text:
\section{Introduction}\label{sec:intro}

The discovery of the Higgs particle at the CERN Large Hadron Collider
(LHC) \cite{Aad:2012tfa,Chatrchyan:2012xdj}, which was previously
predicted by the Standard Model (SM) of Particle Physics
\cite{Glashow:1961tr,Weinberg:1967tq,Salam1968,Englert:1964et,Higgs:1964pj},
generated renewed interest in Beyond the SM (BSM) Higgs Physics. This
is corroborated by the fact that the SM fails to address several key
questions, such as the origin of the observed matter-antimatter
asymmetry and the dark matter relic abundance in the Universe.

There is a plethora of well-motivated New Physics models with
additional Higgs scalars that have been introduced to solve these
problems~\cite{Lee1,Ginzburg:1999fb,Branco1,Delgado:2013zfa,Bonilla:2014xba,
  Darvishi:2016tni}. To distinguish such models, one usually employs
symmetry transformations that leave the particular model
invariant. These symmetries impose constraints over the theoretical
parameters of the models and thus enhance their predictability to be
probed in future experiments.

In this paper we construct potentials of multi-Higgs Doublet Models
($n$HDMs) with $n$ scalar-doublets based on SU(2)$_L$-preserving
accidental symmetries. These symmetries can be realized in two sets:
(i) as continuous symmetries and (ii) as discrete symmetries (Abelian and
non-Abelian symmetry groups). To find continuous symmetries, we
present an algorithmic method that provides the full list of proper,
improper and semi-simple subgroups for any given integer $n$. We also
include all known discrete symmetries in $n$HDM
potentials~\cite{Derman:1977wq,
  Pakvasa:1977in,Pakvasa:1978tx,Yamanaka:1981pa,Brown:1984mq,Segre:1978ji,Ishimori:2012zz,Ivanov:2012fp,Keus:2013hya,Ivanov:2014doa,deMedeirosVarzielas:2019rrp}. Previous
analyses led to a maximum of 13 accidental symmetries for the Two
Higgs Doublet Model (2HDM) potential, where the maximal symmetry group
is
$G^{\bm{\Phi}}_ {\text{2HDM}}=\text{Sp(4)}/ \text{Z}_2 \otimes
\text{SU(2)}_L$~\cite{Pilaftsis:2011ed}.
Here, we extend this theoretical framework to $n$HDM potentials based
on the maximal symmetric group
Sp($2n)/Z_2
\otimes\text{SU(2)}_L$~\cite{Pilaftsis:2016erj,Darvishi:2019ltl}.
The Maximally Symmetric $n$HDM (MS-$n$HDM) can provide natural SM
alignment that exhibits quartic coupling unification up to the
Planck scale~\cite{Dev:2014yca,Pilaftsis:2016erj,Darvishi:2019ltl,Darvishi:2020teg}.

Identifying the symplectic group Sp($2n)$ as the maximal symmetry
group allows us to classify all SU(2)$_L$-preserving accidental
symmetries in $n$HDM potentials.  We introduce {\em{prime invariants}}
to construct accidentally symmetric potentials in terms of fundamental
building blocks that respect the symmetries. In the same context, we
use irreducible representations to derive all potentials that are
invariant under non-Abelian discrete symmetries.

The layout of this paper is as follows. In Section \ref{sec:s1}, we
define the $n$HDMs in the bilinear scalar field formalism. Given that
Sp(2$n)$ is the maximal symmetry of the $n$HDM potential, we adopt
the bi-adjoint representation of this group which becomes relevant to this bilinear formalism.  In Section
\ref{sec:cs}, we start with classifying continuous symmetries for
$n$HDM potentials. As illustrative examples, we classify all
accidentally symmetric potentials for the 2HDM and the Three Higgs
Doublet Model (3HDM). Then, we introduce prime invariants to build
potentials that are invariant under SU(2)$_L$-preserving continuous
symmetries. In Section~\ref{sec:ds}, we discuss possible and known
discrete symmetries for $n$HDM potentials and recover the discrete
symmetries for the 2HDM and the 3HDM cases
\cite{Ivanov:2012fp,Ivanov:2014doa,deMedeirosVarzielas:2019rrp}. This
section also includes our approach to building 3HDM potentials with
the help of irreducible representations of discrete symmetries. Having
discussed the classifications of symmetries, we provide the list of
all SU(2)$_L$-preserving accidental symmetries for the 2HDM and the
3HDM potentials including the relationships among the theoretical
parameters of the scalar potentials. Section \ref{sec:con} contains
our conclusions. Finally, technical details are delegated to
Appendices \ref{V2-3}, \ref{sp-g}, \ref{C}, and \ref{D}.

\section{Bilinear Formalism and the Maximal Symmetry Sp($2n$)}
\label{sec:s1}

The $n$HDMs contain $n$ scalar doublet fields,
$ \phi_{ i} \,\, (i=1,2,\dots,n)$, which all share the same
U(1)$_Y$-hypercharge quantum number, i.e. $Y_{\phi_{ i} }={1/2}$.  The
most general form of the $n$HDM potential may conventionally be
expressed as follows \cite{Botella:1994cs}:
\begin{equation}
V_n=\ \displaystyle\sum_{i,j=1}^{n}\, m_{ij}^2\, ( \phi_i^{\dagger} \phi_j)+ \ \displaystyle\sum_{i,j\, k,l=1}^{n}\,
\lambda_{ijkl}\, ( \phi_i^{\dagger} \phi_j)( \phi_k^{\dagger} \phi_l),
\end{equation}
with $\lambda_{ijkl}=\lambda_{klij}$.  In general, the
SU(2)$_L\times$U(1)$_Y$ invariant $n$HDM potential contains $n^2$
physical mass terms along with $n^2(n^2+1)/2$ physical quartic
couplings.

An equivalent way to write the $n$HDM potential is based on the
so-called bilinear field formalism
\cite{Velhinho:1994np,Maniatis:2006fs,Nishi:2006tg,Ivanov:2006yq,Battye:2011jj}.  To
this end, we first define a $4n$-dimensional ($4n$-D) complex
${\bm{\Phi}}_n$-multiplet as
\begin{equation}
\small
{\bm{\Phi}}_2 =\begin{pmatrix}
\phi_1 \\
\phi_2 \\
i \sigma^2 \phi_1^{*} \\
i \sigma^2 \phi_2^{*}
\end{pmatrix}, \quad
{\bm{\Phi}}_3 =\begin{pmatrix}
\phi_1 \\
\phi_2 \\
\phi_3 \\
i \sigma^2 \phi_1^{*} \\
i \sigma^2 \phi_2^{*} \\
i \sigma^2 \phi_3^{*}
\end{pmatrix},\quad \dots,\quad
{\bm{\Phi}}_n =\begin{pmatrix}
\phi_1 \\
\phi_2 \\
\phi_3 \\
\cdot \\
\cdot \\
\cdot \\
i \sigma^2 \phi_1^{*} \\
i \sigma^2 \phi_2^{*} \\
i \sigma^2 \phi_3^{*} \\
\cdot \\
\cdot \\
\cdot
\end{pmatrix},
\end{equation}
where $\sigma^{1,2,3}$ are the Pauli matrices and
$i \sigma^2 \phi_i^{*}$ (with $i=1,2,\dots,n$) are the U(1)$_Y$
hypercharge-conjugate of $\phi_i$.  Observe that the
${\bm{\Phi}}_n$-multiplet transforms covariantly under an SU(2$)_L$
gauge transformation as,
\begin{equation}
{\bm{\Phi}}_n\to \text{U}_L {\bm \Phi}_n, \quad \text{U}_L\in \text{SU(2)}_L.
\label{su(2)}
\end{equation}
Note that U$_L$ is the 2$\times$2 unitary matrix that may also be
represented as $\sigma^0 \otimes {\bf{1}}_n \otimes \text{U}_L$ in the
${\bm{\Phi}}_n$-space.  Additionally, the ${\bm{\Phi}}_n$-multiplet
satisfies the Majorana-type property \cite{Battye:2011jj},
\begin{equation}
{\bm{\Phi}}_n=C {{\bm \Phi}_n^*},
\end{equation}
where $C=\sigma^2 \otimes {\bf{1}}_n \otimes \sigma^2$
($C=C^{-1}=C^*$) is the charge conjugation operator and ${\bf{1}}_n$
is the $n\times n$ identity matrix.

With the help of the ${\bm{\Phi}}_n$-multiplet, we may now define the
bilinear field
vector~\cite{Ivanov:2006yq,Pilaftsis:2011ed,Battye:2011jj},
\begin{equation}
R_{n}^{A} \equiv {\bm{\Phi}}_n^{\dagger} {\Sigma}^{A}_n {\bm{\Phi}}_n,
\end{equation}
with ${A}=0,1,2,\dots,n(2n-1)-1$. Notice that $n(2n-1)$-vector
$R_{n}^{A}$ is invariant under SU(2)$_L$ transformations thanks to
(\ref{su(2)}).

The matrices ${\Sigma}^{A}_n$ have $4n \times 4n$ elements and
can be expressed in terms of double tensor products as,
\begin{gather}
{\Sigma}^{A}_n=\left(\sigma^0 \otimes t_S^a \otimes \sigma^0,
 \quad \sigma^i \otimes t_A^b \otimes \sigma^0 \right),
\end{gather}
where $t_S^a$ and $ t_A^b$ are the symmetric and anti-symmetric
matrices of the SU($n$) symmetry generators, respectively. Specifically,
for the case of the 2HDM, the following 6 matrices may be defined:
\begin{eqnarray}
{\Sigma_2}^{0,1,3}&=&\dfrac{1}{2}\sigma^0 \otimes \sigma^{0,1,3} \otimes \sigma^0, \quad \quad {\Sigma_2}^{2}=\dfrac{1}{2}\sigma^3 \otimes \sigma^{2} \otimes \sigma^0,
\nonumber \\
{\Sigma_2}^{4}&=&-\dfrac{1}{2}\sigma^2 \otimes \sigma^{2} \otimes \sigma^0, \quad \quad \;
 {\Sigma_2}^{5}=-\dfrac{1}{2}\sigma^1 \otimes \sigma^{2} \otimes \sigma^0.
\end{eqnarray}
Correspondingly, for the 3HDM, we have the following 15 matrices:
\begin{eqnarray}
{\Sigma_3}^{0,1,2,3,7,8}&=&\dfrac{1}{2}\sigma^0 \otimes G^{0,1,4,6,3,8} \otimes \sigma^0,\quad \quad
 {\Sigma_3}^{4,5,6}=\dfrac{1}{2}\sigma^3 \otimes G^{2,5,7} \otimes \sigma^0, \nonumber \\
{\Sigma_3}^{9,10,11}&=&\dfrac{1}{2}\sigma^2 \otimes G^{2,5,7} \otimes \sigma^0, \quad \quad \;\;
{\Sigma_3}^{12,13,14}=\dfrac{1}{2}\sigma^1 \otimes G^{2,5,7} \otimes \sigma^0,
\end{eqnarray}
where $G^i$ are the standard Gell-Mann matrices of SU(3)~\cite{GellMann:1962xb}.
Note that the ${\Sigma}^{A}_n$ matrices satisfy the property,
\bea
C^{-1}{\Sigma}
^{A}_nC=({\Sigma}^{A}_n)^{\mathsf{T}},
\eea
which means that $ {\Sigma}^{A}_n$ are C-even.

