# Title

B. L. Gonçalves Departamento de Física and CFTP, Instituto Superior Técnico,
Universidade de Lisboa, Lisboa, Portugal Centro de Física Teórica e
Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande,
Edifício C8, 1749-016 Lisboa, Portugal Matthew Knauss High Energy Theory
Group, William & Mary, Williamsburg, VA 23187, USA Marc Sher High Energy
Theory Group, William & Mary, Williamsburg, VA 23187, USA

# Lepton Flavor Specific Extended Higgs Model

B. L. Gonçalves Departamento de Física and CFTP, Instituto Superior Técnico,
Universidade de Lisboa, Lisboa, Portugal Centro de Física Teórica e
Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande,
Edifício C8, 1749-016 Lisboa, Portugal Matthew Knauss High Energy Theory
Group, William & Mary, Williamsburg, VA 23187, USA Marc Sher High Energy
Theory Group, William & Mary, Williamsburg, VA 23187, USA

###### Abstract

In extended Higgs models, a discrete symmetry is needed in the quark sector to
avoid tree-level flavor-changing neutral currents. However, this is not
necessarily the case in the lepton sector. We consider a model in which one
Higgs couples to quarks and three others couple to the electron, muon and tau,
respectively. This four-doublet model is presented with the full scalar
potential and the gauge and Yukawa couplings. The constraints from
boundedness, perturbativity and oblique parameters are incorporated as well as
constraints from meson-antimeson mixing, radiative $B$-decays and the diphoton
Higgs decay rate. We also consider bounds from searches for heavy neutral and
charged scalars at the LHC. Since the Standard Model Higgs couplings match
predictions very well, we focus on the alignment limit of the model. It is
shown that for a wide range of parameters, the lightest additional scalar,
pseudoscalar and charged scalar can have substantial decays into electrons and
muons (in contrast to the usual leptonic decays into taus). An interesting
signature in the neutral sector would be the production, through vector boson
fusion, of a pair of scalars, each of which decays into an electron or muon
pair.

##  I Introduction

The Higgs boson was initially discovered Aad:2012tfa ; Chatrchyan:2012xdj
through its decay into gauge bosons. Since then, the coupling of the Higgs to
third generation fermions has also been determined with increasing accuracy
Aad:2015vsa ; Chatrchyan:2014nva ; Aaboud:2018zhk ; CMS:2018nsn ; CMS:2018uxb
.

However, while there is evidence CMS:2020xwi of the Higgs decay into muons,
there remain large uncertainties and the discovery has not yet been made. This
leads one to ask if there are viable models in which the muon and tau couple
to different Higgs bosons. It is often claimed that models in which fermions
of a given charge couple to different Higgs bosons contain tree-level flavor
changing neutral currents (FCNC). However, the seminal papers of Glashow and
Weinberg Glashow:1976nt and of Paschos Paschos:1976ay explicitly referred to
the quark sector. As we will see, FCNC can be avoided in the lepton sector
even if different leptons couple to different Higgs bosons.

The first such model, called the muon-specific Two Higgs Doublet (2HDM) model,
was developed by Abe, Sato and Yagyu Abe:2017jqo (ASY). They use a $Z_{4}$
symmetry, under which the muon and tau have different quantum numbers, and
break this softly. The model has no tree-level FCNC and the Yukawa couplings
for the muon and tau are no longer simply proportional to their masses with
the proportionality coefficient being the same for all flavors: rather, the
ASY model can substantially enhance or suppress the muon interactions of
scalars relative to those with tau leptons. The purpose of their model was to
attempt an explanation of the muon g-2 anomaly, and for the parameters they
considered, the dimuon coupling of the 125 GeV Higgs is not suppressed. Their
model can address the g-2 anomaly, but only for a very narrow region of
parameter-space. A more detailed analysis was carried out in Ref.
Ferreira:2020ukv where the phenomenology of the model was studied.

The ASY muon specific 2HDM used a $Z_{4}$ discrete symmetry in which the left-
handed muon doublet and right-handed singlet have charge $i$ and $\Phi_{1}$
has charge -1. All other fields have charge +1. This then has $\Phi_{1}$
coupling to muons and $\Phi_{2}$ coupling to all other fermions. Ivanov and
Nishi have pointed out Ivanov:2013bka that the actual symmetry group of the
model is a softly broken $Z_{2}$ in which $\Phi_{1}$ and $\mu_{R}$ are
negative and with a $U(1)$ corresponding to muon number. This does not affect
the ASY Lagrangian. In this model, the mass matrix of the charged leptons
breaks into a $2\times 2$ submatrix, corresponding to $e-\tau$ and a $1\times
1$ corresponding to the muon. One might be concerned about how the PMNS matrix
is generated if the muon and muon neutrino mass matrices decouple. However,
even if the charged lepton and neutrino mass matrices are diagonal, one will
still obtain a PMNS matrix using the see-saw (type 1) mechanism. The light
neutrino mass matrix is then
$m_{ij}=(M_{D})_{ik}(M_{N})^{-1}_{kl}(M_{D})_{lj}$ where $M_{D}$ is the
diagonal Dirac neutrino mass matrix and $M_{N}$ is the superheavy Majorana
right-handed neutrino mass matrix. The latter is arbitrary and so the light
neutrino mass matrix is not diagonal, leading to a non-trivial PMNS matrix.
Note that this will not work in the quark sector.

In this paper, we take the ASY model one step further and suppose that each of
the charged leptons couples to a different Higgs doublet, which we will label
as $\Phi_{e},\Phi_{\mu}$ and $\Phi_{\tau}$. This can be achieved with a
$(Z_{4})_{e}\times(Z_{4})_{\mu}\times(Z_{4})_{\tau}$ symmetry in which
$L_{\ell}$ and $\ell_{R}$ have quantum number under $(Z_{4})_{\ell}$ of $i$
and the $\Phi_{\ell}$ has quantum number $-1$. Equivalently, one can replace
the $Z_{4}$ with $Z_{2}\times U(1)$ as discussed above - the Lagrangian in
either case is identical. To achieve a non-trivial PMNS matrix, the symmetry
must be softly broken in the superheavy Majorana neutrino mass matrix. The
simplest implementation of this model would be a 4HDM in which the fourth
Higgs $\Phi_{q}$ couples to the quarks. This is similar to the lepton-specific
model. Certainly one could have one of $\Phi_{\ell}$ be the same as
$\Phi_{q}$, leading to a 3HDM. However, if the $\Phi_{q}$ is $\Phi_{\tau}$,
then the resulting model is very similar to the muon-specific model - the only
difference being the very small interaction of the Higgs with the electron.
For simplicity, we assume they are separate. One could also adopt a type-II
structure, with 5HDM, but that brings in additional complications and the
type-II parameter space is much narrower than the type-I. So, we will focus on
the 4HDM with $\Phi_{q},\Phi_{e},\Phi_{\mu}$ and $\Phi_{\tau}$.

Although there are hundreds of papers that study models with three Higgs
doublets, very few look at models with four. A recent paper with 4HDM in which
each Higgs couples to sets of fermions with similar masses has been proposed
Rodejohann:2019izm and a special ansatz, “singular alignment”, is needed to
suppress FCNC. A supersymmetric model Arroyo-Urena:2019lzv had one doublet
each coupling to up-quarks, down-quarks and leptons, with the fourth needed
for anomaly cancellation. A similar non-supersymmetric model was proposed
Cree:2011uy (with the fourth Higgs needed to relax some tight constraints). An
early discussion that mentions 4HDMs Ivanov:2011ae studied Abelian symmetries
in multidoublet models. There are also many studies of symmetries and vacuum
states of $N$ doublet models. An extremely extensive 2017 review of Ivanov
Ivanov:2017dad , with over 500 references, studied numerous extended scalar
sectors (including two doublet models, $N$ doublet models, singlet and triplet
extensions). Most relevant papers before that time are referred to in this
review. A more recent paper Faro:2020qyp looked at the interesting issue of
non-decoupling in multiscalar models. Related work deMedeirosVarzielas:2021zqs
dealt with large discrete symmetry groups in $N$ doublet models. Additionally,
the “Private Higgs” model of Porto and Zee Porto:2007ed ; Porto:2008hb had one
Higgs doublet for every fermion. In contrast to the model we propose, their
model had numerous discrete symmetries and included several “darkon” scalars.

We see that there are many models with more than two Higgs doublets in the
literature. All of these treat the three charged leptons identically, except
for the “Private Higgs” model which treated quarks and leptons in the same
manner. Yet the lepton sector is one of the most mysterious, given the large
mixing angles and small masses in the neutrino sector. In this paper, we are
treating the lepton and quark sectors differently, but not treating the
charged leptons identically, coupling each lepton to a separate Higgs.

In section II, the model is presented, including the full scalar potential and
the gauge and Yukawa couplings. In section III, we discuss the constraints on
the potential from boundedness and constraints from oblique parameters. In
section IV, two benchmark models are presented. In the first model, the
potential is divided into two $2\times 2$ subsections and in the second, the
full $4\times 4$ model is discussed in the experimentally indicated alignment
limit. Section V contains our results and conclusions.

##  II The Model

###  II.1 Scalar sector

The potential can be written as a sum of quadratic and quartic terms:
$V=V_{2}+V_{4}$. We allow for soft breaking of the discrete symmetry in the
quadratic terms:

$\displaystyle V_{2}=$ $\displaystyle\
m_{qq}^{2}\Phi_{q}^{\dagger}\Phi_{q}+m_{ee}^{2}\Phi_{e}^{\dagger}\Phi_{e}+m_{\mu\mu}^{2}\Phi_{\mu}^{\dagger}\Phi_{\mu}+m_{\tau\tau}^{2}\Phi_{\tau}^{\dagger}\Phi_{\tau}$
(1) $\displaystyle+$ $\displaystyle\
[m_{qe}^{2}(\Phi_{q}^{\dagger}\Phi_{e})+m_{q\mu}^{2}(\Phi_{q}^{\dagger}\Phi_{\mu})+m_{q\tau}^{2}(\Phi_{q}^{\dagger}\Phi_{\tau})$
$\displaystyle+$ $\displaystyle\
m_{e\mu}^{2}(\Phi_{e}^{\dagger}\Phi_{\mu})+m_{e\tau}^{2}(\Phi_{e}^{\dagger}\Phi_{\tau})+m_{\mu\tau}^{2}(\Phi_{\mu}^{\dagger}\Phi_{\tau})]+{\rm
h.c.}$

