#  CFTP/23-001
A viable $A_{4}$ 3HDM theory of quark mass matrices

Iris Brée111<EMAIL_ADDRESS>Sérgio Carrôlo
<EMAIL_ADDRESS>Jorge C. Romão
<EMAIL_ADDRESS>João P. Silva<EMAIL_ADDRESS>
CFTP, Departamento de Física,
Instituto Superior Técnico, Universidade de Lisboa,
Avenida Rovisco Pais 1, 1049 Lisboa, Portugal

###### Abstract

It is known that a three Higgs doublet model (3HDM) symmetric under an exact
$A_{4}$ symmetry is not compatible with nonzero quark masses and/or non-block-
diagonal CKM matrix. We show that a 3HDM with softly broken $A_{4}$ terms in
the scalar potential does allow for a fit of quark mass matrices. Moreover,
the result is consistent with $m_{h}=125\textrm{GeV}$ and the $h\rightarrow
WW,ZZ$ signal. We also checked numerically that, for each point that passes
all the constraints, the minimum is a global minimum of the potential.

##  1 Introduction

The observation in 2012 of a scalar particle with 125GeV by the ATLAS and CMS
collaborations [1, 2] has incentivized experimental searches for beyond the
Standard Model (SM) particles at the LHC. On par with these experimental
endeavors, theoretical efforts in the search for extra scalar particles have
been strengthened since this discovery. A promising framework is found in
N-Higgs doublet models (NHDM).

Such models have many free parameters, which are often curtailed by imposing
some discrete family symmetry. Here, we focus on the implementation of $A_{4}$
in a three Higgs doublet model (3HDM). The $A_{4}$ group is the group of even
permutations on 4 elements. It is the smallest discrete group to contain a
three-dimensional irreducible representation (irrep), which is ideal for
describing the three families of quarks with a minimal number of independent
Yukawa couplings. Thus, NHDM supplemented by the $A_{4}$ discrete symmetry has
long been of interest in flavour physics research.

A number of early articles include: [3], mainly devoted to the leptonic sector
and where the solution to the quark sector is briefly mentioned to include a
fourth Higgs doublet and all quark fields in singlets (which is effectively
the same as the Standard Model quark sector); [4], where $A_{4}$ is broken by
dimension four Yukawa couplings, thus rendering the theory non-renormalizable
or, alternatively, the low energy limit of a broader complete theory; [5],
which requires three Higgs doublets in the down-type quark sector and a
further two in the up-type quark sector, consisting of a 5HDM; and [6], which
is devoted to the leptonic sector, but has the interesting side query that it
might be possible to recover a realistic CKM matrix through soft-breaking of
$A_{4}$.

Quark mass matrices in the context of a 3HDM with Higgs doublets in the
triplet representation of $A_{4}$ were studied in [7] and [8], with the vacuum
expectation value (vev) structure $(e^{i\alpha},e^{-i\alpha},r)$, where
$\alpha$ and $r$ are real constants. Unfortunately, Degee, Ivanov and Keus [9]
proved in 2013 that such a vacuum can never be the global minimum of the
$A_{4}$ symmetric 3HDM. In this beautiful paper, geometric techniques were
used in order to identify all possible global minima (thus, all possible
viable vacua) of the $A_{4}$ symmetric 3HDM. Immediately thereafter, those
minima were used to show that all assignments of the quark fields into irreps
of $A_{4}$, when combined with the possible vevs for the exact $A_{4}$
potential, yield vanishing quark masses and/or a CP conserving CKM matrix,
both of which are forbidden by experiment. This is in fact a consequence of a
much broader theorem, proved in [10, 11]: given any flavour symmetry group,
one can obtain a physical CKM mixing matrix and, simultaneously, non-
degenerate and non-zero quark masses only if the vevs of the Higgs fields
break completely the full flavour group. The idea is that a symmetry will
reduce the number of redundant Yukawa couplings present in the SM, and it
might even predict relations among observables which turn out to be consistent
with experiment.

When studying in detail the extensions of $A_{4}$ to the quark sector found by
Ref. [12], we noticed that, in some of them, if it weren’t for the particular
form of the vevs allowed by the exact $A_{4}$ 3HDM potential, the Yukawa
matrices could allow for massive quarks, and for a realistic CKM matrix. Since
the $A_{4}$ symmetric potential doesn’t allow for minima other than those
shown in [9], here we consider the case where the $A_{4}$ symmetry is softly
broken by the addition of quadratic terms to the potential. Such terms do not
spoil the theory’s renormalizability, but break the $A_{4}$ symmetry.

Our article is organized as follows. We define the notation for the scalar
potential in Sec. 2.1, discuss the Yukawa Lagrangian and the form of the
possible mass matrices in Sec. 2.2, giving all the expressions needed for the
fit in Sec. 2.3. In Sec. 3 we present our fit to the quarks mass matrices,
while in Sec. 4 we discuss the viability of the vacuum found in the fit in
terms of the scalar potential. Sec. 5 is devoted to the implementation of the
theoretical constraints to be imposed, and in Sec. 6 we briefly discuss the
constraints coming from the LHC. The results and conclusions are presented in
Sec. 7 and 8, respectively. The Appendices contain some additional expressions
that are needed for the fits.

##  2 Parameterization for the softly-broken $A_{4}$ 3HDM

###  2.1 Potential and candidates for local minimum

The softly-broken potential of the 3HDM with an $A_{4}$ symmetry is given by

$V_{H}=V_{4,\,A_{4}}+M^{2}_{ij}\left(\phi_{i}^{\dagger}\phi_{j}\right)\,,$ (1)

where $V_{4,\,A_{4}}$ is the quartic potential for the $A_{4}$ symmetric three
Higgs doublet model (3HDM), which is, in the notation of [9],

$\displaystyle V_{4,\,A_{4}}=$
$\displaystyle\frac{\Lambda_{0}}{3}\left(\phi_{1}^{\dagger}\phi_{1}+\phi_{2}^{\dagger}\phi_{2}+\phi_{3}^{\dagger}\phi_{3}\right)^{2}$
$\displaystyle+\Lambda_{1}\left[\left(\textrm{Re}\left\\{\phi_{1}^{\dagger}\phi_{2}\right\\}\right)^{2}+\left(\textrm{Re}\left\\{\phi_{2}^{\dagger}\phi_{3}\right\\}\right)^{2}+\left(\textrm{Re}\left\\{\phi_{3}^{\dagger}\phi_{1}\right\\}\right)^{2}\right]$
$\displaystyle+\Lambda_{2}\left[\left(\textrm{Im}\left\\{\phi_{1}^{\dagger}\phi_{2}\right\\}\right)^{2}+\left(\textrm{Im}\left\\{\phi_{2}^{\dagger}\phi_{3}\right\\}\right)^{2}+\left(\textrm{Im}\left\\{\phi_{3}^{\dagger}\phi_{1}\right\\}\right)^{2}\right]$
$\displaystyle+\frac{\Lambda_{3}}{3}\left[(\phi_{1}^{\dagger}\phi_{1})^{2}+(\phi_{2}^{\dagger}\phi_{2})^{2}+(\phi_{3}^{\dagger}\phi_{3})^{2}-(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})-(\phi_{2}^{\dagger}\phi_{2})(\phi_{3}^{\dagger}\phi_{3})-(\phi_{3}^{\dagger}\phi_{3})(\phi_{1}^{\dagger}\phi_{1})\right]$
$\displaystyle+\Lambda_{4}\left[\textrm{Re}\left\\{\phi_{1}^{\dagger}\phi_{2}\right\\}\,\textrm{Im}\left\\{\phi_{1}^{\dagger}\phi_{2}\right\\}+\textrm{Re}\left\\{\phi_{2}^{\dagger}\phi_{3}\right\\}\,\textrm{Im}\left\\{\phi_{2}^{\dagger}\phi_{3}\right\\}+\textrm{Re}\left\\{\phi_{3}^{\dagger}\phi_{1}\right\\}\,\textrm{Im}\left\\{\phi_{3}^{\dagger}\phi_{1}\right\\}\right]\,.$
(2)

The matrix $M^{2}_{ij}$ is a general hermitian matrix, which can be
parameterized by

$(M^{2}_{ij})=\begin{pmatrix}m^{2}_{11}&m^{2}_{12}e^{i\theta_{12}}&m^{2}_{13}e^{i\theta_{13}}\\\
m^{2}_{12}e^{-i\theta_{12}}&m^{2}_{22}&m^{2}_{23}e^{i\theta_{23}}\\\
m^{2}_{13}e^{-i\theta_{13}}&m^{2}_{23}e^{-i\theta_{23}}&m^{2}_{33}\end{pmatrix}\,,$
(3)

where $m^{2}_{ij}$ are real parameters with the dimension of mass
squared.555In the quadratic terms, the combination
$-\frac{M_{0}}{\sqrt{3}}\left(\phi_{1}^{\dagger}\phi_{1}+\phi_{2}^{\dagger}\phi_{2}+\phi_{3}^{\dagger}\phi_{3}\right)$
is also invariant under $A_{4}$. But, since we are keeping all soft-breaking
terms, we find the notation in (3) more convenient.