Consequently, the vectors $R_{2}^{A}$ and $R_{3}^{A}$ for the 2HDM and
the 3HDM cases are given by
\begin{equation}
\small
R^{A}_2= \begin{pmatrix}
\phi_1^{\dagger} \phi_1 +\phi_2^{\dagger} \phi_2 \\
\phi_1^{\dagger} \phi_2 +\phi_2^{\dagger} \phi_1 \\
-i [\phi_1^{\dagger} \phi_2 -\phi_2^{\dagger} \phi_1] \\
\phi_1^{\dagger} \phi_1 -\phi_2^{\dagger} \phi_2 \\
-[\phi_1^{\mathsf{T}} i \sigma^2 \phi_2 - \phi_2^{\dagger} i \sigma^2 \phi_1^* ]\\
i[\phi_1^{\mathsf{T}} i \sigma^2 \phi_2 + \phi_2^{\dagger} i \sigma^2 \phi_1^* ]
\end{pmatrix},\quad
R^{A}_{3}= \begin{pmatrix}
\phi_1^{\dagger} \phi_1 +\phi_2^{\dagger} \phi_2+\phi_3^{\dagger} \phi_3 \\
\phi_1^{\dagger} \phi_2 +\phi_2^{\dagger} \phi_1 \\
\phi_1^{\dagger} \phi_3 +\phi_3^{\dagger} \phi_1 \\
\phi_2^{\dagger} \phi_3+\phi_3^{\dagger} \phi_2\\
-i [\phi_1^{\dagger} \phi_2 -\phi_2^{\dagger} \phi_1] \\
-i [\phi_1^{\dagger} \phi_3 -\phi_3^{\dagger} \phi_1] \\
-i [\phi_2^{\dagger} \phi_3 -\phi_3^{\dagger} \phi_2] \\
\phi_1^{\dagger} \phi_1 -\phi_2^{\dagger} \phi_2 \\
{1\over\sqrt{3}}[\phi_1^{\dagger} \phi_1 +\phi_2^{\dagger} \phi_2-2 \phi_3^{\dagger} \phi_3] \\
-[\phi_1^{\mathsf{T}} i \sigma^2 \phi_2 - \phi_2^{\dagger} i \sigma^2 \phi_1^*] \\
-[\phi_1^{\mathsf{T}} i \sigma^2 \phi_3 - \phi_3^{\dagger} i \sigma^2 \phi_1^*] \\
-[\phi_2^{\mathsf{T}} i \sigma^2 \phi_3 - \phi_3^{\dagger} i \sigma^2 \phi_2^* ]\\
i[\phi_1^{\mathsf{T}} i \sigma^2 \phi_2 + \phi_2^{\dagger} i \sigma^2 \phi_1^* ]\\
i[\phi_1^{\mathsf{T}} i \sigma^2 \phi_3 + \phi_3^{\dagger} i \sigma^2 \phi_1^* ]\\
i[\phi_2^{\mathsf{T}} i \sigma^2 \phi_3 + \phi_3^{\dagger} i \sigma^2 \phi_2^* ]
\end{pmatrix}.
\end{equation}
With the aid of the $n(2n-1)$-dimensional vectors $R_{n}^{A}$, the
potential $V_n$ for an $n$HDM can be written in the quadratic
form as
\begin{equation}
V_n=-\dfrac{1}{2}M^n_{A} R_n^{A}+\dfrac{1}{4}L^n_{{A}{A'}}R_n^{A}R_n^{{A'}},
\label{VB}
\end{equation}
where $M^n_{A} $ is the $1 \times n(2n-1)$-dimensional mass matrix and
$L^n_{{A}{A'}}$ is a quartic coupling matrix with
$n(2n-1) \times n(2n-1)$ entries.  Evidently, for a U(1$)_Y$-invariant
$n$HDM potential, the first $n^2$ elements of $M^n_{A} $ and
$n^2 \times n^2$ elements of $L^n_{{A}{A'}}$ are only relevant, since
the other U(1$)_Y$-violating components vanish.  The general 2HDM and
3HDM potentials with their corresponding $M^n_{A} $ and
$L^n_{{A}{A^\prime}}$ are presented in Appendix~\ref{V2-3}.

The gauge-kinetic term of the $\bm{\Phi}_n$-multiplet is given by
\begin{equation}
T_n=\frac{1}{2}(D_{\mu}{\bm{\Phi}}_n)^{\dagger} (D^{\mu}{\bm{\Phi}}_n),
\end{equation}
where the covariant derivative in the ${\bm{\Phi}}_n$ space is
\begin{equation}
 D_{\mu}\ =\ \sigma^0 \otimes {\bf{1}}_n \otimes
          (\sigma^0 \partial^0_{\mu} +ig_w W_{\mu}^i \sigma^i /2)\:
+\: \sigma^3 \otimes {\bf{1}}_n \otimes i{g_Y\over 2} B_{\mu} \sigma^0.
\end{equation}
In the limit $g_Y \to 0$, the gauge-kinetic term $T_n$ is invariant
under $\text{Sp}(2n)/Z_2 \otimes \text{SU}(2)_L$ transformations of
the $\bm{\Phi}_n$-multiplet. In general, the maximal symmetry group
acting on the ${\bm{\Phi}}_n$-space in the $n$HDM potentials is
\begin{equation*}
\text{G}_{\mathsf{n\text{-HDM}}}^{{\bm{\Phi}}_n}=\text{Sp}(2n)/Z_2 \otimes \text{SU}(2)_L,
\end{equation*}
which leaves the local SU($2)_L$ gauge kinetic term of ${\bm{\Phi}}_n$
canonical.  The local SU(2$)_L$ group generators can be represented as
$\sigma^0 \otimes {\bf{1}}_n \otimes (\sigma^{1,2,3}/2)$, which commute
with all generators of the Sp($2n$) group.

Let us turn our attention to the Sp(2$n$) generators
${K}^{{B}}_n$, with ${B =0,1,\dots,n(2n+1)-1}$. They satisfy the
important relation~\cite{Pilaftsis:2011ed},
\begin{equation}
C^{-1}{K}^{{B}}_n C=-K_n^{{{B}}*}=-({K}^{{B}}_n)^{\mathsf{T}},
\end{equation}
which implies that ${K}^{{B}}_n$ are C-odd.  The Sp($2n)$ generators
may conveniently be expressed in terms of double tensor products as~\cite{Georgi:1982jb}
\begin{gather}
{K}^{{B}}_n=\left( \sigma^0 \otimes t_A^b \otimes \sigma^0, \quad \sigma^i \otimes t_S^a
\otimes \sigma^0 \right),
\end{gather}
where $t_S^a$ ($ t_A^b$) are the symmetric (anti-symmetric) generators
of the SU($n$) symmetry group. For instance, the 10 generators of
Sp(4) are~\cite{Pilaftsis:2011ed}
\begin{align}
&K^{0,1,3}_2=\dfrac{1}{2}\sigma^3 \otimes \sigma^{0,1,3} \otimes \sigma^0, \, \quad \quad \;\;\;
K^2_2=\dfrac{1}{2}\sigma^0 \otimes \sigma^2 \otimes \sigma^0,\nonumber \\
&K^{4,5,8}_2=\dfrac{1}{2}\sigma^1 \otimes \sigma^{0,3,1} \otimes \sigma^0, \quad \quad
K^{6,7,9}_2=\dfrac{1}{2}\sigma^2 \otimes \sigma^{0,3,1} \otimes \sigma^0.
\end{align}
Likewise, the Sp(6) group has 21 generators, which are
\begin{align}
K^{0,1,2,3,4,5}_3&=\dfrac{1}{2}\sigma^3 \otimes G^{0,1,3,4,6,8}
                   \otimes \sigma^0, \qquad \qquad \quad \quad
K^{6,7,8}_3=\dfrac{1}{2}\sigma^0 \otimes G^{2,5,7} \otimes \sigma^0,
 \nonumber \\
K^{9,10,11,12,13,14}_3&=\dfrac{1}{2}\sigma^1 \otimes G^{0,1,3,4,6,8} \otimes \sigma^0, \quad \quad K^{15,16,17,18,19,20}_3=\dfrac{1}{2}\sigma^2\otimes G^{0,1,3,4,6,8}\otimes \sigma^0.
\end{align}

It is interesting to state the Lie commutators between the
${\Sigma}^{A}_n$ and ${K}^{{B}}_n$ generators,
\begin{equation}
[{K}^{{B}}_n, {\Sigma}^{I}_n]=2if_n^{{B}IJ}{\Sigma}^{J}_n,
\label{commutation}
\end{equation}
where $I,J=1,\dots, n(2n-1)-1$ and $f_n^{{B}IJ}$ is a subset of the
structure constants of the SU(2$n$) group. Employing
(\ref{commutation}), we may define the Sp(2$n$) generators in the
bi-adjoint representation (i.e. the adjoint representation in the bilinear formalism) as
\begin{equation}
(T^{{B}}_{n})_{IJ}\ =\ -i f_n^{{B}IJ}\ =\ \text{Tr} \Big([ {\Sigma}^I_n, K^{{B}}_n]{\Sigma}^{J}_n\Big)\,.
\label{TIJ}
\end{equation}
Note that the dimensionality of the bi-adjoint representation differs from the standard adjoint
representation. The former representation has
$(2n^2-n-1) \times (2n^2-n-1)$ dimensions, whereas the latter has $n(2n+1) \times n(2n+1)$ dimensions
~\cite{Hamermesh,Preskill:1980mz,Georgi:1999wka,Vergados}.
The generators of $T^{{B}}_{n}$ for Sp(4) and Sp(6) corresponding to the 2HDM and
3HDM are presented in Appendix~\ref{sp-g}.

Knowing that Sp(2$n)$ is the maximal symmetry group allows us to
classify all SU(2)$_L$-preserving accidental symmetries of $n$HDM
potentials. These symmetries can be grouped into two categories: (i)
continuous symmetries and (ii) discrete symmetries (Abelian and
non-Abelian symmetry groups). In the next section, we demonstrate the
structure of continuous symmetries and prime invariants for building
$n$HDM potentials.

\section{Continuous Symmetries and Prime invariants}
\label{sec:cs}

The symplectic group Sp($2n)$ acts on the ${\bm{\Phi}}_n$-space, such
that the bilinear vector $R_{n}^{A}$ transforms in the bi-adjoint
representation of Sp(2$n$) defined in (\ref{commutation}) and
(\ref{TIJ}). It is therefore essential to consider the maximal
subgroups of Sp(2$n$).  Then, the accidental maximal subgroups would
be the combinations of smaller symplectic groups, such
as~\cite{Slansky:1981yr}
\begin{equation}
\text{Sp}(2n) \supset \text{Sp}(2p) \otimes \text{Sp}(2q),
\end{equation}
where $p+q=n$. The other maximal subgroup is
\begin{equation}
\text{Sp}(2n) \supset \text{SU}(n) \otimes \text{U}(1).
\end{equation}
The breaking pattern of SU($n$) in the maximal subgroups are
\begin{align}
\text{SU}(n) & \supset \text{SU}(p) \otimes \text{SU}(q) \otimes \text{U}(1), \\
& \supset \text{Sp}(2k), \\
& \supset \text{SO}(n),
\end{align}
with $p+q+1=n$ and $n\leq 2k$. The breaking pattern for SO($n$) is
\begin{align}
\text{SO}(n) & \supset \text{SO}(p) \otimes \text{SO}(q),
\end{align}
where $p+q=n$.
Note that local isomorphisms should also be taken into account, such as
\begin{eqnarray}
&&\text{SO(3)} \cong \text{SU(2)} \cong \text{Sp(2)},
\nonumber \\
&&\text{SO(4)} \cong \text{SU(2)} \otimes \text{SU(2)},
\nonumber \\
&&\text{SO(5)} \cong \text{Sp(4)},
\nonumber \\
&&\text{SO(6)} \cong \text{SU(4)}~\text{etc}.
\nonumber
\end{eqnarray}