and

$\displaystyle V_{4}=$ $\displaystyle\
\lambda_{1}^{q}(\Phi_{q}^{\dagger}\Phi_{q})^{2}+\lambda_{1}^{e}(\Phi_{e}^{\dagger}\Phi_{e})^{2}+\lambda_{1}^{\mu}(\Phi_{\mu}^{\dagger}\Phi_{\mu})^{2}+\lambda_{1}^{\tau}(\Phi_{\tau}^{\dagger}\Phi_{\tau})^{2}$
(2) $\displaystyle+$ $\displaystyle\
\lambda_{3}^{qe}(\Phi_{q}^{\dagger}\Phi_{q})(\Phi_{e}^{\dagger}\Phi_{e})+\lambda_{3}^{q\mu}(\Phi_{q}^{\dagger}\Phi_{q})(\Phi_{\mu}^{\dagger}\Phi_{\mu})+\lambda_{3}^{q\tau}(\Phi_{q}^{\dagger}\Phi_{q})(\Phi_{\tau}^{\dagger}\Phi_{\tau})$
$\displaystyle+$
$\displaystyle\lambda_{3}^{e\mu}(\Phi_{e}^{\dagger}\Phi_{e})(\Phi_{\mu}^{\dagger}\Phi_{\mu})+\lambda_{3}^{e\tau}(\Phi_{e}^{\dagger}\Phi_{e})(\Phi_{\tau}^{\dagger}\Phi_{\tau})+\lambda_{3}^{\mu\tau}(\Phi_{\mu}^{\dagger}\Phi_{\mu})(\Phi_{\tau}^{\dagger}\Phi_{\tau})$
$\displaystyle+$ $\displaystyle\
\lambda_{4}^{qe}(\Phi_{q}^{\dagger}\Phi_{e})(\Phi_{e}^{\dagger}\Phi_{q})+\lambda_{4}^{q\mu}(\Phi_{q}^{\dagger}\Phi_{\mu})(\Phi_{\mu}^{\dagger}\Phi_{q})+\lambda_{4}^{q\tau}(\Phi_{q}^{\dagger}\Phi_{\tau})(\Phi_{\tau}^{\dagger}\Phi_{q})$
$\displaystyle+$ $\displaystyle\
\lambda_{4}^{e\mu}(\Phi_{e}^{\dagger}\Phi_{\mu})(\Phi_{\mu}^{\dagger}\Phi_{e})+\lambda_{4}^{e\tau}(\Phi_{e}^{\dagger}\Phi_{\tau})(\Phi_{\tau}^{\dagger}\Phi_{e})+\lambda_{4}^{\mu\tau}(\Phi_{\mu}^{\dagger}\Phi_{\tau})(\Phi_{\tau}^{\dagger}\Phi_{\mu})$
$\displaystyle+$ $\displaystyle\
\frac{1}{2}\big{[}\lambda_{5}^{qe}(\Phi_{q}^{\dagger}\Phi_{e})^{2}+\lambda_{5}^{q\mu}(\Phi_{q}^{\dagger}\Phi_{\mu})^{2}+\lambda_{5}^{q\tau}(\Phi_{q}^{\dagger}\Phi_{\tau})^{2}$
$\displaystyle+$ $\displaystyle\
\lambda_{5}^{e\mu}(\Phi_{e}^{\dagger}\Phi_{\mu})^{2}+\lambda_{5}^{e\tau}(\Phi_{e}^{\dagger}\Phi_{\tau})^{2}+\lambda_{5}^{\mu\tau}(\Phi_{\mu}^{\dagger}\Phi_{\tau})^{2}+{\rm
h.c.}\big{]}$

Here, we have labeled the quartic couplings to be similar to the standard 2HDM
potential.

If the $m_{{ij}}^{2}$ and $\lambda_{5}^{ij}$ have imaginary components, one
would have CP violation in the scalar sector. There are six CP violating
parameters from the $m_{ij}^{2}$ and another six from the $\lambda_{5}^{ij}$
parameters. Two can be eliminated through rescaling, but that would leave ten
additional parameters. A detailed analysis of CP violation in the 3HDM
Akeroyd:2021fpf considered the effects on the neutron and electron electric
dipole moments as well as CP violating effects in B decays. They also discuss
mixing of scalars and pseudoscalars which would complicate the analysis. In
our model, one would expect very similar effects. For simplicity, we will
assume that these parameters are real and refer the reader to Ref.
Akeroyd:2021fpf for details.

We can write the Higgs doublets as

$\Phi_{i}=\begin{pmatrix}\phi^{+}_{i}\\\
(v_{i}+\phi_{i}+i\chi_{i})/\sqrt{2}\end{pmatrix},(i=q,e,\mu,\tau)$ (3)

where the $v_{i}/\sqrt{2}$ are the vacuum values of the neutral components. To
discuss diagonalizing mass matrices and the various angles involved, we follow
the procedure of Boto, Romão and Silva Boto:2021qgu closely.

Without loss of generality, we can define the angles that rotate the fields
into the Higgs basis in which only one scalar field gets a vev by

$\displaystyle v_{q}$ $\displaystyle=v\cos\beta_{2}\cos\beta_{3}\cos\beta_{4}$
(4) $\displaystyle v_{e}$
$\displaystyle=v\sin\beta_{2}\cos\beta_{3}\cos\beta_{4}$ $\displaystyle
v_{\mu}$ $\displaystyle=v\sin\beta_{3}\cos\beta_{4}$ $\displaystyle v_{\tau}$
$\displaystyle=v\sin\beta_{4}$

giving

$\begin{pmatrix}h_{0}\\\ H_{1}\\\ H_{2}\\\ H_{3}\end{pmatrix}={\cal
O}_{\beta}\begin{pmatrix}\phi_{q}\\\ \phi_{e}\\\ \phi_{\mu}\\\
\phi_{\tau}\end{pmatrix}$ (5)

where

$\mathcal{O}_{\beta}=\begin{pmatrix}c_{\beta_{2}}c_{\beta_{3}}c_{\beta_{4}}&s_{\beta_{2}}c_{\beta_{3}}c_{\beta_{4}}&s_{\beta_{3}}c_{\beta_{4}}&s_{\beta_{4}}\\\
-s_{\beta_{2}}&c_{\beta_{2}}&0&0\\\
-c_{\beta_{2}}c_{\beta_{3}}&-s_{\beta_{2}}s_{\beta_{3}}&c_{\beta_{3}}&0\\\
-c_{\beta_{2}}c_{\beta_{3}}s_{\beta_{4}}&-s_{\beta_{2}}c_{\beta_{3}}s_{\beta_{4}}&-s_{\beta_{3}}s_{\beta_{4}}&c_{\beta_{4}}\end{pmatrix}$
(6)

Here, $h_{0}$ is the field that gets the entire vev, $v$, and $c_{\theta}$
($s_{\theta}$) are $\cos\theta$ ($\sin\theta$).

From this basis, we can now diagonalize the mass matrices of the various
scalars. In the neutral scalar sector, the physical neutral Higgs masses are
given by

$\begin{pmatrix}h_{1}\\\ h_{2}\\\ h_{3}\\\ h_{4}\end{pmatrix}={\cal
O}_{\alpha}\begin{pmatrix}\phi_{q}\\\ \phi_{e}\\\ \phi_{\mu}\\\
\phi_{\tau}\end{pmatrix}$ (7)

where $h_{1}$ is the 125 GeV Higgs particle. For ${\cal O}_{\alpha}$, we use

${\cal O}_{\alpha}={\bf R}_{34}{\bf R}_{24}{\bf R}_{23}{\bf R}_{14}{\bf
R}_{13}{\bf R}_{12}$ (8)

Here, for example, ${\bf R}_{24}$ is given by

${\bf R}_{24}=\begin{pmatrix}1&0&0&0\\\ 0&c_{\alpha_{24}}&0&s_{\alpha_{24}}\\\
0&0&1&0\\\ 0&-s_{\alpha_{24}}&0&c_{\alpha_{24}}\end{pmatrix}$ (9)

and the other R matrices follow. We see that there are six rotation angles.

In the pseudoscalar sector, one has

$\begin{pmatrix}G^{0}\\\ A_{1}\\\ A_{2}\\\ A_{3}\end{pmatrix}={\cal
O}_{\gamma}{\cal O}_{\beta}\begin{pmatrix}\chi_{q}\\\ \chi_{e}\\\
\chi_{\mu}\\\ \chi_{\tau}\end{pmatrix}$ (10)

where ${\cal O}_{\gamma}={\bf P}_{34}{\bf P}_{24}{\bf P}_{23}$ and, as before,
for example

${\bf P}_{24}=\begin{pmatrix}1&0&0&0\\\ 0&c_{\gamma_{24}}&0&s_{\gamma_{24}}\\\
0&0&1&0\\\ 0&-s_{\gamma_{24}}&0&c_{\gamma_{24}}\end{pmatrix}$ (11)

Note that there are only three matrices here, since the Goldstone boson
direction is fixed.

Finally, in the charged sector

$\begin{pmatrix}G^{+}\\\ H_{1}^{+}\\\ H^{+}_{2}\\\
H^{+}_{3}\end{pmatrix}={\cal O}_{\delta}{\cal
O}_{\beta}\begin{pmatrix}\phi^{+}_{q}\\\ \phi^{+}_{e}\\\ \phi^{+}_{\mu}\\\
\phi^{+}_{\tau}\end{pmatrix}$ (12)

where ${\cal O}_{\delta}={\bf Q}_{34}{\bf Q}_{24}{\bf Q}_{23}$ and, as before,
for example

${\bf Q}_{24}=\begin{pmatrix}1&0&0&0\\\ 0&c_{\delta_{24}}&0&s_{\delta_{24}}\\\
0&0&1&0\\\ 0&-s_{\delta_{24}}&0&c_{\delta_{24}}\end{pmatrix}$ (13)

###  II.2 Gauge and Yukawa couplings

####  II.2.1 Gauge couplings

The scalar kinetic Lagrangian, $\mathcal{L}_{\text{k}}$, defined as

$\begin{split}\mathcal{L}_{\text{k}}=\sum_{i=1}^{4}|D_{\mu}\Phi_{i}|^{2}\end{split}$
(14)

with the usual expression for the covariant derivative $D_{\mu}$, contains the
terms relevant to obtain the trilinear couplings of the scalars and gauge
bosons. The couplings $ZZh_{i}$ and $W^{\pm}W^{\mp}h_{i}$ are written in the
form

$\begin{split}\left(\sum_{i=1}^{4}C_{i}h_{i}\right)\left(\frac{g}{2c_{W}}m_{Z}Z_{\mu}Z^{\mu}+gm_{W}W^{-}_{\mu}{W^{+}}^{\mu}\right)\,.\end{split}$
(15)

The $C_{i}$ factors are included in Appendix A. It is possible to check that,
when the set of conditions $\alpha_{1j}=\beta_{j}$ is verified (for
$j=2,3,4$), one gets $C_{1}=1$ together with $C_{k}=0$, for $k\neq 1$, which
defines the alignment limit in this model.

