu\nu}P_{R}\Big{)}_{ij}$
(50)

where

${\mbox{\boldmath$\Gamma$}}^{L}_{V\nu\nu}=C^{L}_{V\nu\nu}\textbf{U}_{L}^{\dagger}\textbf{U}_{L}-C^{R}_{V\nu\nu}\textbf{U}_{R}^{T}\textbf{U}_{R}^{*}$
(51)

and

${\mbox{\boldmath$\Gamma$}}^{R}_{V\nu\nu}=-C^{L}_{V\nu\nu}\textbf{U}_{L}^{T}\textbf{U}^{*}_{L}+C^{R}_{V\nu\nu}\textbf{U}_{R}^{\dagger}\textbf{U}_{R}=-\Big{(}{\mbox{\boldmath$\Gamma$}}^{L}_{V\nu\nu}\Big{)}^{*}$
(52)

for both $V=Z$ and $V=Z^{\prime}$. In order to recover the SM vector and axial
vector couplings of the $Z$ boson and the neutrinos, the right handed mixing
matrices have to vanish and

$\textbf{U}_{L}^{\dagger}\textbf{U}_{L}\to\mathbf{1}_{3}\quad\text{and}\quad\textbf{U}_{L}^{T}\textbf{U}^{*}_{L}\to
0.$ (53)

If one estimates the chiral couplings of the $Z$ and $Z^{\prime}$ bosons in
the presence of sterile neutrinos but with the mixing neglected then one needs
to use the following replacements:

$\begin{split}\textbf{U}_{L}^{\dagger}\textbf{U}_{L}\to\mathbf{1}_{3}\quad\text{and}&\quad\textbf{U}_{L}^{T}\textbf{U}^{*}_{L}\to
0,\\\
\textbf{U}_{R}^{\dagger}\textbf{U}_{R}\to\mathbf{1}_{3}\quad\text{and}&\quad\textbf{U}_{R}^{T}\textbf{U}^{*}_{R}\to
0,\end{split}$ (54)

which we adopted throughout.

##  Appendix C Coefficient functions for hadroproduction and decays of the
Z-prime boson

The theoretical input needed to compute the Drell-Yan pair production cross
section in Eq. (30) is the coefficient functions and branching ratios, which
we present here explicitly. The coefficient functions for the production of a
neutral gauge boson at NLO accuracy in QCD read [32]

$\begin{split}\Delta_{qq}\bigl{(}z,\mu_{R}^{2}\bigr{)}&=\delta(1-z)+\frac{\alpha_{s}(\mu_{R}^{2})}{2\pi}\\\
&\times
C_{F}\biggl{[}\delta(1-z)\biggl{(}\frac{2\pi^{2}}{3}-8\biggr{)}+4\bigl{(}1+z^{2}\bigr{)}\biggl{(}\frac{\ln(1-z)}{1-z}\biggr{)}_{+}-2\frac{1+z^{2}}{1-z}\ln(z)\biggr{]},\\\
\Delta_{gq}\bigl{(}z,\mu_{R}^{2}\bigr{)}&=\frac{\alpha_{s}(\mu_{R}^{2})}{2\pi}\,T_{R}\biggl{[}(1-2z+2z^{2})\ln\biggl{(}\frac{(1-z)^{2}}{z}\biggr{)}+\frac{1}{2}+3z-\frac{7}{2}z^{2}\biggr{]},\end{split}$
(55)

with color factors $C_{F}=4/3$ and $T_{R}=1/2$.

The model considered here is defined in Sect. II. The SM particle spectrum is
extended with the $Z^{\prime}$ boson, a new scalar $s$ and three right handed
neutrinos $N_{i},~{}~{}i=1,2,3$. The mass of the new scalar $M_{s}$ has to be
larger than half the mass of the Higgs boson $M_{h}$, otherwise the decay
width of the Higgs particle becomes too large as compared to the experimental
upper limit $3.2^{+2.8}_{-2.2}\,\mathrm{MeV}$ [39]. This means, that a light
$Z^{\prime}$ $\bigl{(}M_{Z^{\prime}}<M_{Z}\bigr{)}$ can only decay into
fermion pairs:

$\Gamma(Z^{\prime}\to
f+\overline{f})=N_{C}\,\rho\,C_{ff}\,M_{Z^{\prime}}~{}\bigl{(}v_{Z^{\prime},f}^{2}+a_{Z^{\prime},f}^{2}\bigr{)}$
(56)

where $\rho$ is defined in Eq. (16),
$C_{ff}=\frac{G_{F}M_{Z}^{2}}{6\sqrt{2}\pi}\simeq 3.6383\cdot 10^{-3}$, and
the vector and axial vector couplings are given in Sect. II.4. Eq. (56) is
valid for Dirac fermions. The formula for Majorana neutrinos can be obtained
by the replacement $C_{ff}\to\frac{1}{2}C_{ff}$. The invisible branching
fraction of a light $Z^{\prime}$ boson is

$\mathrm{Br}\bigl{(}Z^{\prime}\to\text{inv.}\bigr{)}=\frac{3n_{N}}{3n_{N}+3(1-2\mathrm{c}_{W}^{2}\mathscr{Z})^{2}\,n_{\ell}+(1+2\mathrm{c}_{W}^{2}\mathscr{Z})^{2}\,n_{d}+(1-4\mathrm{c}_{W}^{2}\mathscr{Z})^{2}\,n_{u}},$
(57)

where $n_{f}$ counts the kinematically allowed decays into $n_{f}$ families of
fermion type $f$. The number $n_{N}$ counts the Dirac type neutrinos. The same
formula applies for Majorana neutrinos with the replacement $n_{N}\to
n_{N}/2$. The parameter $\mathscr{Z}$ is defined in Eq. (22) and
$\mathrm{c}_{W}=\cos\theta_{W}$.