Following this procedure, it is straightforward to identify all
accidental continuous symmetries for $n$HDM potentials. For the
simplest case, i.e.~that of the 2HDM, the above breaking chain gives rise to the
following continuous symmetries \cite{Pilaftsis:2011ed},
\vspace{-0.2in}
\begin{align}
\begin{minipage}{0.5\textwidth}
\begin{eqnarray}
&&(a) \ \text{Sp(4)} \cong \text{SO(5)},
\nonumber \\[0.04in]
&&(b) \ \text{Sp(2)} \otimes \text{Sp(2)} \ \cong \text{SO(4)},
\nonumber \\[0.04in]
&&(c) \ \text{Sp(2})_{\text{}} \cong \text{SO(3)},
 \nonumber \\[0.04in]
&&(d) \ \text{SU}(2)_{\text{HF}}\cong \text{O(3)}\otimes \text{O(2)},
 \nonumber
\end{eqnarray}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{eqnarray}
&&(e) \ \text{SO(2})_{\text{HF}}\ \cong Z_2\otimes [\text{O(2)}]^2,
 \nonumber \\[0.04in]
&&(f) \ \text{U(1})_{\text{PQ}}\otimes \text{Sp}(2)_{\text{}} \cong \text{O(2)}\otimes \text{O(3)},
 \nonumber \\[0.04in]
&&(g) \  \text{U(1})_{\text{PQ}}\otimes \text{U(1})_Y
 \cong \text{O(2)}\otimes \text{O(2)},
  \nonumber \\[0.04in]
  &&
 \nonumber
\end{eqnarray}
 \end{minipage}
 \label{s2hdm}
 \nonumber \\
\end{align}
where HF indicates Higgs Family symmetries that only involve the
elements of $\Phi=(\phi_1,\phi_2,\dots,\phi_n)^{\mathsf{T}}$ and not
their complex conjugates.  In the case of 3HDM, the maximal symmetry
is Sp(6), so in addition to all symmetries in (\ref{s2hdm}) we find
\vspace{-0.2in}
\begin{align}
\begin{minipage}{0.5\textwidth}
\begin{eqnarray}
&&(h) \ \text{Sp(6)},
\nonumber \\[0.04in]
&&(i) \ \text{Sp(4)} \otimes \text{Sp(2)},
 \nonumber \\[0.04in]
&& (j) \ \text{Sp(2)} \otimes \text{Sp(2)}\otimes \text{Sp(2)},
 \nonumber
 \end{eqnarray}
 \end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{eqnarray}
&&(k) \ \text{SU(3)}\otimes \text{U(1)},
 \nonumber \\[0.04in]
&&(l) \ \text{SO(3)}.
 \nonumber \\[0.04in]
  &&
 \nonumber
\label{s3hdm}
\end{eqnarray}
 \end{minipage}
 \nonumber \\
\end{align}
We may now construct accidentally symmetric $n$HDM potentials in terms
of fundamental building blocks that respect the symmetries. To this end, we
introduce the invariants $S_n$, $D_n^2$ and~$T_n^2$. In detail, $S_n$
is defined as
\begin{align}
   \label{eq:spn}
 S_n\ =\ {\bm{\Phi}}_n^\dagger{\bm{\Phi}}_n\;,
\end{align}
which is invariant under both the SU($n)_L\otimes$U(1)$_Y$ gauge group and Sp(2$n$).
Moreover, we define the SU(2)$_L$-covariant quantity $D^a_n$ in the HF
space as
\begin{equation}
    \label{su}
D_n^a\ =\ {{\Phi}}^\dagger \sigma^a {{\Phi}}\; .
\end{equation}
Under an SU(2)$_L$ gauge transformation,
$D^a_n \to D_n^{\prime a}= O^{ab}D_n^b$, where $O\in$~SO(3).  Hence,
the quadratic quantity $D_n^2\equiv D^a_n D^a_n$ is both gauge and SU($n$)
invariant. Finally, we define the auxiliary quantity $T_n$ in the HF
space as
\begin{equation}
T_n\ =\ \Phi \Phi^{\mathsf{T}}\;,
\end{equation}
which transforms as a triplet under SU(2)$_L$,
i.e.~$T_n \to T_n'=U_L T_n U_L^{\mathsf{T}}$.  As a consequence, a
proper prime invariant may be defined as
$T_n^2\equiv \text{Tr}(TT^*)$, which is also both gauge and SO($n$)
invariant.

In addition to the above maximal prime invariants, it is useful to
define minimal invariants. For instance, prime invariants that respect Sp(2) can be derived from the doublets $\begin{pmatrix}
  \phi_i \\ i \sigma^2 \phi_i^*\end{pmatrix}$ and $\begin{pmatrix}
  \phi_i \\
  i \sigma^2 \phi_j^* \end{pmatrix}$,
\begin{align}
&S_{ij}=\begin{pmatrix} \phi_i^{\dagger}, -i
  \phi_i^{\mathsf{T}}\sigma^2 \end{pmatrix}\begin{pmatrix} \phi_j \\
i \sigma^2 \phi_j^* \end{pmatrix}=\phi_i^\dagger \phi_j+\phi_j^\dagger \phi_i, \nonumber \\
&S^{\prime}_{ij}=\begin{pmatrix} \phi_i^{\dagger}, -i
  \phi_j^{\mathsf{T}}\sigma^2 \end{pmatrix}\begin{pmatrix} \phi_i \\
i \sigma^2 \phi_j^* \end{pmatrix}=\phi_i^\dagger \phi_i+\phi_j^\dagger \phi_j.
\end{align}

Likewise, to obtain minimal SU(2)$\otimes$U(1) invariants from
$\begin{pmatrix} \phi_i \\ \phi_j \end{pmatrix}$ and $\begin{pmatrix} \phi_i \\
  i \sigma^2 \phi_j^* \end{pmatrix}$, we define
\begin{align}
& D_{ij}^a =\begin{pmatrix} \phi_i^{\dagger},
  \phi_j^{\dagger} \end{pmatrix} \sigma^a \begin{pmatrix} \phi_i \\
 \phi_j \end{pmatrix} =\phi_i^{\dagger} \sigma^a \phi_i
  +\phi_j^{\dagger} \sigma^a \phi_j =D_{ji}^a, \nonumber \\
& D^{\prime a}_{ij}=\begin{pmatrix} \phi_i^{\dagger}, -i
  \phi_j^{\mathsf{T}}\sigma^2 \end{pmatrix} \sigma^a \begin{pmatrix}
  \phi_i \\
i \sigma^2 \phi_j^* \end{pmatrix}=\phi_i^{\dagger} \sigma^a \phi_i -
  \phi_j^{\dagger} \sigma^a \phi_j=-D_{ji}^{\prime a},
\end{align}
where the identity $\sigma^2 \sigma^a \sigma^2=-(\sigma^a){}^{\mathsf{T}}$ has been used.

By analogy, to construct an SO(2)-invariant expression from the doublet
$\begin{pmatrix} \phi_i \\ \phi_j \end{pmatrix}$, we may use quantities such as
\begin{align}
T_{ij}=\phi_i \phi_i^{\mathsf{T}} +\phi_j \phi_j^{\mathsf{T}} =T_{ji}.
\end{align}
Moreover, extra prime invariants can be constructed from
$\begin{pmatrix}\phi_i \\ i \sigma^2 \phi_i^* \end{pmatrix}$ and
$\begin{pmatrix} \phi_j \\ i \sigma^2 \phi_j^* \end{pmatrix}$, e.g.
\begin{align}
S_{ij} &=\phi_i^{\dagger} \phi_j +\underbrace{{ (i \sigma^2 \phi_i^*)}^{\dagger}{ (i \sigma^2 \phi_j^*)}}_{\phi_i^{\mathsf{T}} \phi_j^*}=\phi_i^{\dagger} \phi_j +\phi_j^{\dagger} \phi_i,
\\
iD_{ij}^{\prime\prime a} &=\phi_i^{\dagger}\sigma^a \phi_j + {(i \sigma^2 \phi_i^*)}^{\dagger} \sigma^a (i \sigma^2 \phi_j^*) =\phi_i^{\dagger} \sigma^a \phi_j -\phi_j^{\dagger} \sigma^a \phi_i.
\end{align}
Note that $D_{ij}^{\prime\prime a} D_{ij}^{\prime\prime a}$ depends on $S_{ij}$ and $ S_{ii,jj}$, since
\begin{align}
-{D_{ij}^{\prime\prime 2}}=S_{ij}^2 - (\phi_i^{\dagger} \phi_j)(\phi_j^{\dagger} \phi_j)=S_{ij}^2 -S_{ii} S_{jj}.
\end{align}
Observe that $T_{ij}$ and $S_{ij}$ are not invariant under phase
re-parametrizations, $\phi_i \to e^{i \alpha_i} \phi_i$, and so they
need to be appropriately combined with their complex conjugates.

We are now able to build a symmetric scalar potential
$V_{\rm sym}$ in terms of prime invariants as follows:
\begin{align}
V_{\rm sym}\ =\ -\mu ^2 S_n\: +\:\lambda_S S_n^2\: +\: \lambda_D
  D_n^2\: +\:
  \lambda_T T_n^2\; .
\end{align}
Obviously, the simplest form of the $n$HDM potentials obeys the
maximal symmetry~Sp(2$n$), which has the same form as the SM
potential,
$$
V_{\text{SM}}\ =\ -\mu^2 \left( \phi^{\dagger}\phi \right)\:  +
 \:  \lambda \left( \phi^{\dagger}\phi\right)^2,
$$
with a single mass term and a single quartic coupling. For example,
the 2HDM Sp(4)-invariant potential, the so-called MS-2HDM is
given by~\cite{Dev:2014yca}
\begin{equation}
    \label{eq:MS2HDM}
V_{\text{MS-2HDM}}\ =\ - \mu_1^2 \left( \left| \phi_1 \right|^2\: +\: \left| \phi_2 \right|^2 \right)
 + \lambda_1 \left( \left| \phi_1 \right|^2 + \left| \phi_2 \right|^2
   \right)^2\; ,
\end{equation}
with the obvious relations among the parameters,
\begin{align}
\mu_1^2 = \mu_2^2, \quad m_{12}^2 = 0,\quad
2\lambda_2 = 2\lambda_1= \lambda_3, \quad
\lambda_4 = \text{Re}(\lambda_5) = \lambda_6 = \lambda_7 = 0.
\end{align}
Note that the functional form of the potential in~\eqref{eq:MS2HDM}
is $V= V[S_2] = V[S_{11}+S_{22}]$.

Similarly, the 3HDM potential invariant under Sp(6)${ /Z_2}$ will be a
function of the symmetric block $S_3 = S_{11}+S_{22}+S_{33}$, i.e.
\begin{equation}
V_{\text{MS-3HDM}}\ =\ - \mu_1^2 \left( \left| \phi_1 \right|^2\: +\:
  \left| \phi_2 \right|^2\: +\: \left| \phi_3 \right|^2 \right)
 + \lambda_{11} \left( \left| \phi_1 \right|^2 + \left| \phi_2 \right|^2\: +\: \left| \phi_3 \right|^2
   \right)^2,
\end{equation}
where the non-zero parameters are
\begin{align}
\mu_1=\mu_2=\mu_3,\quad \lambda_{11}=\lambda_{22}=\lambda_{33}=2\lambda_{1122}=2\lambda_{1133}=2\lambda_{2233}.
\nonumber
\end{align}
Remarkably, the MS-$n$HDM potentials
obey naturally the SM-alignment constraints and all quartic couplings
of the MS-$n$HDM potential can vanish
simultaneously~\cite{Dev:2014yca,Darvishi:2019ltl}.

Another example is the SU(3)$\otimes$U(1)-invariant 3HDM
potential. The corresponding symmetric blocks are $S_3 =
S_{11}+S_{22}+S_{33}$ and $D^2_3 = D_{12}^2+D_{13}^2+D_{23}^2$, given
in~\eqref{eq:spn} and~\eqref{su}, respectively.
Therefore, the
SU(3)$\otimes$U(1)-invariant 3HDM potential takes on the form
\begin{align}
V_{\text{SU(3)}\otimes \text{U(1)}}&= - \mu_1^2 \left( |\phi_1|^2\:
                                     +\: |\phi_2|^2\: +\: | \phi_3 |^2
                                     \right)
 + \lambda_{11} \left( |\phi_1|^4 + |\phi_2|^4\: +\: |\phi_3|^4
   \right)\
\nonumber \\ &
+ \lambda_{1122} \left( |\phi_1|^2 |\phi_2|^2
+ |\phi_1|^2  |\phi_3 |^2
 + |\phi_2|^2 \,|\phi_3|^2 \right)
\nonumber \\ &
+ (2 \lambda_{11}-\lambda_{1122}) \left( |\phi_1^{\dagger} \phi_2 |^2
+ |\phi_1^{\dagger} \phi_3|^2
+ |\phi_2^{\dagger} \phi_3|^2\right),
\end{align}
with the following relations between the parameters:
\begin{align}
&\mu_1=\mu_2=\mu_3,\quad \lambda_{11}=\lambda_{22}=\lambda_{33},\quad \lambda_{1122}=\lambda_{1133}=\lambda_{2233},\nonumber \\
&\lambda_{1221}=\lambda_{1331}=\lambda_{2332}=2 \lambda_{11}-\lambda_{1122}.\nonumber
 \end{align}

 This method can be applied to all continuous symmetries of
 $n$HDM potentials. We present all explicit symmetric blocks under all
 continuous symmetries for the 2HDM and the 3HDM potentials in Appendix
 \ref{C}. The complete list of  accidental symmetries for the 2HDM
 and 3HDM potentials, along with the relations among the non-zero
 parameters, are exhibited in Tables~\ref{tab1} and~\ref{tab2}.