####  II.2.2 Yukawa couplings

Following the notation of Branco, et al. Branco:2011iw , the couplings of the
scalar and pseudoscalar Higgs are defined through

$\begin{split}\mathcal{L}_{Y}^{S}=&-\sum_{f\in\left\\{q,e,\mu,\tau\right\\}}\frac{m_{f}}{v}\left(\xi_{h_{1}}^{f}\bar{f}fh_{1}+\xi_{h_{2}}^{f}\bar{f}fh_{2}+\xi_{h_{3}}^{f}\bar{f}fh_{3}+\xi_{h_{4}}^{f}\bar{f}fh_{4}\right)\\\
\mathcal{L}_{Y}^{P}=&-\sum_{f\in\left\\{q,e,\mu,\tau\right\\}}\left(-i\frac{m_{f}}{v}\right)\left(\xi_{A_{1}}^{f}\bar{f}\gamma_{5}fA_{1}+\xi_{A_{2}}^{f}\bar{f}\gamma_{5}fA_{2}+\xi_{A_{3}}^{f}\bar{f}\gamma_{5}fA_{3}\right)\end{split}$
(16)

where $\xi_{h_{j}}^{f}$ and $\xi_{A_{j}}^{f}$ are given by

$\begin{split}\xi_{h_{j}}^{q}=\frac{{\cal{O}_{\alpha}}_{j,1}}{\hat{v}_{1}}\>,\>\xi_{h_{j}}^{e}=\frac{{\cal{O}_{\alpha}}_{j,2}}{\hat{v}_{2}}\>,&\>\xi_{h_{j}}^{\mu}=\frac{{\cal{O}_{\alpha}}_{j,3}}{\hat{v}_{3}}\>,\>\xi_{h_{j}}^{\tau}=\frac{{\cal{O}_{\alpha}}_{j,4}}{\hat{v}_{4}}\\\
\xi_{A_{j}}^{q}=\frac{\left({\cal{O}_{\gamma}\cal{O}_{\beta}}\right)_{j,1}}{\hat{v}_{1}}\>,\>\xi_{A_{j}}^{e}=\frac{\left({\cal{O}_{\gamma}\cal{O}_{\beta}}\right)_{j,2}}{\hat{v}_{2}}\>,&\>\xi_{A_{j}}^{\mu}=\frac{\left({\cal{O}_{\gamma}\cal{O}_{\beta}}\right)_{j,3}}{\hat{v}_{3}}\>,\>\xi_{A_{j}}^{\tau}=\frac{\left({\cal{O}_{\gamma}\cal{O}_{\beta}}\right)_{j,4}}{\hat{v}_{4}}\end{split}$
(17)

using $\hat{v}_{i}\equiv v_{i}/v$. Similarly, the couplings of the charged
Higgs are defined through

$\begin{split}\mathcal{L}_{Y}^{C}=-\sum_{j}\Biggl{[}\Biggr{.}&\left.\sum_{u,d}\frac{\sqrt{2}V_{ud}}{v}\bar{u}\left(m_{u}\xi_{H_{j}^{+}}^{qL}\text{P}_{L}+m_{d}\xi_{H_{j}^{+}}^{qR}\text{P}_{R}\right)dH_{j}^{+}\right.\\\
&\left.+\sum_{l}\frac{\sqrt{2}m_{l}}{v}\xi_{H_{j}^{+}}^{lL}\bar{\nu}_{L}l_{R}H_{j}^{+}\right]+\text{h.c.}\end{split}$
(18)

where $\xi_{H_{j}^{+}}^{f}$ are given by

$\xi_{H_{j}^{+}}^{qLR}=\frac{\left({\cal O}_{\delta}{\cal
O}_{\beta}\right)_{j,1}}{\hat{v}_{1}}\>,\>\xi_{H_{j}^{+}}^{eL}=\frac{\left({\cal
O}_{\delta}{\cal
O}_{\beta}\right)_{j,2}}{\hat{v}_{2}}\>,\>\xi_{H_{j}^{+}}^{\mu
L}=\frac{\left({\cal O}_{\delta}{\cal
O}_{\beta}\right)_{j,3}}{\hat{v}_{3}}\>,\>\xi_{H_{j}^{+}}^{\tau
L}=\frac{\left({\cal O}_{\delta}{\cal O}_{\beta}\right)_{j,4}}{\hat{v}_{4}}$
(19)

A table of general Yukawa couplings are included in Appendix B.

##  III Theoretical constraints on the scalar potential

###  III.1 Bounded from below constraints

In extensions of the scalar sector, one needs to choose quartic parameters
such that the potential is bounded from below (BFB)111 We require that the
potential be bounded at scales where the quartic terms dominate. The case in
which the potential turns over at very high scales due to renormalization
group running will not be considered. In fact, the Standard Model itself would
not satisfy that latter condition. While this is straightforward in the 2HDM,
it can be quite complicated in models with more than two doublets. An added
complication in models with doublets is that there can be an instability in
the charged scalar direction even if there is stability in the neutral scalar
direction (see Ref. Faro:2019vcd for an example). A recent discussion of these
conditions for a three-doublet model can be found in the work of Boto, Romão
and Silva Boto:2022uwv . They showed that while necessary and sufficient
conditions are known for the neutral direction, only sufficient conditions are
known for stability in the charged direction, and they discuss a general
strategy. We will first discuss the neutral directions.

Looking at the neutral direction, the 2HDM potential can be written as
$V_{4}=a_{11}H_{1}^{4}+a_{22}H_{2}^{4}+a_{12}H_{1}^{2}H_{2}^{2}$, where the
matrix is symmetric. The conditions for copositivity (where the potential is
positive for all values of $H_{1}^{2}$ and $H_{2}^{2}$) are given by
$a_{11}\geq 0,a_{22}\geq 0,a_{12}+\sqrt{a_{11}a_{22}}\geq 0$. As shown in
Refs. Klimenko:1984qx ; Kannike:2012pe , for the neutral sector of the 3HDM,
the conditions are

$\displaystyle a_{11}\geq 0,\quad a_{22}$ $\displaystyle\geq 0,\quad
a_{33}\geq 0$ (20) $\displaystyle a_{12}+\sqrt{a_{11}a_{22}}$
$\displaystyle\geq 0$ (21) $\displaystyle a_{13}+\sqrt{a_{11}a_{33}}$
$\displaystyle\geq 0$ (22) $\displaystyle a_{23}+\sqrt{a_{22}a_{33}}$
$\displaystyle\geq 0$ (23)
$\displaystyle\sqrt{a_{11}a_{22}a_{33}}+a_{12}\sqrt{a_{33}}$
$\displaystyle+a_{13}\sqrt{a_{22}}+a_{23}\sqrt{a_{11}}\geq 0$ (24)
$\displaystyle\det{A}$ $\displaystyle\geq 0$ (25)

where $A$ is the matrix with entries $a_{ij}$. Clearly, the first line is
needed for stability along the axes, the next three lines are needed for
stability in the three planes, and the last two lines ensure stability for all
directions. For the 4HDM that we consider, the corresponding conditions must
be satisfied for every three dimensional subspace. The remaining conditions
are extremely complicated, but are given in full in Ref. Klimenko:1984qx . We
have incorporated the conditions in that paper to ensure stability in the
neutral directions.

As shown by Boto, Romão and Silva Boto:2022uwv , even in the 3HDM there are no
straightforward necessary and sufficient conditions for stability in the
charged directions. In the 2HDM, with a quartic potential

$V_{4}=\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}+\frac{1}{2}\lambda_{5}[(\Phi_{1}^{\dagger}\Phi_{2})^{2}+(\Phi_{2}^{\dagger}\Phi_{1})^{2}]$
(26)

the condition for stability is Ivanov:2015mwl ; Song:2022gsz
$\lambda_{3}+\lambda_{4}-|\lambda_{5}|\geq-2\sqrt{\lambda_{1}\lambda_{2}}$.
Rather than attempt a detailed numerical study of stability in the 4HDM case,
we will require that this condition be satisfied for all $2\times 2$ subspaces
of the 4HDM. This requirement is, of course, necessary but may not be
sufficient.

###  III.2 Oblique Parameters

To discuss the S, T, U oblique parameters, we follow the methods and results
in Grimus, et al Grimus:2008nb . To do this, we can write the matrices
$\tilde{\mathcal{U}}$ and $\tilde{\mathcal{V}}$ from Grimus, et al
Grimus:2008nb using our notation in the previous section.
$\tilde{\mathcal{V}}$ is defined through

$\begin{pmatrix}\phi_{1}\,+\,i\chi_{1}\\\ \phi_{2}\,+\,i\chi_{2}\\\
\phi_{3}\,+\,i\chi_{3}\\\
\phi_{4}\,+\,i\chi_{4}\end{pmatrix}=\tilde{\mathcal{V}}\begin{pmatrix}h_{1}&h_{2}&h_{3}&h_{4}&G^{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}^{T}$
(27)

where

$\tilde{\mathcal{V}}\equiv\begin{pmatrix}\mathcal{O}_{\alpha}^{-1}\\\
i\left(\mathcal{O}_{\gamma}\mathcal{O}_{\beta}\right)^{-1}\end{pmatrix}$ (28)

Notice in Eq. (27), our notation slightly differs from Grimus et al
Grimus:2008nb by keeping the Goldstone boson with the pseudoscalar mass
eigenstates.

$\tilde{\mathcal{U}}$ is defined as

$\begin{pmatrix}\phi_{1}^{+}\\\ \phi_{2}^{+}\\\ \phi_{3}^{+}\\\
\phi_{4}^{+}\end{pmatrix}=\tilde{\mathcal{U}}\begin{pmatrix}G^{+}\\\
H_{1}^{+}\\\ H_{2}^{+}\\\ H_{3}^{+}\end{pmatrix}$ (29)

where

$\tilde{\mathcal{U}}\equiv\begin{pmatrix}{\cal O}_{\delta}{\cal
O}_{\beta}\end{pmatrix}$ (30)

We take the values of S, T from Workman:2022ynf with

$\displaystyle\ S$ $\displaystyle=$ $\displaystyle-0.02\pm 0.10$ (31)
$\displaystyle T$ $\displaystyle=$ $\displaystyle 0.03\pm 0.12$

We will not include the detailed calculation of the unitarity and
perturbativity bounds, due to the large number of scalar couplings. Rather, we
will simply require that all of the quartic scalar couplings be less than
$4\pi$.

##  IV Benchmark models

As is clear from examining the scalar potential and the Appendices, the model
contains a large number of free parameters. To focus on the most important
aspects of the model, we will consider two benchmark models. In the first, we
will assume that the $(q\tau)$ sector of the Higgs potential decouples from
the $(\mu e)$ sector. In that case, the $4\times 4$ scalar mass matrices
decouple into two $2\times 2$ matrices which can be trivially diagonalized
analytically. In the second benchmark model, we will take the alignment limit.
In the conventional 2HDMs, this is equivalent to $\cos(\alpha-\beta)=0$, with
$\tan\beta\equiv v_{2}/v_{1}$ and $\alpha$ diagonalizes the scalar mass
matrix. This limit is often chosen since it means that the couplings of the
$125$ GeV Higgs boson are identical to that in the Standard Model (which seems
to be preferred by LHC data). In this case, it is easy to see from Appendices
A and B that the alignment limit corresponds to $\alpha_{1j}=\beta_{j}$, as
previously stated. Since the coupling of the $125$ GeV Higgs is the same as
the Standard Model, there is no need to study Higgs production and tree-level
decays in this case.

###  IV.1 The Model without $(q\tau)$-$(\mu e)$ mixing

In this model, the absence of $(q\tau)$-$(\mu e)$ mixing means that the matrix
that diagonalizes the scalar mass matrix, $\cal{O}_{\alpha}$, is broken into
two $2\times 2$ matrices. The upper $2\times 2$ matrix looks very similar to
the lepton-specific 2HDM. The only difference involves the coupling to the
muon, which is not well-measured. However, in this case, unlike the lepton-
specific model, the value of $v_{q}^{2}+v_{\tau}^{2}$ is not $v^{2}=(246\
\rm{GeV})^{2}$ but will be smaller. As a result, all Yukawa couplings will be
increased. This will affect the decays of the $125$ GeV Higgs boson as well as
the production.