The larger $M_{Z^{\prime}}$, the more decay channels are allowed. For the case
of a heavy $Z^{\prime}$, $M_{Z^{\prime}}\gg M_{h}$, we neglect the finite mass
effects of the particles $Z,W$ and $h$ in the following decay formulas.
However, we keep the full dependence on the unknown $M_{s}$ as it is a free
parameter of the model. The decays into a pair of charged $W$ bosons are [40]

$\begin{split}\Gamma\bigr{(}Z^{\prime}\to W^{+}+W^{-}\bigl{)}&=\varepsilon
M_{Z^{\prime}}\frac{\xi^{2}C_{ff}}{4\rho}\,,\\\ \Gamma\bigr{(}Z^{\prime}\to
Z+W^{+}+W^{-}\bigl{)}&=\varepsilon
M_{Z^{\prime}}\frac{7\xi^{4}C_{ff}^{2}}{320\pi}\mathrm{c}_{W}^{4}\,,\\\
\Gamma\bigr{(}Z^{\prime}\to\gamma+W^{+}+W^{-}\bigl{)}&=\varepsilon
M_{Z^{\prime}}\frac{301\xi^{2}C_{ff}^{2}}{800\pi}\mathrm{c}_{W}^{2}\mathrm{s}_{W}^{2}\,,\end{split}$
(58)

where $\varepsilon=(\xi\mathrm{s}_{Z})^{2}$. The decays into scalar particles
also include two- and three-body ones as

$\begin{split}\Gamma\bigr{(}Z^{\prime}\to Z+S\bigl{)}&=\varepsilon
M_{Z^{\prime}}\frac{\xi^{2}C_{ff}}{4\rho^{2}}\bigl{|}\Gamma_{Z,Z^{\prime},S_{i}}\bigr{|}^{2}\,\bigl{(}1-\zeta_{S}^{2}\bigr{)}^{3}\,,\\\
\Gamma\bigr{(}Z^{\prime}\to Z+S+S\bigl{)}&=\varepsilon
M_{Z^{\prime}}\frac{3\xi^{4}C_{ff}^{2}}{64\pi\rho^{2}}\bigl{|}\Gamma_{Z,Z^{\prime},S_{i},S_{i}}\bigr{|}^{2}\biggl{[}\sqrt{1-4\zeta_{S}^{2}}\biggl{(}1+\frac{26}{3}\zeta_{S}^{2}-\frac{62}{3}\zeta_{S}^{4}+20\zeta_{S}^{6}\biggr{)}\\\
&-4\zeta_{S}^{2}\bigl{(}1-3\zeta_{S}^{2}+6\zeta_{S}^{4}-5\zeta_{S}^{6}\bigr{)}\\\
&\qquad\times\ln\biggl{(}\frac{\bigl{(}1-2\zeta_{S}^{2}\bigr{)}\bigl{(}\sqrt{1-4\zeta_{S}^{2}}+1-2\zeta_{S}^{2}\bigr{)}}{2\zeta_{S}^{4}}-1\biggr{)}\biggr{]},\\\
\Gamma\bigr{(}Z^{\prime}\to Z+s+h\bigl{)}&=\varepsilon
M_{Z^{\prime}}\frac{3\xi^{4}C_{ff}^{2}}{32\pi\rho^{2}}\biggl{(}\mathrm{s}_{S}\mathrm{c}_{S}+\frac{\mathrm{s}_{S}\mathrm{c}_{S}}{\tan\beta^{2}}\biggr{)}^{2}\\\
&\quad\times\biggl{[}1+\frac{1}{3}\zeta_{S}^{2}\biggl{(}10+12\ln\bigl{(}\zeta_{S}^{2}\bigr{)}-18\zeta_{S}^{2}+6\zeta_{S}^{4}-\zeta_{S}^{6}\biggr{)}\biggr{]},\end{split}$
(59)

where $S=h,s$, $\zeta_{S}=M_{S}/M_{Z^{\prime}}$. For a heavy $Z^{\prime}$,
$M_{Z^{\prime}}\gtrsim 1$ TeV, we neglected
$\zeta_{h}\lesssim\mathcal{O}\big{(}10^{-1}\big{)}$. The triple and quartic
vector-scalar vertices are

$\begin{split}\Gamma_{Z,Z^{\prime},h}=\mathrm{c}_{S}+\mathrm{s}_{S}/\tan\beta,&\quad\Gamma_{Z,Z^{\prime},s}=\mathrm{s}_{S}-\mathrm{c}_{S}/\tan\beta,\\\
\Gamma_{Z,Z^{\prime},h,h}=\mathrm{c}_{S}^{2}-\big{(}\mathrm{s}_{S}/\tan\beta\big{)}^{2},&\quad\Gamma_{Z,Z^{\prime},s,s}=\mathrm{s}_{S}^{2}-\big{(}\mathrm{c}_{S}/\tan\beta\big{)}^{2}.\end{split}$
(60)

