\section{Introduction}

Leptophilic gauge bosons are popular candidates for physics beyond the Standard Model (SM).
As opposed to new gauge bosons interacting with quarks, which are strongly constrained by LHC searches~\cite{ATLAS:2017eqx,CMS:2018mgb,ATLAS:2019erb,CMS:2021ctt}, those coupling solely to leptons are only subject to LEP constraints~\cite{ALEPH:2013dgf} and can therefore exist below the TeV scale.
As a consequence, these kind of $Z'$ bosons can have important phenomenological consequences for a plethora of particle physics experiments and observations~\cite{Bauer:2018onh}.

In view of recent experimental developments, $Z'$ bosons interacting with muons are of particular interest.
Most importantly, they can economically solve the long-standing discrepancy between the SM prediction~\cite{Aoyama:2020ynm} and the observed values~\cite{Muong-2:2006rrc,Muong-2:2021vma} of the muon anomalous magnetic moment, as has been shown in~\cite{Foot:1994vd,Gninenko:2001hx,Baek:2001kca,Murakami:2001cs,Fayet:2007ua,Pospelov:2008zw,Ma:2001md,Heeck:2011wj,Carone:2013uh,Harigaya:2013twa,Davoudiasl:2014kua,Altmannshofer:2014cfa,Tomar:2014rya,Lee:2014tba,Allanach:2015gkd,Heeck:2016xkh,Patra:2016shz,Altmannshofer:2016brv,Iguro:2020rby,Holst:2021lzm,Hapitas:2021ilr}.
Slight deviations from SM expectations in lepton flavor universality (LFU) ratios~\cite{HFLAV:2019otj} and an observed deficit of unitarity in the first row of the CKM matrix~\cite{Belfatto:2019swo,Grossman:2019bzp,Shiells:2020fqp,Seng:2020wjq} may also be explained by a new muonic force.
Finally, accumulating evidence for lepton-universality violation in $b\rightarrow s\ell^+\ell^-$ transitions (see e.g.~\cite{Geng:2021nhg} and references therein) serves as a further motivation to study this kind of new physics (NP), although in this case coupling to quarks also need to be invoked.
A global analysis of leptophilic gauge bosons was recently performed in~\cite{Buras:2021btx}.

In this context, the most popular phenomenological model is that of a gauge boson coupling to the $L_\mu - L_\tau$ lepton flavor combination~\cite{He:1990pn,Foot:1990mn,He:1991qd}.
Gauged $U(1)_{L_\mu-L_\tau}$ models are distinguished by being free from gauge anomalies (as are other differences in baryon and/or lepton flavor numbers), and from the stringent experimental constraints on $Z'$ couplings to electrons.

One difficulty is that gauging $L_\mu-L_\tau$ symmetry prevents the usual interactions that generate the masses and mixings of neutrinos through the dimension-5 Weinberg operator.
This means that a fully consistent model requires additional new physics to reproduce the observed values~\cite{Esteban:2020cvm} of the PMNS~\cite{Maki:1962mu,Pontecorvo:1967fh} matrix.
Some proposals in this direction include extra Higgs doublets~\cite{Ma:2001md}, soft-breaking terms~\cite{Bell:2000vh,Choubey:2004hn}, and right-handed neutrinos~\cite{Binetruy:1996cs,Heeck:2011wj,Asai:2017ryy,Araki:2019rmw}.
The common element of all these models is the presence of (often multiple) U(1)$_{L_\mu-L_\tau}$ symmetry-breaking scalars that can significantly complicate the minimal proposal.
From an aesthetic point of view, it also seems rather arbitrary that only the $L_\mu-L_\tau$ difference should be gauged, considering that the SM treats all generations on an equal footing from a structural perspective.

To address these shortcomings, in this paper we propose that SU(3)$_\ell$ of lepton flavor is gauged in a vectorial fashion.
In this setup, the $L_\mu-L_\tau$ gauge boson need only be one of eight new $Z'$ states, all of which are leptophilic. Previous studies have gauged lepton and quark flavors together using horizontal SU(3) family symmetry,
along with additional gauge or discrete symmetries~\cite{Berezhiani:1983hm,King:2001uz,King:2003rf,Alonso:2017bff}, or flavor symmetries for left- and right-handed leptons separately~\cite{Alonso:2016onw}. We are not aware of previous literature treating the  possibility of a single vectorial SU(3)$_\ell$ for lepton flavor.

In the following, we identify the additional particle content needed to spontaneously break SU(3)$_\ell$ to engender the observed lepton and neutrino masses, which would otherwise be forced to be flavor universal.
Further guided by experimental observations, we focus on scenarios where the $L_\mu-L_\tau$ gauge boson is lighter than the other seven and can thus address the $(g-2)_\mu$ tension.
For these purposes, we find it sufficient to add a triplet of vectorlike heavy leptons $E_i$ and three scalars in the $3$, $6$ and $8$ representations, which we denote by $\Phi_3$, $\Phi_6$, and $\Phi_8$, respectively.
The usual triplet of right-handed neutrinos
$N_i$ is also present.
The vacuum expectation values (VEVs) of the scalars give rise to the required lepton and gauge boson masses.
This particle content is summarized in Table \ref{tab1}.

A modest hierarchy in the scalar VEVs, $\langle\Phi_3\rangle \sim\langle\Phi_8\rangle \gg \langle\Phi_6\rangle$ is needed to get the desired spectrum of leptophilic gauge boson masses.
The gauge boson coupling to the $L_\mu-L_\tau$ current is lighter than the rest, which allows it to solve the muon $(g-2)$ anomaly in a way consistent with all other constraints~\cite{Holst:2021lzm,Hapitas:2021ilr}.
At the same time, the Cabibbo angle anomaly (CAA), a 3-5\,$\sigma$ tension in the unitarity of the top row of the CKM matrix~\cite{Belfatto:2019swo,Grossman:2019bzp,Shiells:2020fqp,Seng:2020wjq}, can be resolved by new contributions to $\mu\to e\nu\bar\nu$ decay from two of the heavier new gauge bosons.
A mild 2$\sigma$ hint of nonuniversality in $\tau$ lepton decays~\cite{HFLAV:2019otj} can be similarly ameliorated.

The $\Phi_6$ VEVs further determine the mass matrix of the right-handed neutrinos, and thereby the pattern of light neutrino masses and mixings via a type I seesaw mechanism~\cite{Minkowski:1977sc,Gell-Mann:1979vob,Yanagida:1979as,Glashow:1979nm,Mohapatra:1979ia,Weinberg:1979sa,Witten:1979nr}.
A striking prediction of our scenario is that the lightest active neutrino mass must exceed $\sim 0.1\,\mathrm{meV}$.
We find the right-handed neutrino masses to lie in the GeV to TeV range and to be inversely proportional to the light neutrino masses.
They are therefore heavy neutral leptons
(HNLs), whose mixings with the active states exactly match the flavor composition of the light neutrino mass eigenstates and are therefore completely determined by the PMNS matrix.  We show  that the mixings of the HNLs with active neutrinos can be in a range that is relevant for affecting the primordial abundances of light elements.

The paper is structured as follows. In Section \ref{sect:model} we introduce the particle content of the model and the resulting spectra of charged leptons, heavy vectorlike leptons, light neutrinos, and
new gauge bosons.  Constraints on the gauge bosons
from flavor-sensitive processes are discussed in
Section \ref{sect:pheno}, including the muon $(g-2)$, LEP di-lepton searches, the Cabibbo angle anomaly, and lepton flavor universality limits.  The detailed properties of the
heavy neutral leptons, and their phenomenological implications,
are discussed in Section~\ref{sec:HNL}.
Conclusions are given in Section~\ref{sect:conc}, and a brief discussion of the challenges associated with the construction of a potential in the scalar sector that can give rise to the desired pattern of VEVs is presented in Appendix~\ref{app:VEVs}.

\section{Model and mass spectra}
\label{sect:model}

Our starting point is the gauging of the leptonic SU(3) flavor symmetry, under which both left- and right-handed lepton fields transform simultaneously.
Clearly, our world does not exactly respect such a symmetry, requiring it to be spontaneously broken  at some scale.
For this purpose, we introduce three real scalar fields $\Phi_3$, $\Phi_6$, and $\Phi_8$ in the fundamental, symmetric two-index, and adjoint representations, respectively.
Although $\Phi_6$ by itself would be sufficient to fully
break SU(3)$_\ell$,  this is the minimal scalar sector that we have identified as being phenomenologically viable,\footnote{Other possibilities may exist; it is not
our purpose to exhaust them but rather to construct one instance of a working model.} as we will explain.

Clear observational evidence for breaking of SU(3)$_\ell$ is provided by the hierarchy of lepton masses.
Indeed, with only the SM particle content, the gauge-invariant Higgs coupling $\bar L^i H e_i$ (where $i$ is the SU(3)$_\ell$ index) only allows for a universal charged lepton mass.
To obtain the observed mass splittings through spontaneous breaking of the gauge symmetry in a renormalizable way, we add a triplet of vectorlike charged lepton partners, $E_i$.
Additionally, to generate neutrino masses, a right-handed neutrino triplet $N_i$
is included, which completes the particle content of the model.
The list of particles and gauge charges are shown in Table~\ref{tab1}.

\begin{table}[t]\centering
\setlength\tabcolsep{4pt}
\def1.5{1.2}
\begin{tabular}{|c|c|c|c|}\hline
 & SU(3)$_\ell$ & SU(2)$_L$ & U(1)$_y$\\
\hline
$L_i$ & 3 & 2 & $-1/2$ \\
$e^{c,i}$ & $\bar 3$ & 1 & $+1$ \\
\hline
$E_{\L,i}$ & 3 & 1 & $-1$ \\
$E_R^{c,i}$ & $\bar 3$ & 1 & $+1$ \\
$N^{c,i}$ & $\bar 3$ & 1 & 0 \\
\hline
$\Phi_{3,i}$ & 3 & 1 & 0 \\
$\Phi_{6,ij}$ & 6 & 1 & 0 \\
$\Phi^i_{8,j}$ & 8 & 1 & 0\\
\hline
\end{tabular}
\caption{Field content of the gauged SU(3)$_\ell$ model along with the corresponding gauge charges. The fermionic fields are all listed as left-handed Weyl states.  $L_i$ and $e_i$ denote the SM lepton doublets
and charged singlet leptons, respectively.}
\label{tab1}
\end{table}

The addition of the right-handed neutrino triplet $N_i$ makes SU(3)$_\ell$ anomaly free, since all leptons now come in chiral pairs.
Mixed anomalies involving non-abelian factors likewise cancel because of the tracelessness of the generators, and the SU(3)$_\ell^2\times$U(1)$_y$ one vanishes due to the hypercharges of the SU(3)$_\ell$ triplets adding up to zero.