We define the parameter $\mu_{X}$ as

$\mu_{X}\equiv\frac{\sigma(pp\rightarrow H){\rm BR}(H\rightarrow
X)}{\sigma(pp\rightarrow H)_{\rm SM}{\rm BR}(H\rightarrow X)_{\rm SM}}$ (32)

and look at
$X=gg,\mu\mu,\tau\tau,\bar{c}c,\bar{b}b,\bar{t}t,\gamma\gamma,\gamma Z,WW,ZZ$.
The results are in Figure 1, where we have plotted, in the usual way for
2HDMs, the allowed region in the $\tan\beta-\cos(\beta-\alpha)$ plane. We
require all $\mu_{X}$ to be consistent with unity within 20% at 95% CL, which
is a rough approximation to the precision of current data. 222We are looking
in the context of the lepton-specific 2HDM - but now the combination of vacuum
values, $(v_{q}^{2}+v_{\tau}^{2})^{1/2}$ no longer is equal to the Standard
Model vacuum value, $v$.

We see that if the ratio of $(v_{q}^{2}+v_{\tau}^{2})^{1/2}$ to $v$ is less
than $0.85$, that the entire parameter space practically disappears. Thus much
of the vev is saturated by $v_{q}$ and $v_{\tau}$. Clearly the coupling here
to the muon vanishes and thus in the full model, the muonic decay of the
Standard Model Higgs, if confirmed, will be a strong constraint.

Figure 1: Allowed regions in the $\tan\beta-\cos\left(\beta-\alpha\right)$
plane, in the model without $(q\tau)$-$(\mu e)$ mixing, for different values
of $r\equiv\left(v_{q}^{2}+v_{\tau}^{2}\right)^{1/2}/v$, namely $r=1$ in
orange, $r=0.95$ in purple, $r=0.90$ in blue and $r=0.85$ in cyan.

The shrinking of the parameter-space in the $\cos({\beta-\alpha})<0$ allowed
region occurs mainly due to the combination of $g_{HVV}^{2}$, measured from
Higgs production, and $g_{Hll}^{2}$, measured from Higgs decay. The shrinking
of the parameter-space in the $\cos({\beta-\alpha})>0$ allowed region mainly
occurs due to $g_{HQQ}^{2}$, from Higgs production, now combined with both
$g_{Hqq}^{2}$ and $g_{Hll}^{2}$.

In itself, this benchmark model is phenomenologically unacceptable. Each
$2\times 2$ submatrix will have a zero eigenvalue in the pseudoscalar and in
the charged scalar sectors, leading to two zero eigenvalues in each sector.
Only one can be absorbed by the $W$ and $Z$ gauge bosons. The additional
massless scalars arise due to an additional accidental $SU(2)$ symmetry. Thus,
there must be some off-diagonal terms. We can include these terms but assume
they are small and do a perturbative expansion.

For simplicity, let us add a single off-diagonal term, $\lambda_{5}^{q\mu}$.
This will allow for nonzero masses for the lightest charged and pseudoscalar
Higgs333One can decouple the masses of the charged and pseudoscalar Higgs by
adding a $\lambda_{4}^{q\mu}$ term and can easily satisfy any BFB concerns
with a $\lambda_{3}^{q\mu}$ term.. This term will modify the Yukawa couplings
of the Standard Model 125 GeV Higgs. For the couplings of the quarks, for
example, the Yukawa coupling $g_{Y}\bar{q}q\Phi_{q}$ is $\sqrt{2}m_{q}/v_{q}$.
Writing $\Phi_{q}=V_{11}h_{1}+V_{12}h_{2}+...$, where $h_{1}$ is the 125 GeV
Higgs, one sees that the coupling is modified by a factor of
$\frac{v}{v_{q}}V_{11}$. One can perturbatively calculate the eigenvalues and
eigenvectors of the mass matrix and we find that

$V_{11}=1-\frac{1}{2}\epsilon_{1}^{2}\Bigg{[}\left(\frac{c_{34}s_{12}}{m_{h_{1}}^{2}-m_{h_{3}}^{2}}\right)^{2}+\left(\frac{c_{12}s_{34}}{m_{h_{1}}^{2}-m_{h_{4}}^{2}}\right)^{2}\Bigg{]},$
(33)

where $\epsilon_{1}=\lambda_{5}^{q\mu}v_{q}v_{\mu}$, $c_{ij}=\cos\alpha_{ij}$
($s_{ij}=\sin\alpha_{ij}$) and the masses are the masses of the neutral
scalars. The relevant point here is that $V_{11}$ is reduced, which counters
the effect of the smaller $v_{q}$. In order for the lightest charged Higgs to
have an acceptable mass, there is a minimum value of $\lambda_{5}^{q\mu}$, but
the masses of the neutral scalars can be large enough that the reduction
(proportional to $(v_{\mu}/m_{h_{3}})^{2}$) is quite small.

###  IV.2 The Aligned Model

The full 4HDM has a large number of parameters in the scalar potential: 10
quadratic terms and 22 quartic terms. Not surprisingly, many of these
parameters will have little effect on phenomenology. As noted earlier, the
fact that the $125$ GeV Higgs has decays consistent with the Standard Model
implies that multi-doublet models must be near the alignment limit in which
the Standard Model Higgs interactions are unaffected. From Appendix A, we see
that this will occur if $\alpha_{1j}=\beta_{j}$. Parameters that might be of
phenomenological relevance are then the $\beta_{j}$, $\alpha_{23,24,34}$, the
three $\gamma$ parameters, the three $\delta$ parameters, the four scalar
masses, the three charged masses and the three pseudoscalar masses, in
addition to the SM Higgs vev. Instead of the potential’s couplings, we can
choose to describe the model in terms of the previously mentioned parameters
and six additional parameters, namely the remaining six $m_{ij}^{2}$, giving a
total of 29 parameters444With the addition of the three $\alpha$ parameters
which are defined through the alignment limit, we get 32 parameters, just like
the scalar potential.. As we will see, many of these parameters will not be
relevant for particular processes.

One might wonder about relaxing the alignment limit assumption. Since the
Higgs properties match Standard Model expectations, one would expect
deviations from the alignment limit to be small (of the order of $10\%$ or
less). Since we are not including other similar size effects, such as
radiative corrections to scalar masses, we don’t anticipate any substantial
effects on our plots.

Choosing values for the rotation angles and the squared masses, it is possible
to define the scalar, pseudoscalar, and charged squared-mass matrices as
$M_{s,p,c}^{2}=R^{-1}D_{s,p,c}R$, considering the corresponding $R$ matrix for
each case and $D$ as the diagonal matrix with the squared masses in its
entries. The quartic parameters of the Lagrangian can be expressed in terms of
elements of such matrices, the vevs and the $m_{ij}^{2}$ parameters as the
following:

$\displaystyle\lambda_{1}^{i}=$
$\displaystyle\frac{1}{2v_{i}^{3}}\left(v_{i}M_{s,ii}^{2}+\sum_{j\neq
i}v_{j}m_{ij}^{2}\right)\,,$ (34) $\displaystyle\lambda_{3}^{ij}=$
$\displaystyle\frac{1}{v_{i}v_{j}}\left(M_{s,ij}^{2}-2M_{c,ij}^{2}+m_{ij}^{2}\right)\,,$
$\displaystyle\lambda_{4}^{ij}=$
$\displaystyle\frac{1}{v_{i}v_{j}}\left(2M_{c,ij}^{2}-M_{p,ij}^{2}-m_{ij}^{2}\right)\,,$
$\displaystyle\lambda_{5}^{ij}=$
$\displaystyle\frac{1}{v_{i}v_{j}}\left(M_{p,ij}^{2}-m_{ij}^{2}\right)\,,$

in which $i,j=q,e,\mu,\tau$. In the 2HDM limit, these equations give rise to
the well-known expressions for the $\lambda$ parameters in terms of masses,
angles, the electroweak vev $v$ and the soft-breaking terms $m_{ij}^{2}$
Branco:2011iw ; Branchina:2018qlf . For every possible set of parameters, we
require the following:

  * •

The bounded-from-below conditions are satisfied.

  * •

The perturbativity condition that the absolute values of $\lambda$ parameters
are less than $4\pi$ is maintained.

  * •

The previous condition also applies to Yukawa couplings.

  * •

The values of the S and T parameters are within the range given by Eq. (31).

  * •

Charged Higgs masses must exceed $80$ GeV ALEPH:2013htx .

  * •

Contributions from the charged scalars to the loop-induced Higgs diphoton
decay $h\rightarrow\gamma\gamma$ are compatible with experimental bounds. This
is achieved by checking the value of the diphoton signal strength
$\mu_{\gamma\gamma}$ Bhattacharyya:2014oka ; Djouadi:2005gj for each set of
parameters.

  * •

Bounds coming from new physics contributions to $B$ meson oscillations,
$\Delta{M_{B_{d,s}}}$, as well as $K$ mesons, $\Delta{M_{K}}$, are within the
experimental allowed range for each case Workman:2022ynf ;
FlavourLatticeAveragingGroupFLAG:2021npn . Such nonstandard contributions come
from charged scalars through one-loop processes Chakraborti:2021bpy ;
Enomoto:2015wbn .

  * •

Contributions to $b\rightarrow s\gamma$ Enomoto:2015wbn , again from charged
Higgs particles, are acceptable. In the Type II 2HDM, this gives the strongest
constraint on charged Higgs bosons.

  * •

At the LHC, CMS CMS:2018rmh has searched for a heavy neutral Higgs decaying
into $\tau$ pairs. Although done in the context of the MSSM, the results are
very similar in this model (with adjusted Yukawa couplings, of course) and the
production cross-section times branching ratio varies from 10 pb to 10 fb over
the range of masses from 150 GeV to 1000 GeV. More recently, ATLAS
ATLAS:2020zms has done a similar analysis. Note that one usually assumes that
the decay into top quarks will dominate for masses above 350 GeV, but that
might not be the case here due to the lepton-specific nature of the model. We
impose these experimental bounds on our parameter-space, which, up to small
differences due to form factors, apply to neutral scalars and pseudoscalars.

  * •

Finally, we can consider LHC direct searches for heavy charged Higgs bosons.
Searches fall into two categories - those in which the charged Higgs mass is
greater than $m_{t}+m_{b}$ and those in which it is less.

    * –

If it is greater, then the predominant decay mode will be into $t\bar{b}$,
except for the narrow window of parameter-space in which the charged Higgs in
question has essentially zero overlap with $\Phi_{q}$. The production cross-
section for a charged Higgs mass of $200,300,600$ GeV is Berger:2003sm within
a factor of 2 (scaling the Yukawa coupling appropriately to a lepton-specific
or Type I model) of $0.4,0.1,0.01$ picobarns. ATLAS ATLAS:2021upq has found
bounds from Run II on the product of the production rate and the
$H^{+}\rightarrow t\bar{b}$ branching ratio. Their result is below our
production cross-section by a factor of a few, and thus the model is not yet
constrained by the non-observation at the LHC.

    * –

If the charged Higgs is lighter, then a major decay mode is into
$\tau\nu_{\tau}$. In this case the predominant production mode is through
$t\rightarrow bH^{+}$. Since top production is well understood, searches at
ATLAS ATLAS:2018gfm and CMS CMS:2019bfg place bounds on ${\rm BR}(t\rightarrow
bH^{+}){\rm BR}(H^{+}\rightarrow\tau\nu_{\tau})$. This bound may not be too
restrictive, since a charged Higgs that is either quarkphobic or leptophobic
will not contribute and thus it will depend on mixing angles. Nonetheless, we
have incorporated the results of these searches in bounding our parameter-
space.