The largest contribution from the scalar sector to the $Z^{\prime}$ decay
width is obtained by setting $\zeta_{S}=0$. In that case, we find the sum of
partial decay widths with a scalar in the final state independent of the
scalar mixing angle as

$\sum\Gamma\bigl{(}Z^{\prime}\to Z+\text{ scalar}\bigr{)}\lesssim\varepsilon
M_{Z^{\prime}}\frac{1}{4\rho^{2}}\biggl{[}\xi^{2}C_{ff}\bigl{(}1+\tan^{-2}\beta\bigr{)}+\frac{3}{16\pi}\xi^{4}C_{ff}^{2}\bigl{(}1+\tan^{-4}\beta\bigr{)}\biggr{]}.$
(61)

We use this upper limit (61) in our numerical calculation to take into account
the effect of the scalar sector in the total decay width of $Z^{\prime}$.
Consulting Eq. (16) one can recognize that $\varepsilon\simeq\rho-1$ is a
small parameter for a heavy $Z^{\prime}$.

We computed the fermionic branching fractions of a heavy $Z^{\prime}$ boson
using the V-A couplings obtained in Sect. II.4, and obtained

$\begin{split}\mathrm{Br}\bigl{(}Z^{\prime}\to\ell^{+}\ell^{-}\bigr{)}&=\frac{2-6\mathscr{Z}+5\mathscr{Z}^{2}}{16-32\mathscr{Z}+\mathscr{Z}^{2}\bigl{(}41+C_{w,s}\xi^{2}\bigr{)}},\\\
\mathrm{Br}\bigl{(}Z^{\prime}\to
U\overline{U}\bigr{)}&=\frac{2-10\mathscr{Z}+17\mathscr{Z}^{2}}{48-96\mathscr{Z}+3\mathscr{Z}^{2}\bigl{(}41+C_{w,s}\xi^{2}\bigr{)}},\\\
\mathrm{Br}\bigl{(}Z^{\prime}\to
D\overline{D}\bigr{)}&=\frac{2+2\mathscr{Z}+5\mathscr{Z}^{2}}{48-96\mathscr{Z}+3\mathscr{Z}^{2}\bigl{(}41+C_{w,s}\xi^{2}\bigr{)}},\end{split}$
(62)

where the coupling constant is

$C_{w,s}=C_{ff}\frac{15+7\mathrm{c}_{W}^{4}}{160\pi}\simeq 1.4\times
10^{-4}\,.$ (63)

We note that for $M_{Z^{\prime}}>14~{}\mathrm{TeV}$, the decays into scalars
and $W$ bosons start to dominate and the fermionic branching fractions may
decrease significantly depending on the charge assignment encoded in
$\mathscr{Z}$.

##  Appendix D Kinetic mixing for U(1) extensions

Low energy experiments, such as NA64, BaBar, FASER place a stringent
constraints on models which are extended by a $U(1)$ gauge group introducing a
new gauge boson coupled to the SM fermions, which can be interpreted as a dark
photon $A^{\prime}$ that has kinetic mixing with the known photon. In the dark
photon model the interaction term involving the $A^{\prime}$ coupled to the
electromagnetic current $J^{\mu}_{\rm EM}$ can be written as

$\mathcal{L}_{\rm int}=-\epsilon eA^{\prime}_{\mu}J^{\mu}_{\rm EM}$ (64)

where $\epsilon$ can be viewed as the kinetic mixing parameter. The
experimental exclusion bounds are placed on the parameter plane
$\bigl{(}\epsilon,M_{A^{\prime}}\bigr{)}$. In order to extend those
constraints to the parameters of a more general U(1) extension, one may set
$M_{A^{\prime}}\equiv M_{Z^{\prime}}$ but relating $\epsilon$ to the free
parameters discussed Sect. IV.1. involves some subtlety, discussed below.

The NA64 and BaBar experiments search for dark photon brehmsstrahlung in the
invisible decay channel of the $A^{\prime}$ [22]. Following ref. [28] (see
also [16]) we have

$\frac{\sigma\bigl{(}e^{-}+Z\longrightarrow
e^{-}+Z+A^{\prime}\bigr{)}}{\sigma\bigl{(}e^{-}+Z\longrightarrow
e^{-}+Z+Z^{\prime}\bigr{)}}=\frac{\mathrm{Br}(Z^{\prime}\to\mathrm{inv.})}{\mathrm{Br}(A^{\prime}\to\mathrm{inv.})},$
(65)

where $\mathrm{Br}(Z^{\prime}\to\mathrm{inv.})$ is given in Eq. (57) and and
$\mathrm{Br}(A^{\prime}\to\mathrm{inv.})=1$ in the NA64 experiment. Computing
the cross sections of the $A^{\prime}$ and $Z^{\prime}$ bremsstrahlung
processes yield

$\begin{split}&\frac{\sigma\bigl{(}e+Z\longrightarrow
e+Z+A^{\prime}\bigr{)}}{\sigma\bigl{(}e+Z\longrightarrow
e+Z+Z^{\prime}\bigr{)}}=\epsilon^{2}\,\frac{\bigl{(}4\mathrm{s}_{W}^{2}\mathrm{c}_{W}^{2}\bigr{)}}{v_{Z^{\prime},\ell}^{2}+a_{Z^{\prime},\ell}^{2}\bigl{(}1+f(M_{Z^{\prime}})\bigr{)}}+\mathcal{O}\biggl{(}\frac{m_{e}^{2}}{s}\biggr{)},\end{split}$
(66)

where $v_{Z^{\prime},\ell}$ and $a_{Z^{\prime},\ell}$ are given in Tab. 2,
$\sqrt{s}=100~{}\mathrm{GeV}$ in the NA64 experiment and