\section{Discrete symmetries and Irreducible representations}
\label{sec:ds}

As discussed in Section \ref{sec:s1}, Sp(2$n)$ is the maximal symmetry
group for $n$HDM potentials. This will help us to classify all
SU(2)$_L$-preserving accidental symmetries of such potentials. In
addition to continuous symmetries shown in Section \ref{sec:cs}, there
are also discrete symmetries as subgroups of continuous symmetries.
Known examples of this type are the Standard CP symmetry, the Cyclic
discrete group $Z_n$, the Permutation group $S_n$, or a product of
them possibly combining with continuous symmetries.  In general, these
discrete symmetries can be grouped into Abelian and non-Abelian
symmetry groups.  In this section, we discuss all possible and known
discrete symmetries for $n$HDM potentials, including our approach to
build $n$HDM potentials by employing irreducible representations of
discrete symmetry groups.

\subsection{Abelian Discrete Symmetries}

To start with, let us first consider the Abelian discrete symmetry
groups~\cite{Humphreys,Beardon}
\begin{equation}
\quad Z_2, \quad Z_3, \quad Z_4, \quad Z_2 \times Z_2, \quad \dots, \quad Z_n, \quad \dots,
\end{equation}
where $Z_n=\{1, \omega, \dots, \omega^{(n-1)}\}$ with $\omega^n=1$.
Note that iff $n$ and $m$ have no common prime factor, the product
$Z_n \times Z_m$ is identical to $Z_{n\times m}$. These discrete
symmetries can be imposed to restrict the independent theoretical
parameters of the model. For example, in the 2HDM, the $Z_2$ symmetry is
invoked to avoid flavour changing neutral currents~\cite{Glashow:1976nt} or to ensure the
stability of dark matter~\cite{Silveira:1985rk}.

In addition to these discrete symmetries, there are Generalized CP
(GCP) transformations defined as,
\begin{equation}
\text{GCP}[{{\phi}}_i]=G_{ij} {{\phi}}_{j}^*,
\end{equation}
with $G_{ij} \in \text{SU(}n)\times$U(1), where U(1) can always be
eliminated by U(1)$_Y$ transformation. The GCP transformations realize
different types of CP symmetry. For example, in the case of the 2HDM
(3HDM), there are two types of CP symmetries: (i)~standard CP or CP1:
${{{\phi}}_{1,2}\, (\phi_3)\to {{\phi}}_{1,2} ^*\,(\phi^*_3)}$ and (ii)~non-standard CP or CP2:
${{\phi}}_1 \to {{\phi}}_2^*$, ${{\phi}}_2 \to -{{\phi}}_1^*$
(${{{\phi}}_3 \to
  {{\phi}}_3^*}$)~\cite{Ferreira:2009wh,Ferreira:2010yh}\footnote{In
  some of the
  earlier literature, there has been a third class of CP symmetries
  called CP3 which relies on SO(2)$_{\text{HF}}$ (or
  SO(3)$_{\text{HF}}$ for a 3HDM potential) and the standard CP
  symmetry. However, continuous SO$(n)_{\text{HF}}$ symmetries lead
  automatically to CP-invariant $n$HDM potentials, without introducing
  further restrictions on the theoretical parameters. In other words,
  CP1 is an emergent symmetry that results from SO$(n)_{\text{HF}}$,
  so CP3 should not be regarded as a new CP symmetry.}.  In general,
without continuous group factors, the discrete symmetries for the 2HDM
are CP1, CP2 and $Z_2$. The generators of these discrete symmetries
can be expressed in terms of double tensor products as
\begin{eqnarray}
\Delta_{\text{CP1}}&=&\sigma^2 \otimes \sigma^0 \otimes \sigma^2,
\\
\Delta_{\text{CP2}}&=&i \sigma^2 \otimes \sigma^2 \otimes \sigma^2,
\\
\Delta_{Z_2}&=&\sigma^0 \otimes \sigma^3 \otimes \sigma^0.
\end{eqnarray}
In the bilinear $R_2^{A}$-space, the transformation matrices (or the generating group elements) associated with CP1, CP2 and $Z_2$ discrete symmetries are given by
\begin{align}
D_{\text{CP1}}&=\text{diag} ({\bf{1,-1,1}},1,-1), \\
D_{\text{CP2}}&=\text{diag} ({\bf{-1,-1,-1}},1,-1),\\
D_{Z_2}&=\text{diag} ({\bf{-1,-1,1}},-1,-1).
\end{align}
where the U(1)$_Y$-conserving elements are denoted in boldface.

Let us turn our attention to the 3HDM potential. In this case, the corresponding CP1 discrete symmetry can be represented as
\begin{equation}
\Delta_{\text{CP1}}=\sigma^2 \otimes G^0 \otimes \sigma^2,
\end{equation}
resulting in the following transformation matrix in the $R_3^{A}$-space,
\begin{align}
\small
D_{\text{CP1}} &=\text{diag} ({\bf{1,1,1,-1,-1,-1,1,1}},1,1,1,-1,-1,-1).
\end{align}
On the other hand, CP2 discrete symmetries may be given by
\begin{align}
 \Delta_{\text{CP2}}&=\begin{pmatrix}0&-i \bar{G}^2 \otimes \sigma^2 \\ i\bar{G}^{2^*} \otimes \sigma^2&0 \end{pmatrix},
\end{align}
with~\cite{Ivanov:2015mwl}
\begin{align}
\bar{G}^{2} ={\begin{pmatrix}0&-1&0\\ 1&0&0\\0&0& e^{i \alpha} \end{pmatrix}},
\end{align}
where the phase $e^{i \alpha}$ is an arbitrary phase factor. This
results in the following transformation matrix in the bilinear
$R_3^{A}$-space (dots stand for zero elements)
\begin{align}
&\setlength{\arraycolsep}{0.3pt}
\renewcommand{\arraystretch}{0.01}
D_{\text{CP2}}=\left(\scriptsize
\begin{array}{cccccccccccccc}
\bf{-1 }&. & . & . & . & . & . &. & . & . & . & . & . & . \\
. &. &\bm{-\cos{\bm\alpha}}  & . & . &\bm{\sin{\bm\alpha}} & . & . & . & . & . & . & . & .\\
. &\bm{\cos{\bm\alpha}}  & . & .  &\bm{-\sin{\bm\alpha}} & . & . & . & . & . & . & . & . & . \\
. &. & . & \bf{-1} & . & . & . & . & . & . & . & . & . & . \\
. &. &\bm{\sin{\bm\alpha}} & . & . &\bm{\cos{\bm\alpha}} & . & . & . & . & . & . & . & . \\
. &\bm{-\sin{\bm\alpha}}& . & . & \bm{-\cos{\bm\alpha}}  & . & . & . & . & . & . & . & . & . \\
. &. & . & . & . & . &\bf{-1 }& . & . & . & . & . & . & . \\
. &. & . & . & . & . & . &\bf{1} & . & . & . & . & . & . \\
. & . &. & . &. & . & . & . & 1 & . & . &. & . & . \\
 . & . & . & . & . & . & . & . & . & . &-\cos\alpha & . &. &\sin\alpha \\
 . & . & . & . & . & . & . & . & . &\cos\alpha & . & . &-\sin\alpha&.\\
 . & . & . & . & . & . & . & . & .& . & . &-1 & . & . \\
 . & . & . & . & . & . & . & . & . & .&-\sin\alpha & . & . & \cos\alpha \\
 . & . & . & . & . & . & . & . & . & \sin\alpha& .& . & -\cos\alpha & . \\
\end{array}
\right).
\end{align}
Note that the CP2 transformation matrix $D_{\text{CP2}}$ in  the bilinear
$R_3^{A}$-space is non-diagonal, contrary to the 2HDM case. We must
remark here that in the case of 2HDM,
$\Delta_{\text{CP2}}^2 =-\bf{1}_8\neq \bf{1}_8$ and
$\Delta_{\text{CP2}}^4 =\bf{1}_8$ and in the bilinear space
$D_{\text{CP2}}^2 =\bf{1}_6$. However, in the case of the 3HDM, we have
$\Delta_{\text{CP2}}^2 \neq -\bf{1}_{12}$ and
$\Delta_{\text{CP2}}^4= \bf{1}_{12}$ and
$D_{\text{CP2}}^4 =\bf{1}_{14}$, in agreement with a property termed
CP4 in~\cite{Ivanov:2015mwl}. Without loss of generality, we
set $\alpha=0$.

In addition, there are several Abelian discrete symmetries for the 3HDM potential~\cite{Ivanov:2011ae},~i.e.
\begin{equation}
\quad Z_2, \quad Z^{\prime}_2, \quad Z_3, \quad Z_4, \quad Z_2 \times Z_2.
\end{equation}
The generators of these Abelian discrete symmetries are given by
\begin{align}
\Delta_{{Z_2}}&=\sigma^0 \otimes {z_2} \otimes \sigma^0,
\qquad \qquad \quad \,\,
\Delta_{{Z^{\prime}_2}}=\sigma^0 \otimes {z^{\prime}_2} \otimes \sigma^0,
\nonumber \\
\Delta_{{Z_3}}&=\text{diag}[{z_3} \otimes \sigma^0,{z_3}^* \otimes \sigma^0],
\quad
\Delta_{{Z_4}}=\text{diag}[{z_4} \otimes \sigma^0,{z_4}^* \otimes \sigma^0],
\label{22p34}
\end{align}
where
\begin{align}
{z_2}&=\text{diag}[1,-1,1],
\quad
{z^{\prime}_2}=\text{diag}[1,-1,-1],
\nonumber \\
{z_3}&=\text{diag}[\omega^2,\omega,1],
\quad
{z_4}=\text{diag}[i,-i,1],
\end{align}
with $\omega=e^{i2\pi/3}$.  In the bilinear $R_3^{A}$-space, as a
result of flipping the sign of the one or two doublets, the following
diagonal transformation matrices for the discrete symmetries $Z_2$ and
$Z_2^{\prime}$ may be derived:
\begin{align}
\small
D_{{Z_2}}&=\text{diag} ({ {\bm{-1}}, {\bm{1}}, {\bm{-1}}, {\bm{-1}}, {\bm{1}}, {\bm{-1}}, {\bm{1}}, {\bm{1}}},-1,1,-1,-1,1,-1), \\
D_{{Z^{\prime}_2}}&=\text{diag} ({{\bm{-1}}, {\bm{-1}}, {\bm{1}}, {\bm{-1}}, {\bm{-1}}, {\bm{1}}, {\bm{1}}, {\bm{1}}},-1,-1,1,-1,-1,1).
\label{z22}
\end{align}
In the same way, the transformation matrices for $Z_3$ and $Z_4$ may
be represented by the non-diagonal matrices
\begin{align}
&\setlength{\arraycolsep}{1pt}
\renewcommand{\arraystretch}{0.9}
D_{{Z_3}}={1\over 2}\left(\scriptsize
\begin{array}{cccccccccccccc}
 {\bm{-1}} &. & . &{\bm{\sqrt{3}}} & . & . & . & . & . & . & . & . & . & . \\
. & {\bm{-1}}& . & . & {-\bm{\sqrt{3}}}& . & . & . & . & . & . & . & . & .\\
. & . & {\bm{-1}} & .  & . &{\bm{\sqrt{3}}} & . & . & . & . & . & . & . & . \\
{-\bm{\sqrt{3}}} &. & . & {\bm{-1}} & . & . & . & . & . & . & . & . & . & . \\
. &{\bm{\sqrt{3}}}& . & . & {\bm{-1}} & . & . & . & . & . & . & . & . & . \\
. &. &{-\bm{\sqrt{3}}}& . &. & {\bm{-1}}& . & . & . & . & . & . & . & . \\
. &. & . & . & . & . & {\bm{2}} & . & . & . & . & . & . & . \\
. &. & . & . & . & . & . & {\bm{2}}& . & . & . & . & . & . \\
. & . &. & . &. & . & . & . & 2 & . & . &. & . & . \\
 . & . & . & . & . & . & . & . & . & -1 &. & . &\sqrt{3} &. \\
 . & . & . & . & . & . & . & . & . &. & -1 & . & . &-\sqrt{3}\\
 . & . & . & . & . & . & . & . & .& . & . &2 & . & . \\
 . & . & . & . & . & . & . & . & . &-\sqrt{3}& . & . & -1 & . \\
 . & . & . & . & . & . & . & . & . & . &\sqrt{3}& . & . & -1\\
\end{array}
\right),
\end{align}
\begin{align}
&\setlength{\arraycolsep}{3pt}
\renewcommand{\arraystretch}{0.6}
D_{{Z_4}}={1\over 2}\left(\scriptsize
\begin{array}{cccccccccccccc}
 {\bm{-1}} &. & . &. & . & . & . & . & . & . & . & . & . & . \\
. &. & . & . & {\bm{1}}& . & . & . & . & . & . & . & . & .\\
. & . &. & .  & . & {\bm{-1}} & . & . & . & . & . & . & . & . \\
 &. & . & {\bm{-1}}& . & . & . & . & . & . & . & . & . & . \\
. & {\bm{-1}}& . & . &. & . & . & . & . & . & . & . & . & . \\
. &. & {\bm{1}} & . &. & .& . & . & . & . & . & . & . & . \\
. &. & . & . & . & . & {\bm{1}} & . & . & . & . & . & . & . \\
. &. & . & . & . & . & . & {\bm{1}}& . & . & . & . & . & . \\
. & . &. & . &. & . & . & . & 1 & . & . &. & . & . \\
 . & . & . & . & . & . & . & . & . & . &. & . &-1 &. \\
 . & . & . & . & . & . & . & . & . &. & . & . & . &1\\
 . & . & . & . & . & . & . & . & .& . & . &1 & . & . \\
 . & . & . & . & . & . & . & . & . &1& . & . & . & . \\
 . & . & . & . & . & . & . & . & . & . &-1& . & . &.\\
\end{array}
\right).
\end{align}