We will primarily focus on the lightest neutral scalar (other than the $125$
GeV Higgs), the lightest pseudoscalar and the lightest charged scalar. Results
from these scalars will also apply to the heavier scalars by appropriate
choice of mixing angles (with the exception of heavy scalar decays into
lighter scalars, which we will not consider). The lepton-specific 2HDM has one
scalar coupling to quarks and another to leptons. The primary difference
between our model and the lepton-specific model is that different scalars
couple to the muon and the electron (note that the muon-specific model
Abe:2017jqo ; Ferreira:2020ukv has the same scalar coupling to the quarks and
the $\tau$, which is more like an extension of the type I 2HDM). As a result,
we will focus on decays involving muons and electrons.

We first consider the decay of the lightest neutral scalar (other than the
$125$ GeV Higgs, which has Standard Model couplings in the alignment limit)
into electrons, muons and taus. Since the heavier masses aren’t relevant in
the analysis, the parameter-space is substantially reduced. We consider two
mass regions, in which the scalar mass is below and above $350$ GeV,
respectively. In the latter case, decays to top quarks can be substantial,
even if the mixing angles are small.

As noted above, given the masses, soft-breaking mass parameters and mixing
angles, the quartic couplings are determined. We scan the full parameter space
and check each of the conditions above. Typically, we find several million
parameter sets that are acceptable. The results are plotted in Figure 2. Note
that in the Standard Model the branching ratio of the dimuon decay of the
Higgs is $2\times 10^{-4}$ and this level (and somewhat below) is certainly
experimentally accessible. One can see that for a scalar mass below $350$ GeV,
the dielectron decay branching ratio can be much, much larger than the
Standard Model and the dimuon decay branching ratio can approach unity. Above
$350$ GeV, the opening of the top decay channel, even if the mixing angle is
very small, substantially reduces the leptonic branching ratios.

Figure 2: These scatterplots show allowed points for $h_{2}$ decays. Results
are shown for $h_{2}$ masses below $350$ GeV and above that mass scale (at
which point the $\bar{t}t$ channel opens up). The upper figures plot $ee$ and
$\mu\mu$ decays and the lower figures plot $\mu\mu$ and $\tau\tau$ decays. The
decay branching ratio of the SM Higgs to $\mu\mu$ is approximately $2\times
10^{-4}$.

It is not surprising that this can occur. If one chose parameters such that
there was no mixing at all between $\Phi_{ee}$ and the other scalars, then the
only decay of the $\Phi_{ee}$ would be into electrons. This would require
extreme fine-tuning, since no symmetry will eliminate mixing in the quartic
sector of the potential and even very small values of the quartic mixing terms
would allow for other decays that could dominate. Nonetheless, we see many
sets of parameters for which the dielectron and dimuon decays of this lightest
neutral scalar (other than the Standard Model Higgs) can be substantial.

In Figure 2, we also show the branching ratios to muons and to taus. Again,
one can see that the absolute branching ratio to dimuons can be substantially
more than that into two taus. Thus, we find that searches for heavy neutral
Higgs bosons decaying into leptons, which generally focus on tauonic decays,
should also study muonic and electronic decays.

Figure 3: These scatterplots show allowed points for $H^{\pm}$ decays. Results
are shown for $H^{\pm}$ masses below $180$ GeV and above that mass scale (at
which point the $\bar{t}b$ channel opens up). The upper figures plot $e\nu$
and $\mu\nu$ decays and the lower figures plot $\mu\nu$ and $\tau\nu$ decays.

Since we are in the alignment limit, there is no three-point coupling of these
scalars to two gauge bosons. They could be produced in a collider through $WW$
or $ZZ$ fusion to two $\Phi$s. The signature would be two electron-positron or
muon pairs each coming from a $\Phi$. The electron-positron pair rate will be
smaller, but more distinctive. While four lepton events have been searched for
ATLAS:2021kog , we know of no analysis of this particular signature. An
approximate production cross-section can be obtained by comparison with the
inert doublet model Yang:2021hcu which has a similar production process.
Typical production cross-sections at the LHC are approximately $0.5$ fb. With
an integrated luminosity of $3\ {\rm ab}^{-1}$, this means that branching
fractions of $O(10^{-3})$ or less will be difficult to detect until the next
generation colliders.

We have also studied the decays of the pseudoscalar into leptons and find very
similar results. For the charged Higgs decays, we show the branching ratios to
$e\nu$ versus $\mu\nu$ decays in Figure 3 as well as the branching ratios for
$\mu\nu$ versus $\tau\nu$ decays . Here, we consider mass ranges below and
above $180$ GeV, at which point the $t\bar{b}$ opens up. Note that there are
more points in the region above $180$ GeV since below that mass a much higher
proportion of points are experimentally excluded. There is a large number of
points in which the electronic decays are substantial and the muonic decay
branching ratios can approach unity.

In Appendix C we show several benchmark points. These points satisfy all of
the various constraints listed earlier in this section. For point S1, one can
see that the $h_{2}\rightarrow\mu\mu$ branching ratio is almost $47\%$ and the
electronic branching ratio is over $0.25\%$. Clearly, the signature would most
likely be two muon pairs, each coming from a neutral scalar, most of the other
decays being tau pairs or $\bar{b}b$, with an occasional electron-positron
pair. In benchmark point S2, the dimuon decay of the scalar is smaller than
that of the electron. Here, one would see the ditau decays dominate, but the
electron-positron decays might be measurable.

We also consider some benchmark points for the lightest charged Higgs, looking
at the region in which the mass is below $180$ GeV so the top-bottom channel
is not available. For point C1, the decay into muons is slightly bigger than
the decay into taus, and the electronic decay is $0.2\%$. For C2, the muon
decay is the smallest of three branching ratios and the electron decay is as
high as $1.7\%$. Again, this shows that decays into muons and electrons might
be much, much higher than in traditional 2HDMs.

##  V Conclusion

It is often believed that all fermions of a given charge must couple to the
same Higgs multiplet in order to avoid tree-level flavor-changing neutral
currents. However this is only true in the quark sector and need not be true
in the lepton sector. The quark mass matrix cannot be diagonal without
eliminating CKM mixing, however the lepton mass matrix can be diagonal, since
PMNS mixing can come from the superheavy Majorana neutrino sector. We have
studied a 4HDM in which one scalar doublet couples to quarks and the other
three couple to the electron, muon and tau families, respectively.

There are numerous constraints on such a model, including bounded from below
constraints, perturbativity, $S$ and $T$ parameters, the diphoton decay of the
Higgs, limits from meson-antimeson oscillations, radiative $b$ decays and
various LHC constraints from heavy scalar searches. Scanning the parameter
space, we find numerous acceptable points in which the dielectron and dimuon
decays of the lightest neutral scalar (other than the $125$ GeV Higgs) can be
much, much larger than expected. The results for the lightest pseudoscalar and
charged scalar are also presented.

Generally, searches for heavier Higgs bosons focus (in the lepton sector) on
decays into $\tau$s. However, this model shows that decays into electrons and
muons can be substantial (and certainly easier to detect). An interesting
signature at either a linear collider or a hadron collider arises from vector
boson fusion into two such Higgs bosons, each of which decays into an electron
or muon pair. We know of no bounds on such a process and hope to see searches
in the near future.

## Acknowledgments

The work of MS and MK was supported by the National Science Foundation under
Grant PHY-1819575. The work of BLG is supported by Fundação para a Ciência e a
Tecnologia (FCT, Portugal) through the PhD grant SFRH/BD/139165/2018 and the
projects UIDB/00777/2020, UIDP/00777/2020, UIDB/00618/2020, UIDP/00618/2020,
CERN/FIS-PAR/0019/2021 and CERN/FIS-PAR/0025/2021. BLG thanks the Fulbright
Commission in Portugal and William & Mary for support. MK thanks Pitt-PACC at
the University of Pittsburgh for their hospitality. We thank Igor Ivanov for
clarifying the symmetry group of the model, Arnab Dasgupta for coding help and
suggestions and for useful discussions, and Pedro Ferreira for a helpful
discussion of the lepton-specific 2HDM.

##  Appendix A Gauge Couplings

Trilinear Gauge Couplings $ZZh_{i}$ and $W^{\pm}W^{\mp}h_{i}$
---
$C_{1}$ |  $\text{c}_{12}\text{c}_{13}\text{c}_{14}\text{c}_{2}\text{c}_{3}\text{c}_{4}+\text{c}_{13}\text{c}_{14}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{2}+\text{c}_{14}\text{c}_{4}\text{s}_{13}\text{s}_{3}+\text{s}_{14}\text{s}_{4}$
$C_{2}$ |  $-\text{c}_{12}\text{c}_{2}\text{c}_{3}\text{c}_{4}(\text{c}_{24}\text{s}_{13}\text{s}_{23}+\text{c}_{13}\text{s}_{14}\text{s}_{24})-\text{c}_{23}\text{c}_{24}\text{c}_{3}\text{c}_{4}\text{s}_{12-2}-\text{c}_{24}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{13}\text{s}_{23}\text{s}_{2}-\text{c}_{13}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{14}\text{s}_{24}\text{s}_{2}+\text{c}_{13}\text{c}_{24}\text{c}_{4}\text{s}_{23}\text{s}_{3}-\text{c}_{4}\text{s}_{13}\text{s}_{14}\text{s}_{24}\text{s}_{3}+\text{c}_{14}\text{s}_{24}\text{s}_{4}$
$C_{3}$ |  $-\text{c}_{12}\text{c}_{3}\text{c}_{4}[\text{c}_{13}\text{c}_{24}\text{c}_{2}\text{s}_{14}\text{s}_{34}+\text{s}_{23}(-\text{c}_{2}\text{s}_{13}\text{s}_{24}\text{s}_{34}+\text{c}_{34}\text{s}_{2})+\text{c}_{23}(\text{c}_{34}\text{c}_{2}\text{s}_{13}+\text{s}_{24}\text{s}_{34}\text{s}_{2})]+\text{c}_{34}\text{c}_{4}[\text{c}_{2}\text{c}_{3}\text{s}_{12}\text{s}_{23}+\text{c}_{23}(-\text{c}_{3}\text{s}_{12}\text{s}_{13}\text{s}_{2}+\text{c}_{13}\text{s}_{3})]+\text{s}_{34}[\text{c}_{23}\text{c}_{2}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{24}+\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{13}\text{s}_{23}\text{s}_{24}\text{s}_{2}-\text{c}_{24}\text{c}_{4}\text{s}_{13}\text{s}_{14}\text{s}_{3}-\text{c}_{13}\text{c}_{4}(\text{c}_{24}\text{c}_{3}\text{s}_{12}\text{s}_{14}\text{s}_{2}+\text{s}_{23}\text{s}_{24}\text{s}_{3})+\text{c}_{14}\text{c}_{24}\text{s}_{4}]$
$C_{4}$ |  $-\text{c}_{2}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{23}\text{s}_{34}-\text{c}_{13}\text{c}_{24}\text{c}_{34}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{14}\text{s}_{2}+\text{c}_{34}\text{c}_{3}\text{c}_{4}\text{s}_{12}\text{s}_{13}\text{s}_{23}\text{s}_{24}\text{s}_{2}+\text{c}_{12}\text{c}_{3}\text{c}_{4}[-\text{c}_{13}\text{c}_{24}\text{c}_{34}\text{c}_{2}\text{s}_{14}+\text{c}_{34}\text{s}_{24}(\text{c}_{2}\text{s}_{13}\text{s}_{23}-\text{c}_{23}\text{s}_{2})+\text{s}_{34}(\text{c}_{23}\text{c}_{2}\text{s}_{13}+\text{s}_{23}\text{s}_{2})]-\text{c}_{24}\text{c}_{34}\text{c}_{4}\text{s}_{13}\text{s}_{14}\text{s}_{3}-\text{c}_{13}\text{c}_{34}\text{c}_{4}\text{s}_{23}\text{s}_{24}\text{s}_{3}+\text{c}_{23}\text{c}_{4}[\text{c}_{34}\text{c}_{2}\text{c}_{3}\text{s}_{12}\text{s}_{24}+\text{s}_{34}(\text{c}_{3}\text{s}_{12}\text{s}_{13}\text{s}_{2}-\text{c}_{13}\text{s}_{3})]+\text{c}_{14}\text{c}_{24}\text{c}_{34}\text{s}_{4}$

Table 1: $C_{i}$-factors of the trilinear gauge couplings $ZZh_{i}$ and
$W^{\pm}W^{\mp}h_{i}$ as defined in Eq. (15) in the main text. Here
$\text{c}_{ij}=\cos\alpha_{ij}$ ($\text{s}_{ij}=\sin\alpha_{ij}$) and
$\text{c}_{i}=\cos\beta_{i}$ ($\text{s}_{i}=\sin\beta_{i}$). In this notation,
$\text{s}_{ij-k}$ stands for $\sin(\alpha_{ij}-\beta_{k})$.