With the field content listed in Table~\ref{tab1}, the most general renormalizable Lagrangian that can be constructed (omitting kinetic terms) is
\begin{eqnarray}\label{eq:Lagrangian}
{\cal L} =
    &-& \bar E_\L^i(\mu_{{\sss E}\E}\delta^j_i + y_{{\sss E}\E}\Phi^j_{8,i}) E_{{\sss R},j}\nonumber\\
    &-& \bar E_\L^i(\mu_{{\sss E} e}\delta_i^j + y_{{\sss E} e}\Phi_{8,i}^j)e_j \nonumber\\
    &-&\bar L^i H(y_{\L e} e_i + y_{\L{\sss E}} E_{{\sss R},i}) -	y_{\L{\sss N}} \bar L^i \tilde H N_i\nonumber\\
	&-& {y_{\sss N}\over 2} \bar N^i\Phi_{6,ij}N^{c,j} + {\rm h.c.} \nonumber \\
	&-& V(H,\, \Phi_3,\, \Phi_6,\, \Phi_8),
\end{eqnarray}
where $V$ denotes the scalar potential, $\mu_i$ are
constants with dimensions of mass, and $y_i$ are dimensionless couplings.
In the following sections we describe the consequences of nonvanishing vacuum expectation values for the scalar fields.
The required values of the VEVs are constructed in a bottom-up fashion, making use of experimental measurements and constraints as guiding principles.
Obtaining the desired symmetry breaking pattern from a
specific scalar potential is likely to be a challenging task, beyond the scope of the present work.
We however take some preliminary steps in this direction
in Appendix~\ref{app:VEVs}.

In the following, it will be convenient to order the gauge indices associated with the lepton flavors as $(1,2,3) = (\tau,\mu,e)$.
This allows the $\mu-\tau$ gauge boson to be associated with the third generator $T_3$ of SU(3)$_\ell$ (using the standard form of the Gell-Mann matrices) instead of a linear combination of $T_3$ and $T_8$.

\subsection{Charged lepton masses}

In the absence of spontaneous breaking of SU(3)$_\ell$, charged leptons get a universal mass from electroweak
symmetry breaking.  We will use the $\Phi_8$ scalar field VEV, in conjunction with the vectorlike lepton partners $E_i$,
to split masses between the different lepton generations.\footnote{Nonzero $\mu_{{\sss E}\E}$ or $\mu_{{\sss E} e}$ values in Eq.~\eqref{eq:Lagrangian} are also essential, since in the $\mu_{{\sss E}\E},\,\mu_{{\sss E} e}\rightarrow 0$ limit the charged lepton masses become universal again.}\ \
The Lagrangian~\eqref{eq:Lagrangian} gives rise to
mass mixing between the SM leptons and the heavy $E_i$
partners.
For simplicity, we take $\langle\Phi_8\rangle$ to be diagonal, in which case the $6\times 6$ Dirac mass matrix for the charged leptons becomes block diagonal
(no flavor mixing),
with three blocks of the form
\begin{equation} \label{eq:Dirac_mass_matrix}
    (\bar e_\L^i,\, \bar E_\L^i)\left({y_{\L e}\bar v\atop \tilde \mu^i_{{\sss E} e}}\,{ y_{\L{\sss E}} \bar v\atop \tilde \mu^i_{{\sss E}\E}}
    \right)\left({e_{{\sss R},i}\atop E_{{\sss R},i}}\right),
\end{equation}
where $\bar v \cong 174\,$GeV is the complex Higgs VEV and we define
\begin{eqnarray}
    \tilde \mu^i_{{\sss E} e} &=& \mu_{{\sss E} e} + y_{{\sss E} e}\langle\Phi_8\rangle^i_i\,,\nonumber\\
    \tilde \mu^i_{{\sss E}\E} &=& \mu_{{\sss E}\E} + y_{{\sss E}\E}\langle\Phi_8\rangle^i_i\,.
    \label{tildedef}
\end{eqnarray}
Diagonalizing each of the $2\times2$ blocks separately under the assumption $y_{\L e}\bar{v},\,y_{\L{\sss E}}\bar{v}\ll \tilde\mu^i_{{\sss E}\E}$ or $\tilde\mu^i_{{\sss E} e}$ leads to three heavy and three light mass eigenstates with masses
\begin{eqnarray}
    (m_{\sss E}^i)^2 &=& (\tilde\mu^i_{{\sss E} e})^2 + (\tilde\mu^i_{{\sss E}\E})^2,\nonumber\\
    (m_e^i)^2 &=& \bar{v}^2\,\frac{(y_{\L e}\tilde\mu_{
    {\sss E}\E}^i - y_{\L{\sss E}}\tilde\mu_{
    {\sss E} e}^i)^2}{(\tilde\mu^i_{{\sss E} e})^2 + (\tilde\mu^i_{{\sss E}\E})^2}.\label{eq:charged_lepton_masses}
\end{eqnarray}
The contributions to the light charged lepton masses in the perturbative limit are depicted in Fig.\ \ref{fig:diag}.
In the limit of large $\mu_{{\sss E}\E}$ or $\mu_{{\sss E} e}$, the heavy lepton effects do not decouple; lepton masses generally continue to be split by the heavy-$E_i$ effects even for arbitrarily heavy $E_i$.

\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{figures/diag.pdf}
\caption{Extra contribution to the charged lepton masses from integrating out the heavy vectorlike lepton, that splits them relative to the flavor-independent mass  $y_{\L e}\bar v$.}
 \label{fig:diag}
\end{figure}

Since we assume that $\langle\Phi_8\rangle$ is diagonal, mixing with the heavy states does not cause any flavor-violating effects in the charged lepton sector.
However, the mixing of the light states with the heavy ones is constrained by electroweak precision data (EWPD).
Diagonalizing  the  mass matrix in Eq.~\eqref{eq:Dirac_mass_matrix} yields mass eigenstates $(e^i_{\L,{\sss R}},\,E^i_{\L,{\sss R}})'$, given by
\begin{equation}
\left({e^i_{\L,{\sss R}}\atop  E^i_{\L,{\sss R}}}\right) =
\left({c_{\theta^i_{\L,{\sss R}}}\atop -s_{\theta^i_{\L,{\sss R}}}}\,{s_{\theta^i_{\L,{\sss R}}}\atop
\phantom{-}c_{\theta^i_{\L,{\sss R}}}}\right)\left( e_{\L,{\sss R}}^i \atop E_{\L,{\sss R}}^i \right)'\,.
\end{equation}
To second order in $\bar{v}/\tilde\mu$, the mixing angles are given by
\begin{eqnarray}
\tan 2\theta^i_\L &=& \frac{2\bar v\,(y_{\L e}\tilde \mu^i_{{\sss E} e} + y_{\L{\sss E}}\tilde \mu^i_{{\sss E}\E})}{(\tilde \mu^i_{{\sss E} e})^2 + (\tilde \mu^i_{{\sss E}\E})^2}\,,\nonumber\\
\tan 2\theta^i_{\sss R} &=& 2 \frac{\tilde\mu^i_{{\sss E}\E} \tilde\mu^i_{{\sss E} e} + \bar{v}^2 y_{\L{\sss E}}y_{\L e}}{(\tilde\mu^i_{{\sss E}\E})^2 - (\tilde\mu^i_{{\sss E} e})^2} \,.\label{eq:charged_lepton_mixings}
\end{eqnarray}
In the limit where $\tilde \mu^i_{{\sss E}\E}$ or $\tilde\mu_{{\sss E} e}^i\gg \bar v$, the angle $\theta^i_\L$ is suppressed, but $\theta^i_{\sss R}$ can be relatively large.
EWPD constraints require that $\theta_{{\sss R}}(\tau,\mu,e) \lesssim (0.03,\,0.02,\,0.02)$~\cite{delAguila:2008pw}.\footnote{The precise upper limits are model-dependent; we have chosen the strongest limits corresponding to models that give dominant right-handed mixing.}\ \
To sufficiently suppress $\theta_{\sss R}^i$, there must be a hierarchy $\tilde \mu^i_{{\sss E} e} \ll \tilde \mu^i_{{\sss E}\E}$ or $\tilde \mu^i_{{\sss E} {\sss E}} \ll \tilde \mu^i_{{\sss E} e}$ for each value of $i$.

To give a concrete example, consider the
case $\tilde \mu^i_{{\sss E} e} \lesssim 10^{-2}\, \tilde \mu^i_{{\sss E}\E}$; the derivation for the opposite possibility is completely analogous.
In this limit, the masses of the heavy charged leptons are simply
\begin{equation}
	m^i_{{\sss E}} \simeq \tilde \mu^i_{{\sss E}\E}\,,
\end{equation}
while the mass of the SM charged leptons becomes
\begin{equation}
	m^i_e \cong \bar{v} \left( y_{\L e} - y_{\L{\sss E}} \frac{\tilde \mu^i_{{\sss E} e}}{\tilde \mu^i_{{\sss E}\E}} \right).
\end{equation}
Taking  for simplicity $y_{\L{\sss E}}=1$ and $y_{\L e} = 0$ ($y_{\L e}\lesssim 10^{-6} y_{\L{\sss E}}$ is the sufficient condition), the ratios
\begin{equation}\label{eq:mu_tilde_ratios}
	\frac{\tilde \mu^i_{{\sss E} e}}{\tilde \mu^i_{{\sss E}\E}} \simeq - \frac{m_e^i}{\bar{v}} \sim (10^{-2},\, 10^{-3},\, 10^{-6})
\end{equation}
are determined by the measured charged lepton masses.
Using Eq.~\eqref{tildedef} and taking into account the fact that $\Phi_8$ is traceless,
one sees that achieving the hierarchies in Eq.~\eqref{eq:mu_tilde_ratios} requires some cancellations between terms in the numerator or denominator.
For example,  take  $\langle\Phi_8\rangle =
\mathrm{diag}(-\mu_{{\sss E}\E},\,\mu_{{\sss E}\E},\,0)$.
Then  Eq.~\eqref{eq:mu_tilde_ratios} is realized by choosing
\begin{align}
    \mu_{{\sss E} e} &= \frac{m_e}{\bar{v}} \, \mu_{{\sss E}\E} \simeq 3\times 10^{-6}\, \mu_{{\sss E}\E} \,,\nonumber\\
    y_{{\sss E}\E} &= \frac{m_\tau+m_\mu}{m_\tau-m_\mu} \simeq 1.1\,,\nonumber\\
        y_{{\sss E} e} &= \frac{2}{\bar{v}} \frac{m_\tau m_\mu}{m_\tau-m_\mu}\simeq 1.3\times 10^{-3}\,.\label{eq:BM_parameters}
\end{align}
The required order-of-magnitude enhancement of $m_\tau$ relative to $m_\mu$ is achieved by a $\sim 10\%$ cancellation in the denominator of Eq.~\eqref{eq:mu_tilde_ratios} for $i=\tau$, while the smallness of $m_e$ is due to $\mu_{{\sss E} e}\ll
\mu_{{\sss E}\E}$ and $\langle\Phi_8\rangle_e^e = 0$.