Now, with the help of the $D$-transformation matrices, we can
construct accidentally symmetric $n$HDM potentials. For example, in
the case of $Z_4$ symmetry, applying the transformation matrix
$D_{{Z_4}}$ on the U(1)$_Y$-conserving elements of $R_3^{A}$ yields
\begin{align}
&R_3^1\to -R_3^1, &R_3^5\to -R_3^2,
\nonumber \\
&R_3^2\to +R_3^5,  &R_3^6\to +R_3^3,
\nonumber \\
&R_3^3\to -R_3^6, &R_3^7\to +R_3^7,
\nonumber \\
&R_3^4\to -R_3^4, &R_3^8\to +R_3^8.
\end{align}
Therefore, all possible combinations $R_3^i R_3^j$ ($i,j=1,\dots, 8$) that
respect the $Z_4$ symmetry may be obtained as
\begin{align}
&(R_3^1)^2, \quad (R_3^4)^2, \quad R_3^1 R_3^4, \quad
(R_3^7)^2, \quad (R_3^8)^2, \quad R_3^7 R_3^8,
\nonumber \\
&R^{1,4}\big[(R^2_3 \pm R_3^3) - (R_3^5 \mp R_3^6)\big], \quad
(R_3^5 \pm R_3^6)^2+(R_3^2 \mp R_3^3)^2\;.
\end{align}
These combinations lead to the following $Z_4$-invariant 3HDM potential:
\begin{eqnarray}
V_{Z_4} =&-& \mu_1^2 \left| \phi_1 \right|^2
- \mu_2^2 \left| \phi_2\right|^2
- \mu_3^2 \left| \phi_3\right|^2
\nonumber \\
&+& \lambda_{11} \left| \phi_1 \right|^4
+ \lambda_{22} \left| \phi_2\right|^4
+ \lambda_{33} \left| \phi_3\right|^4
\nonumber \\
&+& \lambda_{1122} \left| \phi_1\right|^2 \left|\phi_2\right|^2
+ \lambda_{1133}  \left| \phi_1\right|^2 \left|\phi_3\right|^2
+ \lambda_{2233}  \left| \phi_2\right|^2 \left|\phi_3\right|^2
\nonumber \\
&+& \lambda_{1221} |\phi_1^{\dagger} \phi_2 |^2
+ \lambda_{1331} |\phi_1^{\dagger} \phi_3 |^2
+ \lambda_{2332} |\phi_2^{\dagger} \phi_3 |^2
\nonumber \\ &+& {\lambda_{1323} } (\phi_1^{\dagger} \phi_3) (\phi_2^{\dagger} \phi_3)
+ {\lambda_{1323}^* (\phi_3^{\dagger} \phi_1) (\phi_3^{\dagger} \phi_2)} \nonumber \\
&+&
{\lambda_{1212} \over 2} (\phi_1^{\dagger} \phi_2)^2 +
    {\lambda_{1212}^* \over 2} (\phi_2^{\dagger} \phi_1)^2,
\label{z4}
\end{eqnarray}
where the complex phase of $\lambda_{1212}$ can be rotated away by a field redefinition.

In the same fashion, we can use this procedure to construct $n$HDM
potentials invariant under all Abelian discrete
symmetries. Tables~\ref{tab1} and~\ref{tab2} give the parameter
relations of the 2HDM and 3HDM potentials constrained by these
symmetries.

\subsection{Non-Abelian Discrete Symmetries}

Non-Abelian discrete symmetries constitute another class of discrete
symmetries, which may be thought of as combinations of Abelian
discrete symmetries. The most familiar non-Abelian groups can be
summarized as follows~\cite{Frampton:1994rk,Ishimori:2012zz}:

(i) \textit{Permutation group} $S_N$. The best known non-Abelian
discrete groups are the permutation groups. The order of this group is
$N!$. An $S_2$ group is an Abelian symmetry group and consists of a
permutation in the form $(x_1, x_2)\to (x_2, x_1)$. Thus, the lowest
order non-Abelian group is $S_3$.

(ii) \textit{Alternating group} $A_N$. This group consists of only
even permutations of $S_N$ and thus, its order is $N!/2$. The smallest
non-Abelian group of this class is $A_4$ since $A_3 \cong Z_3$.

(iii) \textit{Dihedral group} $D_N$. This group is also denoted as
$\Delta(2N)$ and its order is $2N$. The $D_N$ group is isomorphic to
$Z_N\rtimes Z_2$ that consists of the cyclic rotation $Z_N$ and its
reflections. Note that $D_3 \cong S_3$.

(iv) \textit{Binary Dihedral group} $Q_{2N}$. This group, which is
also called \textit{Quaternion group}, is a double cover of $D_N$
symmetry group and its order is 4$N$.

(v) \textit{Tetrahedral group} $T_N$. This group is of order $3N$ and
isomorphic to $Z_N\rtimes Z_3$, where $N$ is any prime number. The
smallest non-Abelian discrete symmetry of this type is $T_7$. This
would imply that a $T_N$-symmetric $n$HDM potential will also be
symmetric under $Z_7$.

(vi) \textit{Dihedral-like groups}. These generic groups obey the
following isomorphisms:
\begin{align}
&\Sigma(2 N^2)\cong (Z_N \times {Z'}_N) \rtimes Z_2, &\quad
                                                       \Delta(3N^2)
                                                       \cong (Z_N
                                                       \times {Z'}_N)
                                                       \rtimes Z_3,
                                                       \nonumber \\
& \Sigma(3 N^3)\cong (Z_N \times {Z'}_N \times {Z''}_N) \rtimes Z_3,
                                                     &\quad
                                                       \Delta(6N^2)
                                                       \cong
                                                       (Z_N\times
                                                       {Z'}_N) \rtimes
                                                       S_3.
\end{align}
The simplest groups of this type are $\Sigma(2) \cong Z_2$, $\Delta(6)
\cong S_3$, $\Delta(24) \cong S_4$, $\Delta(12) \cong A_4$, and
$\Sigma(24) \cong Z_2 \times \Delta(12)$.

({vii}) \textit{Crystal-like groups} $\Sigma (M \phi)$, with
$\phi=1,2,3$. These groups are of order $M$ and are given by
\begin{align}
&\Sigma (60\phi),\quad \Sigma (168 \phi ),\quad \Sigma (36\phi), \quad \Sigma (72\phi), \quad \Sigma (216\phi), \quad \Sigma (360\phi),
\end{align}
where $\Sigma (60)\cong A_5$ and $\Sigma (216)\cong A_4$.

The decomposition of tensor products of 3-dimensional irreducible representations of these groups are given in Appendix \ref{D}. Note that imposing many of these symmetry groups lead to identical potentials or to potentials that are invariant under continuous symmetries. These non-Abelian discrete symmetries can be the symmetry of $n$HDM potentials for sufficiently large $n$, as discussed in Section~\ref{sec:cs}.

In the case of the 3HDM, the complete list of non-Abelian discrete
symmetries has been reported
in~\cite{Ishimori:2012zz,Ivanov:2012fp,Keus:2013hya,Ivanov:2014doa,deMedeirosVarzielas:2019rrp}. There
are two non-Abelian discrete symmetries as subgroups of SO(3), namely
$D_3$ and $D_4$.  In addition, there are non-Abelian discrete
symmetries as subgroups of SU(3),
\begin{equation}
A_4, \, S_4, \, \left\{ \Sigma(18), \, \Delta(27), \, \Delta(54) \right\}, \, \Sigma(36),
\end{equation}
where the symmetry groups stacked in curly brackets produce identical potentials.
Here, we discuss the cases $D_3$ and $A_4$, while the description of the rest of these types of symmetries for the 3HDM potential may be found in Appendix \ref{D}.

Let us start with the smallest non-Abelian discrete group $D_3 \cong S_3$. The irreducible representations of the $D_3$ symmetry group can be expressed by two singlets, \textbf{1} and \textbf{1$^{\prime}$}, and one doublet \textbf{2}. The $2\otimes 2$ tensor product of this group decomposes as
\begin{eqnarray}
D_3: \quad \textbf{2} \otimes \textbf{2}=\textbf{1} \oplus \textbf{1$^{\prime}$} \oplus \textbf{2}.
\end{eqnarray}
Moreover, the generators of $D_3$ discrete symmetry group should
satisfy the conditions: ${g_1^3=1}$, ${g_2^2=1}$ and
$g_1 \cdot g_2 = g_2 \cdot (g_1 \cdot g_1)$.  Thus, two generators of
$D_3$ in terms of double tensor products are given by
\begin{equation}
\Delta_{D_3}^1=\text{diag}[{g_1} \otimes \sigma^0,{g_1}^* \otimes \sigma^0], \quad
\Delta_{D_3}^2=\sigma^0 \otimes {g_2} \otimes \sigma^0,
\end{equation}
where
\begin{eqnarray} \small
g_1 =
	\begin{pmatrix}
	\omega^2 & 0 & 0 \\
	0 & \omega & 0 \\
	0 & 0 & 1
	\end{pmatrix},
\quad
g_2 =
	\begin{pmatrix}
	0 & -1 & 0 \\
	-1 & 0 & 0 \\
	0 & 0 & 1
	\end{pmatrix}.
\end{eqnarray}
Imposing $D_3$ on the U(1)-conserving part of the $R^{A}_3$ vector will lead to the following linear decomposition: \vspace{-0.2in}
\begin{flalign}
\begin{minipage}{0.3\textwidth}
\begin{eqnarray}
\textbf{1} &:& \;
	\begin{pmatrix}
	R_3^0
	\end{pmatrix},
\nonumber \\
\textbf{1$^{\prime}$} &:& \;
	\begin{pmatrix}
	R_3^7
	\end{pmatrix},
\nonumber \\
	\textbf{1$^{\prime\prime}$} &:& \;
	\begin{pmatrix}
	R_3^8
	\end{pmatrix},
	\nonumber
	\end{eqnarray}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\begin{eqnarray}
\textbf{2} &:& \;
	\begin{pmatrix}
	R_3^1 \\
	R_3^4
	\end{pmatrix}.
\nonumber \\
	\textbf{2$^{\prime}$} &:& \;
	\begin{pmatrix}
	-R_3^2-R_3^3 \\
	R_3^5-R_3^6
	\end{pmatrix},
\nonumber
\end{eqnarray}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\begin{eqnarray}
\textbf{2$^{\prime\prime}$ } &:& \;
	\begin{pmatrix}
	R_3^2+ R_3^6 \\
	R_3^3+ R_3^5
	\end{pmatrix},
\nonumber \\
\textbf{2$^{\prime\prime\prime}$} &:& \;
	\begin{pmatrix}
	R_3^2-R_3^6 \\
	R_3^3-R_3^5
	\end{pmatrix}.
\nonumber
\end{eqnarray}
\end{minipage}
\label{DS3}
\end{flalign}
Note that in the bilinear space, there are three singlets \textbf{1},
\textbf{1$^{\prime}$} and \textbf{1$^{\prime \prime}$}, and two
doublets, \textbf{2} and \textbf{2$^{\prime}$}.  Thus, given the
irreducible representations in~\eqref{DS3}, we may parametrize the
$D_3$-invariant 3HDM potential as follows:
\begin{eqnarray} \small
V_{D_3} = &&
-M_1\,{\bf{ 1}} -M_2\,{\bf {1^{\prime\prime}}}
	 + \Lambda_{0}\,{\bf{1^{\mathsf{2}}}}
    +\Lambda_{1}\,{\bf{{1^{\prime}}^ \mathsf{2}}}
    +\Lambda_{2}\,{\bf{{1^{\prime \prime}}^ \mathsf{2}}}
  \nonumber   \\
   &&
   +\Lambda_{3}\,{\bf{1}}\cdot{ \bf{1^{\prime\prime}}}
    +\Lambda_{4}\, {\bf{{{2^{\mathsf{T}}}}\cdot \bf{2}}}
   +\Lambda_{5}\, {\bf{{{2^{\prime}}^\mathsf{T}}\cdot \bf{2^{\prime}}}}
    \nonumber   \\
   &&
    +\Lambda_{6}\, {\bf{{{2}^\mathsf{T}}\cdot \bf{2^{\prime}}}}
   +\Lambda_{7}\, {\bf{{{2^{\prime \prime}}^\mathsf{T}}\cdot \bf{2^{\prime \prime}}}}
    +\Lambda_{8}\, {\bf{{{2^{\prime \prime\prime}}^\mathsf{T}}\cdot \bf{2^{\prime\prime \prime}}}}.
\end{eqnarray}
This can be rewritten as
\begin{eqnarray} \small
V_{D_3} =
&-& \mu_1^2 \left(|\phi_1|^2+|\phi_2|^2\right)
- \mu_3^2 |\phi_3|^2
\nonumber \\
&+& \lambda_{11} \left( |\phi_1|^4
+ |\phi_2|^4 \right)
+ \lambda_{33} |\phi_3|^4
\nonumber \\
&+& \lambda_{1122} |\phi_1|^2 |\phi_2|^2
+ \lambda_{1133}\left(|\phi_1|^2 |\phi_3|^2 + |\phi_2|^2 |\phi_3|^2 \right)
\nonumber \\
&+& \lambda_{1221} |\phi_1^{\dagger} \phi_2|^2
+ \lambda_{2332} \left( |\phi_2^{\dagger} \phi_3|^2 + |\phi_1^{\dagger} \phi_3|^2 \right)
\nonumber \\
&+& \lambda_{2131} \left( (\phi_2^{\dagger} \phi_1) (\phi_3^{\dagger} \phi_1) - (\phi_1^{\dagger} \phi_2) (\phi_3^{\dagger} \phi_2) \right.
\nonumber \\
&+& \left. (\phi_1^{\dagger} \phi_2) (\phi_1^{\dagger} \phi_3) - (\phi_2^{\dagger} \phi_1) (\phi_2^{\dagger} \phi_3) \right)
\nonumber \\
&+& \lambda_{1323} (\phi_1^{\dagger} \phi_3) (\phi_2^{\dagger} \phi_3) + \lambda^*_{1323} (\phi_3^{\dagger} \phi_1) (\phi_3^{\dagger} \phi_2),
\end{eqnarray}
where $\lambda_{1323}$ is complex while all other couplings are real.