Additionally, in the notation of [13], the exact $A_{4}$ potential can be
written as

$\displaystyle V_{A_{4}}=$
$\displaystyle\frac{r_{1}+2r_{4}}{3}\left[(\phi_{1}^{\dagger}\phi_{1})+(\phi_{2}^{\dagger}\phi_{2})+(\phi_{3}^{\dagger}\phi_{3})\right]^{2}+\frac{2(r_{1}-r_{4})}{3}\left[(\phi_{1}^{\dagger}\phi_{1})^{2}+(\phi_{2}^{\dagger}\phi_{2})^{2}\right.$
$\displaystyle\left.+(\phi_{3}^{\dagger}\phi_{3})^{2}-(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})-(\phi_{2}^{\dagger}\phi_{2})(\phi_{3}^{\dagger}\phi_{3})-(\phi_{3}^{\dagger}\phi_{3})(\phi_{1}^{\dagger}\phi_{1})\right]$
$\displaystyle+2r_{7}\left(|\phi_{1}^{\dagger}\phi_{2}|^{2}+|\phi_{2}^{\dagger}\phi_{3}|^{2}+|\phi_{3}^{\dagger}\phi_{1}|^{2}\right)$
$\displaystyle+\Big{[}c_{3}\left[(\phi_{1}^{\dagger}\phi_{2})^{2}+(\phi_{2}^{\dagger}\phi_{3})^{2}+(\phi_{3}^{\dagger}\phi_{1})^{2}\right]+h.c.\Big{]}\,.$
(4)

The relation between the two notations is

$\displaystyle r_{1}=\frac{1}{3}(\Lambda_{0}+\Lambda_{3})\,,$ $\displaystyle
r_{4}=\frac{1}{6}(2\Lambda_{0}-\Lambda_{3})\,,$ $\displaystyle
r_{7}=\frac{1}{4}(\Lambda_{1}+\Lambda_{2})\,,$
$\displaystyle\text{Re}(c_{3})=\frac{1}{4}(\Lambda_{1}-\Lambda_{2})\,,$
$\displaystyle\text{Im}(c_{3})=-\frac{1}{4}\Lambda_{4}\,.$ (5)

We consider that the scalar fields can take complex vacuum expectation values
(vevs), to be determined later. Thus, we write,

$\phi_{i}=\begin{bmatrix}\varphi^{+}_{i}\\\
\frac{|v_{i}|e^{i\rho_{i}}}{\sqrt{2}}+\frac{1}{\sqrt{2}}\left(x_{i}+ix_{i+3}\right)\end{bmatrix}.$
(6)

Because CP is spontaneously violated, the unrotated neutral fields have no
definite CP, and for convenience we label them $x_{i},i=1,\ldots,6$. We can
also use the gauge freedom to absorb one of the phases in the vevs, that we
choose to be $\rho_{1}$. Therefore we have the vector of vevs defined as

$\vec{v}=(|v_{1}|,|v_{2}|e^{i\rho_{2}},|v_{3}|e^{i\rho_{3}})\,.$ (7)

This vev contributes with four free parameters to our model, because one of
the parameters is constrained by the mass of the gauge bosons to match the
observed SM values,

$|v_{1}|^{2}+|v_{2}|^{2}+|v_{3}|^{2}\equiv v^{2}\simeq(246\textrm{GeV})^{2}.$
(8)

The vev can also be parameterised as

$\vec{v}=v\left(\cos(\beta_{1})\cos(\beta_{2}),\cos(\beta_{2})\sin(\beta_{1})e^{ip_{2}},\sin(\beta_{2})e^{ip_{3}}\right).$
(9)

Of the quantities arising out of the scalar potential, the vevs are the only
relevant to the quark mass matrices. This leads many authors to just proclaim
some vevs, without checking whether they can indeed be the global minima of a
realistic Higgs potential. We will perform this crucial verification below, in
Section 4.

###  2.2 Yukawa Lagrangian

As in Refs. [7, 12], we consider that the Higgs doublets are in the 3 of
$A_{4}$ as well as the three left-handed $SU(2)$ doublets $Q_{Lj}$ of
hypercharge 1/6. There are three right-handed $SU(2)$ singlets $n_{R,j}$ of
hypercharge $-1/3$ and three right-handed $SU(2)$ singlets $p_{R,j}$ of
hypercharge $2/3$. Our assignments for the singlets are as follows

$\displaystyle n_{R1},p_{R1}\to\textbf{1},\quad
n_{R2},p_{R2}\to\textbf{1}^{\prime},\quad
n_{R3},p_{R3}\to\textbf{1}^{\prime\prime}\ \text{ of }A_{4}\,.$ (10)

Then, the $A_{4}$ transformations on the fields are generated by [7, 12]

$\displaystyle
T:\left\\{\begin{matrix}[l]\phi_{1}\to\phi_{2}\to\phi_{3}\to\phi_{1},\\\\[2.84526pt]
Q_{L1}\to Q_{L2}\to Q_{L3}\to Q_{L1},\\\\[2.84526pt] n_{R1}\to
n_{R1},n_{R2}\to\omega n_{R2},n_{R3}\to\omega^{2}n_{R3},\\\\[2.84526pt]
p_{R1}\to n_{R1},p_{R2}\to\omega
p_{R2},p_{R3}\to\omega^{2}p_{R3},\end{matrix}\right.$ (11)

and

$\displaystyle
S:\left\\{\begin{matrix}[l]\phi_{1}\to\phi_{1},\phi_{2}\to-\phi_{2},\phi_{3}\to-\phi_{3},\\\\[2.84526pt]
Q_{L1}\to Q_{L1},Q_{L2}\to-Q_{L2},Q_{L3}\to-Q_{L3}.\\\\[2.84526pt]
\end{matrix}\right.$ (12)

One can easily verify that the scalar potential in Eq. (4) is invariant under
the previous transformations. Now we write the $A_{4}$ invariant Yukawa
Lagrangian for quarks. We have

$\displaystyle-\mathcal{L}_{\rm Yukawa}=$
$\displaystyle\sqrt{2}\,\hat{a}\left(\overline{Q_{L1}}\phi_{1}+\overline{Q_{L2}}\phi_{2}+\overline{Q_{L3}}\phi_{3}\right)n_{R1}$
$\displaystyle+\sqrt{2}\,\hat{b}\left(\overline{Q_{L1}}\phi_{1}+\omega\,\overline{Q_{L2}}\phi_{2}+\omega^{2}\,\overline{Q_{L3}}\phi_{3}\right)n_{R2}$
$\displaystyle+\sqrt{2}\,\hat{c}\left(\overline{Q_{L1}}\phi_{1}+\omega^{2}\,\overline{Q_{L2}}\phi_{2}+\omega\,\overline{Q_{L3}}\phi_{3}\right)n_{R3}$
$\displaystyle+\sqrt{2}\,\hat{a}^{\prime}\left(\overline{Q_{L1}}\tilde{\phi}_{1}+\overline{Q_{L2}}\tilde{\phi}_{2}+\overline{Q_{L3}}\tilde{\phi}_{3}\right)p_{R1}$
$\displaystyle+\sqrt{2}\,\hat{b}^{\prime}\left(\overline{Q_{L1}}\tilde{\phi}_{1}+\omega\,\overline{Q_{L2}}\tilde{\phi}_{2}+\omega^{2}\,\overline{Q_{L3}}\tilde{\phi}_{3}\right)p_{R2}$
$\displaystyle+\sqrt{2}\,\hat{c}^{\prime}\left(\overline{Q_{L1}}\tilde{\phi}_{1}+\omega^{2}\,\overline{Q_{L2}}\tilde{\phi}_{2}+\omega\,\overline{Q_{L3}}\tilde{\phi}_{3}\right)p_{R3}+\text{h.c.},$
(13)

where, as usual,

$\tilde{\phi}_{j}\equiv i\,\sigma_{2}\phi^{*}_{j}\,,$ (14)

and we define

$\hat{a}=ae^{i\,\alpha},\ \hat{b}=be^{i\,\beta},\ \hat{c}=ce^{i\,\gamma},\
\hat{a}^{\prime}=a^{\prime}e^{i\,\alpha^{\prime}},\
\hat{b}^{\prime}=b^{\prime}e^{i\,\beta^{\prime}},\
\hat{c}^{\prime}=c^{\prime}e^{i\,\gamma^{\prime}}\,,$ (15)

where $a,b,c,a^{\prime},b^{\prime},c^{\prime}$ are real and positive. This
choice of invariant Lagrangian corresponds to the case I identified in Ref.
[12] (see the next section).

###  2.3 Yukawa matrices, masses and CKM

We aim to fit six quark masses and four CKM matrix elements to the currently
accepted SM values for these observables. Therefore, we’re interested in
softly-broken $A_{4}$ symmetric models with up to ten parameters. Ref. [12]
has studied all of the possible extensions of $A_{4}$ to the fermion sector.
Using their results, we can check which of them can accommodate non-vanishing
quark masses, CKM mixing angles and CP violation by considering a general vev
$\vec{v}$. We take the Jarlskog invariant as a measure of CP violation [14].
Out of all possibilities, we are left with five of them, which we list in
Table 1. There, $A$ are real constants, $\Omega$ are constants in the
$[0,2\pi[$ interval, $\omega=e^{i\frac{2\pi}{3}}$ ($\omega^{3}=1$) and T is
the transpose of the matrix.