Having fixed the charged lepton masses, one is still free to choose $\mu_{{\sss E}\E}$ and therefore the absolute mass scale of the charged lepton partners.
If they are sufficiently light, they may be directly accessible at the Large Hadron Collider (LHC).
They can be pair-produced by the Drell-Yan process, and their main decays are $E_i\to L_i + h$ via the
$y_{\L{\sss E}}$ interaction.  Ref.\ \cite{Kumar:2015tna} showed that electroweak-singlet vectorlike leptons (VLL's) are very difficult to observe at the LHC;  indeed, existing
constraints \cite{ATLAS:2015qoy,CMS:2019hsm} have focused on the search for doublet VLL's.
VLL models similar to ours were studied in
Ref.\ \cite{Bell:2019mbn}, where current LHC limits were found to be as low as  $m_{\sss E}^i \gtrsim 300\,$GeV. This leaves a large range of allowed VLL masses for future discovery, consistent with our requirement $\langle\Phi_8\rangle\gtrsim 1$~TeV, that will be derived in Section~\ref{sect:pheno}.

In the following sections, we adopt the parameter values described above as our principal benchmark model (BM1).
However, other possibilities for fixing the constants in the Lagrangian~\eqref{eq:Lagrangian} exist.
There are eight free parameters: two dimensionful scales $\mu_{{\sss E}\E}$ and $\mu_{{\sss E} e}$, four Yukawa couplings $y_{{\sss E}\E}$, $y_{{\sss E} e}$, $y_{\L{\sss E}}$, and $y_{\L e}$, and two independent entries of $\langle\Phi_8\rangle$.
Reproducing the observed charged lepton masses imposes three constraints through Eq.~\eqref{eq:charged_lepton_masses}.
In addition, the mixings in Eq.~\eqref{eq:charged_lepton_mixings} must be sufficiently small.
In order to explore the parameter space and find other viable models, we employed a Markov Chain Monte Carlo method.
Two further examples of parameters that satisfy all the phenomenological requirements but differ qualitatively from BM1 are given in Table \ref{tab3}.

In the second example (BM2), also expressed in terms of the unconstrained parameter $\mu_{{\sss E}\E}$, the heavy lepton masses are all approximately $m_{\sss E}^i\cong \mu_{{\sss E}\E}$.
Satisfying the EWPD bounds on mixing leads to the largest Yukawa coupling, $y_{\L{\sss E}} = 3.6$, being close to unitarity limits.
In this example, since $\mu_{{\sss E}\E}\gg\langle\Phi_8\rangle$, the heavy leptons are far above the TeV scale and therefore not directly accessible in current collider experiments.

In the third example (BM3) and similarly to BM1, the VLL are of order $\langle\Phi_8\rangle$, making them potentially accessible at the LHC.
In this case (as in BM1), all Yukawa couplings are well below perturbative unitarity bounds.  A qualitatively different feature of BM3 is that $\tilde\mu_{{\sss E}\E}^\tau \ll
\tilde\mu_{{\sss E} e}^\tau$, while in the others
$\tilde\mu_{{\sss E}\E}^i \gg
\tilde\mu_{{\sss E} e}^i$ for all flavors.
Moreover, the VVL-charged lepton mixing angles fall well below the experimental limits in this model.

\iffalse
\begin{table*}[t]\centering
\begin{tabular}{|c||c|c|c|c||c|c|c|c||c|c|c|c|}\hline
& \multicolumn{4}{c||}{Input parameters} & \multicolumn{4}{c||}{Derived parameters} & \multicolumn{4}{c|}{Phenomenological quantities} \\
\hline
 & $\phi_\mu$ & $\phi_e$ & $\mu_{{\sss E}\E}$ & $y_{\L{\sss E}}$ & $\mu_{{\sss E} e}$ & $y_{{\sss E}\E}$ & $y_{{\sss E} e}$ & $y_{\L e}$ & $m^i_E$ &
$\theta_{{\sss R}}^{\tau}$ & $\theta_{{\sss R}}^{\mu}$ & $\theta_{{\sss R}}^e$\\
\hline
BM1 & $\phi_8$ & $0$ & $\phi_8$ & $1$ & $3\times 10^{-6}\,\phi_8$ & $1.1$ & $1.3\cdot 10^{-3}$ & $\lesssim 10^{-6}$ & $(1,\,1,\,1)\,\phi_8$ & $1\cdot 10^{-2}$ & $6\cdot 10^{-4}$ & $3\cdot 10^{-6}$\\
\hline
BM2 & $0.014\,\mu_{{\sss E}\E}$ & $0.016\,\mu_{{\sss E}\E}$ & $\mu_{{\sss E}\E}$ & $3.6$ & $0.023\,\mu_{{\sss E}\E}$ & $0.32$ & $-0.055$ & $0.56$ & $(1,\,1,\,1)\,\mu_{{\sss E}\E}$ & $0.021$ & $5.6\cdot 10^{-4}$ & $0.024$ \\
 \hline
BM3 & $-3.0\,\mu_{{\sss E}\E}$ & $-3.2\,\mu_{{\sss E}\E}$ & $\mu_{{\sss E}\E}$ & $-0.011$ & $1.55\,\mu_{{\sss E}\E}$ & $0.16$ & $-0.48$ & $-8.5\cdot 10^{-4}$ & $(4.5,1.5,1.5)\mu_{{\sss E}\E}$ & $-1\cdot 10^{-3}$ & $6\cdot 10^{-3}$ & $6\cdot 10^{-3}$ \\
 \hline
\end{tabular}
\caption{Input parameters and derived parameters and quantities relevant for phenomenology (columns 10-13) for three benchmark models that can reproduce the observed charged lepton properties.
The mixing angles $\theta_{\sss R}^i$ between right-handed charged leptons and heavy vectorlike leptons of mass $m_E^i$ are consistent with EWPD constraints~\cite{delAguila:2008pw}.
The traceless octet VEV is parametrized as $\langle\Phi_8\rangle = \mathrm{diag}(-\phi_\mu-\phi_e,\,\phi_\mu,\,\phi_e)$.
In model BM1, we have taken $\mu_{{\sss E}\E}=\phi_8$ to simplify the presentation, full expressions avoiding this assumption can be found in Eq.~\eqref{eq:BM_parameters}.}
\label{tab3}
\end{table*}
\fi

\begin{table}[t]\centering
\setlength\tabcolsep{6pt}
\def1.5{1.4}
\begin{tabular}{|c| c c c|}\cline{2-4}
\multicolumn{1}{c|}{} & BM1 & BM2 & \multicolumn{1}{c|}{BM3} \\ \cline{1-4}
\multicolumn{4}{|c|}{Input parameters} \\ \hline
$\phi_\mu$ & $\mu_{{\sss E}\E}$ & $0.014\,\mu_{{\sss E}\E}$ & $-3.0\,\mu_{{\sss E}\E}$ \\ \hline
$\phi_e$ & $0$ & $0.016\,\mu_{{\sss E}\E}$ & $-3.2\,\mu_{{\sss E}\E}$ \\ \hline
$y_{\L{\sss E}}$ & $1$ & $3.6$ & $-0.011$ \\ \hline
\multicolumn{4}{|c|}{Derived parameters} \\ \hline
$\mu_{{\sss E} e}$ & $3\times 10^{-6}\,\mu_{{\sss E}\E}$ & $0.023\,\mu_{{\sss E}\E}$ & $1.55\,\mu_{{\sss E}\E}$ \\ \hline
$y_{{\sss E}\E}$ & $1.1$ & $0.32$ & $0.16$ \\ \hline
$y_{{\sss E} e}$ & $1.3\times 10^{-3}$ & $-0.055$ & $-0.48$ \\ \hline
$y_{\L e}$ & $\lesssim 10^{-6}$ & $0.56$ & $-8.5\times 10^{-4}$ \\ \hline
\multicolumn{4}{|c|}{Phenomenological quantities} \\ \hline
$m^i_E$ & $(1,\,1,\,1)\,\mu_{{\sss E}\E}$ & $(1,\,1,\,1)\,\mu_{{\sss E}\E}$ & $(4.5,\,1.5,\,1.5)\,\mu_{{\sss E}\E}$ \\ \hline
$\theta_{{\sss R}}^{\tau}$ & $1\times 10^{-2}$ & $0.021$ & $-1\times 10^{-3}$ \\ \hline
$\theta_{{\sss R}}^{\mu}$ & $6\times 10^{-4}$ & $5.6\times 10^{-4}$ & $6\times 10^{-3}$ \\ \hline
$\theta_{{\sss R}}^e$ & $3\times 10^{-6}$ & $0.024$ & $6\times 10^{-3}$ \\ \hline
\end{tabular}
\caption{Input parameters, derived parameters and quantities relevant for phenomenology for three benchmark models that can reproduce the observed charged lepton properties.
The mixing angles $\theta_{\sss R}^i$ between right-handed charged leptons and heavy vectorlike leptons of mass $m_E^i$ are consistent with EWPD constraints~\cite{delAguila:2008pw}.
The traceless octet VEV is parametrized as $\langle\Phi_8\rangle = \mathrm{diag}(-\phi_\mu-\phi_e,\,\phi_\mu,\,\phi_e)$.
The dimensionful parameter $\mu_{{\sss E}\E}$, which controls the scale of the octet VEV and the VLL masses, is left as freely adjustable parameter.
}
\label{tab3}
\end{table}

\subsection{Neutrino masses}

The right-handed neutrinos $N_i$ get a Majorana mass matrix from the
sextet VEVs, $M_N = y_{\sss N}\langle\Phi_6\rangle$, while the Dirac
neutrino mass matrix is proportional to the unit matrix, with
coefficient $m_D = y_{\L{\sss N}}\bar v$.  The light neutrino mass matrix is
therefore $m_\nu = -m_D^2 M_N^{-1}$ from the seesaw mechanism, and we can solve for the sextet
VEVs in terms of the PMNS mixing matrix $U_{\rm\scriptscriptstyle PMNS}$ and the light neutrino
mass eigenvalues $D = {\rm diag}(m_1,m_2,m_3)$:
\begin{equation}\label{eq:sextet_VEV}
	\langle\Phi_6\rangle = -{m_D^2\over y_{\sss N}}\,U_{\rm\scriptscriptstyle PMNS}\, D^{-1}\, U_{\rm\scriptscriptstyle PMNS}^{\sss T}\,.
\end{equation}
Recall that we have interchanged the first and third rows of $U_{\rm\scriptscriptstyle PMNS}$ in
our ordering convention for the lepton flavors.  This allows us
to refer to the $L_{\mu-\tau}$ gauge boson as the one associated with
the Gell-Mann matrix $\lambda_3$.
A quantitative analysis of the right-handed neutrino mass and mixing spectrum is deferred to Sec.~\ref{sec:HNL}.