Another example of a non-Abelian discrete symmetry for the 3HDM
potential is $A_4$, which is a subgroup of~SU(3). This symmetry group
consists of three singlets, $\textbf{1}$, $\textbf{1}^{\prime}$ and
$\textbf{1}^{\prime\prime}$, and one triplet $\textbf{3}$. The
$\textbf{3} \otimes \textbf{3}$ tensor product of $A_4$ decomposes:
\begin{eqnarray}
A_4: \quad \textbf{3} \otimes \textbf{3}=\textbf{1} \oplus \textbf{1}^{\prime} \oplus \textbf{1}^{\prime\prime} \oplus \textbf{3} \oplus\textbf{3}^{\prime}.
\end{eqnarray}
The generators of the $A_4$ discrete symmetry group in terms of double tensor products are
\begin{equation}
\Delta_{A_4}^1=\sigma^0 \otimes {g_1} \otimes \sigma^0, \quad
\Delta_{A_4}^2=\sigma^0 \otimes {g_2} \otimes \sigma^0,
\end{equation}
where
\begin{eqnarray} \small
g_1 =
	\begin{pmatrix}
	0 & 1 & 0 \\
	0 & 0 & 1\\
	1& 0& 0
	\end{pmatrix}\,,
\quad
g_2 =
	\begin{pmatrix}
	1 & 0 & 0 \\
	0 & -1 & 0 \\
	0 & 0 & -1
	\end{pmatrix},
\end{eqnarray}
satisfy the conditions: $g_1^3=g_2^2=(g_1 \cdot g_2)^3=1$.
The $A_4$-symmetric blocks in the bilinear-space can be represented as \vspace{-0.2in}
\begin{flalign}
\begin{minipage}{0.40\textwidth}
\begin{eqnarray} \small
\textbf{1} &:& \;
	\begin{pmatrix}
	\phi_1^{\dagger} \phi_1 +\phi_2^{\dagger} \phi_2+\phi_3^{\dagger} \phi_3
	\end{pmatrix},
\nonumber \\
 \textbf{1}^{\prime} &:& \;
	\begin{pmatrix}
	\phi_1^{\dagger} \phi_1 + \omega^2 \phi_2^{\dagger} \phi_2 + \omega \phi_3^{\dagger} \phi_3
	\end{pmatrix},
\nonumber \\
\textbf{1}^{\prime\prime} &:& \;
	\begin{pmatrix}
	\phi_1^{\dagger} \phi_1 + \omega \phi_2^{\dagger} \phi_2 + \omega^2 \phi_3^{\dagger} \phi_3
	\end{pmatrix},
\nonumber
\end{eqnarray}
\end{minipage}
\begin{minipage}{0.40\textwidth}
\begin{eqnarray}\small
\textbf{3}: \;
	\begin{pmatrix}
	R^1_3 \\
	R^2_3 \\
	R^3_3 \\
	\end{pmatrix},
\quad
\textbf{3}^{\prime}: \;
	\begin{pmatrix}
	R^4_3 \\
	-R^5_3 \\
	R^6_3 \\
	\end{pmatrix}.
	\nonumber
\end{eqnarray}
\end{minipage}
\end{flalign}
Thus, an $A_4$-invariant 3HDM potential may be written as
\begin{eqnarray}
V_{A_4} =&& -M\,{\bf{1}}
	  +\Lambda_{0}\,{\bf{1^{\mathsf{2}}}}
    +\Lambda_{1}\,{\bf{{1^{\prime}}^\ast}\cdot \bf{1^{\prime\prime}}}
	  +\Lambda_{2}\,{\bf{3^{\mathsf{T}}}\cdot \bf{3}}
	  \nonumber\\
  &&  + \Lambda_{3}\,{\bf{3^{\prime \mathsf{T}}}\cdot \bf{3^{\prime}}}
    + \Lambda_{4}\,{\bf{3^{\prime \mathsf{T}}}\cdot \bf{3}}.
\end{eqnarray}
Equivalently, the  $A_4$-symmetric potential can be rewritten as follows:
\begin{align}
V_{A_4} = &
- \mu_1^2 \left(|\phi_1|^2
+|\phi_2|^2
+ |\phi_3|^2\right)
+\lambda_{11} \left( |\phi_1|^4
+ |\phi_2|^4
+ |\phi_3|^4 \right)
 	\nonumber \\ &
+ \lambda_{1122} \left(|\phi_1|^2 |\phi_2|^2
+ |\phi_1|^2 |\phi_3|^2
 +|\phi_2|^2 |\phi_3|^2 \right)
 	\nonumber \\ &
+ \lambda_{1221} \left( |\phi_1^{\dagger} \phi_2|^2
+ |\phi_1^{\dagger} \phi_3|^2
+ |\phi_2^{\dagger} \phi_3|^2\right)
	\nonumber \\ &
+ {\lambda_{1212} \over 2} \left((\phi_1^{\dagger} \phi_2)^2 +
(\phi_1^{\dagger} \phi_3)^2 +
(\phi_2^{\dagger} \phi_3)^2 \right)
	\nonumber \\ &
+ {\lambda_{1212}^* \over 2} \left( (\phi_2^{\dagger} \phi_1)^2 +
(\phi_3^{\dagger} \phi_1)^2 +
(\phi_3^{\dagger} \phi_2)^2 \right).
\label{pa4}
\end{align}

In a similar way, the remaining 3HDM potentials that are invariant under non-Abelian discrete symmetries may be obtained. These are presented in Appendix~\ref{D}.

In Tables~\ref{tab1} and~\ref{tab2}, we present all
SU(2)$_L$-preserving accidental symmetries for the 2HDM and the 3HDM
potentials.  The 2HDM potential has a total number of 13 accidental
symmetries~\cite{Battye:2011jj}, of which 6 preserve
U(1)$_Y$~\cite{Ivanov:2007de,Ferreira:2009wh,Ferreira:2010yh} and 7
are custodially symmetric~\cite{Pilaftsis:2011ed}. Given the isomorphism
of the Lie algebras: SO(5) $\sim$ Sp(4), the maximal symmetry group of
the 2HDM in the original $\bm{\Phi}$-field space is
$G^{\bm{\Phi}}_{\text{2HDM}} = [\text{Sp(4)}/\text{Z}_2] \otimes
\text{SU(2)}_L$~\cite{Pilaftsis:2011ed}.

For the case of the 3HDM potential, we find that there exists a total
number of $40$ SU(2)$_L$-preserving accidental symmetries, of which
$18$ preserve U(1)$_Y$ and $22$ are custodially symmetric.  The
maximal symmetry group of the 3HDM potential in the original
$\bm{\Phi}$-field space is
$G^{\bm{\Phi}}_{\text{3HDM}} = [\text{Sp(6)}/\text{Z}_2] \otimes
\text{SU(2)}_L$~\cite{Pilaftsis:2011ed}.
Note that the $40$ accidental symmetries are subgroups of Sp(6).

\section{Conclusions} \label{sec:con}

The $n$HDM potentials may realize a large number of SU(2)$_L$-preserving
accidental symmetries as subgroups of the symplectic group
Sp(2$n)$. We have shown that there are {\em{two}} sets of symmetries:
(i) continuous symmetries and (ii) discrete symmetries (Abelian and
non-Abelian symmetry groups). For the continuous symmetries, we have
offered an algorithmic method that provides the full list of proper,
improper and semi-simple subgroups for any given integer $n$. We have
also included all known discrete symmetries in $n$HDM potentials.

Having defined the bi-adjoint representation of the Sp($2n$)
symmetry group, we introduced prime invariants and irreducible
representations in the bilinear field space to construct the scalar
sector of $n$HDM potentials. These quantities have been systematically
used to construct accidentally symmetric $n$HDM potentials by
employing fundamental building blocks that respect the symmetries.

Using the method presented in this paper, we have been able to
classify all symmetries and the relations among the theoretical
parameters of the scalar potential for the following: (i) the 2HDM and (ii) the
3HDM. For the 2HDM potential, we recover the maximum number of $13$
accidental symmetries. For the 3HDM potential, we derive {\em{
    for the first time}} the complete list of $40$ accidental
symmetries.

Our approach can be systematically applied to $n$HDM potentials, with
$n>3$, once all possible discrete symmetries have been identified.

\section{Acknowledgements}
\noindent
The work of A.P. and N.D. is supported in part by the
Lancaster-Manchester-Sheffield Consortium\- for Fundamental Physics,
under STFC research grant ST/P000800/1.
\\The work of N.D. is also supported in part by the Polish National Science Centre
HARMONIA grant under contract UMO- 2015/18/M/ST2/00518 (2016-2020).

redpajama_set_name: RedPajamaArXiv


Participants are supposed to take care of their own accommodation. We recommend the following hotels for your accommodation during the conference. However, if none of them is your choice, please find other hotels near the conference venue on Booking or Agoda.


Apparently, the remake rights of the film were obtained by T-Series in Bollywood and the production house sent a notice to the makers of Agnyaathavaasi regarding the film ahead of the release.
Pawan Kalyan's 'Agnyaathavasi' released yesterday, 10th January, 2018 worldwide amidst much hype however, started receiving poor reviews after the first few shows. Many critics were disappointed with the film's slow pace and screen-writing, which eventually led to disappointment after there huge expectations on the combination of Trivikram Srinivas and Pawan Kalyan. While Pawan Kalyan received a thumbsup for his performance in the film, many were surprised and disappointed with the writing of Trivikram Srinivas in Agnyaathavaasi who is widely regarded for his witty writing skills. Days prior to the release, Agnyaathavaasi found itself amidst plagiarism controversy and it was being said that the film was a remake of French film 'Largo Winch' which is directed by Jérôme Salle. Apparently, the remake rights of the film were obtained by T-Series in Bollywood and the production house sent a notice to the makers of Agnyaathavaasi regarding the film ahead of the release. However, the issue was amicably settled after the makers of Agnyaathavaasi agreed to pay an agreed amount as 'settlement'.
Largo Winch director Jérôme Salle, on learning about this, had tweeted saying that he will watch the movie to know more about the plagiarism issue. And on the day of the release, Jérôme Salle watched the film and later tweeted saying just a settlement with T-Series will not be enough regarding the plagiarism issue since Agnyaathavaasi is released worldwide. This may not be good news for the makers of Agnyaathavaasi who are already in a spot of bother after the film failed to impress a major segment of the audience.
With such reaction coming from Jérôme Salle, this will add to the woes of the makers of Agnyaathavaasi who already are worried about the performance of the film in cinema halls.