$f(M)=2\biggl{(}\frac{m_{e}^{2}}{M^{2}}\biggr{)}\biggl{(}1-\log^{-1}\biggl{(}\frac{M^{2}}{4s}\biggr{)}\biggr{)}+\mathcal{O}\biggl{(}\frac{m_{e}^{4}}{M^{4}}\biggr{)}.$
(67)

collects the finite mass effects of the electron and the $Z^{\prime}$ boson.
Assembling the pieces for Eq. (65), gives the matching relation

$\epsilon=\frac{\sqrt{v_{Z^{\prime},\ell}^{2}+a_{Z^{\prime},\ell}^{2}\bigl{(}1+f(M_{Z^{\prime}})\bigr{)}}}{2\mathrm{s}_{W}\mathrm{c}_{W}}\sqrt{\mathrm{Br}(Z^{\prime}\to\mathrm{inv.})}.$
(68)

The axial vector couplings can be neglected for a light $Z^{\prime}$ boson,
hence the matching relation reduces to

$\epsilon=\frac{|\mathrm{s}_{Z}|}{2\mathrm{s}_{W}\mathrm{c}_{W}}\left|2\mathrm{c}_{W}^{2}-\frac{1}{\mathscr{Z}}\right|\sqrt{\mathrm{Br}(Z^{\prime}\to\mathrm{inv.})}$
(69)

or in terms of $g_{z}$ it is given as

$\epsilon=\frac{|z_{N}\,g_{z}|}{e}\left|2\mathrm{c}_{W}^{2}\mathscr{Z}-1\right|\sqrt{\mathrm{Br}(Z^{\prime}\to\mathrm{inv.})}$
(70)

where $\mathrm{Br}(Z^{\prime}\to\mathrm{inv.})$ is given in (57). This formula
reproduces the corresponding one given in Ref. [28] for
$M_{Z^{\prime}}<m_{\pi}=130\,\mathrm{MeV}$ and $\mathscr{Z}=2$ (with
$z_{N}=1/2$ and $z_{\phi}=1$), which is the superweak extension of the SM.

In the FASER experiment, predominantly a light neutral meson ($m^{0}=\pi^{0}$,
$\eta^{0}$ or $\omega^{0}$) decays into a neutral gauge boson pair, which may
include the $m^{0}\rightarrow\gamma+A^{\prime}$ production channel. They
searched for a dark photon in the
$\gamma+A^{\prime}\rightarrow\gamma+e^{-}+e^{+}$ decay channel [24]. The
partial rate of the neutral pion into a photon and dark photon is then given
as

$\Gamma\bigl{(}\pi^{0}\to
A^{\prime}+\gamma\bigr{)}=2\biggl{(}1-\frac{M_{A^{\prime}}^{2}}{m_{\pi}^{2}}\biggr{)}^{3}~{}\bigl{(}2~{}\mathrm{tr}(\text{gens.})~{}\bigr{)}^{2}~{}\Gamma\bigl{(}\pi^{0}\to\gamma+\gamma\bigr{)},$
(71)

where the first factor of $2$ is due to the symmetry factor difference between
the $A^{\prime}+\gamma$ and $\gamma+\gamma$ final states and the factor
containing $M_{A^{\prime}}$ is due to the polarization sum of a massive vector
boson times the phase space factor. The trace of the generators is

$2~{}\mathrm{tr}(\text{gens.})=2N_{C}\mathrm{tr}\bigl{(}\tau^{a}Q\,Q\bigr{)}~{}\epsilon=\epsilon,$
(72)

with the matrices

$\tau^{a}=\begin{pmatrix}1/2&0\\\ 0&-1/2\end{pmatrix},\quad\text{and}\quad
Q=\begin{pmatrix}2/3&0\\\ 0&-1/3\end{pmatrix}.$ (73)

The trace of the generators in $U(1)$ extensions considered in this paper is

$2~{}\mathrm{tr}(\text{gens.})=2N_{C}\mathrm{tr}\bigl{(}\tau^{a}Q\,\mathbf{v}_{Z^{\prime},q}\bigr{)}=\biggl{|}\frac{v_{Z^{\prime},\ell}}{2\mathrm{s}_{W}\mathrm{c}_{W}}\biggr{|},$
(74)

where

$\mathbf{v}_{Z^{\prime},q}=\begin{pmatrix}v_{Z^{\prime},u}&0\\\
0&v_{Z^{\prime},d}\end{pmatrix}=\frac{\mathrm{s}_{Z}}{3}\begin{pmatrix}\frac{1}{\mathscr{Z}}-4\mathrm{c}_{W}^{2}&0\\\
0&\frac{1}{\mathscr{Z}}+2\mathrm{c}_{W}^{2}\end{pmatrix}.$ (75)

In order to match the exclusion bounds of FASER one has to solve two equations
for the two parameters $(M_{A^{\prime}},\epsilon)$. The first one expresses
the equality of the signal events in the two models:

$\Gamma\bigl{(}\pi^{0}\to A^{\prime}+\gamma\bigr{)}\,\mathrm{Br}(A^{\prime}\to
e^{+}e^{-})=\Gamma\bigl{(}\pi^{0}\to
Z^{\prime}+\gamma\bigr{)}\,\mathrm{Br}(Z^{\prime}\to e^{+}e^{-})$ (76)

where $\mathrm{Br}(A^{\prime}\to e^{+}e^{-})=1$ in the FASER experiment. The
second equation,

$M_{A^{\prime}}\Gamma_{A^{\prime}}=M_{Z^{\prime}}\Gamma_{Z^{\prime}}\,,$ (77)

ensures that the decay length of the dark photon $A^{\prime}$ and that of the
$Z^{\prime}$ boson are the same as those are required to decay in the detector
itself. For $M_{A^{\prime}}\ll m_{\pi}$ the matching relations yield

$\begin{split}M_{Z^{\prime}}&=\mathrm{Br}(Z^{\prime}\to
e^{+}e^{-})M_{A^{\prime}},\\\
|z_{N}\,g_{z}|&=\frac{e\epsilon}{\sqrt{\mathrm{Br}(Z^{\prime}\to
e^{+}e^{-})}}\frac{1}{|1-2\mathrm{c}_{W}^{2}\mathscr{Z}|}.\end{split}$ (78)

In the $B-L$ extension considered in Ref. [24] one has $\mathscr{Z}=0$,
$\mathrm{Br}(Z^{\prime}\to e^{+}e^{-})=2/5$ and $z_{N}=1$ as they use a
different normalization for the $z$ charges.

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#  Unsupervised Change Detection of
Extreme Events Using ML On-Board

Vít Růžička1, Anna Vaughan2, Daniele De Martini1, James Fulton3,
Valentina Salvatelli4, Chris Bridges5, Gonzalo Mateo-Garcia6, Valentina
Zantedeschi7
1University of Oxford, 2University of Cambridge, 3University of Edinburgh,
4Microsoft,
5University of Surrey, 6University of Valencia, 7INRIA, University College
London
Correspondence to<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce RaVÆn, a lightweight, unsupervised approach for
change detection in satellite data based on Variational Auto-Encoders (VAEs)
with the specific purpose of on-board deployment. Applications such as
disaster management enormously benefit from the rapid availability of
satellite observations. Traditionally, data analysis is performed on the
ground after all data is transferred – downlinked – to a ground station.
Constraint on the downlink capabilities therefore affects any downstream
application. In contrast, RaVÆn pre-processes the sampled data directly on the
satellite and flags changed areas to prioritise for downlink, shortening the
response time. We verified the efficacy of our system on a dataset composed of
time series of catastrophic events – which we plan to release alongside this
publication – demonstrating that RaVÆn outperforms pixel-wise baselines.
Finally we tested our approach on resource-limited hardware for assessing
computational and memory limitations.

##  1 Introduction

Satellite observations of the Earth’s surface provide a vital data source for
diverse environmental applications, including disaster management, landcover
change detection, and ecological monitoring. Currently, sensors on satellites
collect and downlink all data points for further processing on the ground;
yet, limitations in downlink capacity and speed result in delayed data
availability and inefficient use of ground stations. These limitations affect
time-sensitive applications such as disaster management where data are
required with as low latency as possible to inform decision making in real
time. This problem is set to worsen as the sensing resolution and number of
satellites in orbit increase together with further restrictions on radio-
frequency spectrum sharing licensing.

A solution is to apply processing on-board to identify the most useful data
for a particular scenario, and prioritise this for rapid downlink. There has
been considerable recent interest in using machine learning on-board for this
processing. Existing work has focused on the deployment of supervised
classifiers for applications such as identifying clouds [1] or floods [2].
Supervised learning has a significant drawback: only events of a particular
type determined at training time will be flagged and prioritised for downlink,
with no generalisation to new event types, imager specifications, and lighting
or local features.

In this work we propose a new fully unsupervised novelty-detection model
suitable for deployment on remote sensing platforms. We use a Variational
AutoEncoder (VAE) [3] to generate a latent representation of incoming sensor
data over a particular region. A novelty score is assigned to this data using
the distance in the latent space between representations from consecutive
passes. This offers a substantial advantage over existing supervised methods
as any change between passes can be detected on-board at inference time,
regardless of the availability of training data.

We evaluate the performance of this model detecting changes in land-surface
observations from the Sentinel-2 Multispectral Instrument on time series of
natural disaster events where obtaining data with low-latency is of
importance, including floods, landslides, wildfires and hurricanes. The
unsupervised novelty detection model is demonstrated to assign higher novelty
scores to regions of known change and outperforms computer-vision image
differencing baselines. We further demonstrate via experiments on constrained
Xilinx Pynq hardware that this model is suitable for deployment on a remote
sensing platform.

###  1.1 Application Context

Our proposed method RaVÆn facilitates intelligent decision making on board of
a satellite for rapid disaster response. With this application in mind it has
been tested on constrained hardware to similar to that available on remote
sensing platforms. Using this system we can detect extreme events and rapidly
prioritize data over the affected regions for downlink. Vital data will
therefore be available with lower latency to inform decision making during
rapidly evolving events. In the rest of this section we frame our work in the
context of machine learning literature.