Case | $M_{d}$ | $M_{u}$
---|---|---
I | $\begin{pmatrix}ae^{i\alpha}v_{1}&be^{i\beta}v_{1}&ce^{i\gamma}v_{1}\\\ ae^{i\alpha}v_{2}&\omega be^{i\beta}v_{2}&\omega^{2}ce^{i\gamma}v_{2}\\\ ae^{i\alpha}v_{3}&\omega^{2}be^{i\beta}v_{3}&\omega ce^{i\gamma}v_{3}\\\ \end{pmatrix}$ | $\begin{pmatrix}A\rightarrow A^{\prime},&A\in\\{a,b,c\\}\\\ \Omega\rightarrow\Omega^{\prime},&\Omega\in\\{\alpha,\beta,\gamma\\}\\\ v_{i}\rightarrow v_{i}^{*},&i\in\\{1,2,3\\}\end{pmatrix}$
II | $\text{I}_{d}^{T}$ | $\text{I}_{u}^{T}$
III | $\begin{pmatrix}0&(ae^{i\alpha}-be^{i\beta})v_{3}&(ae^{i\alpha}+be^{i\beta})v_{2}\\\ (ae^{i\alpha}+be^{i\beta})v_{3}&0&(ae^{i\alpha}-be^{i\beta})v_{1}\\\ (ae^{i\alpha}-be^{i\beta})v_{2}&(ae^{i\alpha}+be^{i\beta})v_{1}&0\end{pmatrix}$ | $\begin{pmatrix}A\rightarrow A^{\prime},&A\in\\{a,b\\}\\\ \Omega\rightarrow\Omega^{\prime},&\Omega\in\\{\alpha,\beta\\}\\\ v_{i}\rightarrow v_{i}^{*},&i\in\\{1,2,3\\}\end{pmatrix}$
IV | $\text{I}_{d}$ | $\text{III}_{u}$
V | $\text{III}_{d}$ | $\text{I}_{u}$
Table 1: Extensions of $A_{4}$ to the Yukawa sector with non-vanishing
determinant, and non-zero $J$ for general, complex valued, vevs
$(v_{1},v_{2},v_{3})$. In the Table, $\text{I}_{d}$ stands for the matrix
$M_{d}$ for case I and similarly for the other entries.

In the table above, we have used the convention where the quarks’ mass terms
are written as

$-\mathcal{L}_{\text{Yukawa}}\supset\overline{n_{L}}M_{d}n_{R}+\overline{p_{L}}M_{u}p_{R}+\text{h.c.}\,,$
(16)

where h.c. stands for the hermitian conjugate.

In the Yukawa sector, there are ten observables, six masses, three mixing
angles and one Jarlskog invariant, therefore, we would prefer to look for a
case with ten parameters, or less. All possible neutral vevs of the 3HDM are
consistent with the parameterization in Eq. (9), which consists of four free
parameters that we can fit; two angles, and two phases. Looking at the cases
in Table 1, we will see that it is possible to reduce the number of free
parameters by performing both basis transformations to right-handed quarks and
global $U(1)_{Y}$ rephasings, both of which have no effect on the physical
predictions of the theory.

For case I, the down quark mass matrices read

$\displaystyle
M_{d}=\begin{pmatrix}ae^{i\alpha}v_{1}&be^{i\beta}v_{1}&ce^{i\gamma}v_{1}\\\
ae^{i\alpha}v_{2}&\omega be^{i\beta}v_{2}&\omega^{2}ce^{i\gamma}v_{2}\\\
ae^{i\alpha}v_{3}&\omega^{2}be^{i\beta}v_{3}&\omega ce^{i\gamma}v_{3}\\\
\end{pmatrix}\,=D_{v}WD_{a}D_{\alpha},$ (17)

where (remember that the $v_{i}$ are complex)

$D_{v}=\textrm{diag}(v_{1},v_{2},v_{3})\,,\,\,D_{a}=\textrm{diag}(a,b,c)\,,\,\,D_{\alpha}=\textrm{diag}(e^{i\alpha},e^{i\beta},e^{i\gamma})\,,\,\,W=\begin{pmatrix}1&1&1\\\
1&\omega&\omega^{2}\\\ 1&\omega^{2}&\omega\end{pmatrix}\,.$ (18)

We see that we can perform a unitary transformation to the right-handed quarks
that removes all three phases $\alpha$, $\beta$, $\gamma$. The same holds for
$M_{u}$, by performing the substitution $A\rightarrow A^{\prime}$,
$\Omega\rightarrow\Omega^{\prime}$ and $v_{i}\rightarrow v_{i}^{*}$. We note
that the case I matrices were also used by Ref. [15] as the mass matrices for
the charged leptons.

In this work, we study this case, that corresponds to the Lagrangian in Eq.
(2.2). Then, given that $D_{\alpha}D_{\alpha}^{\dagger}=\mathds{1}$ and
$D_{a}D_{a}^{\dagger}=D_{a^{2}}=\text{diag}(a^{2},b^{2},c^{2})$, we find

$\displaystyle H_{d}$ $\displaystyle\equiv$ $\displaystyle
M_{d}M_{d}^{\dagger}=D_{v}S_{d}D_{v}^{\dagger}\,,$ $\displaystyle H_{u}$
$\displaystyle\equiv$ $\displaystyle
M_{u}M_{u}^{\dagger}=D_{v}^{\dagger}S_{u}D_{v}\,,$ (19)

where $S_{d}=WD_{a^{2}}W^{\dagger}$ and $a^{2}\rightarrow a^{\prime 2}$ for
the up quark case. This matrix can now be explicitly written out using
appropriate parameters as

$S_{d}=\begin{pmatrix}\Sigma_{d}&Z_{d}e^{i\phi_{d}}&Z_{d}e^{-i\phi_{d}}\\\
Z_{d}e^{-i\phi_{d}}&\Sigma_{d}&Z_{d}e^{i\phi_{d}}\\\
Z_{d}e^{i\phi_{d}}&Z_{d}e^{-i\phi_{d}}&\Sigma_{d}\end{pmatrix},$ (20)

where $\Sigma_{d}$ and $Z_{d}$ are real, and

$\displaystyle\Sigma_{d}$ $\displaystyle\equiv$ $\displaystyle
a^{2}+b^{2}+c^{2}\,,$ $\displaystyle Z_{d}\ e^{i\phi_{d}}$
$\displaystyle\equiv$ $\displaystyle a^{2}+\omega^{2}b^{2}+\omega c^{2}\,,$
(21)

with corresponding primes for the up case. For completeness, the specific
forms for $H_{d}$ and $H_{u}$ found after using the parameterizations in Eqs.
(9) and (21) are written in Appendix A. The eigenvalues of the matrices
$H_{d}$ and $H_{u}$ will be fitted for the (square of the) quark masses,
$(m_{d}^{2},m_{s}^{2},m_{b}^{2})$ and $(m_{u}^{2},m_{c}^{2},m_{t}^{2})$,
respectively

We now turn to the Cabibbo-Kobayashi-Maskawa (CKM) matrix. As found by Branco
and Lavoura [16], the absolute values of the CKM matrix can be obtained
through calculating the traces of appropriate powers of the matrices $H_{u}$
and $H_{d}$. They observe that

$\textrm{Tr}\left(H_{u}^{a}H_{d}^{b}\right)\equiv
L_{ab}=\sum_{k,i}U_{ki}(D_{u}^{a})_{kk}(D_{d}^{b})_{ii}\,,$ (22)

where $U_{ki}=|V_{ki}|^{2}\,$ and $V$ is the CKM matrix. The CKM matrix is
unitary and therefore $U$ only has four independent entries. Consequently, in
order to compute $U$, it is only necessary to resort to

$\displaystyle L_{11}=$ $\displaystyle U_{ki}(D_{u})_{kk}(D_{d})_{ii}\,,$
$\displaystyle L_{12}=$ $\displaystyle U_{ki}(D_{u})_{kk}(D_{d}^{2})_{ii}\,,$
$\displaystyle L_{21}=$ $\displaystyle U_{ki}(D_{u}^{2})_{kk}(D_{d})_{ii}\,,$
$\displaystyle L_{22}=$ $\displaystyle
U_{ki}(D_{u}^{2})_{kk}(D_{d}^{2})_{ii}\,.$ (23)

These equations are linear in $U_{ik}$ and are, therefore, invertible for this
variable. Thus, by picking $U_{11}$, $U_{21}$, $U_{13}$, and $U_{23}$
(respectively, $U_{ud}$, $U_{cd}$, $U_{ub}$, and $U_{cb}$), we are able to
obtain a unique solution for the magnitudes of the CKM elements as a function
of $L_{ab}$ and the quark masses. Namely,