##  Appendix B General Yukawa Couplings

General Yukawa Neutral Scalar
---
$\xi_{h}^{\text{ud}}$ |  $\text{c}_{12}\text{c}_{13}\text{c}_{14}\;/\;\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h}^{e}$ |  $\text{s}_{12}\text{c}_{13}\text{c}_{14}\;/\;\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h}^{\mu}$ |  $\text{s}_{13}\text{c}_{14}\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{h}^{\tau}$ |  $\text{s}_{14}\;/\;\text{s}_{4}$
$\xi_{h_{2}}^{\text{ud}}$ |  $-\left(\text{c}_{23}\text{c}_{24}\text{s}_{12}+\text{c}_{12}\left(\text{c}_{24}\text{s}_{13}\text{s}_{23}+\text{c}_{13}\text{s}_{14}\text{s}_{24}\right)\right)\;/\;\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h_{2}}^{e}$ |  $\left(\text{c}_{12}\text{c}_{23}\text{c}_{24}-\text{s}_{12}\left(\text{c}_{24}\text{s}_{13}\text{s}_{23}+\text{c}_{13}\text{s}_{14}\text{s}_{24}\right)\right)\;/\;\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h_{2}}^{\mu}$ |  $\left(\text{c}_{13}\text{c}_{24}\text{s}_{23}-\text{s}_{13}\text{s}_{14}\text{s}_{24}\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{h_{2}}^{\tau}$ |  $\text{c}_{14}\text{s}_{24}\;/\;\text{s}_{4}$
$\xi_{h_{3}}^{\text{ud}}$ |  $\left(\text{s}_{12}\left(\text{c}_{34}\text{s}_{23}+\text{c}_{23}\text{s}_{24}\text{s}_{34}\right)\\!-\\!\text{c}_{12}\left(\text{c}_{13}\text{c}_{24}\text{s}_{14}\text{s}_{34}+\text{s}_{13}\left(\text{c}_{23}\text{c}_{34}\\!-\\!\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\right)/\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h_{3}}^{e}$ |  $-\\!\left(\text{c}_{12}\left(\text{c}_{34}\text{s}_{23}\\!+\\!\text{c}_{23}\text{s}_{24}\text{s}_{34}\right)\\!+\\!\text{s}_{12}\left(\text{c}_{13}\text{c}_{24}\text{s}_{14}\text{s}_{34}\\!+\\!\text{s}_{13}\left(\text{c}_{23}\text{c}_{34}\\!+\\!\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\right)\\!/\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h_{3}}^{\mu}$ |  $\left(-\text{c}_{24}\text{s}_{13}\text{s}_{14}\text{s}_{34}+\text{c}_{13}\left(\text{c}_{23}\text{c}_{34}-\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{h_{3}}^{\tau}$ |  $\text{c}_{14}\text{c}_{24}\text{s}_{34}\;/\;\text{s}_{4}$
$\xi_{h_{4}}^{\text{ud}}$ |  $\left(\text{s}_{12}\left(\text{c}_{23}\text{c}_{34}\text{s}_{24}-\text{s}_{23}\text{s}_{34}\right)\\!-\\!\text{c}_{12}\left(\text{c}_{13}\text{c}_{24}\text{c}_{34}\text{s}_{14}\\!-\\!\text{s}_{13}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\right)/\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h_{4}}^{e}$ |  $-\\!\left(\text{c}_{12}\left(\text{c}_{23}\text{c}_{34}\text{s}_{24}\\!-\\!\text{s}_{23}\text{s}_{34}\right)\\!+\\!\text{s}_{12}\left(\text{c}_{13}\text{c}_{24}\text{c}_{34}\text{s}_{14}\\!-\\!\text{s}_{13}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}\\!+\\!\text{c}_{23}\text{s}_{34}\right)\right)\right)\\!/\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{h_{4}}^{\mu}$ |  $-\left(\text{c}_{24}\text{c}_{34}\text{s}_{13}\text{s}_{14}+\text{c}_{13}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{h_{4}}^{\tau}$ |  $\text{c}_{14}\text{c}_{24}\text{c}_{34}\;/\;\text{s}_{4}$

Table 2: General Yukawa couplings of the scalar Higgs particles to quarks and
charged leptons, as defined in Eqs. (16) and (17) in the main text. Here
$\text{c}_{ij}=\cos\alpha_{ij}$ ($\text{s}_{ij}=\sin\alpha_{ij}$) and
$\text{c}_{i}=\cos\beta_{i}$ ($\text{s}_{i}=\sin\beta_{i}$).

General Yukawa Pseudoscalar
---
$\xi_{A_{1}}^{\text{q}}$ |  $-\left(\text{c}_{23}\text{c}_{24}\text{s}_{2}+\text{c}_{2}\left(\text{c}_{24}\text{s}_{3}\text{s}_{23}+\text{c}_{3}\text{s}_{4}\text{s}_{24}\right)\right)\;/\;\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{A_{1}}^{e}$ |  $\left(\text{c}_{2}\text{c}_{23}\text{c}_{24}-\text{s}_{2}\left(\text{c}_{24}\text{s}_{3}\text{s}_{23}+\text{c}_{3}\text{s}_{4}\text{s}_{24}\right)\right)\;/\;\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{A_{1}}^{\mu}$ |  $\left(\text{c}_{3}\text{c}_{24}\text{s}_{23}-\text{s}_{3}\text{s}_{4}\text{s}_{24}\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{A_{1}}^{\tau}$ |  $\text{s}_{24}\text{c}_{4}\;/\;\text{s}_{4}$
$\xi_{A_{2}}^{\text{q}}$ |  $\left(\text{s}_{2}\left(\text{c}_{34}\text{s}_{23}+\text{c}_{23}\text{s}_{24}\text{s}_{34}\right)-\text{c}_{2}\left(\text{c}_{3}\text{c}_{24}\text{s}_{4}\text{s}_{34}+\text{s}_{3}\left(\text{c}_{23}\text{c}_{34}-\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\right)\\!/\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{A_{2}}^{e}$ |  $-\\!\left(\text{c}_{2}\left(\text{c}_{34}\text{s}_{23}+\text{c}_{23}\text{s}_{24}\text{s}_{34}\right)\\!+\\!\text{s}_{2}\left(\text{c}_{3}\text{c}_{24}\text{s}_{4}\text{s}_{34}+\text{s}_{3}\left(\text{c}_{23}\text{c}_{34}\\!-\\!\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\right)\\!/\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{A_{2}}^{\mu}$ |  $\left(-\text{c}_{24}\text{s}_{3}\text{s}_{4}\text{s}_{34}+\text{c}_{3}\left(\text{c}_{23}\text{c}_{34}-\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{A_{2}}^{\tau}$ |  $\text{c}_{24}\text{s}_{34}\text{c}_{4}\;/\;\text{s}_{4}$
$\xi_{A_{3}}^{\text{q}}$ |  $\left(\text{s}_{2}\left(\text{c}_{23}\text{c}_{34}\text{s}_{24}-\text{s}_{23}\text{s}_{34}\right)-\text{c}_{2}\left(\text{c}_{3}\text{c}_{24}\text{c}_{34}\text{s}_{4}-\text{s}_{3}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\right)\\!/\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{A_{3}}^{e}$ |  $-\\!\left(\text{c}_{2}\left(\text{c}_{23}\text{c}_{34}\text{s}_{24}-\text{s}_{23}\text{s}_{34}\right)\\!+\\!\text{s}_{2}\left(\text{c}_{3}\text{c}_{24}\text{c}_{34}\text{s}_{4}\\!-\\!\text{s}_{3}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\right)\\!/\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{A_{3}}^{\mu}$ |  $-\left(\text{c}_{24}\text{c}_{34}\text{s}_{3}\text{s}_{4}+\text{c}_{3}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{A_{3}}^{\tau}$ |  $\text{c}_{24}\text{c}_{34}\text{c}_{4}\;/\;\text{s}_{4}$

Table 3: General Yukawa couplings of the pseudoscalar Higgs particles to
quarks and charged leptons, as defined in Eqs. (16) and (17) in the main text.
Here $\text{c}_{ij}=\cos\gamma_{ij}$ ($\text{s}_{ij}=\sin\gamma_{ij}$) and
$\text{c}_{i}=\cos\beta_{i}$ ($\text{s}_{i}=\sin\beta_{i}$).

