# Symmetries of stationary points of the $G$-invariant potential and
the framework of the auxiliary group

R. Krishnan<EMAIL_ADDRESS>
https://orcid.org/0000-0002-0707-3267 Saha Institute of Nuclear Physics, 1/AF
Bidhannagar, Kolkata 700064, India

###### Abstract

We classify the constraints on a stationary point of the potential invariant
under a finite group into intrinsic and extrinsic based on whether they are
independent of the coefficients in the potential or not. We find that the
symmetry group of a set of stationary points can be larger than that of the
potential and the stabilizer under this group generates intrinsic constraints.
By applying these findings in the framework of the auxiliary group, we show
that the constraints that can only be obtained extrinsically in an elementary
theory can be generated intrinsically in an effective theory111A video
presentation of this paper is available here..

I. Introduction

Discrete symmetries implemented using finite groups have been used extensively
King (2015); Petcov (2018) in particle physics to study flavor. Scalar fields
called flavons, which transform as multiplets under a finite group, are often
introduced. Their potential is constructed, and one of the stationary points
(SPs) of the potential is chosen as their vacuum expectation value (VEV),
leading to spontaneous symmetry breaking (SSB). The VEV plays a central role
in determining the structure of the fermion mass matrices. SSB of finite
groups appears in the context of dark matter fields Hirsch  _et al._ (2010);
Haba  _et al._ (2011) and inflatons Cohn and Stewart (2000); Carone  _et al._
(2014); Schimmrigk (2015); Chigusa and Nakayama (2019) in cosmology and
various fields Gaeta (2006); Michel and Zhilinskii (2001) in solid-state
physics also. Therefore, understanding the symmetries of SPs of the potential
is crucial in a wide variety of settings.

II. Constraints on SPs

Consider a field $\phi$ transforming as a multiplet of a $d$-dimensional
orthogonal real representation of a finite group $G_{\\!f}$. The invariants of
the group action constructed using $\phi$ constitute the potential. Let the
number of linearly independent invariants up to order $p$ be $N$. The most
general potential of order $p$ is given by

$\mathcal{V}^{p}=c_{\alpha}\mathcal{I}_{\alpha},$ (1)

where $\mathcal{I}_{\alpha}$ are the invariants and $c_{\alpha}$ are the
corresponding real coefficients. Invariants of mass dimension up to four are
renormalizable. Since higher-order invariants are suppressed, we have $p\geq
4$. Let $G_{\mathclap{f}}^{p}$ be the largest group under which $\phi$
transforms as an orthogonal real representation such that the potential
$\mathcal{V}^{p}$ remains invariant. In general, we have
$G_{\mathclap{f}}^{p}\geq G_{\\!f}$. In our analysis, we assume that $p$ is
sufficiently high so that $G_{\mathclap{f}}^{p}$ is a finite group.

The set of coefficients $c=(c_{1},..,c_{N})$ and the field
$\phi=(\phi_{1},..,\phi_{d})$ form the topological spaces $\mathbb{R}^{N}$ and
$\mathbb{R}^{d}$, respectively. Let $\mathcal{C}$ be an open subset of
$\mathbb{R}^{N}$ with $c\in\mathcal{C}$ such that we can define a map

$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}:\mathcal{C}\rightarrow\mathbb{R}^{d}:\quad
c_{\alpha}\,\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}=0,$
(2)

where $\partial_{i}$ is the derivative with respect to the component
$\phi_{i}$. Eq.(2) implies that
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ is a SP of
$\mathcal{V}^{p}$ constructed with the coefficients $c$. Moreover, we require
the SP to be nondegenerate Bott (1982), i.e. the Hessian matrix $\quad
c_{\alpha}\,\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}$
is assumed to be nonsingular. This requirement ensures that there exists an
open subset in $\mathbb{R}^{d}$ such that
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ is the only SP in it for a
given $c$, and as a result the map
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ is uniquely defined.

Through SSB, $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ can become the
VEV of $\phi$. In flavor models, constraints on
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ result in constraints on
the mass matrices leading to interesting predictions about the observables.
Since these constraints are at the heart of such models, let us study them in
detail. If

$\mathcal{F}(\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c))=0\quad\forall
c\in\mathcal{C},$ (3)

where $\mathcal{F}$ is a function that does not explicitly depend on $c$, we
call (3) an intrinsic constraint. An intrinsic constraint remains valid
irrespective of any change in the coefficients (within $\mathcal{C}$ where the
map $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ is uniquely defined).
Its genesis is totally in the group theoretical structure of the system.