\subsection{Gauge boson masses}

The mass-squared matrix of gauge bosons is given by
\begin{eqnarray}
\label{gbmasses}
	{M^2_{ab}\over g'^2} &=& \sfrac14\Phi_3^\dagger \{\lambda^a,\lambda^b\} \Phi_3
	- \sfrac12{\rm tr}\left([\Phi_8,\lambda^a][\Phi_8,\lambda^b]\right)\nonumber\\
	&+&
\sfrac12{\rm tr}\left([\Phi_6^\dagger\lambda^a +
\lambda^{a*}\Phi_6^\dagger][\Phi_6\lambda^{b*} + \lambda^b\Phi_6]\right)\,,
\end{eqnarray}
where the VEVs of the scalar fields are understood.
In the approximation that $\langle\Phi_6\rangle=0$, while
$\langle\Phi_3\rangle\sim \langle\Phi_8\rangle$, all gauge bosons get masses except for $A_3$,
which corresponds to the $L_{\mu-\tau}$ gauge boson in our numbering
scheme.  Thus $A_3$ gets its mass solely from $\langle\Phi_6\rangle$.
The overall magnitude of $\langle\Phi_6\rangle$ is determined by the
smallest light neutrino mass $m_{\nu_1}$; hence one prediction of the model
is that $m_{\nu_1}$ cannot be arbitrarily small.

The various contributions of the scalar VEVs to the mass eigenvalues $M_i^2/g'^2$ are summarized in
Table~\ref{tab:gauge_boson_masses}.
Before turning on the $\Phi_6$ VEV, the gauge boson mass matrix is diagonal, and the eigenvalues can be labelled with the number of the corresponding generator.

We first consider the contribution of the $\Phi_8$ VEV. Following the discussion in the previous section, in  benchmark scenario BM1 it takes the form
\begin{equation} \label{eq:Phi8_VEV}
    \langle\Phi_8\rangle \equiv \phi_8\,{\rm diag}(-1,\,1,\,0)\,,
\end{equation}
with $\phi_8 = \mu_{{\sss E}\E}$.\footnote{More generally, $\phi_8/\mu_{{\sss E}\E}$ could differ from 1, and the relations
(\ref{eq:BM_parameters}) would be modified by factors of $\phi_8/\mu_{{\sss E}\E}$.  We allow for this more general form in the following.}\ \
The resulting contributions to the different gauge boson masses are shown in the third column of Table~\ref{tab:gauge_boson_masses}.
Purely diagonal $\langle\Phi_8\rangle$ VEVs leave the $A_3$ and $A_8$ gauge bosons massless.
To generate mass for $A_{8}$,  we introduce the VEV
\begin{equation} \label{eq:Phi3_VEV}
    \langle\Phi_3\rangle = \phi_3\,(0,0,1)^T ,
\end{equation}
which  further lifts the $A_{4,5,6,7}$ masses,  leaving only $A_3$ massless.
The numerical values of the contributions to $M_i^2/g'^2$ are shown in the fourth column of Table~\ref{tab:gauge_boson_masses}.
As we will see in the next section, phenomenological constraints on the leptophilic gauge bosons lead to the requirements $\phi_8\gtrsim 1$~TeV and $\phi_3\gtrsim 10$~TeV.

The $\Phi_6$ VEV contributes to all of the masses and causes mixing between the gauge bosons.
We assume that it is much smaller than the other two VEVs so that the mixing, as well as the shifts to the heavy masses, can be neglected, or computed perturbatively.
Therefore the $\langle\Phi_6\rangle_{3,3}$
element directly determines $M_3$, and thereby  the new physics contribution to $(g-2)_\mu$.

\begin{table}[t]\centering
\setlength\tabcolsep{5pt}
\def1.5{1.5}
\begin{tabular}{|c|c|c|c|c|}\hline
\begin{tabular}[c]{@{}c@{}}Gauge\\boson\end{tabular} & \begin{tabular}[c]{@{}c@{}}Flavor\\structure\end{tabular} & $\Phi_8$ & $\Phi_3$ & $\Phi_6$\\
\hline
$A_{1,2}$ &$\phantom{1\over\sqrt{3}}$\scalebox{0.6}{ $\begin{pmatrix} 0 & * & 0\\ * & \phantom{-}0\phantom{-} & 0\\ 0 & 0 & 0\end{pmatrix}\begin{array}{c}
\tau\\ \mu\\ e\end{array}$}
    & $4\,\phi_8^2$ & \ding{55} & \ding{51} \\
\hline
$A_{3}$ & $\phantom{1\over\sqrt{3}}$
\scalebox{0.6}{$ \begin{pmatrix} 1 & 0 & 0\\ 0 & -1\phantom{-} & 0\\ 0 & 0 & 0\end{pmatrix}\quad\;\;\;$}
& \ding{55} & \ding{55}  & \ding{51}\\
\hline
$A_{4,5}$ & $\phantom{1\over\sqrt{3}}$\scalebox{0.6}{$\begin{pmatrix} 0 & 0 & *\\ 0 & \phantom{-}0\phantom{-} & 0\\ * & 0 & 0 \end{pmatrix}$\quad\;} & $\phi_8^2$ & $\sfrac12\, \phi_3^2$ & \ding{51}\\
\hline
$A_{6,7}$ & $\phantom{1\over\sqrt{3}}$\scalebox{0.6}{$\begin{pmatrix} 0 & 0 & 0\\ 0 & \phantom{-}0\phantom{-} & *\\ 0 & * & 0 \end{pmatrix}$\quad\;} & $\phi_8^2$ & $\sfrac12\, \phi_3^2$ &\ding{51}\\
\hline
$A_{8}$ & ${1\over\sqrt{3}}$\scalebox{0.6}{ $\begin{pmatrix} 1 & \;\;0\; & 0\\ 0 & \;\;1\; & 0\\ 0 & \;\;0\; & -2\end{pmatrix}\quad\;\;$} & \ding{55} & $\sfrac23\, \phi_3^2$ & \ding{51}\\
\hline
\end{tabular}
\caption{Flavor structure of the couplings of the 8 leptophilic gauge bosons in the SU(3)$_\ell$ model, together with the contributions to $M_{i}^2/g'^2$, their squared masses divided by $g^2$, arising from the various scalar VEVs (see Eq.~\ref{eq:Phi8_VEV}).
The contributions from  $\Phi_6$ are small compared to those of $\Phi_3$ and $\Phi_8$ by construction, and they lead to mixing between the tabulated Lagrangian states.}
\label{tab:gauge_boson_masses}
\end{table}

Using Eq.~(\ref{eq:sextet_VEV}), the light gauge boson mass can be written as
\begin{equation}
    M_3 \sim g' {m_D^2\over y_{\sss N}\, m_{\nu_1}} \equiv
        g' \phi_6\,,
\end{equation}
where the exact proportionality factor depends on the absolute scale of neutrino masses and varies between $\sim 0.5$ and $1.5$ within the range of interest.
It will be shown in the next section that for values of $g'$ and $M_3$ that can explain the muon anomalous magnetic moment,  $\phi_6 = \mathcal{O}(20-200)\,$GeV.
If $y_{\sss N}\sim 1$, this is also the scale of the sterile neutrino masses,  whose mass matrix is $m_{\sss N} = y_{\sss N}\langle\Phi_6\rangle$.
In the following sections, we discuss the phenomenological consequences of the new particles predicted by the model.

\section{Phenomenology of the leptophilic gauge bosons}
\label{sect:pheno}
The phenomenology of gauge bosons associated with lepton number gauge symmetries has been widely studied; see e.g.~\cite{Foldenauer:2016rpi,Buras:2021btx} and references therein.
In the SU(3)$_\ell$ model presented here, neglecting the small mixing between gauge bosons,  there are six purely flavor-violating currents, mediated by $A_{1,2}$ for $\mu\leftrightarrow\tau$, $A_{4,5}$ for $e\leftrightarrow\tau$, and $A_{6,7}$ for $e\leftrightarrow\mu$.
The flavor-conserving $A_3$ couples to $\mu-\tau$, and $A_8$ to the $2e-\mu-\tau$ current.
These leptophilic gauge bosons have implications for a number of different observables.

\subsection{Muon anomalous magnetic moment}
\label{amusect}

By construction, $A_3$ is much lighter than the rest of the vector bosons, due to the symmetry-breaking pattern of the model.
This enables it to explain the current $4.2\sigma$ discrepancy between experimental measurements~\cite{Muong-2:2006rrc,Muong-2:2021vma} and SM predictions~\cite{Aoyama:2020ynm} of the anomalous magnetic moment of the muon.\footnote{A recent lattice evaluation of $(g-2)_{\rm SM}$~\cite{Borsanyi:2020mff} may help alleviate this tension, but it is in disagreement with the other SM predictions~\cite{Aoyama:2020ynm}.}\ \
A gauge boson $A'$ coupling to the muonic vector current with strength $g'$ contributes to $a_\mu = (g-2)_\mu/2$ at the one-loop level~\cite{Baek:2001kca} with
\begin{equation}
\Delta a_\mu = \frac{g'^2}{4\pi^2}\int_0^1 \mathop{\mathrm{{d}} x} \frac{m_\mu^2\, x\, (1-x)^2}{m_\mu^2 (1-x)^2 + m_{A'}^2 \,x}.
\end{equation}
This has the right sign to address the experimental anomaly and is phenomenologically viable as long as $A'$ couples negligibly to electrons, which is the case for $A_3$.
Such a $L_\mu-L_\tau$ gauge boson was searched for in the $e^+e^- \rightarrow 4\mu$ channel at BaBar~\cite{BaBar:2016sci} and through neutrino trident production at CCFR~\cite{Altmannshofer:2014pba}.
Taking the ensuing constraints into account, the $A_3$ gauge boson can explain the $(g-2)_\mu$ discrepancy as long as its mass lies in the $10\lesssim m_{A_3}\lesssim 200$~MeV range and the gauge coupling is $3\times 10^{-4}\lesssim g' \lesssim 10^{-3}$ \cite{Holst:2021lzm,Hapitas:2021ilr} as
shown in Fig.~\ref{fig:gminus2}.