General Yukawa Charged
---
$\xi_{H_{1}^{+}}^{\text{qLR}}$ |  $-\left(\text{c}_{23}\text{c}_{24}\text{s}_{2}+\text{c}_{2}\left(\text{c}_{24}\text{s}_{3}\text{s}_{23}+\text{c}_{3}\text{s}_{4}\text{s}_{24}\right)\right)\;/\;\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{H_{1}^{+}}^{\text{eL}}$ |  $\left(\text{c}_{2}\text{c}_{23}\text{c}_{24}-\text{s}_{2}\left(\text{c}_{24}\text{s}_{3}\text{s}_{23}+\text{c}_{3}\text{s}_{4}\text{s}_{24}\right)\right)\;/\;\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{H_{1}^{+}}^{\text{$\mu$L}}$ |  $\left(\text{c}_{3}\text{c}_{24}\text{s}_{23}-\text{s}_{3}\text{s}_{4}\text{s}_{24}\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{H_{1}^{+}}^{\text{$\tau$L}}$ |  $\text{s}_{24}\text{c}_{4}\;/\;\text{s}_{4}$
$\xi_{H_{2}^{+}}^{\text{qLR}}$ |  $\left(\text{s}_{2}\left(\text{c}_{34}\text{s}_{23}+\text{c}_{23}\text{s}_{24}\text{s}_{34}\right)-\text{c}_{2}\left(\text{c}_{3}\text{c}_{24}\text{s}_{4}\text{s}_{34}+\text{s}_{3}\left(\text{c}_{23}\text{c}_{34}-\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\right)\\!/\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{H_{2}^{+}}^{\text{eL}}$ |  $-\\!\left(\text{c}_{2}\\!\left(\text{c}_{34}\text{s}_{23}+\text{c}_{23}\text{s}_{24}\text{s}_{34}\right)\\!+\\!\text{s}_{2}\left(\text{c}_{3}\text{c}_{24}\text{s}_{4}\text{s}_{34}+\text{s}_{3}\left(\text{c}_{23}\text{c}_{34}-\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\right)\\!/\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{H_{2}^{+}}^{\text{$\mu$L}}$ |  $\left(-\text{c}_{24}\text{s}_{3}\text{s}_{4}\text{s}_{34}+\text{c}_{3}\left(\text{c}_{23}\text{c}_{34}-\text{s}_{23}\text{s}_{24}\text{s}_{34}\right)\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{H_{2}^{+}}^{\text{$\tau$L}}$ |  $\text{c}_{24}\text{s}_{34}\text{c}_{4}\;/\;\text{s}_{4}$
$\xi_{H_{3}^{+}}^{\text{qLR}}$ |  $\left(\text{s}_{2}\left(\text{c}_{23}\text{c}_{34}\text{s}_{24}-\text{s}_{23}\text{s}_{34}\right)-\text{c}_{2}\left(\text{c}_{3}\text{c}_{24}\text{c}_{34}\text{s}_{4}-\text{s}_{3}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\right)\\!/\text{c}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{H_{3}^{+}}^{\text{eL}}$ |  $\left(\text{c}_{2}\left(\text{s}_{23}\text{s}_{34}-\text{c}_{23}\text{c}_{34}\text{s}_{24}\right)-\text{s}_{2}\left(\text{c}_{3}\text{c}_{24}\text{c}_{34}\text{s}_{4}-\text{s}_{3}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\right)\\!/\text{s}_{2}\text{c}_{3}\text{c}_{4}$
$\xi_{H_{3}^{+}}^{\text{$\mu$L}}$ |  $-\left(\text{c}_{24}\text{c}_{34}\text{s}_{3}\text{s}_{4}+\text{c}_{3}\left(\text{c}_{34}\text{s}_{23}\text{s}_{24}+\text{c}_{23}\text{s}_{34}\right)\right)\;/\;\text{s}_{3}\text{c}_{4}$
$\xi_{H_{3}^{+}}^{\text{$\tau$L}}$ |  $\text{c}_{24}\text{c}_{34}\text{c}_{4}\;/\;\text{s}_{4}$

Table 4: General Yukawa couplings of the charged Higgs particles to quarks and
leptons, as defined in Eqs. (18) and (19) in the main text. Here
$\text{c}_{ij}=\cos\delta_{ij}$ ($\text{s}_{ij}=\sin\delta_{ij}$) and
$\text{c}_{i}=\cos\beta_{i}$ ($\text{s}_{i}=\sin\beta_{i}$).

##  Appendix C Benchmark Points

Scalar benchmark points |  S1 |  S2
---|---|---
$\beta_{2}/\pi,\,\,\beta_{3}/\pi,\,\,\beta_{4}/\pi$ |  $0.05,\,\,0.16,\,\,0.18$ |  $0.04,\,\,0.14,\,\,0.21$
$\alpha_{23}/\pi,\,\,\alpha_{24}/\pi,\,\,\alpha_{34}/\pi$ |  $-0.09,\,\,-1.00,\,\,-0.70$ |  $-0.02,\,\,-0.05,\,\,0.10$
$\gamma_{23}/\pi,\,\,\gamma_{24}/\pi,\,\,\gamma_{34}/\pi$ |  $0.50,\,\,0.59,\,\,0.80$ |  $0.16,\,\,0.52,\,\,0.39$
$\delta_{23}/\pi,\,\,\delta_{24}/\pi,\,\,\delta_{34}/\pi$ |  $0.08,\,\,-0.26,\,\,-0.96$ |  $0.62,\,\,-0.93,\,\,-0.95$
$m_{h_{2}},\,\,m_{h_{3}},\,\,m_{h_{4}}\,(\text{GeV})$ |  $269,\,\,396,\,\,483$ |  $175,\,\,359,\,\,360$
$m_{A_{1}},\,\,m_{A_{2}},\,\,m_{A_{3}}\,(\text{GeV})$ |  $439,\,\,454,\,\,484$ |  $265,\,\,351,\,\,369$
$m_{H_{1}^{\pm}},\,\,m_{H_{2}^{\pm}},\,\,m_{H_{3}^{\pm}}\,(\text{GeV})$ |  $438,\,\,441,\,\,443$ |  $289,\,\,352,\,\,370$
$m^{2}_{qe},\,\,m^{2}_{q\mu},\,\,m^{2}_{q\tau}\,(\text{GeV}^{2})$ |  $-17700,\,71700,\,-340000$ |  $16000,\,-34600,\,-168000$
$m^{2}_{e\mu},\,\,m^{2}_{e\tau},\,\,m^{2}_{\mu\tau}\,(\text{GeV}^{2})$ |  $-18600,\,20700,\,-53600$ |  $14000,\,-31200,\,-57400$
$BR(h_{2}\rightarrow ee)$ |  $2.72\times 10^{-3}$ |  $1.63\times 10^{-4}$
$BR(h_{2}\rightarrow\mu\mu)$ |  $4.68\times 10^{-1}$ |  $7.85\times 10^{-6}$
$BR(h_{2}\rightarrow\tau\tau)$ |  $1.22\times 10^{-1}$ |  $7.42\times 10^{-1}$

Table 5: Benchmark points for the leptonic decays of the lightest neutral
scalar (other than the Standard Model Higgs) from Figure 2, for a $h_{2}$-mass
range below $350~{}\text{GeV}$.

Charged benchmark points |  C1 |  C2
---|---|---
$\beta_{2}/\pi,\,\,\beta_{3}/\pi,\,\,\beta_{4}/\pi$ |  $0.05,\,\,0.05,\,\,0.09$ |  $0.10,\,\,0.16,\,\,0.11$
$\alpha_{23}/\pi,\,\,\alpha_{24}/\pi,\,\,\alpha_{34}/\pi$ |  $0.09,\,\,0.54,\,\,0.34$ |  $0.20,\,\,0.88,\,\,0.72$
$\gamma_{23}/\pi,\,\,\gamma_{24}/\pi,\,\,\gamma_{34}/\pi$ |  $-0.04,\,\,0.66,\,\,0.60$ |  $0.68,\,\,0.50,\,\,-0.52$
$\delta_{23}/\pi,\,\,\delta_{24}/\pi,\,\,\delta_{34}/\pi$ |  $-0.98,\,\,0.00,\,\,-0.36$ |  $1.00,\,\,0.00,\,\,0.77$
$m_{h_{2}},\,\,m_{h_{3}},\,\,m_{h_{4}}\,(\text{GeV})$ |  $127,\,\,187,\,\,208$ |  $180,\,\,237,\,\,240$
$m_{A_{1}},\,\,m_{A_{2}},\,\,m_{A_{3}}\,(\text{GeV})$ |  $131,\,\,179,\,\,244$ |  $161,\,\,172,\,\,173$
$m_{H_{1}^{\pm}},\,\,m_{H_{2}^{\pm}},\,\,m_{H_{3}^{\pm}}\,(\text{GeV})$ |  $164,\,\,172,\,\,229$ |  $158,\,\,181,\,\,234$
$m^{2}_{qe},\,\,m^{2}_{q\mu},\,\,m^{2}_{q\tau}\,(\text{GeV}^{2})$ |  $-14800,\,\,-17400,\,\,6210$ |  $57000,\,\,-127000,\,\,-15100$
$m^{2}_{e\mu},\,\,m^{2}_{e\tau},\,\,m^{2}_{\mu\tau}\,(\text{GeV}^{2})$ |  $5880,\,\,22100,\,\,9060$ |  $-75600,\,\,-9570,\,\,81300$
$BR(H_{1}^{\pm}\rightarrow e^{\pm}\nu_{e})$ |  $2.24\times 10^{-3}$ |  $1.68\times 10^{-2}$
$BR(H_{1}^{\pm}\rightarrow\mu^{\pm}\nu_{\mu})$ |  $5.36\times 10^{-1}$ |  $6.91\times 10^{-3}$
$BR(H_{1}^{\pm}\rightarrow\tau^{\pm}\nu_{\tau})$ |  $4.55\times 10^{-1}$ |  $5.23\times 10^{-1}$

Table 6: Benchmark points for the leptonic decays of the lightest charged
scalar from Figure 3, for a $H_{1}^{\pm}$-mass range below $180~{}\text{GeV}$.

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# Emotion Detection from EEG
using Transfer Learning

Sidharth Sidharth12 1Sidharth Sidharth, Ashish Abraham Samuel and Ranjana H
are co-first authors2Department of Electronics and Communication Engineering,
College of Engineering Trivandrum<EMAIL_ADDRESS>Ashish Abraham
Samuel12  Ranjana H12  Jerrin Thomas Panachakel2  Sana Parveen K2,

###### Abstract

The detection of emotions using an Electroencephalogram (EEG) is a crucial
area in brain-computer interfaces and has valuable applications in fields such
as rehabilitation and medicine. In this study, we employed transfer learning
to overcome the challenge of limited data availability in EEG-based emotion
detection. The base model used in this study was Resnet50. Additionally, we
employed a novel feature combination in EEG-based emotion detection. The input
to the model was in the form of an image matrix, which comprised Mean Phase
Coherence (MPC) and Magnitude Squared Coherence (MSC) in the upper-triangular
and lower-triangular matrices, respectively. We further improved the technique
by incorporating features obtained from the Differential Entropy (DE) into the
diagonal, which previously held little to no useful information for
classifying emotions. The dataset used in this study, SEED EEG (62 channel
EEG), comprises three classes (Positive, Neutral, and Negative). We calculated
both subject-independent and subject-dependent accuracy. The subject-dependent
accuracy was obtained using a 10-fold cross-validation method and was 93.1%,
while the subject-independent classification was performed by employing the
leave-one-subject-out (LOSO) strategy. The accuracy obtained in subject-
independent classification was 71.6%. Both of these accuracies are at least
twice better than the chance accuracy of classifying 3 classes. The study
found the use of MSC and MPC in EEG-based emotion detection promising for
emotion classification. The future scope of this work includes the use of data
augmentation techniques, enhanced classifiers, and better features for emotion
classification.

###### Index Terms:

Brain-computer interface, emotion detection, transfer learning,
electroencephalogram, mean phase coherence, magnitude squared coherence,
SEED_EEG

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##  I Introduction

Emotion detection from biological signals has become a highly significant area
of study in the realms of neuroscience and signal processing research [1].
Emotion detection could be done in various ways. Such as through facial
expressions, paralinguistic analysis, physiological signals, behavioral
signals, and biochemical signals.