Suppose we impose $(N-n)$ constraints of the form $f_{i}(c)=0$, where $f_{i}$
are functions with $i=1,..,N-n$, on the coefficients. Let $\mathcal{C}_{n}$ be
the $n$-dimensional subspace of $\mathcal{C}$ that satisfies these
constraints. Let $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}$ be the
restriction of $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ to the
domain $\mathcal{C}_{n}$. If (3) is not valid, i.e., $\exists
c\in\mathcal{C}:\mathcal{F}(\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c))\neq
0$, and if

$\mathcal{F}(\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n})=0\quad\forall
c\in\mathcal{C}_{n},$ (4)

then we call (4) an extrinsic constraint. Rather than being the result of the
group theoretical structure, an extrinsic constraint is an artifact of the
restriction. As an example, consider the model described in Altarelli and
Feruglio (2005) which consists of two triplets and a singlet of $A_{4}$, i.e.,
$\phi=((\varphi_{1},\varphi_{2},\varphi_{3}),(\varphi^{\prime}_{1},\varphi^{\prime}_{2},\varphi^{\prime}_{3}),\xi)$.
The authors set certain cross-couplings between the two triplets to vanish,
i.e., they imposed constraints of the form $f_{i}(c)=0$, so as to obtain the
desired SP. This SP, which is nothing but the restriction of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ to the lower dimensional
domain $\mathcal{C}_{n}$, was given by
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}=((v(c),v(c),v(c)),(v^{\prime}(c),0,0),u(c))$
where $v(c),v^{\prime}(c)$ and $u(c)$ were defined in $\mathcal{C}_{n}$. This
corresponds to four independent extrinsic constraints:
$\varphi^{\mathrlap{\circ}}_{1\\!}(c)_{n}-\varphi^{\mathrlap{\circ}}_{2\\!}(c)_{n}=0$,
$\varphi^{\mathrlap{\circ}}_{1\\!}(c)_{n}-\varphi^{\mathrlap{\circ}}_{3\\!}(c)_{n}=0$,
$\varphi^{\mathrlap{\hskip-0.20999pt\prime\hskip-0.20999pt\circ}}_{2}(c)_{n}=0$
and
$\varphi^{\mathrlap{\hskip-0.20999pt\prime\hskip-0.20999pt\circ}}_{3}(c)_{n}=0$.

Neutrino flavor models widely impose constraints on the coefficients to obtain
extrinsic constraints. They include the ones like Altarelli and Feruglio
(2005); Ma (2006a); Bazzocchi and Morisi (2009); Chakraborty  _et al._ (2020)
which use them to decouple the SPs of irreducible multiplets (irreps) that are
associated with different fermion mass terms and others like King and Malinsky
(2007); King and Luhn (2010); Krishnan  _et al._ (2018) which use them to
obtain constraints within the SP of a single irrep. Even though some of these
models give arguments (which are perhaps not very convincing) to justify their
imposition, we advocate the group theoretically elegant approach in which all
constraints are obtained intrinsically. However, such an approach will be
quite difficult because the experimental data points towards a complete
breaking of $G_{\\!f}$ while intrinsic constraints have always been found to
be associated with the unbroken part (the stabilizer) of
$G_{\mathclap{f}}^{p}$.

III. Intrinsic constraints generated from the

stabilizer under $G_{\mathclap{f}}^{p}$

The stabilizer Michel and Zhilinskii (2001) (often referred to as the residual
symmetry group) of $\phi$ under $G_{\mathclap{f}}^{p}$ is given by

$H_{\mathclap{f}}^{p}=\\{h\in
G_{\mathclap{f}}^{p}:h_{ij}\phi_{j}=\phi_{i}\\}.$ (5)

In this section, we will show that the stabilizer of a SP generates intrinsic
constraints. We define

$\Lambda_{i\alpha}\hskip
11.00008pt\mathllap{(c)}=\left(\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,i}}(c+\delta
c)-\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,i}}(c)\right)/\delta
c_{\alpha},$ (6)

where $\delta c$ denotes a change of $c$ with $(c+\delta c)\in\mathcal{C}$.
Applying (2) at $(c+\delta c)$ gives $(c_{\alpha}+\delta
c_{\alpha})\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c+\delta
c)}=0$. Taylor expanding this equation around $c$ and using (6), we obtain

$\begin{split}&(c_{\alpha}+\delta
c_{\alpha})\left(\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}+\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}\Lambda_{j\beta}\hskip
11.00008pt\mathllap{(c)}\delta c_{\beta}\right.\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.\vphantom{\big{|}_{\phi}}+\mathcal{O}(\delta
c^{2})\right)=0.\end{split}$ (7)