\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{figures/SU3_gminus2_parameter_space.pdf}
\caption{Region of the gauge coupling versus mass parameter space for which a massive dark photon coupled to the $L_\mu-L_\tau$ current can explain the observed value of the muon $g-2$ factor.
The blue-dotted region shows the $2\sigma$ preferred values obtained from the recently reported measurement~\cite{Muong-2:2021vma} and SM calculations~\cite{Aoyama:2020ynm}. The other shaded areas correspond to constraints from Babar~\cite{BaBar:2016sci} (orange), CCFR~\cite{Altmannshofer:2014pba} (green), and cosmology~\cite{Escudero:2019gzq} (purple).}
 \label{fig:gminus2}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{figures/gauge_bosons_combined.pdf}
\caption{$95\%$ CL contours on the model parameters $\phi_8$ and $\phi_3$ from LEP dilepton analyses and lepton flavor universality constraints as labelled in the figure. The blue-dotted region shows the $2\,\sigma$ preferred values to address the CKM unitarity deficit, also known as the Cabibbo Angle Anomaly (CAA). The dotted orange line shows the edge of the preferred region for the slight $\sim 2\,\sigma$ excess in the LFU ratio~\eqref{eq:luv_2}.}
 \label{fig:lepton_universality}
\end{figure}

\subsection{Di-lepton searches at LEP}

The LEP experiments searched for $e^+e^-\to \ell^+\ell^-$
events beyond the Standard Model \cite{ALEPH:2013dgf}, parametrizing the new physics (NP) contributions by contact interactions of the form
\begin{equation}
    {4\pi\over\Lambda^2} J_{e,\alpha}
    \left[\sfrac12 J_e^\alpha + J_\mu^\alpha + J_\tau^\alpha
    \right],
\end{equation}
where $J_i^\mu$ is the vector current for charged leptons of flavor $i$.
Different limits on $\Lambda$ were derived depending on the combinations of final states observed.
However, the derived limits do not directly apply to states such as $A_8$ that couple with different strengths to different flavors.

However, Ref.\ \cite{ALEPH:2013dgf}  provides the observed differential distributions for the $e^+e^-\rightarrow \ell\ell$ processes studied by LEP, enabling us to
derive limits on nonabelian leptophilic gauge bosons
such as those in the present model.
For dimuon and ditau final states we use the data for the total cross-section $\sigma_{\rm T}$ and forward-backward asymmetry $A_{\rm fb}$ as a function of $\sqrt{s}$, while for the $e^+e^-$ final state we use the full averaged differential cross section given in~\cite{ALEPH:2013dgf}.
Employing a $\chi^2$ test statistic on the SM versus SM + NP models leads to the following bounds on the individual gauge boson masses at $95\%$ CL:
\begin{eqnarray}
    {M_{4,5}\over g'} &>& 4.6\,{\rm TeV},\nonumber\\
    {M_{6,7}\over g'} &>& 5.2\, {\rm TeV},\nonumber\\
    {M_8\over g'} &>& 8.9\,{\rm TeV}.
    \label{LEPbounds2}
\end{eqnarray}

We further construct the combined $\chi^2$ for all the aforementioned channels, including the effects of all relevant gauge bosons parametrized by the two model parameters $\phi_3$ and $\phi_8$.
Doing so leads to the $95\%$ CL constraints on the VEVs corresponding to the yellow contours in Fig.~\ref{fig:lepton_universality}.
As shown in Eq.~(\ref{LEPbounds2}), the strongest limits are on $A_8$, it being the only state coupling diagonally to the electron current.
Since the mass of $A_8$ comes solely from $\langle\Phi_3\rangle$, this constrains $\phi_3$ independently of the value of $\phi_8$, which explains the shape of the exclusion contour in Fig.~\ref{fig:lepton_universality}.

\subsection{Lepton flavor universality violation}

Vector bosons with flavor off-diagonal couplings can induce lepton decays of the form $\ell_i\rightarrow\ell_j \bar{\nu}\nu$, which interfere with the SM amplitudes and spoil the flavor-universal nature of these weak decays.
Following the prescription of~\cite{Buras:2021btx}, these effects can be measured using the amplitude ratios
\begin{equation}
    R(\ell_i\rightarrow\ell_j) = \frac{\mathcal{A}(\ell_i\rightarrow\ell_j \bar{\nu}\nu)}{\mathcal{A}(\ell_i\rightarrow\ell_j \bar{\nu}\nu)_{\rm SM}} = 1 + 2 \frac{g'^2}{g_2^2} \frac{m_W^2}{m_{A'}^2}\,,
\end{equation}
where $g_2$ denotes the SU(2)$_L$ gauge coupling and we have expanded to first order in $g'^2$, {\it i.e.,} keeping only SM-NP interference terms.
These can be confronted with experimental measurements of the ratios of partial widths in purely leptonic decays~\cite{HFLAV:2019otj},\footnote{The $\tau-\mu$ universality ratio can also be measured in semi-hadronic decays, but with a larger uncertainty that does not significantly strengthen the constraints.}
\begin{align}\label{eq:luv_1}
    \frac{\mathcal{A}(\tau\rightarrow\mu \bar{\nu}\nu)}{\mathcal{A}(\tau\rightarrow e \bar{\nu}\nu)} &= 1.0018\pm 0.0014 , \\
    \label{eq:luv_2}
    \frac{\mathcal{A}(\tau\rightarrow\mu \bar{\nu}\nu)}{\mathcal{A}(\mu\rightarrow e \bar{\nu}\nu)} &= 1.0029\pm 0.0014 , \\
    \label{eq:luv_3}
    \frac{\mathcal{A}(\tau\rightarrow e \bar{\nu}\nu)}{\mathcal{A}(\mu\rightarrow e \bar{\nu}\nu)} &= 1.0010\pm 0.0014 ,
\end{align}
with the correlations given in Ref.~\cite{HFLAV:2019otj}.
In our model, the $R$-ratios are completely determined by the $\Phi_3$ and $\Phi_8$ VEVs, which can be parametrized using $\phi_3$ and $\phi_8$ as defined in Eqs.~\eqref{eq:Phi8_VEV} and~\eqref{eq:Phi3_VEV}.
Applying the experimental constraints in Eqs.~\eqref{eq:luv_1}-\eqref{eq:luv_3} at $95\%$ CL leads to the exclusion limits shown in Fig.~\ref{fig:lepton_universality}.

\subsection{Cabibbo angle (or CKM) anomaly}

In addition to LFU violation, a modification of the $\mu\rightarrow e \bar{\nu}\nu$ decay rate changes the inferred value of the Fermi constant, which affects the determination of $V_{ud}$ from $\beta$ decays.
The latter measurements currently exhibit a $\sim 3 \sigma$~\cite{Belfatto:2019swo,Grossman:2019bzp,Shiells:2020fqp,Seng:2020wjq} tension with the SM predictions, which could be eased by a NP contribution at the level of~\cite{Buras:2021btx}
\begin{equation}
R(\mu\rightarrow e) = 1.00075\pm 0.00025\,.
\end{equation}
This experimental anomaly was originally formulated as a breakdown of unitarity in the first row of the CKM matrix, and is thus commonly referred to as the Cabibbo angle anomaly (CAA).
Since the sum $|V_{ui}|^2$ appears to be less than unity, one interpretation is that weak decays of nuclei are suppressed relative to those of leptons.
The additional contributions of $A_{6,7}$ gauge interactions to muon decays can explain the anomaly if the effective Fermi constant $G_\mu$ inferred from muon decays is increased by a factor of $(1+\delta_\mu)$, with $\delta_\mu = 7\times 10^{-4}$~\cite{Belfatto:2019swo}.
It can be achieved in the present model with
\begin{equation}
    {M_{6,7}\over g'}\cong 13\,{\rm TeV}\,
    \label{CAAeq}
\end{equation}
due to the additional neutral current contribution to $\mu\to e\nu_\mu\bar\nu_e$.
This is consistent with the LEP and LFU bounds, as can be seen in Fig.~\ref{fig:lepton_universality}, where the $2\sigma$ preferred region to explain the CAA anomaly corresponds to the blue-dotted contour.
If a NP origin of the anomaly were to be confirmed, it could be explained by the nonabelian model with VEVs $\phi_3\sim10-20$~TeV and $\phi_8\sim1-15$~TeV.\footnote{The LFU ratio~\eqref{eq:luv_2} also deviates by $2\sigma$ from the SM prediction, which would single out an even more restricted range $\phi_8\sim1-5$~TeV for the $\Phi_8$ VEV as indicated by the orange dotted line in Fig.~\ref{fig:lepton_universality}.}

\subsection{Gauge boson mixing}
The $\Phi_6$ VEV generates mixing between all the gauge boson Lagrangian states in the model, with mixing angles of order $\sin\xi_{ij}\sim(\phi_6/\phi_{3,8})^2\lesssim 10^{-4}$ for the VEVs of interest here.
To see that these are phenomenologically harmless, we consider the most sensitive observable probing transitions enabled by them, $\mu\rightarrow e \gamma$, whose branching ratio is constrained to be $\mathrm{Br}(\mu\rightarrow e\gamma)\leq 4.2\times 10^{-13}$ \cite{SINDRUMII:2006dvw,BaBar:2009hkt,MEG:2016leq}.
This process is generated at one loop via $A_{1,2}-A_{4,5}$ mixing (internal $\tau$ lepton) or $A_{6,7}-A_{8}$ mixing (internal electron and muon).
Following Ref.\ \cite{Buras:2021btx}, we derive the bounds
\begin{align}
\frac{M_{i,j}}{g'} &\geq 3.7\,\mathrm{TeV}\,\sqrt{ \frac{\sin\xi_{ij}}{10^{-4}} }\,,\nonumber\\
\frac{M_{k,8}}{g'} &\geq 48\,\mathrm{GeV}\,\sqrt{ \frac{\sin\xi_{k8}}{10^{-4}} }\,,
\end{align}
where $i=1,2$, $j=4,5$, and $k=6,8$.
The first limit is stronger than the second one by a factor of $\sqrt{m_\tau/m_e}\sim 60$ due to a cancellation of the $\mu$-exchange contribution in the latter.
In any case, neither of the two compete with the previously discussed LEP or LFU bounds.