#### Anomaly detection

The use of VAEs has been explored for unsupervised anomaly detection in [4],
where the model reconstruction error has been used as anomaly score. Our
approach differs in that, instead of basing our predictions on the
reconstruction error of a single input, which has been shown in [5] to be an
unreliable indicator in the unsupervised context, we consider a sequence of
input images from the same location. Our problem is better framed as change
detection.

#### Change detection

The need for annotations of supervised change detection techniques, such as
siamese networks in [6], can be reduced using active learning approaches as
demonstrated in [7], but then it still lacks in terms of generality. The main
challenge of unsupervised change detection is being able to distinguish
changes of interest from spurious change due to noise. Many existing
approaches [8, 9, 10] achieve this by combining dimensionality reduction
techniques, such as Principal Component Analysis, and clustering, such as
$k$-means, to detect only relevant change between images of consecutive
passes. Approaches based on neural networks (see [11] for a review) rely
instead on supervised auxiliary tasks, such as semantic segmentation, to
extract informative features that are then used to detect change in a time
series. Our method leverages neural networks without requiring supervision at
any stage.

##  2 Data

As part of this study, we compile a new dataset to evaluate the proposed
unsupervised change detection models. Images are taken from the Sentinel-2
multi-spectral imager (MSI)
instrument111https://sentinels.copernicus.eu/web/sentinel/user-
guides/sentinel-2-msi/overview and include the ten highest resolution channels
with all channels interpolated to the highest resolution of 10m. Training data
are taken from the WorldFloods dataset of [2] locations (Figure 1(a)), with a
total of 233 scenes and a time series of five images per scene.

(a)

(b)

Figure 1: Locations used for training (a) and validation (b) images.

The validation set, consists of Sentinel-2 time series for four classes of
disasters: hurricanes, fire burn scars, landslides and floods (Figure 1(b)).
We identified events in each of these classes through an extensive search of
Sentinel-2 records aided by the Copernicus EMS
system222https://emergency.copernicus.eu/mapping/list-of-activations-rapid.
Each event in the validation set consists of a time series of five images
where the first four images are taken before the disaster while the fifth
image is taken afterwards. To mitigate the effects of cloud cover, we
discarded validation images with greater than 20% cloud cover. Events are only
included where all images are within 180 days before and 90 days after the
event. For each event a change mask was hand annotated to mark corresponding
to the difference between the final two images in the time series, for example
Figure 2. We emphasize that these labels are used for evaluation only. The
distribution of data in the validation dataset is further detailed in Appendix
A.3.

Figure 2: Example of validation sample – in this case, a hurricane event – and
its corresponding ground-truth mask (which contains labels of change and
clouds).

##  3 Methodology

#### Preprocessing

Tiles of 32x32 pixels are extracted from the Sentinel-2 scenes described above
and used as inputs to the considered models. These are further normalized by
applying a log transform and scaling to constrain them to the $[0,1]$
interval.

#### Model

We use a VAE (illustrated in Figure 6 of the appendix) to learn an meaningful
embedding space for change detection. The VAE is chosen as it guarantees a
meaningful distance metric on the embedding space. The model is trained to
reconstruct the original input tile from its compressed representation
distribution in the embedding space. Once trained we use this model as a
feature extractor to encode individual tiles, e.g. $x_{t}^{a,b}$ and
$x_{t-1}^{a,b}$, and compare their latent representations to measure the
change between time-steps $t$ and $t-1$. Parameters $a,b$ correspond to the
indexed location in the original image. Comparing compressed representations
of images improves robustness to noise and reduces the computational and
memory requirements of storing images from previous passes, which is a
critical in a constrained environment. We denote by
$\mathcal{N}_{t}^{a,b}=\mathcal{N}(\mu_{t}^{a,b},\sigma_{t}^{a,b})$ the
encoded Gaussian distribution of tile $x_{t}^{a,b}$ with
$\mu_{t}^{a,b},\sigma_{t}^{a,b}\in\mathbb{R}^{n}$. In the following, we fix
the latent size to $n=128$ after the initial experiments indicated that larger
latent sizes did not yield improved results.

#### Change detection novelty score

We compute a change score for tiles $x_{t-1}^{a,b},x_{t}^{a,b}$ using either
the KL divergence between encoded distributions $\mathcal{N}_{t-1}^{a,b}$ and
$\mathcal{N}_{t}^{a,b}$, or the Euclidean or cosine distance between mean
encoded vectors $\mu_{t-1}^{a,b}$ and $\mu_{t}^{a,b}$, that we generally note
as $\mathrm{D}_{\mathrm{VAE}}(x_{t-1}^{a,b},x_{t}^{a,b})$. We extend the
methodology to use a longer time series of $k$ past images by retaining the
minimal distance as:

$\mathrm{S}_{\mathrm{VAE}}(x_{t}^{a,b})=\min_{i=1...k}\mathrm{D}_{\mathrm{VAE}}(x_{t-i}^{a,b},x_{t}^{a,b})$
(1)

To compare the performance of this approach to simpler on-board processing
methods that does not make use of machine learning, we compare our method to a
baseline which compares tiles directly in the input space using the Euclidean
or the cosine distance and after applying the same data pre-processing as for
the VAE.

The proposed workflow outlined in this section is summarised in Figure 3.

Figure 3: Example of RaVÆn, the suggested decision making process on board of
the satellite (for a time window of $k=1$).