Electroencephelography (EEG) is a method of measuring the electrical activity
of the brain and is widely used for emotion detection. It possesses numerous
advantages over other methods. Being a non-invasive method, it does not
deliver an unpleasant experience to the user. It can provide a high temporal
and spatial resolution (subjective to the number of electrodes used) [2] of
brain activity that enables the detection of rapid changes in an emotional
state in a specific section of the brain. It is easily portable. Moreover, EEG
is a direct measurement of brain activity, which provides an objective measure
of the emotional state.
The study [3] demonstrated the utility of coherence as a connectivity index
between EEG channels in understanding the relationship between brain activity
and emotional states. Previous research in the field of emotion detection from
EEG signals has employed various techniques for feature extraction, including
the Short Time Fourier Transform (STFT)[4], Differential Entropy (DE)[5],
Power Spectral Density (PSD)[6], and Continuous Wavelet Transform (CWT)[7].
Emotion is a multi-faceted and intricate process that engages various regions
of the brain. As such, utilizing simultaneous feature extraction from
multichannel EEG data is crucial to identify an emotion accurately. Few works
which include [8] have already employed simultaneous feature extraction from
EEG for emotion detection. [8] employed the Riemannian geometry properties of
the Symmetric Positive Definite (SPD) matrix obtained from the original EEG
data to classify emotions.
In [9], mean phase coherence (MPC) and magnitude squared coherence (MSC) was
used as features for decoding imagined speech and also employed transfer
learning along with data augmentation to overcome the data scarcity issue.
In our work, we used MPC and MSC as the main features employed in emotion
classification. The dataset used was SEED_EEG[10]. It classifies emotions into
three categories; positive, negative, and neutral. We also leveraged the
information extracted from Differential Entropy (DE) provided in the dataset.
We adopted ResNet-50 based transfer learning model as the classifier to tackle
the problem of data insufficiency. The concept of using MPC and MSC as
features in classifying emotions using transfer learning is the novel method
presented in this work.

##  II Dataset

The dataset used for this study is SEED (SJTU Emotion EEG Dataset). We
specifically utilized the SEED_EEG data which contains the EEG of 15
subjects(7 males and 8 females). They were shown approximately 4 minutes of
movie clips that were chosen and edited to induce positive, negative, or
neutral emotions. Each subject underwent the experiment 3 times within an
interval of one week, giving a total of 45 trials. The data were downsampled
to 200 Hz following which a bandpass filter of 0-75 Hz was applied. 62-channel
EEG was obtained via the international 10 - 20 system EEG cap using brain-like
computing and machine learning(BCMI) methods. The 62-channel EEG data is
converted into three frequency bands - alpha (8 to 13 Hz), beta(13 to 30 Hz),
and gamma(30 to 70Hz) by using a bandpass filter. MPC and MSC features are
extracted for each band and entered into an array such that the element at
position (i, j) is the normalized MPC value between the $i^{th}$ and $j^{th}$
channel in the upper triangle of the matrix. Similarly, position (i, j) in the
lower triangular matrix corresponds to normalized MSC. The three bands alpha,
beta, and gamma correspond to the RGB values of an image matrix. Thus the data
input to the classifier is a 62$\times$62 image. The diagonals of the image
generated using this method contained no usable information. To add relevant
information to the diagonals, the Differential Entropy values provided in the
SEED_EEG dataset [11] are introduced as the 62 diagonal elements in each
image.

##  III Feature extraction and Methodology

The main features used in classifying emotions in this work are

  1. 1.

Mean phase coherence (MPC)

  2. 2.

Magnitude squared coherence (MSC)

MPC represents the synchronisation between any two channels of the EEG.[12,
13] MPC across the $i^{th}$ and $k^{th}$ channel is given by:

$MPC_{i,k}=\frac{1}{N}\Bigg{|}\sum_{n=0}^{N-1}e^{-j(\phi_{i}({n})-\phi_{k}({n}))}\Bigg{|}$
(1)

where $\phi_{i}$(n) and $\phi_{k}$(n) denote the phase of $i^{th}$ and
$k^{th}$ channel and N denotes the length of time series data contained in the
channel. The phase of each channel is computed using Hilbert Transform.

Magnitude squared coherence(MSC) demonstrates the linear relationship between
a pair of signals. The MSC between $X_{i*}$ and $X_{k*}$ is given by:

$MSC_{i,k}({\omega})=\frac{|S_{i,k}|^{2}}{S_{i,i}({\omega})S_{k,k}({\omega})}$
(2)

where $S_{i,i}({\omega})$ and$S_{k,k}({\omega})$ denote the power spectral
densities and $S_{i,k}({\omega})$ is the cross power spectral density.
Values of MPC and MSC lies in [0,1]. MPC and MSC for all the combinations of
the channels were calculated. MPC was obtained for each of the three bands
separately, while MSC was summed and normalized. Since the coherence (i, j) is
the same as (j, i), We stacked these features into a matrix of dimension
62$\times$62 such that the upper triangle of the matrix was filled with MPC
values and the lower triangle with MSC values. Fig 1. shows the block diagram
of the method employed to generate the dataset. The diagonal elements of these
matrices contained information about the coherence between the same channels.
Hence they carried no information in particular. In [11], the authors
concluded that DE is well suited for emotion recognition. Motivated by this,
we took advantage of the DE feature provided along with SEED.
Let $X$ be a random variable with probability distribution $f$, the DE $h(X)$
is defined as

$h(X)=-\int_{-\infty}^{\infty}f(X)\log f(X)dx\\\ $ (3)

We obtained the mean of the DE values for every channel and stacked it into
the diagonal of the matrices. The matrices were then converted to images such
that alpha($\alpha$), beta($\beta$), and gamma($\gamma$) bands correspond to
RGB values. The images were then fed to the ResNet-50-based transfer learning
model. We performed subject-dependent and subject-independent classifications
of emotions.
In subject-dependent classification, to avoid overfitting due to training on
the limited availability of data, 10-fold cross-validation was employed. We
added a global spatial average pooling layer, a fully-connected layer, and a
logistic layer to the model as shown in Fig 3.. These layers were randomly
initialized and trained on the subject-dependent classification task.
In the subject-independent classification process, as shown in Fig 2., we
incorporated a 2D convolution layer, a Dense layer, a Flatten layer and
finally a Dense layer which generated a probability distribution over the 3
classes.

##  IV Experiments

Two experiments were conducted on the SEED dataset. The first experiment was a
subject-dependent classification

Figure 1: Method used for generating the dataset which is fed to the
Classifiers.(BPF refers to band pass filter) Figure 2: Transfer learning
approach using ResNet-50 for subject-independent classification

and the second was a subject-independent classification.
1) Subject-dependent experiment:
Subject-dependent classification is a method where the classifier is trained
and tested on data that is specific to certain subjects or individuals. In
this scenario, a 10-fold cross-validation technique was employed to train and
evaluate the classifier. The data was divided into 10 equal parts, with 9
parts being used for training and the remaining part being used for testing.
This approach allows for a robust evaluation of the classifier’s performance,
as it is trained and tested on different subsets of the data, simulating the
scenario where the model is trained on data from some subjects and tested on
data from other subjects. The evaluation metric used was accuracy. Mean
accuracy was obtained by averaging the accuracy scores of all the folds.
2) Subject-independent experiment:
Subject independent classification refers to a scenario where the classifier
is trained and tested on data from different subjects or individuals. In this
case, the classifier is not specifically tailored to the characteristics of
the subjects for which it is trained and is expected to generalize well to new
data from unseen subjects. The experimental setup involved dividing the
dataset into a training set and a testing set in a specific way. A leave-one-
subject-out (LOSO) cross-validation approach was used, where out of the 15
subjects, one subject was designated as the testing set while the remaining 14
subjects were used as the training set. This process was repeated for each
subject so that each subject appears in the testing set exactly once. The mean
accuracy obtained from each iteration was calculated, providing an overall
measure of the classifier’s performance on the dataset.

##  V Results

We conducted two types of experiments on the SEED dataset: subject-dependent
and subject-independent. In the

Figure 3: Transfer learning approach using ResNet-50 for subject-dependent
classification

subject-dependent experiment, the classifier achieved an accuracy of 93.1%
with a standard deviation of 3%. In the subject-independent experiment, the
accuracy obtained was 71.6% with a standard deviation of 7.6% as shown in Fig
4. The error bars in the graph shows the standard deviation. The results of
these experiments were compared against various previous models, and the
comparison is presented in tables 1 and 2.

TABLE I: Comparison with existing literature for subejct-dependent emotion classification using SEED_EEG Paper | Model | Accuracy
---|---|---
Liu, Junxiu et al.[14] | DNN and SAE | 96.7%
Song, Zheng et al.[15] | DGCNN | 90.4%
Asghar, Adeel et al.[16] | DNN | 91.3%
Our Model | Transfer learning | 93.1%
TABLE II: Comparison with existing literature for subject independent emotion classification using SEED$\\_$EEG Paper | model | Accuracy
---|---|---
Sunhee Hwang et al.[17] | Multi-task DNN | 75.3%
Cimtay and Ekmekcioglu.[18] | SVM and DNN | 78.3%
Li, Jingkong et al.[19] | SOGNN | 86.8%
Our Model | Transfer learning | 71.6%

Cohen’s Kappa $\kappa$ is defined as

$\kappa:=\frac{p_{cl}-p_{ch}}{100-p_{ch}}$ (4)

Here, $p_{cl}$ is the accuracy of the classifier and Here, $p_{ch}$ is the
chance level accuracy. The value of $\kappa$ lies in the range [-1,1] where
values closer to 0 indicate that the performance

Figure 4: Accuracy for subject-dependent and subject-independent
classifications

of the classifier is only as good as a random guess.
The chance accuracy for classifying three classes is 33.33%.
The $\kappa$ value for the subject-independent classification accuracy is
0.574 and 0.897 for subject-dependent classification.
Our results show that the classifier performed at least twice as well as
chance accuracy in both subject-dependent and subject-independent experiments.

##  VI conclusion

In this work, a novel combination of features is proposed for classifying
emotions from EEG data. To overcome the limitation of limited data
availability, transfer learning using a pre-trained ResNet-50 model is
employed. The achieved accuracy in subject-dependent classification is 93.1%,
and in subject-independent classification is 71.6%. These results are
significantly better than the chance accuracy of classifying three classes.
The innovation in this work is the use of MPC, MSC and DE as the primary
features for EEG emotion classification and compactly arranging them in a
three-dimensional array. This three-dimensional array, resembling a colour
image in the RGB space is fed to a ResNet-50 model, pretrained on ImageNet
dataset, enabling the model to effectively classify emotions.

##  VII Future scope

The future direction of this research includes exploring the potential of data
augmentation techniques to enhance the model’s performance and generalization
ability. By implementing data augmentation, the limitation of limited data
availability can be mitigated to some extent. Furthermore, experimenting with
different classifiers and a combination of enhanced features can lead to an
improvement in classification accuracy. The significance of this research is
substantial as it holds the potential to be applied in the field of
neuroprosthetics to aid individuals dealing with emotional disorders. In
summary, this study presents a novel combination of features for the
classification of emotions from EEG data, and the outcomes demonstrate the
promise of the proposed method for further research in this area.

##  VIII Ethics Statement

Publicly available dataset (SEED_EEG) was used in this
study(https://bcmi.sjtu.edu.cn/home/seed/seed.html). The studies involving
human participants were reviewed and approved by Ethics Committee of South
China Normal University. The patients/participants provided their written
informed consent to participate in this study. Written informed consent was
obtained from the individual(s) for the publication of any potentially
identifiable images or data included in this article.

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