\section{Phenomenology of the heavy neutral leptons}\label{sec:HNL}

The spectrum of heavy neutral leptons $N_i$, and their mixing with the light neutrinos $\nu_i$, is largely dictated by the measured $\nu_i$ masses and mixings, once the lightest mass $m_{\nu, l}$ is specified.
In particular, there is a strict relation between pairs of $m_{\nu_i}$ and $M_{N_i}$ eigenvalues,
\begin{equation}
    m_{\nu_i}M_{N_i} = m_D^2, \quad {i=1,2,3}\,,
    \label{mMrel}
\end{equation}
where $m_D$ is the universal neutrino Dirac mass.
Because of SU(3)$_\ell$ symmetry, $m_D$ is just a number, not a matrix.
This implies that the logarithmic range of $N_i$ masses is the same as that of the light neutrinos, as is represented schematically in Fig.~\ref{fig:mass_hierarchy}.
The mixing angles between $N_i$ and the $\nu_i$ are diagonal in the basis of mass eigenstates,
and are approximately
\begin{equation}
    U_i = {m_D\over M_{N_i}} = \sqrt{\frac{m_{\nu_i}}{M_{N_i}}}\,.
\end{equation}
Thus, heavier HNLs are more weakly mixed and the flavor mixing pattern of each one exactly matches that of the corresponding active neutrino, as is illustrated in Fig.~\ref{fig:mass_hierarchy}.
We now proceed to quantify this picture more precisely.

\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{figures/Mass_hierarchy.pdf}
\caption{Schematic representation of the mass and mixing hierarchies of the active neutrinos and the heavy neutral leptons in the gauged SU(3)$_\ell$ model, for both normal and inverted orderings.
The masses of the HNLs are predicted to be inversely proportional to the light neutrino ones, so that they are distributed following the reflected pattern represented in the figure.
The mixings of each HNL mass eigenstate with the light neutrino flavors, represented by colors, exactly match the flavor admixture of the corresponding active neutrino, which is dictated by the PMNS matrix.}
 \label{fig:mass_hierarchy}
\end{figure}

\subsection{HNL masses}
There are three parameters that determine the HNL mass and mixing spectrum, together with the $\Phi_6$ contribution to gauge boson masses: the Yukawa couplings $y_N$ and $y_{\L{\sss N}}$ and the mass of the lightest active neutrino $m_{\nu,l}$.
This state is labeled as $\nu_1$ in the normal
mass hierarchy (NO), and $\nu_3$ in the inverted one (IO),
as shown in Fig.\ \ref{fig:mass_hierarchy}.
The choice of hierarchy as well as the relative signs of active neutrino masses affects the HNL mixings.

\begin{figure}[t]
\centering
\centerline{
\includegraphics[width=0.49\textwidth]{figures/Ratio_HNL_masses.pdf}}
\caption{Ratios of the absolute value of the lighter two HNL masses to the heaviest one, as a function of the mass of the lightest active neutrino.
For normal ordering, there is a single HNL that is substantially lighter than the other two, while for inverted ordering (IO) there are two nearly degenerate HNLs that are lighter than the third one.  For IO, the curves terminate when the cosmological limit on $\sum_i |m_{\nu,i}|$ becomes saturated. }
 \label{fig:fs}
\end{figure}

From Eq.~\eqref{eq:sextet_VEV}, the sextet VEV can be written as
\begin{equation}
    \langle \Phi_6 \rangle = - \frac{y_{\L{\sss N}}^2}{y_N} \, \frac{\bar{v}^2}{m_{\nu,l}} \, U_{\rm\scriptscriptstyle PMNS}\, \hat{D}^{-1}\, U_{\rm\scriptscriptstyle PMNS}^T\,,
\end{equation}
where $\hat{D}$ is the diagonal active neutrino mass matrix rescaled by $1/m_{\nu,l}$ and $U_{\rm\scriptscriptstyle PMNS}$ is the PMNS matrix with columns arranged according to our flavor ordering scheme $(\tau,\,\mu,\,e)$.
The $\Phi_6$ VEV determines the ratio $M_{3}/g'$ for the $\mu-\tau$ gauge boson via Eq.~\eqref{gbmasses}, and thus its ability to explain the $(g-2)_\mu$ anomaly.
As was noted in the previous section,
\begin{equation}
    \frac{M_3}{g'}\in (18\,\mathrm{GeV},\,200\,\mathrm{GeV})
\end{equation}
is the preferred range at the $2\sigma$ level, corresponding to $g'\in(3.7,\,13)\times10^{-3}$.
The precise value of $M_3$ predicted in this way has a weak dependence on the relative sign of the active neutrino masses, which amounts to a factor of $\sim 2$ for $m_{\nu,l}=0.03$~eV and disappears for  $m_{\nu,l}\ll 0.03$\,eV.

\begin{figure*}[t]
\centering
\centerline{
\includegraphics[width=0.49\textwidth]{figures/HNL_mixing_NO.pdf}\hfil
\includegraphics[width=0.49\textwidth]{figures/HNL_mixing_IO.pdf}}
\caption{Dark blue: predictions from the gauged SU(3)$_\ell$ lepton flavor model for the mass and mixing of light HNLs.
The mixings are normalized to the muon \emph{(left)} or electron \emph{(right)} one as indicated in the insets.
We show bounds (grey) from terrestrial searches as compiled in~\cite{Alekhin:2015byh} as well as the BBN limit (green) computed following~\cite{Boyarsky:2020dzc,Bondarenko:2021cpc}.
For the normal  mass hierarchy \emph{(left)}, there is a single light HNL with suppressed $e$ mixing and we therefore show terrestrial searches based on the $\mu$ mixing. For inverted hierarchy \emph{(right)}, there are two quasi-degenerate light HNLs with fairly universal mixings and the leading terrestrial constraints therefore come from $e$ mixing.}
 \label{fig:HNL_mixing_BBN}
\end{figure*}

The mass matrix for the HNLs is in turn determined by the sextet VEV as
\begin{eqnarray}\label{eq:HNL_mass}
    M_N &=& y_N\langle\Phi_6\rangle\nonumber\\
    &=& - y_{\L{\sss N}}^2  \frac{\bar{v}^2}{m_{\nu,l}} \, U_{\rm\scriptscriptstyle PMNS}\, \hat{D}^{-1}\, U_{\rm\scriptscriptstyle PMNS}^T\,.
\end{eqnarray}
Using the top line in Eq.~\eqref{eq:HNL_mass}, the $(g-2)_\mu$-preferred values for $\langle\Phi_6\rangle$ result in the prediction that the scale of HNL masses lies in the $\mathcal{O}(1-100) \mathrm{GeV}$ range for $\mathcal{O}(1)$ values of $y_N$.

Eq.~\eqref{eq:HNL_mass} implies that the PMNS matrix also represents the unitary transformation that diagonalizes $M_N$.
As a consequence, each HNL mass eigenstate is strictly inversely proportional to the corresponding active neutrino mass,
\begin{equation}
    M_{N_i} = - y_{\L{\sss N}}^2  \frac{\bar{v}^2}{m_{\nu_i}}\,.
\end{equation}
This means that in the normal ordering case, there is one light and two heavy HNLs, while for inverted ordering there are two light and one heavy HNLs.
This is schematically represented in Fig.~\ref{fig:mass_hierarchy}, while the quantitative mass ratios are shown in Fig~\ref{fig:fs} as a function of $m_{\nu,l}$.
The largest considered value of $m_{\nu, l}=0.03\, (0.015)$~eV in the NO (IO) case, roughly saturates the cosmological constraint on the sum of the absolute value of neutrino masses\footnote{A recent evaluation~\cite{DiValentino:2021hoh} finds a tighter limit $\sum |m_{\nu_i}|\leq0.09$~eV, which would compromise the viability of the IO scenario but would not significantly affect the NO one.} $\sum |m_{\nu_i}|\leq0.12$~eV~\cite{Planck:2018vyg}.

\subsection{HNL mixings}

\begin{figure*}[t]
\centering
\centerline{
\includegraphics[width=0.49\textwidth]{figures/HNL_yN_yLN_no.pdf}\hfil
\includegraphics[width=0.49\textwidth]{figures/HNL_yN_yLN_io.pdf}}
\caption{Regions of the $y_N$-$y_{\L{\sss N}}$ parameter space for which the gauged SU(3)$_\ell$ model can address the $(g-2)_\mu$ anomaly while satisfying all other phenomenological constraints, for various values of the lightest neutrino mass as labeled in the figure.
The blue-dotted regions show where the $A_3$ gauge boson mass and coupling $g'$ are consistent with the $2\sigma$ preferred values shown in Fig~\ref{fig:gminus2}.
The green region is the BBN exclusion from Fig.~\ref{fig:HNL_mixing_BBN} and the orange one
is forbidden by perturbative unitarity of the HNL Yukawa coupling, $y_N\leq\sqrt{4\pi}$.
Left (right) panel corresponds to the  NO (IO) active neutrino mass hierarchy, respectively.
}
\label{fig:HNL_yN_yLN}
\end{figure*}

Transforming to the HNL mass eigenbasis (denoted by primes), it is straightforward to show that the mixing between the light neutrino flavors and the HNLs is given by
\begin{eqnarray}
    \nu_\alpha &\cong&\phantom{-} \nu'_\alpha + U_{\alpha i}\, N_i'\nonumber\\
    N_i &\cong& -U_{\alpha i}\,\nu'_\alpha + N'_i
    \,,
\end{eqnarray}
where
\begin{eqnarray}
    U_{\alpha i} = U^{\mathrm{\scriptscriptstyle PMNS}}_{\alpha i} \frac{m_{\nu_i}}{y_{\L{\sss N}}\bar{v}}\,.
\end{eqnarray}
The mixing structure is completely determined by the PMNS matrix: the mixing pattern of the HNL mass eigenstate $i$ is proportional to the flavor composition of the corresponding active neutrino mass eigenstate (see Fig.~\ref{fig:mass_hierarchy} for a graphical representation).
The approximate scale of the mixing is $U =  (m_\nu/M_N)^{1/2}\sim 10^{-6}-10^{-5}$ for HNLs in the $1-100$~GeV mass range.

While the HNL masses and mixings predicted by the
SU(3)$_\ell$ model are out of reach for present and upcoming terrestrial experiments, they have significant implications for cosmology.
In particular, HNLs decaying too close to the time of Big Bang Nucleosynthesis (BBN) can affect the predictions for light element yields~\cite{Dolgov:2000pj,Dolgov:2000jw}.
The strongest current constraints come from the hadronic decays of the HNLs, which inject light mesons into the plasma that can act to imbalance the proton-to-neutron ratio at BBN~\cite{Boyarsky:2020dzc}.