$\displaystyle U_{11}$ $\displaystyle=$
$\displaystyle\left({m_{b}}^{2}-{m_{s}}^{2}\right)\left({m_{c}}^{2}-{m_{t}}^{2}\right)\frac{a_{11}}{\textrm{det}}\,,$
$\displaystyle U_{21}$ $\displaystyle=$
$\displaystyle\left({m_{b}}^{2}-{m_{s}}^{2}\right)\left({m_{u}}^{2}-{m_{t}}^{2}\right)\frac{a_{21}}{\textrm{det}}\,,$
$\displaystyle U_{13}$ $\displaystyle=$
$\displaystyle\left({m_{d}}^{2}-{m_{s}}^{2}\right)\left({m_{c}}^{2}-{m_{t}}^{2}\right)\frac{a_{13}}{\textrm{det}}\,,$
$\displaystyle U_{23}$ $\displaystyle=$
$\displaystyle\left({m_{d}}^{2}-{m_{s}}^{2}\right)\left({m_{u}}^{2}-{m_{t}}^{2}\right)\frac{a_{23}}{\textrm{det}}\,,$
(24)

where

$\displaystyle a_{11}$ $\displaystyle=$
$\displaystyle{L_{11}}\left({m_{b}}^{2}+{m_{s}}^{2}\right)\left({m_{c}}^{2}+{m_{t}}^{2}\right)-{L_{12}}\left({m_{c}}^{2}+{m_{t}}^{2}\right)-{L_{21}}\left({m_{b}}^{2}+{m_{s}}^{2}\right)+{L_{22}}$
(25)
$\displaystyle+m_{b}^{2}\left(-m_{c}^{2}m_{t}^{2}\left(m_{d}^{2}+m_{s}^{2}\right)-m_{s}^{2}m_{u}^{2}\left(m_{c}^{2}+m_{t}^{2}\right)+m_{s}^{2}m_{u}^{4}\right)+m_{c}^{2}m_{d}^{2}m_{t}^{2}\left(m_{d}^{2}-m_{s}^{2}\right)\,,$
$\displaystyle a_{21}$ $\displaystyle=$
$\displaystyle-{L_{11}}\left({m_{b}}^{2}+{m_{s}}^{2}\right)\left({m_{t}}^{2}+{m_{u}}^{2}\right)+{L_{12}}\left({m_{u}}^{2}+{m_{t}}^{2}\right)+{L_{21}}\left({m_{b}}^{2}+{m_{s}}^{2}\right)-{L_{22}}$
(26)
$\displaystyle+m_{b}^{2}\left(m_{c}^{2}m_{s}^{2}\left(m_{t}^{2}+m_{u}^{2}-m_{c}^{2}\right)+m_{t}^{2}m_{u}^{2}\left(m_{d}^{2}+m_{s}^{2}\right)\right)+m_{d}^{2}m_{t}^{2}m_{u}^{2}\left(m_{s}^{2}-m_{d}^{2}\right)\,,$
$\displaystyle a_{13}$ $\displaystyle=$
$\displaystyle-{L_{11}}\left({m_{d}}^{2}+{m_{s}}^{2}\right)\left({m_{t}}^{2}+{m_{c}}^{2}\right)+{L_{12}}\left({m_{c}}^{2}+{m_{t}}^{2}\right)+{L_{21}}\left({m_{d}}^{2}+{m_{s}}^{2}\right)-{L_{22}}$
(27)
$\displaystyle+m_{b}^{2}m_{c}^{2}m_{t}^{2}\left(m_{d}^{2}+m_{s}^{2}-m_{b}^{2}\right)+m_{d}^{2}m_{s}^{2}\left(m_{c}^{2}\left(m_{t}^{2}+m_{u}^{2}\right)+m_{u}^{2}\left(m_{t}^{2}-mu^{2}\right)\right)\,,$
$\displaystyle a_{23}$ $\displaystyle=$
$\displaystyle{L_{11}}\left({m_{d}}^{2}+{m_{s}}^{2}\right)\left({m_{t}}^{2}+{m_{u}}^{2}\right)-{L_{12}}\left({m_{u}}^{2}+{m_{t}}^{2}\right)-{L_{21}}\left({m_{d}}^{2}+{m_{s}}^{2}\right)+{L_{22}}$
(28)
$\displaystyle+m_{t}^{2}m_{u}^{2}\left(m_{b}^{4}-m_{b}^{2}\left(m_{d}^{2}+m_{s}^{2}\right)-m_{d}^{2}m_{s}^{2}\right)+m_{c}^{4}m_{d}^{2}m_{s}^{2}-m_{c}^{2}m_{d}^{2}m_{s}^{2}\left(m_{t}^{2}+m_{u}^{2}\right)\,,$

and

$\textrm{det}=\left({m_{b}}^{2}-{m_{d}}^{2}\right)\left({m_{c}}^{2}-{m_{u}}^{2}\right)\left({m_{d}}^{2}-{m_{s}}^{2}\right)\left({m_{u}}^{2}-{m_{t}}^{2}\right)\left({m_{b}}^{2}-{m_{s}}^{2}\right)\left({m_{c}}^{2}-{m_{t}}^{2}\right)\,.$
(29)

In these equations, the $L_{ij}$ are obtained by evaluating the left hand side
of Eq. (22). Finally, we note that knowing these four CKM magnitudes, we can
determine the Jarslkog invariant [14], up to its sign. Thus, given some phase
convention, we are also able to determine the phases of all CKM matrix
elements.

##  3 The fit to the quark mass matrices

###  3.1 The fitting procedure

We have implemented a $\chi^{2}$ analysis of the model, through a minimization
performed using the CERN Minuit library [17]. The observables employed in this
analysis, labeled by $i=1,...,11$ are specified in Table 2, where
$\overline{X}_{i}$ represents the experimental mean value of the observable
$X_{i}$ and $\sigma_{i}$ is the experimental error, which, when both left and
right bounds are stated, is assumed to be the largest of the two.

Observable | Experimental value | Model prediction
---|---|---
$m_{u}$ [MeV] | $2.16\pm 0.50$ | $2.15$
$m_{c}$ [MeV] | $1270\pm 20$ | $1271.6$
$m_{t}$ [GeV] | $172.69\pm 0.30$ | $172.68$
$m_{d}$ [MeV] | $4.67\pm 0.50$ | $4.66$
$m_{s}$ [MeV] | $93.4\pm 8.6$ | $92.08$
$m_{b}$ [MeV] | $4180\pm 30$ | $4180.39$
$|V_{11}|$ | $0.97435\pm 0.00016$ | $0.97434$
$|V_{21}|$ | $0.22486\pm 0.00067$ | $0.22479$
$|V_{13}|$ | $0.00369\pm 0.00011$ | $0.00369$
$|V_{23}|$ | $0.04182\pm 0.00085$ | $0.04178$
$J$ | $(3.08\pm 0.15)\times 10^{-5}$ | $3.09\times 10^{-5}$
Table 2: Experimental values and fit results.

The data on the quark masses as well as for the CKM matrix elements and the
Jarlskog invariant experimental values were obtained from [18]. As mentioned,
$|J|$ is fixed by $|V_{11}|$, $|V_{21}|$, $|V_{13}|$, and $|V_{23}|$. However,
using it in the fit speeds the numerical convergence onto a good solution.

The $\chi^{2}$ function depends on the 10 parameters of our model,

$\beta_{1},\beta_{2},\rho_{2},\rho_{3},\Sigma_{d},\Sigma_{u},Z_{d},Z_{u},\phi_{d},\phi_{u}$
(30)

and is written as

$\chi^{2}(\textrm{{p}})=\sum_{i=1}^{11}\left(\frac{P_{i}(\textrm{{p}})-\overline{X}_{i}}{\sigma_{i}}\right)^{2}\,,$
(31)

where $P_{i}(\textrm{{p}})$ is our model’s prediction for each of the 11 (10 +
$J$) observables. The fit is complicated by the fact that the masses (squared)
are obtained from the eigenvalues of $H_{d},H_{u}$ but the elements of the CKM
also depend on the masses, see Eq. (2.3). So, we start by calculating the
eigenvalues of $H_{d}$ and $H_{u}$, which depend only on the parameters in Eq.
(30). Then, we evaluate the $L_{ij}$ from the left hand side of Eq. (22), and
finally the CKM elements are obtained from Eq. (2.3). In Appendix A we give
the explicit expressions for the matrices $H_{d}$ and $H_{u}$.

###  3.2 Results of the fit

We have found an excellent fit of our model to the data, given in the second
column of Table 2. This fit results in $\chi^{2}=0.058$, for the parameters

$\displaystyle\beta_{1}=$ $\displaystyle 1.4260868\ \textrm{radians}\,,$
$\displaystyle\beta_{2}=$ $\displaystyle 1.5424328\ \textrm{radians}\,,$
$\displaystyle\rho_{2}=$ $\displaystyle 4.2784971\ \textrm{radians}\,,$
$\displaystyle\rho_{3}=$ $\displaystyle 5.3682785\ \textrm{radians}\,,$
$\displaystyle\Sigma_{d}=$ $\displaystyle 0.2889178\times 10^{-3}\,,$
$\displaystyle\Sigma_{u}=$ $\displaystyle 0.4927455\,,$ $\displaystyle Z_{d}=$
$\displaystyle 0.1816577\times 10^{-3}\,,$ $\displaystyle Z_{u}=$
$\displaystyle 0.4758317\,,$ $\displaystyle\phi_{d}=$
$\displaystyle-1.7324779\ \textrm{radians}\,,$ $\displaystyle\phi_{u}=$
$\displaystyle 0.20644967\times 10^{-1}\ \textrm{radians}\,.$ (32)

This fit also leads to the data in the third column of Table 2, as well as to
the vevs

$|v_{i}|=\left(1.00604,6.90357,245.901\right)\textrm{(GeV)}.$ (33)

We notice that the vevs obey $v_{1}<v_{2}<<v_{3}$. This hierarchy of vevs is
related to the hierarchy of the quark masses. This was also obtained in Ref.
[7], although their model is not consistent, as their vev structure is not
that of [9] for the symmetric $A_{4}$ potential they consider.