Chicken Leg Socks--Inspired by Baba Yaga?
Today has become an unofficial Baba Yaga day! After sharing the Lego Baba Yaga in today's earlier post, I wanted to share these funny socks that immediately inspired me to think of Halloween costumes and Baba Yaga possibilities throughout the year.
Not much to say about them, but they made me smile and imagine so I thought I'd share. And if you are by chance unfamiliar with Baba Yaga, go read some of her tales on SurLaLune.
Sorry, the image loader isn't working. I've added my own image captures just in case. They can be acquired in sets or just one color, see them all on the Amazon page where I found them.
I was not expecting to see a Baba Yaga cottage and such a great one at that! Again, the Russian fairy tale influence is strong right now. Love it! Some impressive engineering to get that house to sit on those chicken legs, too.
There are some other great builds in the gallery of entries. Be sure not to miss them. I don't envy the judges!
Vassa in the Night by Sarah Porter was released in late 2016. The novel is a retelling of the Russian fairy tale "Vasilisa the Beautiful," which is itself a Russian Cinderella story. You can read a version of the tale here. The book follows the strong recent trend of Russian fairy tale novels that's been enduring for several years now actually. The Russian versions of popular tales tend to be darker and atmospheric so I can understand their appeal for modern audiences.
Peau d'âne by Hélène Druvert is one of my recent discoveries, published in France in 2015. The book is available at Amazon.co.uk and Amazon.fr but not in Amazon.com.
I have a penchant for Peau d'âne picture books which pretty much only come in French since the story is beloved in France but avoided by publishers in other countries due to the possible controversies of the tale of which there are a few. (Not familiar with Peau d'âne? Read Donkeyskin at the SurLaLune primary site.) Which always makes the books even more appealing to me because they are rare to find. This is now one of my favorite renditions. This version also uses minimal text to tell the story, letting the reader take it at face value.
Druvert has a beautiful illustration style--using silhouettes to tell her stories. Every other page in her pictures books--for she has illustrated a few titles--are cut out pages, intricately die cut and designed to overlay the pages between them as you turn the pages. The pages are sturdy on heavy stock paper but still fragile since they are die cut. The results are beautiful.
So in the images below--which you can click to see larger--the left side of the image is the back side of the die cut page.
Other books by Druvert include, with the first two linking to Amazon.com and the second linking to Amazon.co.uk. The Mary Poppins and Paris titles also have the die cut pages. I do not know for sure about the other two.
Rose & Thorn by Sarah Prineas was released in the US in October. It is a Sleeping Beauty retelling and is also the second in a series, following last year's Ash & Bramble, which is a Cinderella retelling.
This beauty isn't sleeping! Discover the true story of Sleeping Beauty in Sarah Prineas's bold YA fairy-tale retelling filled with thrilling adventure and romance, perfect for fans of the Lunar Chronicles and the Girl of Fire and Thorns trilogy.
After the spell protecting her is destroyed, Rose seeks safety in the world outside the valley she had called home. She's been kept hidden all her life to delay the three curses she was born with—curses that will put her into her own fairy tale and a century-long slumber. Accompanied by Griff, the handsome and mysterious Watcher, and Quirk, his witty and warmhearted partner, Rose tries to escape from the ties that bind her to her story. But will the path they take lead them to freedom, or will it bring them straight into the fairy tale they are trying to avoid?
Set in the world of Sarah Prineas's Ash & Bramble fifty years later, Rose & Thorn is a powerful retelling of the classic "Sleeping Beauty" tale where the characters fight to find their own happy ever after.
The Star-Touched Queen by Roshani Chokshi is one of those books that has been on my radar for a while for my TBR pile and climbed higher on the list when I attended her panel at the Southern Festival of Books last October. I thought I had already posted about the book, but apparently I haven't. So lucky you if this sparks your interest because the ebook is also on sale for a short time for $2.99, down from the $10ish it was previously.
Another bonus is that this book incorporates India (as in the country, not Native American) folklore and mythology. Chokshi loves rich language--and it was revealed she is a fairy tale and SurLaLune reader before she knew I was in the room--so I have many reasons to recommend her here, too! I love this current trend of Indian folklore inspired fiction with books like this and Renée Ahdieh's The Wrath and the Dawn to name a few. I read a lot of Indian folklore as I research so it has a filing cabinet all its own in my brain that I love to see explored through other sources.


text:
International Business Means International Trade Essay
In my opinion, I think that international business is better referred to as international trade. This is because international trade involves all the people around the world. International trade is means by exchange of goods and services between countries. Type of trading also raises matters of price, supply and demand, also affects and is affected by global events that occur. Political changes taking place in Asia can cause an increase in labor costs. Thus, it increases the manufacturing costs for the American sneaker company based in Malaysia, which in turn will cause an increase in the price paid to buy the shoes at a local shopping center.
Markets can also be extended to international ...view middle of the document...
If a country is unable to produce a product that they want, they can get it from other countries using the trading process. This is known as a specialization in international trade.
Besides improving efficiency, it also makes the countries to take part in a global full fill, except that it also encourages foreign investors to invest money in companies and other assets. Economy had to grow more rapidly and easier, and faster to become a participant in a competitive economy. On the government side, FDI is a way in which currencies and foreign expertise can enter the country. This will increase the level of employment and his theory, lead to a growth in gross national product. On behalf of the investors, FDI offers company expansion and growth, and therefore get a higher yield.
As with other theories, there are opposing views. International trade has two contrasting views regarding the level of control placed on trade: free trade and protectionism. Free trade is the simpler of the two theories: a laissez-faire approach, with no restrictions on trade. The main idea is that supply and demand factors, operating on a global scale, will ensure that...
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1004 words - 5 pages . (2005). Free trade versus strategic trade as a choice between two 'second best' policies: A symmetric versus asymmetric information analysis. International Economic Journal, 19(3), 417-446. doi:10.1080/10168730500199640 Manu, F. A. (2009). Import substitution and export promotion: A continuing dilemma for developing countries. Journal Of International Business & Economics, 9(1), 100-104. Matschke, X., & Schöttner, A. (2013). Antidumping as
882 words - 4 pages energy efficient products are being made and shared with each other because of the trade implications. This will actually mean less depletion of natural resources in the longer run. This also brings more uniform standards for tackling environmental issues. The only problem is that the positives of international trade keep up with long term negative implications of growth. Income Inequality: With increased use of international business tools such
International Trade Essay 1582 Words
1582 words - 7 pages As Ron Wyden states, "Trade wars are not started by countries appealing to respected, independent trade authorities. Rather, trade wars begin when one country decides to violate international trade rules to undercut another country's industries." International trade comes from when one country is needy and another country can provide something another country needs. Global Trade, during the recession, fell 20 percent in 2009. In a study
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FreeTvOnline: Breaking U.S News and Latest World Events
Iran approved the law on intensifying nuclear activities
December 2, 2020 Jane Lester
Iranian authorities adopted a law on intensifying nuclear activities. This was reported by the Iranian agency Fars.
The corresponding decree of the parliament, adopted the day before, was approved by the Constitutional Guardian Council. The document will be handed over to the government for execution.
According to the law, the Atomic Energy Organization of Iran must begin producing at least 120 kg of 20% of enriched uranium annually at the Fordow nuclear site two months after the approval of this bill. Uranium will be stored domestically.
Including enrichment and enriched uranium production capacities will be increased to at least 500 kg per month. Iranian legislators will examine the details of the bill after approval of its general provisions.
Earlier that day, Democrat Joe Biden, who announced his victory in the U.S. presidential election, announced his intention to work to return Washington to the Joint Comprehensive Action Plan on Iran's nuclear program. In May 2018, the USA unilaterally withdrew from the treaty.
On November 27, Iranian nuclear physicist Mohsen Fahrizade was killed in Iran. The attack was linked to another attempt by the administration of U.S. President Donald Trump and the government of Israeli Prime Minister Benjamin Netnyahu to increase pressure on Iran and make it difficult to negotiate the return of Tehran in the nuclear deal.
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France issued 110,000 fines for curfew violations
Polish President spoke in favor of sanctions against Russia and Gazprom
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Chebotko, a cross-country skiing medalist, died in a car accident in Kazakhstan
Pentagon buys data on people's movements without a court order
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Russian man pleads guilty to cybercrime in the U.S.
CNN star Larry King died
Psaki could not clarify Biden's position on Uighurs in China
TikTok was partially blocked in Italy after the death of a child
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Early Access Papers
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Spatial variation in the importance of different prey types in the diet of red foxes
Cameron A. Graham, Martine Maron, Clive A. McAlpine
Journal: Australian Zoologist
Australian Zoologist (2017) 38 (4): 610–628.
DOI: https://doi.org/10.7882/AZ.2017.025
... A. Graham1,3, Martine Maron1, Clive A. McAlpine1,2 1. The University of Queensland, Landscape Ecology and Conservation Group, School of Geography, Planning and Environmental Management, Brisbane, Australia. 4068 2. The University of Queensland, The Ecology Centre, Brisbane, Australia. 4068 3. Corresponding...
Modelling the potential range of the koala at the Last Glacial Maximum: future conservation implications
Christine Adams-Hosking, Patrick Moss, Jonathan Rhodes, Hedley Grantham, Clive McAlpine
... of Queensland, Landscape Ecology and Conservation Group, School of Geography, Planning, and Environmental Management, Brisbane, Qld 4072 Australia. b The University of Queensland, The Ecology Centre, Brisbane, Qld 4072 Australia. Correspondence: Christine Adams-Hosking, The University of Queensland, School...
Long-term ecological research in Australia: innovative approaches for future benefits
Jean-Marc Hero, J. Castley, Mikalah Malone, Ben Lawson, William Magnusson
... (terrestrial and aquatic) that are systematically spaced in grids (e.g. 5 km × 5 km) and modules (e.g. 5 km × 1 km) within a hierarchical long-term ecological research (LTER) network. Modules and grids sample biodiversity and biophysical variation in an unbiased manner across the landscape. Infrastructure...
Managing biodiversity in arid Australia: a landscape view
By J. A. Kerle, M. R. Fleming, J. N. Foulkes
Book: Animals of Arid Australia: Out on their own?
Series: Other RZS NSW Publications
Publisher: Royal Zoological Society of New South Wales
DOI: 10.7882/FS.2007.040
EISBN: 978-0-9803272-0-5
... The Australian arid zone is a large complex area. In order to maintain its biodiversity, it is essential that landscape-wide and long-term analyses form the backdrop to the development of management strategies. From the current knowledge of the biodiversity and ecology of the arid zone...
Endangered ecological communities and landscape conservation in NSW: successes and failures in the Sydney Basin
By Tony D. Auld, Mark Tozer
Book: Threatened species legislation: Is it just an Act?
EISBN: 0-9586085-9-8
... We examine the nature and extent of listings of endangered ecological communities (EECs) under the NSW Threatened Species Conservation Act 1995 ( TSC Act ), including the role of endangered ecological communities in conservation from the landscape to species level. We ask how well...
Landscape legacies: Koala habitat change in Noosa Shire, South-east Queensland
L.M. Seabrook, C. A. McAlpine, S. R. Phinn, J. Callaghan, D. Mitchell
...L.M. Seabrook; C. A. McAlpine; S. R. Phinn; J. Callaghan; D. Mitchell Present day Australian landscapes are legacies of our colonial history, while future landscapes will be legacies of ecological processes and human impacts occurring today. This paper investigates the legacies of European...
Developing a dynamic landscape approach to the study of arid-zone wildlife: challenges and opportunities
By Clive McAlpine, Stuart Phinn, Tony Pople, Norbert Menke, Bronwyn Price
.... In this chapter, we focus on three key challenges in developing a dynamic landscape approach to the study of arid-zone wildlife: (i) conceptualising landscape structure and dynamics; (ii) dealing with the problem of ecological scale and spatial autocorrelation; and (iii) the challenge of seasonal variability. We...
Positive Dingo Management: how not to throw out the baby with the bath water
By Paul D. Meek, Jim Shields
Book: A Symposium on the Dingo
... of wild dogs while simultaneously protecting important dingo genetic stock and providing for the continued ecological role of dingoes across the landscape. Innovative and strategic approaches are needed to create management outcomes with regards to pest management. State Forests of NSW has developed...
Fauna of the grassland-forest landscape mosaics of the Bunya Mountains, eastern Australia
Geoffrey Smith, Roderick Fensham, Daniel Ferguson, Luke Hogan, Michael Mathieson
...): 467-477. http://dx.doi.org/10.2307/1565965 Fairfax, R.J., Fensham, R.J, Butler, D.W., Quinn, K., Sigley, B. and Holman, J. 2009. Effects of multiple fires on tree invasion in montane grasslands. Landscape Ecology 24(10): 1363-1373. http://dx.doi.org/10.1007/s10980-009-9388-y Fensham, R.J...
Fossil features of mulga Acacia aneura landscapes: possible imprinting by extinct Pleistocene fauna
James Noble
.... Spatial organisation of landscapes and its function in semi-arid woodlands, Australia Landscape Ecol. 10 51 63 Ludwig, J., Tongway, D., Freudenberger, D., Noble, J. and Hodgkinson, K. (eds), 1997. Landscape Ecology, Function and Management. Principles from Australia's Rangelands, CSIRO...
Ecology of Honeyeaters (Meliphagidae) in Western Australian Eucalypt Woodlands I: Resource Allocation Among Species in the Great Western Woodland During Spring
Harry F. Recher, Mike C. Calver, William E. Davis, Jr.
... the habitats they occupy thereby affecting the composition of avian communities and the distribution of species on a landscape scale. Such behaviour has cascading effects on ecosystems, changing the abundances and kinds of plant- eating arthropods. A comprehensive knowledge of the ecology of honeyeaters...
Application and efficiency of beeswax casting and digital photogrammetry to study the morphology of a Varanus sp. foraging excavation
Tamra Chapman
...-arid Australian woodland The Rangeland Journal 30 327 337 Eldridge, D.J. and James, A.I. 2009. Soil-disturbance by native animals plays a critical role in maintaining healthy Australian landscapes. Ecological Management and Restoration 10: S27-S34. Soil-disturbance by native animals plays...
A history of a contested ideal: national parks for fauna conservation
Daniel Lunney
... deepened with insights from the fields of landscape ecology and conservation biology in the latter part of the 20th century. Butler reflects that there are now far fewer places on Earth which could be described as intact or wild and where wildlife is secure from the pressures of humanity. His North...
How do we sleep while our beds are burning? Impacts of the 2019–20 mega-fires on a rainforest dependent species – the Golden-tipped Bat Phoniscus papuensis
G. Madani, C. Turbill, A. Lloyd, L. Gonsalves, T. Brassil ...
... can also run the risk of creating ecological traps for target animals if they increase predation risk or draw animals into areas with insufficient food resources (Cowan et al. 2021). However, artificial roosts have also been demonstrated to improve fitness, enhance persistence in the landscape...
Interacting impacts of drought and fire on bird populations—insights from a long-term study in the Warrumbungles
Helen C Stevens, David M Watson
...Helen C Stevens; David M Watson ABSTRACT The interacting effects of drought and fire on ecological communities are poorly understood. Long-term studies in the Warrumbungle Mountains, central-west New South Wales, subject to drought and fire during the past 21 years, enabled their separate...
Looking to the future: what next for the dingo?
Thomas Newsome
... experiments in both conservation and managed agricultural landscapes. Without such efforts, I see a future for the dingo that continues to be steeped in controversy and debate. © 2021 Royal Zoological Society of New South Wales 2021 dingo predator ecology Australia ABSTRACT Looking to the future...
Megafire: the Darwinian guillotine?
Dale G. Nimmo, Chris J. Jolly, Alexandra J. R. Carthey
...Dale G. Nimmo; Chris J. Jolly; Alexandra J. R. Carthey ABSTRACT Southern Australia's 2019–20 wildfire season was unprecedented, but the ecological toll remains poorly understood. Estimates of three billion animals being affected by the fires attracted global attention, but how many of those animals...
Non-breeding habitat requirements of the giant burrowing frog, Heleioporus australiacus (Anura: Myobatrachidae) in south-eastern Australia
Trent Penman, Francis Lemckert, Chris Slade, Michael Mahony
... Du, L. and Burel, F. 2003. Response of the small mammal community to changes in western French agricultural landscapes. Landscape Ecology 18: 265-278. Response of the small mammal community to changes in western French agricultural landscapes Landscape Ecology 18 265 278 Gillespie, G.R...
Biodiversity offsets: Lessons from the American experience
Shelley Burgin
...., Crane M., Michael D. and MacGregor C. 2007. Reptile and arboreal marsupial response to replanted vegetation in agricultural landscapes. Ecological Applications 17, 609-619. Reptile and arboreal marsupial response to replanted vegetation in agricultural landscapes Ecological Applications 17 609 619...
Bassian bats in the Eyrean sub-region? Preliminary acoustic data from the Bogan and Macquarie Rivers in semi-arid NSW
Vaughan Monamy, Jennifer Taylor, Leroy Gonsalves, Murray Ellis
.... 2008.The Mammals of Australia. New Holland Publishers, Sydney. The Mammals of Australia Verboom, B. and Huitema, H. 1997. The importance of linear landscape elements for the pipistrelle Pipistrellus pipistrellus and the serotine bat Eptesicus serotinus. Landscape Ecology 12: 117-125. http...
Royal Zoological Society of New South Wales