##  4 Experimental Setting and Reproducibility

We use different environments for training the VAE and for inference. For
development (training and validation of the model), we use a n1-standard-16
instance on Google Cloud Platform with two NVIDIA Tesla V100 GPUs. In addition
we measure the performance of the models on the Xilinx Pynq FPGA board with
limited compute power, 650MHz ARM Cortex-A9 CPU and 512MB RAM which emulates
the resources available on a typical small satellite. We report the metrics
for the methodology deployed on real hardware in Appendix A.2.

#### Model architecture design choices

To minimize the size of the model and maximize efficiency on constrained
devices we conducted a parametric search over both the number of layers and
number of units per layer in both the encoder network $E$ and the decoder
network $D$. More precisely we tested three different model architecture
configurations (small, medium and large). Details and results of this
experiment are included in Appendix A.2, where we show their maintained
performance while reaching fast processing speeds of only $\sim$2 seconds on
the Xilinx Pynq FPGA board. In the following, we report results for the large
VAE configuration.

##  5 Results

Figure 4 shows a qualitative comparison between the VAE model developed in
this study and the image differencing baseline. The before image shows a river
that floods and therefore changes colour in the after image. The labels and
the change scores from our large VAE and baseline methods are shown in the
bottom row of the figure. In this case, the scores were calculated using a
history of $k=3$ frames, although only the most recent before frame is shown
for brevity. In this example, our method – the cosine embedding – produces a
change map that is crisper than the cosine baseline; for instance, the small
flooded canal can be seen in the cosine embedding image but not in the
baseline. In a similar fashion, Figure 5 shows a qualitative comparison in the
case of a burnt-area detection.

The change-score maps, like those in Figures 4 and 5, were produced for every
image in the evaluation set. We use these maps and our labels to calculate the
area under the precision-recall curve. We produce the curve tile-wise, so that
each individual tile across each image is treated as a positive or negative
example of change, rather than treating the full image as one example. This
means our quality metric is sensitive to the fact that our evaluation images
are not equal; they have different number of tiles and different ratios of
positive pixels (as reported in Table 5). We also ignore tiles that have
clouds in the after image or in the most recent image before the event. We
produce a precision-recall curve for each of the four different event types in
our evaluation set.

Figure 4: Comparison of the change detected using the baseline and the large
VAE method on an example of a flooding river. Two images immediately before
and immediately after a change are shown, along with the human labels of
change and the calculated change scores. Both methods used a history of $k=3$
frames. Figure 5: Additional comparison of the change detected using the
baseline and the large VAE method on an example of a fire disaster. Both
methods used a history of $k=3$ frames. The cosine baseline prediction seems
to more closely copy the details present in the image, making it susceptible
to small, noisy variations between the two images.

Table 1 reports the results of our change detection experiments for all
disaster types. We found that cosine distance, applied on the input space or
on the embeddings, generally provides the best scores. Surprisingly KL-
divergence is the lowest-performing metric, and is beaten by both cosine and
Euclidean embedding scores in all events, even though these methods do not use
the variance values calculated by the VAE. Metrics based on the VAE embedding
outperforms the baseline on floods, hurricanes and fires, and reaches similar
performance on landslides.

Table 2 shows the effects of including a longer frame history. When three
previous images are provided instead of just one, both the embedding and
baseline perform better except for the case of landslide dataset where the
cosine baseline with memory 1 beats memory 3 with a small margin. The table
also shows that our method of detecting significant change based on the
embedding space outperforms the baseline in every dataset when $k=3$.

Table 1: Area under the precision-recall curve for baseline and VAE methods with time window $k=1$ (averaged over 5 runs). Detection method | Dataset
---|---
| Landslides | Floods | Hurricanes | Fires
Cosine baseline | 0.629 | 0.378 | 0.513 | 0.818
Euclidean baseline | 0.267 | 0.326 | 0.351 | 0.770
Cosine embedding | 0.599 $\pm$ 0.012 |  0.448 $\pm$ 0.011 |  0.676 $\pm$ 0.014 |  0.833 $\pm$ 0.008
Euclidean embedding | 0.266 $\pm$ 0.004 |  0.450 $\pm$ 0.007 | 0.478 $\pm$ 0.019 | 0.800 $\pm$ 0.011
KL-Divergence | 0.258 $\pm$ 0.022 | 0.247 $\pm$ 0.018 | 0.301 $\pm$ 0.035 | 0.731 $\pm$ 0.016
Table 2: Area under the precision-recall curve for the best performing metrics from table 1 with and without an extended history $k$ (averaged over 5 runs). Detection method | k | Dataset
---|---|---
|  | Landslides | Floods | Hurricanes | Fires
Cosine baseline | 1 | 0.629 | 0.378 | 0.513 | 0.818
| 3 | 0.622 | 0.378 | 0.570 | 0.865
Cosine embedding | 1 | 0.599 $\pm$ 0.012 |  0.448 $\pm$ 0.011 | 0.676 $\pm$ 0.014 | 0.833 $\pm$ 0.008
| 3 |  0.759 $\pm$ 0.024 |  0.443 $\pm$ 0.009 |  0.726 $\pm$ 0.011 |  0.913 $\pm$ 0.008

##  6 Conclusion

In conclusion, we introduce a new method RaVÆn for unsupervised change
detection in remote sensing data using a VAE. Our method is evaluated on a new
dataset of remote sensing images of disasters which we aim to release with our
work. The proposed model outperforms a classical computer vision baseline in
all of the tested disaster classes. This demonstrates that RaVÆn is a robust
change detection method and suitable for application in improving data
acquisition for disaster response.