##  4 Viability of the vacuum found in the fit

We start by defining the three doublets as in Eq. (6). Next we define the
physical eigenstates for the charged Higgs as
$(G^{+},S^{+}_{2},S^{2}_{3})^{T}$, and for the neutral states we have
$(G^{0},S^{0}_{2},S^{0}_{3},S^{0}_{4},S^{0}_{5},S^{0}_{6})^{T}$, identifying
the would-be Goldstone bosons $G^{+}\equiv S^{+}_{1}$ and $G^{0}\equiv
S^{0}_{1}$. With these conventions, and following the definitions in [19], we
define the $3\times 3$ matrix $\tilde{U}$ by

$\varphi_{i}^{+}\equiv\sum_{j=1}^{3}\tilde{U}_{ij}S^{+}_{j}\,,$ (34)

and the $3\times 6$ matrix $\tilde{V}$ by

$x_{i}+ix_{i+3}=\sum_{j=1}^{6}\tilde{V}_{ij}S^{0}_{j}\,.$ (35)

These matrices666From the point of view of a simultaneous fit of the Yukawa
and scalar sectors, it is a pity that these matrices $\tilde{V}$ and
$\tilde{U}$ have in the literature the same notation as the CKM matrix $V$ and
$U_{ki}=|V_{ki}|^{2}$. are then related to the diagonalization matrices of the
charged and neutral scalars, to which we now turn.

###  4.1  The minimization of the potential

In our procedure we already know the values of the vevs. So, we use the
stationarity equations to solve for the soft parameters, and leave the quartic
parameters of the potential $\Lambda_{i}$ as free parameters. In this way we
can solve for $m^{2}_{11},m^{2}_{22},m^{2}_{33}$ as well as for
$\text{Im}(m^{2}_{12}),\text{Im}(m^{2}_{13})$, leaving as free parameters the
$\Lambda_{i}$ and
$\text{Re}(m^{2}_{12}),\text{Re}(m^{2}_{13}),\text{Re}(m^{2}_{23}),\text{Im}(m^{2}_{23})$.
When evaluating the scalar mass matrices (see below) the conditions have to be
applied to ensure that we are at the minimum. For completeness we write these
conditions in Appendix B.

###  4.2 The charged mass matrix

The charged mass matrix is obtained from the second derivatives at the
minimum,

$\mathcal{M}^{2}_{C}=\left.\frac{\partial^{2}V_{H}}{\partial\varphi_{i}^{+}\partial\varphi_{j}^{-}}\right|_{\text{Min}}.$
(36)

The matrix $\mathcal{M}^{2}_{C}$ is an hermitian matrix, with real eigenvalues
and satisfying, with our usual conventions,

$R_{\rm ch}\mathcal{M}^{2}_{C}R^{\dagger}_{\rm
ch}=\text{diag}(0,m^{2}_{S^{+}_{2}},m^{2}_{S^{+}_{3}})\equiv\mathcal{M}^{2}_{D_{ch}}\,,$
(37)

where $R_{\rm ch}$ is an unitary matrix that satisfies,

$S^{+}_{i}=\sum_{j=1}^{3}\left(R_{\rm ch}\right)_{ij}\varphi^{+}_{j}\,.$ (38)

This can be seen from

$\displaystyle\mathcal{L}_{\rm mass}=$
$\displaystyle-\varphi^{-}_{i}\left(\mathcal{M}^{2}_{C}\right)_{ij}\varphi^{+}_{j}=-\varphi^{-}_{i}\left(R^{\dagger}_{\rm
ch}R_{\rm ch}\mathcal{M}^{2}_{C}R^{\dagger}_{\rm ch}R_{\rm
ch}\right)_{ij}\varphi^{+}_{j}=-\varphi^{-}_{i}\left(R^{\dagger}_{\rm
ch}\mathcal{M}^{2}_{D_{ch}}R_{\rm ch}\right)_{ij}\varphi^{+}_{j}$
$\displaystyle=$
$\displaystyle-S^{-}_{i}\left(\mathcal{M}^{2}_{D_{ch}}\right)_{ij}S^{+}_{j}\,,$
(39)

where we have used Eq. (38).

We have checked both algebraically and numerically that we have a zero
eigenvalue corresponding to $G^{+}$ and we require that all other masses
squared are positive, a condition for a local minimum.

###  4.3 The neutral mass matrix

Since in our case CP is not conserved, we denote the unrotated neutral scalars
by $x_{i},i=1,\ldots,6$, as in Eq. (6). We therefore obtain the neutral mass
matrix as,

$\mathcal{M}^{2}_{N}=\left.\frac{\partial^{2}V_{H}}{\partial x_{i}\partial
x_{j}}\right|_{\text{Min}}.$ (40)

This is a symmetric real matrix diagonalized by an orthogonal $6\times 6$
matrix,

$R_{\rm neu}\mathcal{M}^{2}_{N}R^{T}_{\rm
neu}=\text{diag}(0,m^{2}_{S^{0}_{2}},m^{2}_{S^{0}_{3}},m^{2}_{S^{0}_{4}},m^{2}_{S^{0}_{5}},m^{2}_{S^{0}_{6}})\equiv\mathcal{M}^{2}_{D_{\rm
neu}}\,,$ (41)

with

$S^{0}_{i}=\sum_{j=1}^{6}\left(R_{\rm neu}\right)_{ij}x_{j}\,.$ (42)

As for the case of the charged scalars, we have checked both algebraically and
numerically that we have a zero eigenvalue corresponding to $G^{0}$ and we
require that all other masses squared are positive, a condition for a local
minimum.

##  5 Theoretical Constraints

After having shown that a solution exists for the vevs and parameters in the
Yukawa sector that correctly fits the quarks masses and the CKM entries, we
have to show that this is compatible with the scalar potential analysis. In
particular we have to show that the vevs correspond to a local minimum of the
potential and that both the theoretical constraints as well as those coming
from LHC are satisfied. In this section we analyze the theoretical
constraints.

###  5.1 Perturbative Unitarity

This problem was already solved in [13], so we take the potential in the form
of Eq. (4). From Ref. [13] we have the following expression for the
eigenvalues $\lambda_{i}$777We use $\lambda_{i}$ instead of $\Lambda_{i}$, in
order to not confuse with the notation of Eq. (2).

$\displaystyle\lambda_{1}=$ $\displaystyle
2\left(2\text{Re}(c_{3})+r_{1}\right)$ (43) $\displaystyle\lambda_{2}=$
$\displaystyle
2\left(\sqrt{3}\,|\text{Im}(c_{3})|-\text{Re}(c_{3})+r_{1}\right)$ (44)
$\displaystyle\lambda_{3}=$ $\displaystyle
2\left(-\sqrt{3}\,|\text{Im}(c_{3})|-\text{Re}(c_{3})+r_{1}\right)$ (45)
$\displaystyle\lambda_{4}=$ $\displaystyle 2(r_{4}+r_{7})$ (46)
$\displaystyle\lambda_{5}=$ $\displaystyle 2(r_{4}-r_{7})$ (47)
$\displaystyle\lambda_{6}=$ $\displaystyle 2(r_{1}+2r_{7})$ (48)
$\displaystyle\lambda_{7}=$ $\displaystyle 2(r_{1}-r_{7})$ (49)
$\displaystyle\lambda_{8}=$ $\displaystyle 2(r_{4}+|c_{3}|)$ (50)
$\displaystyle\lambda_{9}=$ $\displaystyle 2(r_{4}-|c_{3}|)$ (51)
$\displaystyle\lambda_{10}=$ $\displaystyle 6r_{1}+8r_{4}+4r_{7}$ (52)
$\displaystyle\lambda_{11}=$ $\displaystyle 6r_{1}-2(2r_{4}+r_{7})$ (53)
$\displaystyle\lambda_{12}=$ $\displaystyle 6|c_{3}|+2r_{4}+4r_{7}$ (54)
$\displaystyle\lambda_{13}=$ $\displaystyle-6|c_{3}|+2r_{4}+4r_{7}$ (55)

Perturbative unitarity is satisfied if

$|\lambda_{i}|<8\pi,\quad\forall i.$ (56)

###  5.2 The BFB conditions

For the $A_{4}$ symmetric potential, the conditions for boundedness from below
along the neutral directions (BFB-n) have been conjectured in [20], and proved
to hold in [21]. These are

$\displaystyle\Lambda_{0}+\Lambda_{3}\geq 0\,,$ (57)
$\displaystyle\frac{4}{3}(\Lambda_{0}+\Lambda_{3})+\frac{1}{2}(\Lambda_{1}+\Lambda_{2})-\Lambda_{3}-\frac{1}{2}\sqrt{(\Lambda_{1}-\Lambda_{2})^{2}+\Lambda_{4}^{2}}\geq
0\,,$ (58)
$\displaystyle\Lambda_{0}+\frac{1}{2}(\Lambda_{1}+\Lambda_{2})+\frac{1}{2}(\Lambda_{1}-\Lambda_{2})\cos{(2k\pi/3)}+\frac{1}{2}\Lambda_{4}\sin{(2k\pi/3)}\geq
0\,\ \ (k=1,2,3)\,.$ (59)