Following the prescription of~ Ref.\ \cite{Bondarenko:2021cpc} together with the width calculations of Ref.~\cite{Bondarenko:2018ptm}, we obtain the constraints shown in Fig.~\ref{fig:HNL_mixing_BBN} for normal and inverted mass hierarchies.
For NO, there is a single light HNL whose mixings match those of $\nu_3$ (see Fig.~\ref{fig:mass_hierarchy}).
As a consequence, its $e$ mixing is suppressed by a factor of $\sim 5$ compared to the $\mu$ and $\tau$ mixings.
Then the BBN bound gives $M_{N_3}\gtrsim2.4$~GeV or $U^2_{\mu3}\lesssim 1\times 10^{-11}$, which translates into $y_{\L{\sss N}}\gtrsim 6.4\times10^{-8}$.
IO has two light HNL with quasi-degenerate masses and mixings dictated by the flavor composition of $\nu_1$ and $\nu_2$ (see Fig.~\ref{fig:mass_hierarchy}).
The BBN bound arising from the combined effect of both HNLs leads to $M_{N_1}\sim M_{N_2}\gtrsim0.95$~GeV and $U^2_{e1}\lesssim 4\times 10^{-11}$, implying that $y_{\L{\sss N}}\gtrsim 4.0\times10^{-8}$.

The BBN constraint on $y_{\L{\sss N}}$, together with the bound $y_N\leq\sqrt{4\pi}$ arising from perturbative unitarity in $N\bar{N}\rightarrow N\bar{N}$ scattering, rule out
regions of the $y_{\L{\sss N}}$ versus $y_{\sss N}$ parameter space, including some relevant for explaining the  $(g-2)_\mu$ anomaly.
They are shown in Fig.~\ref{fig:HNL_yN_yLN} for several
choices of the lightest active neutrino mass and for the two possible neutrino mass orderings.
Here we assume that $m_{\nu_3}$ has opposite sign to $m_{\nu_1}$ and $m_{\nu_2}$, which gives the model the greatest latitude for consistency.
Other choices of relative signs differ by factors  $\lesssim 2$ for the extent of $(g-2)_\mu$ preferred regions.

Several conclusions can be drawn from  Fig.~\ref{fig:HNL_yN_yLN}.
The minimum value for $m_{\nu,l}$ that is compatible with the experimental measurement of $(g-2)_\mu$ is $m_{\nu,l}\simeq 10^{-4}$~eV.
Consequently, the maximum possible mass for the heaviest HNL is $M_{N_1}\simeq 1.2$~TeV for NO and $M_{N_3}\simeq 1$~TeV for IO.
At the other extreme, values of $y_{\L{\sss N}}$ as large as $10^{-6}$ are possible if the mass of the lightest neutrino saturates the cosmological upper bound.
This reveals a novel connection between the scale of neutrino masses and the anomalous magnetic moment of the muon within the SU(3)$_\ell$ paradigm.
It constitutes a unique prediction of the proposal, which could exclude it definitively in light of more precise experimental data.

\section{Conclusions}
\label{sect:conc}

In this work, we have proposed a framework for gauging the vectorial SU(3) family symmetry of lepton flavor in a minimal way.
The construction can consistently reproduce the observed patterns of charged lepton masses and neutrino masses and mixings.
The new particle content of the model, in addition to the eight leptophilic gauge bosons, consists of a right-handed neutrino triplet and a triplet of vectorlike isosinglet charged lepton partners. In addition, three scalar multiplets in the fundamental, symmetric two-index, and adjoint representations of SU(3)$_\ell$ provide the symmetry breaking required to reproduce the observed lepton masses and mixings, and allowed gauge boson masses.
The charges and interactions are collected in Table~\ref{tab1} and Eq.~\eqref{eq:Lagrangian}, respectively.

One of the new leptophilic gauge bosons, the one associated with the $L_\mu-L_\tau$ current, is taken to lie in the $10-200$~MeV range,
being parametrically lighter than the others.
This allows it to address the current discrepancy between experimental measurements and SM predictions of the anomalous magnetic moment of the muon.
Moreover, the Cabibbo angle anomaly can be explained by
the pair of vector bosons that couple off-diagonally to $e\mu$, with masses in the $1-10$~GeV range, while complying with all other LEP and lepton-flavor universality constraints.

Interestingly, the relative lightness of the $L_\mu-L_\tau$ gauge boson is shown to be linked to a low ($\sim 1$-$100\,$GeV) mass scale for the right-handed neutrinos that participate in the seesaw mechanism.
As shown schematically in Fig.~\ref{fig:mass_hierarchy}, the masses of the HNL eigenstates are inversely proportional to the light neutrino ones, and their mixings with the active flavors match the flavor composition of the corresponding light neutrino mass eigenstates.
These two connections lead us to predict that the lightest active neutrino cannot be arbitrarily light, but must rather have a mass larger than $\sim 0.1\,$~meV.

Once fit to the observed lepton and neutrino properties
(including $(g-2)_\mu$), the gauged SU(3)$_\ell$ scenario is highly predictive and makes potentially testable forecasts for existing and upcoming astrophysical and particle physics experiments.
Focusing on the HNL sector, there are three main avenues that complement each other in testing the parameter space in Fig.~\ref{fig:HNL_yN_yLN}: (\emph{i}) more precise experimental measurements and SM predictions for the muon anomalous magnetic moment;
(\emph{ii}) refined measurements and theoretical calculations of the light-element yields and the number of relativistic degrees of freedom at Big Bang Nucleosynthesis, and (\emph{iii}) improved determinations of the absolute scale of neutrino masses from cosmology or terrestrial experiments like KATRIN~\cite{Aker:2021gma}.
In the charged lepton sector, electroweak precision constraints on lepton mixing with heavy vectorlike partners, needed for realistic lepton mass generation, constitute a powerful test of the model.
These lepton partners might be discoverable at the High-Luminosity LHC or other future colliders.
In contrast, although they are energetically accessible, the new scalar fields in the model are weakly coupled to the SM states, making their possible detection difficult.

Although it is not mandatory for the model, it is possible to introduce an additional fermionic vectorlike triplet $\Psi_i$ as a dark matter candidate.
As in the $U(1)_{\L_\mu-\L_\tau}$ model of Refs.~\cite{Foldenauer:2018zrz, Holst:2021lzm}, the observed relic density could be achieved through resonantly enhanced
$\Psi_{\mu/\tau}\bar\Psi_{\mu/\tau}\to \mu^+\mu^- + \nu_{\mu,\tau}\bar\nu_{\mu,\tau}$ annihilations.
The obstacle in the SU(3)$_\ell$ model is that the $\Psi_e$ component would not be sufficiently
suppressed unless its mass also happens to
allow for resonantly enhanced (co)annihilations through the exchange of the heavier gauge bosons.
This would require additional model building, which we do not pursue here.

 A remaining challenge for future study is to construct a suitable scalar potential leading to the required $\Phi_3$, $\Phi_6$, and $\Phi_8$ VEVs. Although the preliminary study in Appendix~\ref{app:VEVs} makes it plausible that the desired symmetry breaking pattern can be achieved, further investigation is required
 to prove it.

It is in principle possible to construct a similar model for quark masses and mixing based on a spontaneously broken SU(3)$_B$ symmetry.
This extension, combined with the present proposal including mixing between the SU(3)$_\ell$ and SU(3)$_B$ gauge bosons, might explain the accumulating evidence for lepton-universality violation in $b\rightarrow s\ell^+\ell^-$ decays~\cite{Geng:2021nhg}.
Work in this direction is underway.

\newpage
{\bf Acknowledgments.}  We thank B.\ Grinstein, A.\ Crivellin, W.\ Altmannshofer, and F. Kahlhoefer for helpful correspondence.
This work was supported by NSERC (Natural Sciences
and Engineering Research Council, Canada).
G.A. is supported by the McGill Space Institute through a McGill Trottier Chair Astrophysics Postdoctoral Fellowship.

\bibliographystyle{utphys}


\section{Introduction}

Generative Adversarial Models (GANs)\cite{Goodfellow:2014:GAN:2969033.2969125} have been found to produce images of very high quality on some datasets \cite{DBLP:conf/iclr/KarrasALL18, DBLP:journals/corr/abs-1812-04948}. However, their results on other datasets, while impressive, still lag behind \cite{DBLP:journals/corr/abs-1809-11096}. This raises the question whether GANs are indeed the right choice to model some distributions. This paper aims to test the distribution learning ability of GANs by evaluating them on synthetic datasets.

\subsection{Related works and Contributions}
It has been proposed in recent work that \say{a high number of classes is what makes ImageNet \cite{imagenet_cvpr09} synthesis difficult for GANs} \cite{pmlr-v70-odena17a}. Indeed, GANs have been able to produce very high quality images on CelebA \cite{DBLP:journals/corr/abs-1812-04948, DBLP:conf/iclr/KarrasALL18} while results on Imagenet are not so impressive \cite{DBLP:journals/corr/abs-1809-11096}. Because distributions of natural images are complex, in this work, we focus our attention on synthetically generated datasets. We study the learnability of commonly encountered distributions in low dimensional space and the impact of discontinuity. Additionally, we evaluate a specific aspect of learning high dimensional image distributions, counting similar objects in a scene. This constitutes an important part of learning latent space representations of images since for an image to be semantically well-formed, certain objects must obey certain numerical constraints (for example, number of heads on an animal).

Our evaluation is performed on synthetic point and image datasets. To our knowledge, the only instance of synthetic image datasets used for GAN evaluation have been to learn manifolds of convex polygons (specifically triangles) \cite{Lucic:2018:GCE:3326943.3327008}. Although, we also use polygons as a testbed for our experiments, we focus on learning a manifold with multiple polygons where their number is fixed.

Our contributions are as follows:
\begin{enumerate}
    \item We show via experiments on synthetic datasets that commonly found distributions are learnable by GANs. We also highlight that distributions with gaps in support may be difficult to learn without using a mixture of generators.
    \item We empirically evaluate whether GANs can learn semantic constraints on objects in images in high dimensional space. Specifically, we test a GAN's ability to count an object that is repeated in an image.
    \item We underline a possible tension between generalization ability of GANs and their learning capabilities.
\end{enumerate}

\section{Experimental Setup}
In this section, we describe the setup of our experiments which include details about the datasets generated, architectures used and the reasoning behind them.