## Acknowledgments and Disclosure of Funding

This work has been enabled by Frontier Development Lab (FDL) Europe, a public
partnership between the European Space Agency (ESA) Phi-Lab (ESRIN) and ESA
Mission Operations (ESOC), Trillium Technologies and the University of Oxford;
the project has been also supported by Google Cloud, D-Orbit and Planet. The
authors would like to thank all FDL faculty members, Atılım Güneş Baydin and
Yarin Gal (University of Oxford), Chedy Raissi (INRIA), Brad Neuberg (Planet)
and Nicolas Longépé (ESA ESRIN) for discussions and comments throughout the
development of this work.

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##  Appendix A Appendix

###  A.1 Model architecture

Figure 6: Illustration of the VAE model, $n$ denotes the latent dimension.
Number and type of the used layers can be adjusted given the limitations of
the used hardware as is explored in more detail in A.2.

The encoder of our VAE was composed of a series of downsampling blocks. Each
downsampling block first had a 2D convolutional layer with kernel size 3,
stride size 2, and zero padding of 1, such that the dimensions are halved in
the spatial domain. Following this layer, the block also had a sequence of
extra 2D convolutional layers (the number extra depth referred to in table 4).
Skip connections were used so that the extra depth convolutional layers formed
a residual block. The network could then easily learn to skip these non-
downsampling layers. In the residual block, the number of hidden channels and
image size were conserved. Each convolution layer used leaky ReLU activations
and batch normalisation. Following a given number of downsampling blocks, the
result was flattened and further reduced in dimension using a fully connected
layer which output the mean and log variance. The decoder was essentially the
encoder in reverse. The upsampling method used was nearest neighbour
upsampling followed by a single convolution. This method was preferred over
transpose convolution to avoid checkerboard artefacts Odena et al. [2016],
Wang et al. [2020].

The main model presented in this paper is denoted as large on Table 4, it used
3 downsampling blocks with 32, 64, 128 channels on each successively smaller
scale. The network was further squeezed via the fully connected layer to a
latent dimension of 128. We used extra depth value of 2, so after each
downscale convolution there was a residual block of 2 additional convolutional
layers.

###  A.2 Model timings

Table 3: Area under the precision recall curve and timings for different sizes of model (averaged over 5 runs). The AUPRC results are for the cosine similarity of the embedding with a history of 3 frames. | Dataset
---|---
| Landslides | Floods | Hurricanes | Fires
Small model |  0.748 $\pm$ 0.014 |  0.445 $\pm$ 0.014 |  0.748 $\pm$ 0.002 | 0.907 $\pm$ 0.002
Medium model |  0.758 $\pm$ 0.007 | 0.428 $\pm$ 0.004 |  0.738 $\pm$ 0.018 |  0.912 $\pm$ 0.003
Large model |  0.759 $\pm$ 0.024 |  0.443 $\pm$ 0.009 | 0.726 $\pm$ 0.011 |  0.913 $\pm$ 0.001

It is strongly desired that this change detection method could be run onboard
a satellite, so that images can be filtered or prioritised before they reach
the ground. Therefore the time it takes to run change detection is of the
utmost importance. We need to design our models so that the low power
hardware, available onboard a mission, can keep up with the incoming stream of
data. We therefore tested the accuracy of our model with different numbers of
channels, and different numbers of layers.

Table 3 shows the accuracy of a few variations of model size and the time it
took to process a 574$\times$509 px image (approx. 5km$\times$5km at
Sentinel-2 10m resolution) whilst running on the CPU of a Xilinx PYNQ. We see
the the results of all tested models are comparable and that it is reasonable
to aim for the smallest model, which takes only 2.06 seconds to process the
patch. Running onboard the PYNQ means that there is considerable potential to
speed up this runtime by a large factor by deploying directly on the FPGA
module rather than using the board’s CPU.

Table 4: Differences in the architecture for different models of table 3. Note that during inference, we only need the encoder network of the VAE model. We also only need to process the newly acquired image to obtain their latent representation, while the latent vectors of the previous image can be loaded. | Total | Encoder | Runtime | Extra depth | Hidden channels
---|---|---|---|---|---
| parameters | parameters | (seconds) |  |
| (millions) | (millions) |  |  |
Small model | 0.443 | 0.285 | 2.06 | 0 | 16, 32, 64
Medium model | 0.979 | 0.617 | 4.86 | 0 | 32, 64, 128
Large model | 1.500 | 1.005 | 13.98 | 2 | 32, 64, 128

###  A.3 Dataset description

We describe the statistics of the manually annotated validation dataset in
Table 5. While each type of the event is represented by similar amount of
locations, the affected area varies dramatically with different disasters.
Namely the area of burn scars in the Fire dataset has both the largest area of
effect and also the largest proportion of changed pixels to all not cloudy
pixels (reported as positive ratio).

Table 5: Validation dataset details. Each location is captured in 4 time-steps before the event and once after the event (it is however counted only once into the cumulative square km). Note that only the last pair of images is labelled with changed and not changed pixel map - and that the reported percentage only considers this map. | Number of | Cumulative | Positive rate
---|---|---|---
| locations | area (km2) | (%)
Landslides | 5 | 108 | 10.48
Floods | 4 | 1301 | 6.74
Hurricanes | 5 | 1622 | 24.31
Fires | 5 | 3485 | 53.79