However, as shown in [22, 20], a potential which is BFB-n is not necessarily
BFB along the charge breaking directions (BFB-c). Necessary BFB-c conditions
have yet to be found for the $A_{4}$ 3HDM, but sufficient conditions have been
proposed in [23] following the technique developed in [24]. They are,

$A_{d}\geq 0\,,\ \ \ \ A_{o}\geq-A_{d}/2\,,$ (60)

where

$\displaystyle A_{d}=$ $\displaystyle a\ =\
\frac{2}{3}(\Lambda_{0}+\Lambda_{3})\,,$ $\displaystyle A_{o}=$ $\displaystyle
b+\textrm{min}(0,c)-d$ $\displaystyle=$
$\displaystyle\frac{1}{3}(2\Lambda_{0}-\Lambda_{3})+\frac{1}{2}(\Lambda_{1}+\Lambda_{2})+\min(0,-\frac{1}{2}(\Lambda_{1}+\Lambda_{2}))-\frac{1}{2}\sqrt{(\Lambda_{1}-\Lambda_{2})^{2}+\Lambda_{4}^{2}}\,.$
(61)

It is important to remark that, since these are sufficient, but not necessary,
conditions, some good points in parameter space may be excluded by this
restriction.

###  5.3 The oblique parameters $S,T,U$

For this we use the notation and results from [19], which require the matrices
$\tilde{U}$ and $\tilde{V}$. Comparing Eq. (38) with the definition in Eq.
(34), we conclude that

$\tilde{U}=R_{\rm ch}^{\dagger}\,,$ (62)

where the matrix $R_{\rm ch}$ is obtained from the numerical diagonalization
of Eq. (37). Similarly, comparing Eq. (42) with the definition of $\tilde{V}$
in Eq. (35), we get,

$\tilde{V}_{ij}=\left(R^{T}_{\rm neu}\right)_{ij}+i\left(R^{T}_{\rm
neu}\right)_{i+3,j}.$ (63)

Having $\tilde{U}$ and $\tilde{V}$, we can construct the needed matrices
$\text{Im}\left(\tilde{V}^{\dagger}\tilde{V}\right)$ ,
$\tilde{U}^{\dagger}\tilde{U}$, $\tilde{V}^{\dagger}\tilde{V}$ and
$\tilde{U}^{\dagger}\tilde{V}$, and implement the procedure of [19].

###  5.4 Global minimum

After finding a set of $m_{i,j}$ and $\Lambda_{i}$ which reproduce the vevs in
Eq. (33) necessary for a good fit of the quark mass matrices, and after
performing the previous theoretical checks on the scalar potential, we must
still ensure that our minimum is indeed the global minimum. This step is
almost never taken in studies of quark mass matrices, since there are no exact
analytical formulae for it. Moreover, one must check that there are no lower
minima both along the neutral directions and along the charge breaking
directions. We follow the strategy discussed in Ref. [23]. Take a specific set
of $m^{2}_{ij}$ and $\Lambda_{i}$. Then we parameterize the scalar doublets as
[22, 23],

$\langle\phi_{1}\rangle=\sqrt{r_{1}}\,\begin{pmatrix}0\\\\[2.84526pt]
1\end{pmatrix}\,,\quad\langle\phi_{2}\rangle=\sqrt{r_{2}}\,\begin{pmatrix}\sin(\alpha_{2})\\\\[2.84526pt]
\cos(\alpha_{2})e^{i\beta_{2}}\end{pmatrix}\,,\quad\langle\phi_{3}\rangle=\sqrt{r_{3}}e^{i\gamma}\,\begin{pmatrix}\sin(\alpha_{3})\\\\[2.84526pt]
\cos(\alpha_{3})e^{i\beta_{3}}\end{pmatrix}\,,$ (64)

where we have already used the gauge freedom. Now we let the vevs run free,
for both charge conserving and charge violating directions. We give one seed
point and perform a minimization of the potential using the CERN Minuit
library [17]. We obtain not only the value of the potential at the minimum,
but also the values of $r_{i},\alpha_{2},\beta_{2},\alpha_{3},\beta_{3}$ and
$\gamma$. Then, we take one more (randomly generated) seed point and repeat
the minimization. Finally, we take the minimum as the global one if it is
found as the global minimum in each of 200 searches with randomly generated
seed points. We have done this verification for every point that passed all
the constraints. In all cases, we found that the local minimum was also a
global minimum. In particular we always found that

$\sin(\alpha_{2})=\sin(\alpha_{3})=0,$ (65)

showing that we do not have charged breaking directions888 To cross check our
numerical procedure we also considered points that violated the BFB
conditions. And, indeed for these points, our algorithm showed that the
potential was not BFB and could have charge breaking directions as well. and,
comparing with Eq. (6), we verified numerically that,

$\frac{|v_{i}|}{\sqrt{2}}=\sqrt{r_{i}},\quad
e^{i\,\rho_{2}}=\cos(\alpha_{2})\,e^{i\,\beta_{2}},\quad
e^{i\,\rho_{3}}=\cos(\alpha_{3})\,e^{i\,(\beta_{3}+\gamma)}\,.$ (66)

##  6 Simple LHC Constraints

Up to now we have implemented the theoretical constraints on the model. The
next step is to implement the LHC constraints. To do this completely one would
have to implement all the decays of the neutral and charged Higgs as well as
their branching ratios. One would also have to worry about the electric dipole
moments (EDM) and the flavour-changing neutral couplings (FCNC), as the model
does not have a structure of couplings of the Higgs to the fermions that
automatically ensures vanishing FCNC [25, 26, 27]. This lies beyond the scope
of the present work. Nonetheless, we can implement easily the constraints that
come from $h\rightarrow WW/ZZ$ in the $\kappa$ formalism, where the deviation
from the coupling of the SM Higgs boson to a pair of $W$’s (or $Z$’s) is
measured by $\kappa_{V}$. In our model,

$\kappa_{V}=R^{\rm neu}_{21}v_{1}+R^{\rm neu}_{22}v_{2}\cos(\rho_{2})+R^{\rm
neu}_{23}v_{3}\cos(\rho_{3})+R^{\rm neu}_{25}v_{2}\sin(\rho_{2})+R^{\rm
neu}_{26}v_{3}\sin(\rho_{3}),$ (67)

where $R^{\rm neu}$ is matrix defined in Eq. (41). We take the experimental
constraint from ATLAS [28],

$\kappa_{W}=1.0206{\,}^{+0.05172}_{-0.05087},\quad\kappa_{Z}=0.99{\,}^{+0.06136}_{-0.05214}\,.$
(68)

##  7 Results

In this section we present the results of the analysis of the scalar potential
after imposing that we have a good solution for the fit of the quarks masses
and CKM entries, as explained in Section 3.

###  7.1 Scanning strategy

We start by imposing the vevs obtained in the fit.

$v_{1}=1.00604\,\text{(GeV)},\quad
v_{2}=6.90357\,e^{i\,4.278497}\,\text{(GeV)},\quad
v_{3}=245.901\,e^{i\,5.368278}\,\text{(GeV)}.$ (69)

Now we vary the free parameters of the potential in the following ranges,

$\log_{10}|\Lambda_{i}|\in[-3,1],\quad\log_{10}|\text{Im}(m^{2}_{23})|\in[-1,7]\text{GeV}^{2},\quad\log_{10}|\text{Re}(m^{2}_{ij})|\in[-1,7]\text{GeV}^{2},$
(70)

where in the last equation we use

$m^{2}_{ij}\in\left\\{m^{2}_{12},m^{2}_{13},m^{2}_{23}\right\\}.$ (71)

We randomly scan as in Eq. (70), and then:

  1. 1.

Apply the theoretical constraints that only depend on the $\Lambda_{i}$, that
is BFB and perturbative unitarity.

  2. 2.

Then obtain the eigenvalues for the charged and neutral scalars. Verify that
all the masses squared are positive, and that we have a zero eigenvalue
corresponding to the Goldstone bosons, $G^{0}$ and $G^{+}$.

  3. 3.

Verify the S, T and U oblique parameters.

  4. 4.

Apply the LHC constraint on $\kappa_{V}$.

  5. 5.

Check numerically that the vev is indeed a global minimum.