\subsection{Datasets}
We generate two kinds of datasets (each with 5000 examples) for our evaluation : point datasets, where each sample is a point in $R^n$ and image datasets with each image containing a fixed number of polygons. We use 4 point datasets in our evaluation : Mixtures of Gaussians, Concentric Circles, S-shape curves and Swiss rolls. The first two are 2D while the latter two are in 3D space. This choice was made to enable us to visualize the learned distribution. For each of these four settings, we experimented with three variants, each containing a different amount of noise.

\begin{figure}
\centering
\begin{minipage}{.45\linewidth}
  \includegraphics[width=\linewidth]{Unknown-4.png}
  \caption{ S-Curve Distribution}
  \label{img4}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.45\linewidth}
  \includegraphics[width=\linewidth]{Unknown-5.png}
  \caption{Swiss Roll Distribution}
  \label{img5}
\end{minipage}
\end{figure}

To evaluate high dimensional learning, we generate image datasets which are mixtures of polygons. We created three datasets with each image containing: 1 square of size 4x4 (called Squares 1-4), 3 squares of size 4x4 (called Squares 3-4) and a mixture of two triangles and two circles (called CT2). All the datasets contain images of size 28x28 with the third one containing 3 channels. Additionally, for the first two datasets, all squares are non-overlapping and have edges which are axis-aligned. CT2, on the other hand, contains overlapping polygons. In each dataset, the number of objects is fixed and the only varying quantity is their position (which varies with a Gaussian distribution). For the square datasets, even the rotation and shape of the objects is held constant. Hence, the only source of variation is their position. Some examples are shown in Figures \ref{img1}, \ref{img2}, \ref{img3}.

\begin{figure}
\centering
\begin{minipage}{.25\linewidth}
  \includegraphics[width=\linewidth]{Unknown-7.png}
  \caption{Image from the Squares 1-4 dataset}
  \label{img1}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.25\linewidth}
  \includegraphics[width=\linewidth]{Unknown-8.png}
  \caption{Image from the Squares 3-4 dataset}
  \label{img2}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.25\linewidth}
  \includegraphics[width=\linewidth]{Unknown-9.png}
  \caption{Image from the CT2 dataset}
  \label{img3}
\end{minipage}
\end{figure}

The main objective behind creating images with a fixed number of objects was to test whether GANs can learn to count the number of objects in a scene. More specifically, since GANs have shown impressive image generation capabilities on centered image datasets\cite{DBLP:conf/iclr/KarrasALL18}, we want to measure whether that performance can transfer to datasets with objects occuring at varying locations in the scene. Since the only varying quantity in the square datasets is the position, we would expect GANs with true distribution learning abilities to be able to produce images with the exact numbers of squares at different positions in the image.

Additional details about our data generation process can be found in the Appendix.

\subsection{Architectures}
We use two sets of architectures to train our models. For point datasets, we use a Vanilla GAN with a 3 layer MLP for both the generator and discriminator and another model with the same architecture with Wasserstein loss (enforced via gradient penalty) \cite{Gulrajani:2017:ITW:3295222.3295327}.

For image datasets, we use one model with a DCGAN-inspired \cite{DBLP:journals/corr/RadfordMC15} architecture and another model with the same architecture with Wasserstein loss (enforced via gradient penalty). We do not evaluate Vanilla GANs on our image datasets because they do not seem to be competitive with the other models in our experiments. Further architectural details (including choice of hyperparameters) are described in the Appendix.

\subsection{Experimental Details}
We used the Google Colaboratory environment (12GB RAM Nvidia Tesla K80) for all our experiments in this paper. All models with Wasserstein loss are trained with RMSProp \cite{Tieleman2012} with a learning rate of 0.00005. All others are trained with the ADAM \cite{DBLP:journals/corr/KingmaB14} optimizer with a learning rate of 0.0002. We train models for up to 150k training steps and stop earlier if the model reaches convergence earlier. Increasing training steps beyond 150k were not found to significantly improve sample quality. For each generator update, we update the discriminator 5 times for WGAN-GP inspired architectures. Additional details about our experiments and architectures can be found in the Appendix.

\section{Results and Analysis}
\subsection{Point Data}

\begin{figure}
\centering
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-10.png}
  \caption{Original distribution of Concentric circles}
  \label{img6}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{conc-circ.png}
  \caption{Learned Distribution after 150k steps. \textbf{Note:} this distribution gets better after more iterations but we show the one after 150k for homogeneity}
  \label{img7}
\end{minipage}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-12.png}
  \caption{Original distribution of 3 blobs}
  \label{img8}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-11.png}
  \caption{Learned Distribution after 150k steps}
  \label{img9}
\end{minipage}
\end{figure}

On mixtures of Gaussians and concentric circles, we find that both architectures seem to perform equally well. They seem to approach approximate convergence as both their discriminator's accuracy oscillates around 50\%. Since both datasets contain disconnected components, we find that both models are not able to model this discontinuity and as a result still produce samples which lie in between clusters of data. This may explain the oscillatory behaviour observed. Examples of 1000 samples from the real and fake distributions are shown in Figures \ref{img7}, \ref{img9}.

\begin{figure}
\centering
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-4.png}
  \caption{Original distribution of the shape S (minimal noise)}
  \label{img10}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-14.png}
  \caption{Learned Distribution after 140k steps (minimal noise)}
  \label{img11}
\end{minipage}
\end{figure}

\begin{figure}
\centering
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-5.png}
  \caption{Original distribution of the Swiss roll (minimal noise)}
  \label{img12}
\end{minipage}
\hspace{.05\linewidth}
\begin{minipage}{.4\linewidth}
  \includegraphics[width=\linewidth]{Unknown-13.png}
  \caption{Learned Distribution after 150k steps (minimal noise)}
  \label{img13}
\end{minipage}
\end{figure}

The inability to model discontinuity is understandable since the latent space is continuous and the neural network can be considered to be a continuous function approximator so the output has to be continuous as well. This virtually guarantees that some samples from the model will be necessarily "bad".

Next, we evaluate the S-curve and Swiss roll distributions. Traditionally, mixtures of Gaussians have been the toy distribution of choice for GAN evaluation. However, we find that both distributions are learned fairly faithfully but increasing noise can cause separate surfaces in the distribution to be merged. For example, with increased noise, the S shape in the S-curve can become an 8 or the Swiss roll may look like a circle (samples in Appendix).

This phenomenon, in our experiments, seems to be worse (in terms of losing shape) for Swiss rolls than S-curves. We hypothesize this may be due to overlapping surfaces being closer in the Swiss roll distribution making it more sensitive to noise. This dependence of sensitivity to noise on the underlying distribution suggests that GANs may not be suitable to modelling certain distributions in noisy environments and alternative generative models may need to be explored.

\subsection{Image Datasets}

During training, the first model (without Wasserstein loss) seems to converge after about 74k steps while the other one doesn't seem to converge. Inspite of this difference, we do not find a major difference in image quality from the two models after 150k steps.

We find that both models fail to learn to count on the Squares 3-4 dataset. Our models can potentially produce anything between 0 and 5 squares. Some random samples can be seen in \ref{img14}, \ref{img15}.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{Unknown-15.png}
    \vspace*{-1.2cm}
    \caption{Samples after 150k steps from DCGAN (converged)}
    \label{img14}

    \vspace*{0.5cm}

    \centering
    \includegraphics[width=\linewidth]{Unknown-16.png}
    \vspace*{-1.2cm}
    \caption{Samples after 150k steps from WGAN-GP}
    \label{img15}
\end{figure}

This raises an important question about the learning capability of GANs. Is what we are seeing just poor learning or should it be instead viewed as generalization? For example, in natural image datasets, say faces, we might see GANs produce images with completely new hair placement. When these images look "natural" to the observer, they might see it as generalization. However, when that image does not look plausible, we would call it poor learning. As it stands, there seems to be no clear demarcation between poor learning outcomes and generalization.

Since the only source of variation in our image datasets is location, we would like to see the GAN learn that there are 3 squares in each image and then generalize their position. However, currently, we have no way of enforcing this constraint on the number or shape of objects. Quantifying this apparent tradeoff between learning and generalization is an interesting avenue for further work. In fact, there might not even be tradeoff
and GANs may fundamentally be unable to learn distributions of such a nature. We leave the evaluation of these hypotheses as future work.

We do not focus our discussion excessively on shape because for more natural datasets, shape is more variable and is not rigid, like in our case. For example, we know that cats have 4 legs, but, the legs may not look the same in every image. A natural objection would be that a similar reasoning would be applicable to the count of objects i.e. some of these legs may be occluded in the image and, therefore, it may appear to a GAN that a cat could have 3 legs. Thus, GANs producing cats with less than 4 legs could be justified.

That is precisely why we have chosen non-overlapping squares for our experiment. Since there are no occlusions, the GAN should learn that in every image, there are supposed to be exactly 3 squares. Even in real world datasets, it may not be possible to collect a dataset which has cats whose legs have the same shape. However, it is possible to collect a dataset with cats having four legs in all images (no occlusions).

Interestingly, both models seem to learn that there is only one square in the 1-4 dataset i.e. when trained on the Squares 1-4 dataset, most samples seem to include just 1 square of size which visually looks close to 4x4. Admittedly, this visual test of similarity may not be enough to ascertain whether the shape and size of the square corresponds exactly to a 4x4 square. However, during training with images of 1 square, we observed that if we increase the size of the square i.e. instead of using 4x4 use 16x16 (Squares 1-16), the likelihood of having multiple squares in the image reduces.

We also observe that all squares generated seem to be axis aligned. Therefore, GANs seem to have no problem learning orientation but cannot seem to enforce counting constraints.

We also investigated whether a GAN can leverage the fact that it has learnt what a square looks like on Squares 1-4. In this experiment, we transferred the weights from that trained model and fine-tuned it on the Squares 3-4 dataset. We did not observe a noticeable improvement in image quality. If anything, image quality tends to get worse.

Samples from models trained on CT2 and additional samples from our models are included in the Appendix. We observed that the GANs did not sufficiently reproduce circles and triangles when trained on CT2. Since CT2 contains overlaps and multiple types of polygons, we consider this to be a more challenging dataset to model.

\section{Conclusion}

In this paper, we present the phenomenon of GANs being unable to count. We support this hypothesis with experiments on synthetic datasets where the count of similar objects in a scene is kept constant while their location is varied. We find that in their current form GANs are unable to learn semantic constraints even in the absence of noise introduced by natural image datasets. We also emphasize the fine line between generalization and good learning outcomes in GANs.
Additionally, we conduct experiments on non-image data where we conclude that GANs tend to have difficulty learning discontinuous distributions which might necessitate the usage of mixtures of generators. A thorough evaluation of such an approach is left as future work.