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Terralynn Tiffer scored 10 points and Hannah Dudle, Sara Anderson and Andrea Clutska all added eight points for Carson in the first round of the State Farm Holiday Classic. Dudley and Clutska each had seven rebounds, Kendra Krupp had six boards and Katie McFarren added three assists. Carson will play at 10:45 a.m. today at Damonte Ranch.
Elizabeth Cagle had 10 rebounds and four assists, Lindsay Burroughs and Kaitlin Sinclair each had five rebounds and Breann Perry added six assists for Carson in the State Farm Classic. Carson's freshmen will also play at 10:45 a.m. today at Damonte.


In Amandola v Roman Catholic Diocese of Rockville Ctr., 2015 NY Slip Op 06099 [130 AD3d 761], the New York Supreme Court, Appellate Division, Second Department denied a defendant's motion for summary judgment to dismiss a cause of action for failure to provide adequate supervision in its school.
A plaintiff brings suit against his school for failure to provide adequate supervision after being injured by his classmates on multiple occasions. The plaintiff commenced this action against his school because his school failed to intervene after students repeatedly assaulted the plaintiff on school grounds and in the classroom.
The defendants moved for summary judgment to dismiss the cause of action alleging negligent supervision of students and negligent hiring and retention of employees. The defendants argued that they did not have notice of the conduct on the part of the offending students in order to be held liable for the plaintiff's injury.
The Supreme Court denied the defendants' motion to dismiss the cause of action alleging negligent supervision of students and negligent hiring and retention of the teacher because the defendants did not adequately prove that they lacked notice and knowledge of the prior instances of assault.
The plaintiff in this case sustained injuries in his former school when a fellow student assaulted him on multiple occasions. The plaintiff further alleged that he was repeatedly kicked by a group of students in a classroom. The plaintiff claimed that the school failed to adequately supervise the students who allegedly injured the plaintiff and failed to intervene in a timely and appropriate manner.
Under established case law, courts have held that schools have a duty to adequately supervise students and will be held liable for foreseeable injuries that are proximately related to the absence of the supervision. Schools are held to a high standard to oversee potential dangerous conduct engaged by students and any foreseeable injuries that result from the dangerous conduct.
In order for a defendant to successfully win on a motion for summary judgment to dismiss a cause of action in a case like this, the defendant must prove that there are no triable issues of fact to merit further litigation on the cause of action. Here, the defendants argued that they lacked sufficient knowledge or notice of the dangerous conduct that caused plaintiff's injuries. However, the Court held that the defendant's motion failed to eliminate all triable issues of fact that the defendants did not have sufficient knowledge or notice of the dangerous conduct. The Court found that the defendant's moving papers also failed to eliminate all triable issues of fact as to whether they had knowledge of the student's dangerous tendency to injure the plaintiff based on past altercations with the plaintiff. Finally, the Court held that defendants failed prove there were no questions of fact as to whether the teacher failed to take steps to intervene to prevent the plaintiff's injuries at the very hands of his classmates. As such, the Appellate Court upheld the Supreme Court's denial of defendants' motion for summary judgment.
If you or a loved one has been a victim of negligent supervision, please contact the experienced personal injury lawyers at Gallivan & Gallivan for a free consultation today.


Last Thursday, we logged two years of homeschooling in the books. That means we are officially on summer break! But that doesn't mean we'll abandon all learning for the next two months.
The kids will be doing a couple of pages in activity books right after breakfast to practice the skills they've learned in First Grade and Pre-K. They'll also be reading for at least 20 minutes a day for our library's summer reading program. The science experiments we didn't get to during the year will be done once a week until they're completed also.
As promised, here is a look back at some of the homeschool highlights from our last quarter. We had so much fun exploring space, traveling back in time to Ancient Rome, seeing how donuts are made, and connecting letters of the alphabet with real life objects!
A lot of learning, crafts, and projects were packed into the last 9 weeks. While we used workbooks along with regular books too, whenever there was an opportunity to do something hands-on I grabbed it! After all, that's how my kids learn best.
Reading time at the library! Our local library has been a wonderful resource for us, providing educational videos, stacks of readers & picture books, and a great place to just read.
Playing the xylophone while learning about the letter "X".
Making their own maps at a nearby park for Pioneer Day. My seven-year-old was able to take what he had learned about geography and draw a pretty accurate map of the world!
Matching letter sounds to different toy animals. I think she'll be ready to start blending those sounds together next year when she tries out some simple phonics readers.
Showing off their donut creations during our field trip to Krispy Kreme! Combining learning and eating was a win-win. 🙂 In spite of a sugar high when we got back home, they drew pictures of their favorite parts of the trip and talked about what they learned.
Taking advantage of the warmer weather by relocating school to the backyard. One of my favorite parts of homeschooling is that we don't actually have to stay inside our home to do it!
So that's our last quarter of this homeschool year in a nutshell. While it wasn't always easy or fun, I am so thankful for the ability to educate my kids at home. They certainly learned a lot, but the most important thing was all of the memories that we made together!
Great recap! You were able to give you raids a lot of unique experiences so that they could have fun while learning! I love it!
My favorite part of homeschooling is that we can do it anywhere as well. Lets me be creative about our learning. And vacation often doubles as school! We spent a day in an aquarium last week, we listened to three talks about the animals, watched them eat and saw the octopus get a tentacle massage. It was very cool. She had to read the signs and follow a "dive log" for our day, learn different facts and earn stamps. She learned a lot about the sea and conservation=school day for kindergarten. We have technically one more day of kindergarten left, but I may stretch it some so that I can make sure she truly met all the goals for the year.
Homeschooling post are so inspirational. I really love the matching the letter to the animal activity! Thank you for sharing this!
Great job completing your second you're homeschooling! It's my second year too! I'm definitely curious about the ancient Rome lapbook and I'm going to check out that post now. We are doing similar things over the summer although I'm trying out all year homeschooling to see how that works this year.
I love reading your posts. Looking forward to see what you come up with this summer!
Wow…what a fun filled learning quarter you had! I wish I can do even half of these things 🙂 Thanks for sharing and for the motivation.


Shamrock Shufflers could be in for a cool and soggy run Sunday.
Clouds will move in to the Chicago area early Sunday morning after a sunny Saturday. There will be a chance for rain when the race kicks off at 8:30 a.m., and the possibility for showers will increase throughout the morning and early afternoon.
The temperature for race day is expected to be below the average, although it will be warmer than Saturday. The high for Sunday is 45 degrees, which is about 7 degrees below the average for this time of year.
Saturday's temperatures will be cooler, with a high of only 36 degrees, despite the sunshine.
The March cold snap is expected to break early next week. By Monday, highs will be in the 50s and low 60s again with a chance for rain on Monday, Tuesday and Thursday.
In the past, the weather for the Shamrock Shuffle varied from the frigid and snowy to the hot and humid. In 2009, runners finished the 8K course in the snow and slush. Just two years later, they suffered through temperatures in the mid-80s.