###  7.2 The scalar spectrum

We found that there is a strong correlation in the scalar masses. If we denote
the masses of the neutral scalars by
$(m_{G^{0}}=0,m_{S^{0}_{2}},m_{S^{0}_{3}},m_{S^{0}_{4}},m_{S^{0}_{5}},m_{S^{0}_{6}})$,
and $(m_{G^{+}}=0,m_{H^{+}_{1}},m_{H^{+}_{2}})$ for the charged scalars, we
find numerically that

$m_{S^{0}_{3}}\simeq m_{S^{0}_{4}}\simeq m_{H^{+}_{1}},\quad
m_{S^{0}_{5}}\simeq m_{S^{0}_{6}}\simeq m_{H^{+}_{2}}.$ (72)

This is true even if we do not require $m_{S^{0}_{2}}=125$ GeV, and specially
true after implementing the constraints of perturbative unitarity, BFB and
STU. But, as we want to reproduce the LHC results, we also required that [18]

$m_{S_{2}^{0}}=125.25\pm 0.17\quad\text{GeV}.$ (73)

In the following figures we show the correlation among the masses. Included in
red are the points generated before the theoretical cuts were applied, and in
green the points remaining after the constraints were implemented.

|
---|---
Figure 1: Left panel: Relation between $m_{S_{3}^{0}}$ and $m_{S_{4}^{0}}$. Right panel: Relation between $m_{S_{3}^{0}}$ and $m_{H_{1}^{+}}$. Color conventions: No cuts (red); with cuts (green) |
---|---
Figure 2: Left panel: Relation between $m_{S_{5}^{0}}$ and $m_{S_{6}^{0}}$. Right panel: Relation between $m_{S_{5}^{0}}$ and $m_{H_{2}^{+}}$. Color conventions: No cuts (red); with cuts (green) |
---|---
Figure 3: Left panel: Relation between $m_{S_{3}^{0}}$ and $m_{S_{5}^{0}}$.
Right panel: Relation between $m_{H_{1}^{+}}$ and $m_{H_{2}^{+}}$. Color
conventions: No cuts (red); with cuts (green)

###  7.3 The $\kappa_{V}$ constraint

We can now implement the $\kappa_{V}$ constraint on the model. In the
following figures, in red are points without cuts, in green with cuts but no
$\kappa_{V}$ constraint, and finally in blue points remaining after this
constraint is applied. We took the ATLAS result of Eq. (68) at $2\sigma$.
While the theoretical constraints cut around 99% of the points, the
$\kappa_{V}$ constraint only cuts 9% of the remaining points. In Fig. 4 we
show the relation between $\kappa_{V}$ and $\Lambda_{0,4}$ for the three sets
of points as discussed above.

|
---|---
Figure 4: Left panel: Relation between $\kappa_{V}$ and $\Lambda_{0}$. Right
panel: Relation between $\kappa_{V}$ and $\Lambda_{4}$. Color conventions: No
cuts (red), with theoretical cuts (green), and after the $\kappa_{V}$
constraint (blue).

In fact it is not obvious from Fig. 4 that the $\kappa_{V}$ constraint only
cuts about 8% of the points. This is because there is a very large number of
points with $|\kappa_{V}|\lesssim 1$, even without theoretical cuts, and this
is even more so after imposing the theoretical cuts. In this figure, we have
155483 points in the green region, but from these 144128 are in the blue
region. That is, after theoretical cuts, 92% of the points also satisfy the
$\kappa_{V}$ constraint. In Fig. 5 we show the relation between $\Lambda_{0}$
and $\Lambda_{3,4}$ for the same sets of points.

|
---|---
Figure 5: Left panel: Relation between $\Lambda_{0}$ and $\Lambda_{4}$. Right
panel: Relation between $\Lambda_{0}$ and $\Lambda_{3}$ Color conventions: No
cuts (red), with theoretical cuts (green), and after the $\kappa_{V}$
constraint (blue).

We see that, while for $(\Lambda_{0},\,\Lambda_{4})$ there is not much
difference before and after the $\kappa_{V}$ constraint, the same is not true
for $(\Lambda_{0},\,\Lambda_{3})$, where the constraints impose a linear
relation between those two parameters.

##  8 Conclusions

It is known that the 3HDM symmetric under an exact $A_{4}$ symmetry is not
compatible with non-zero quark masses and/or non-block-diagonal CKM matrix
[12]. In this work, we studied a 3HDM with $A_{4}$ softly broken. This allows
us to evade the above result, by enlarging the structure of the possible
vacua.

We obtained an excellent fit of the quarks mass matrices, including the CP-
violating Jarlskog invariant. This leads to a unique solution for the vevs. We
showed that, with the solution for the vevs obtained from the fit, it is
possible to have a local minimum of the potential. We enforce this by imposing
that all squared masses are positive. As in our scheme the scalar masses are
not input parameters, we have to restrict one of the neutral scalars to have
the mass of the known Higgs boson.

We have implemented the BFB, perturbative unitarity and the oblique parameters
$S,T,U$ theoretical constraints. From LHC, we have considered the observed
Higgs mass and the $\kappa_{V}$ constraint.999The detailed study of other LHC
constraints as well as those coming from FCNC and the EDM lies beyond the
scope of the present work, and is left for a future publication. After
imposing the other constraints, we found that most of the points are close to
the alignment required to respect the experimental $\kappa_{V}$ constraint. We
have discovered a strong correlation among the masses of the scalars, even
before applying the theoretical constraints, especially for moderate to large
scalar masses.

One important point is that we have numerically checked for all the points
that pass our constraints, that for a given set of parameters of the
potential, our minimum is the true global minimum.

Acknowledgments

This work is supported in part by FCT (Fundação para a Ciência e Tecnologia)
under Contracts CERN/FIS-PAR/0002/2021, CERN/FIS-PAR/0008/2019,
UIDB/00777/2020, and UIDP/00777/2020; these projects are partially funded
through POCTI (FEDER), COMPETE, QREN, and the EU. The work of I.B. was
supported by a CFTP fellowship with reference BL210/2022-IST-ID and the work
of S. C. by a CFTP fellowship with reference BL255/2022-IST-ID.

##  Appendix A The matrices $H_{d}$ and $H_{u}$

$\displaystyle H_{d}(1,1)=$
$\displaystyle\Sigma_{d}v^{2}\cos^{2}(\beta_{1})\cos^{2}(\beta_{2})$ (74)
$\displaystyle H_{d}(1,2)=$ $\displaystyle
v^{2}Z_{d}\cos(\beta_{1})\cos^{2}(\beta_{2})\cos(\rho_{2}-\phi_{d})\sin(\beta_{1})$
$\displaystyle-i\,v^{2}Z_{d}\cos(\beta_{1})\cos^{2}(\beta_{2})\sin(\beta_{1})\sin(\rho_{2}-\phi_{d})$
(75) $\displaystyle H_{d}(1,3)=$ $\displaystyle
v^{2}Z_{d}\cos(\beta_{1})\cos(\beta_{2})\cos(\rho_{3}+\phi_{d})\sin(\beta_{2})$
$\displaystyle-i\,v^{2}Z_{d}\cos(\beta_{1})\cos(\beta_{2})\sin(\beta_{2})\sin(\rho_{3}+\phi_{d})$
(76) $\displaystyle H_{d}(2,1)=$ $\displaystyle(H_{d}(1,2))^{*}$ (77)
$\displaystyle H_{d}(2,2)=$
$\displaystyle\Sigma_{d}v^{2}\cos^{2}(\beta_{2})\sin^{2}(\beta_{1})$ (78)
$\displaystyle H_{d}(2,3)=$ $\displaystyle
v^{2}Z_{d}\cos(\beta_{2})\cos(\rho_{2}-\rho_{3}+\phi_{d})\sin(\beta_{1})\sin(\beta_{2})$
$\displaystyle+i\,v^{2}Z_{d}\cos(\beta_{2})\sin(\beta_{1})\sin(\beta_{2})\sin(\rho_{2}-\rho_{3}+\phi_{d})$
(79) $\displaystyle H_{d}(3,1)=$ $\displaystyle(H_{d}(1,3))^{*}$ (80)
$\displaystyle H_{d}(3,2)=$ $\displaystyle(H_{d}(2,3))^{*}$ (81)
$\displaystyle H_{d}(3,3)=$ $\displaystyle\Sigma_{d}v^{2}\sin^{2}(\beta_{2})$
(82)

$\displaystyle H_{u}(1,1)=$
$\displaystyle\Sigma_{u}v^{2}\cos^{2}(\beta_{1})\cos^{2}(\beta_{2})$ (83)
$\displaystyle H_{u}(1,2)=$ $\displaystyle
v^{2}Z_{u}\cos(\beta_{1})\cos^{2}(\beta_{2})\cos(\rho_{2}-\phi_{u})\sin(\beta_{1})$
$\displaystyle-i\,v^{2}Z_{u}\cos(\beta_{1})\cos^{2}(\beta_{2})\sin(\beta_{1})\sin(\rho_{2}-\phi_{u})$
(84) $\displaystyle H_{u}(1,3)=$ $\displaystyle
v^{2}Z_{u}\cos(\beta_{1})\cos(\beta_{2})\cos(\rho_{3}+\phi_{u})\sin(\beta_{2})$
$\displaystyle-i\,v^{2}Z_{u}\cos(\beta_{1})\cos(\beta_{2})\sin(\beta_{2})\sin(\rho_{3}+\phi_{u})$
(85) $\displaystyle H_{u}(2,1)=$ $\displaystyle(H_{u}(1,2))^{*}$ (86)
$\displaystyle H_{u}(2,2)=$
$\displaystyle\Sigma_{u}v^{2}\cos^{2}(\beta_{2})\sin^{2}(\beta_{1})$ (87)
$\displaystyle H_{u}(2,3)=$ $\displaystyle
v^{2}Z_{u}\cos(\beta_{2})\cos(\rho_{2}-\rho_{3}+\phi_{u})\sin(\beta_{1})\sin(\beta_{2})$
$\displaystyle+i\,v^{2}Z_{u}\cos(\beta_{2})\sin(\beta_{1})\sin(\beta_{2})\sin(\rho_{2}-\rho_{3}+\phi_{u})$
(88) $\displaystyle H_{u}(3,1)=$ $\displaystyle(H_{u}(1,3))^{*}$ (89)
$\displaystyle H_{u}(3,2)=$ $\displaystyle(H_{u}(2,3))^{*}$ (90)
$\displaystyle H_{u}(3,3)=$ $\displaystyle\Sigma_{u}v^{2}\sin^{2}(\beta_{2})$
(91)

##  Appendix B The minimization conditions

$\displaystyle m^{2}_{11}=$
$\displaystyle-\frac{\sec(\rho_{2})\sec(\rho_{3})}{24v_{1}^{2}}\left[-12\text{Im}(m^{2}_{23})v_{2}v_{3}\sin(2(\rho_{2}-\rho_{3}))+\cos(\rho_{2}-\rho_{3})\left(4\Lambda_{0}v_{1}^{2}v^{2}\right.\right.$
$\displaystyle\hskip
28.45274pt\left.\left.+6\Lambda_{1}v_{1}^{2}v_{2}^{2}+6\Lambda_{1}v_{1}^{2}v_{3}^{2}+3\Lambda_{1}v_{2}^{2}v_{3}^{2}-3\Lambda_{2}v_{2}^{2}v_{3}^{2}+2\Lambda_{3}v_{1}^{2}\left(2v_{1}^{2}-v_{2}^{2}-v_{3}^{2}\right)\right)\right.$
$\displaystyle\hskip
28.45274pt\left.+4\Lambda_{0}v_{1}^{2}v_{2}^{2}\cos(\rho_{2}+\rho_{3})+4\Lambda_{0}v_{1}^{2}v_{3}^{2}\cos(\rho_{2}+\rho_{3})+4\Lambda_{0}v_{1}^{4}\cos(\rho_{2}+\rho_{3})\right.$
$\displaystyle\hskip
28.45274pt\left.+6\Lambda_{1}v_{1}^{2}v_{2}^{2}\cos(\rho_{2}+\rho_{3})+6\Lambda_{1}v_{1}^{2}v_{3}^{2}\cos(\rho_{2}+\rho_{3})-3\Lambda_{1}v_{2}^{2}v_{3}^{2}\cos(3(\rho_{2}-\rho_{3}))\right.$
$\displaystyle\hskip
28.45274pt\left.+3\Lambda_{2}v_{2}^{2}v_{3}^{2}\cos(3(\rho_{2}-\rho_{3}))-2\Lambda_{3}v_{1}^{2}v_{2}^{2}\cos(\rho_{2}+\rho_{3})-2\Lambda_{3}v_{1}^{2}v_{3}^{2}\cos(\rho_{2}+\rho_{3})\right.$
$\displaystyle\hskip
28.45274pt\left.+4\Lambda_{3}v_{1}^{4}\cos(\rho_{2}+\rho_{3})+3\Lambda_{4}v_{1}^{2}v_{2}^{2}\sin(\rho_{2}-\rho_{3})+3\Lambda_{4}v_{1}^{2}v_{2}^{2}\sin(\rho_{2}+\rho_{3})\right.$
$\displaystyle\hskip
28.45274pt\left.+3\Lambda_{4}v_{1}^{2}v_{3}^{2}\sin(\rho_{2}-\rho_{3})-3\Lambda_{4}v_{1}^{2}v_{3}^{2}\sin(\rho_{2}+\rho_{3})-3\Lambda_{4}v_{2}^{2}v_{3}^{2}\sin(\rho_{2}-\rho_{3})\right.$
$\displaystyle\hskip
28.45274pt\left.+3\Lambda_{4}v_{2}^{2}v_{3}^{2}\sin(3(\rho_{2}-\rho_{3}))-12\text{Re}(m^{2}_{23})v_{2}v_{3}\cos(2(\rho_{2}-\rho_{3}))\right.$
$\displaystyle\hskip
28.45274pt\left.+24\text{Re}(m^{2}_{13})v_{1}v_{3}\cos(\rho_{2})+24\text{Re}(m^{2}_{12})v_{1}v_{2}\cos(\rho_{3})+12\text{Re}(m^{2}_{23})v_{2}v_{3}\right]$
(92)

$\displaystyle m^{2}_{22}=$
$\displaystyle-\frac{1}{12v_{2}}\Big{[}3\sec(\rho_{2})\left(-4\text{Im}(m^{2}_{23})v_{3}\sin(\rho_{3})+v_{2}v_{3}^{2}(\Lambda_{1}-\Lambda_{2})\cos(\rho_{2}-2\rho_{3})-\Lambda_{4}v_{2}v_{3}^{2}\sin(\rho_{2}-2\rho_{3})\right.$
$\displaystyle\hskip
28.45274pt\left.+4\text{Re}(m^{2}_{23})v_{3}\cos(\rho_{3})+4\text{Re}(m^{2}_{12})v_{1}\right)+v_{2}\left(4\Lambda_{0}v^{2}+6\Lambda_{1}v_{1}^{2}+3\Lambda_{1}v_{3}^{2}+3\Lambda_{2}v_{3}^{2}\right.$
$\displaystyle\hskip
28.45274pt\left.-2\Lambda_{3}v_{1}^{2}+4\Lambda_{3}v_{2}^{2}-2\Lambda_{3}v_{3}^{2}+3\Lambda_{4}v_{1}^{2}\tan(\rho_{2})\right)\Big{]}$
(93)

$\displaystyle m^{2}_{33}=$
$\displaystyle-\frac{1}{12v_{3}}\Big{[}3\sec(\rho_{3})\left(4\text{Im}(m^{2}_{23})v_{2}\sin(\rho_{2})+v_{2}^{2}v_{3}(\Lambda_{1}-\Lambda_{2})\cos(2\rho_{2}-\rho_{3})-\Lambda_{4}v_{2}^{2}v_{3}\sin(2\rho_{2}-\rho_{3})\right.$
$\displaystyle\hskip
28.45274pt\left.+4\text{Re}(m^{2}_{23})v_{2}\cos(\rho_{2})+4\text{Re}(m^{2}_{13})v_{1}\right)+v_{3}\left(4\Lambda_{0}\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\right)+6\Lambda_{1}v_{1}^{2}+3\Lambda_{1}v_{2}^{2}\right.$
$\displaystyle\hskip
28.45274pt\left.+3\Lambda_{2}v_{2}^{2}-2\Lambda_{3}v_{1}^{2}-2\Lambda_{3}v_{2}^{2}+4\Lambda_{3}v_{3}^{2}-3\Lambda_{4}v_{1}^{2}\tan(\rho_{3})\right)\Big{]}$
(94)

$\displaystyle\text{Im}(m^{2}_{12})=$
$\displaystyle\frac{1}{4v_{1}}\Big{[}\sec(\rho_{2})\left(4\text{Im}(m^{2}_{23})v_{3}\cos(\rho_{2}-\rho_{3})-\Lambda_{1}v_{2}v_{3}^{2}\sin(2(\rho_{2}-\rho_{3}))-\Lambda_{1}v_{1}^{2}v_{2}\sin(2\rho_{2})\right.$
$\displaystyle\hskip
14.22636pt\left.+\Lambda_{2}v_{2}v_{3}^{2}\sin(2(\rho_{2}-\rho_{3}))+\Lambda_{2}v_{1}^{2}v_{2}\sin(2\rho_{2})-\Lambda_{4}v_{2}v_{3}^{2}\cos(2(\rho_{2}-\rho_{3}))\right.$
$\displaystyle\hskip
14.22636pt\left.+\Lambda_{4}v_{1}^{2}v_{2}\cos(2\rho_{2})-4\text{Re}(m^{2}_{23})v_{3}\sin(\rho_{2}-\rho_{3})-4\text{Re}(m^{2}_{12})v_{1}\sin(\rho_{2})\right)\Big{]}$
(95)

$\displaystyle\text{Im}(m^{2}_{13})=$
$\displaystyle-\frac{1}{4v_{1}}\Big{[}\sec(\rho_{3})\left(4\text{Im}(m^{2}_{23})v_{2}\cos(\rho_{2}-\rho_{3})-\Lambda_{1}v_{2}^{2}v_{3}\sin(2(\rho_{2}-\rho_{3}))+\Lambda_{1}v_{1}^{2}v_{3}\sin(2\rho_{3})\right.$
$\displaystyle\hskip
14.22636pt\left.+\Lambda_{2}v_{2}^{2}v_{3}\sin(2(\rho_{2}-\rho_{3}))-\Lambda_{2}v_{1}^{2}v_{3}\sin(2\rho_{3})-\Lambda_{4}v_{2}^{2}v_{3}\cos(2(\rho_{2}-\rho_{3}))\right.$
$\displaystyle\hskip
14.22636pt\left.+\Lambda_{4}v_{1}^{2}v_{3}\cos(2\rho_{3})-4\text{Re}(m^{2}_{23})v_{2}\sin(\rho_{2}-\rho_{3})+4\text{Re}(m^{2}_{13})v_{1}\sin(\rho_{3})\right)\Big{]}$
(96)

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