A minimal set of points that remains invariant under the action of a group $G$
is called a $G$-orbit. SPs always form orbits with respect to the symmetry
group of the potential Michel and Zhilinskii (2001), i.e., the SPs of
$\mathcal{V}^{p}$ exist in $G_{\mathclap{f}}^{p}$-orbits,

$(c_{\alpha}+\delta
c_{\alpha})\,\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c+\delta
c)}=0\quad\forall\,g\in G_{\mathclap{f}}^{p}.$ (8)

Taylor expanding (8) and using (6), we obtain

$\displaystyle(c_{\alpha}+\delta
c_{\alpha})\left(\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}+\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}g_{jk}\Lambda_{k\beta}\hskip
11.00008pt\mathllap{(c)}\delta c_{\beta}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\left.\vphantom{\big{|}_{\phi}}\mathcal{O}(\delta
c^{2})\right)=0\quad\forall\,g\in G_{\mathclap{f}}^{p}.$ (9)

Comparing the terms linear in $\delta c$ in (7) and (Symmetries of stationary
points of the $G$-invariant potential and  the framework of the auxiliary
group), we obtain

$\begin{split}&\partial_{i}\mathcal{I}_{\beta}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}+c_{\alpha}\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}\Lambda_{j\beta}\hskip
11.00008pt\mathllap{(c)}\\\
&\quad\quad=\partial_{i}\mathcal{I}_{\beta}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}+c_{\alpha}\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}g_{jk}\Lambda_{k\beta}\hskip
11.00008pt\mathllap{(c)}\end{split}$ (10)

Let us assume that $H_{\mathclap{f}}^{p}$ is the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ under
$G_{\mathclap{f}}^{p}$ for a specific value of $c$, i.e., $\exists
c\in\mathcal{C}:h\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)=\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)\,\,\forall
h\in H_{\mathclap{f}}^{p}$. Replacing $g$ with $h$ in (10) results in

$c_{\alpha}\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}\left(\vphantom{\big{|}_{\phi}}\Lambda_{j\beta}\hskip
11.00008pt\mathllap{(c)}-h_{jk}\Lambda_{k\beta}\hskip
11.00008pt\mathllap{(c)}\right)=0.$ (11)

Since the Hessian matrix
$c_{\alpha}\partial_{j}\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}$
is assumed to be nonsingular, we obtain

$h_{jk}\Lambda_{k\beta}\hskip 11.00008pt\mathllap{(c)}=\Lambda_{j\beta}\hskip
11.00008pt\mathllap{(c)}\quad\forall\,h\in H_{\mathclap{f}}^{p}.$ (12)

The action of $h$ on $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c+\delta
c)$ is given by

$h_{ij}\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,j}}(c+\delta
c)=h_{ij}(\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,j}}(c)+\Lambda_{j\beta}\hskip
11.00008pt\mathllap{(c)}\delta
c_{\beta})=\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,i}}(c+\delta
c),$ (13)

i.e., if $H_{\mathclap{f}}^{p}$ is the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ for a specific value of
$c$, it will be the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c+\delta c)$ for all $(c+\delta
c)\in\mathcal{C}$. This results in intrinsic constraints of the form

$(h-I)_{ij}\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,j}}(c)=0\,\,\forall
h\in H_{\mathclap{f}}^{p},\,c\in\mathcal{C}$ (14)

where $I$ is the identity.

IV. Symmetry group of a set of SPs

SPs form $G_{\mathclap{f}}^{p}$-orbits, i.e.,
$c_{\alpha}\,\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}=0\quad\forall
g\in G_{\mathclap{f}}^{p},c\in\mathcal{C}$. Let us examine whether a union of
such $G_{\mathclap{f}}^{p}$-orbits can have a symmetry group larger than
$G_{\mathclap{f}}^{p}$ for all $c\in\mathcal{C}$. We define

$\tilde{G}_{\mathclap{f}}^{p}=\\{g:\,c_{\alpha}\,\partial_{i}\mathcal{I}_{\alpha}\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)}=0\quad\forall
c\in\mathcal{C}\\}.$ (15)

Given a SP $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$, (15) implies
the existence of a set of SPs
$g\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ with
$g\in\tilde{G}_{\mathclap{f}}^{p}$ for all $c\in\mathcal{C}$. We expect
$\tilde{G}_{\mathclap{f}}^{p}\geq G_{\mathclap{f}}^{p}$. Let us examine a
scenario where
$\tilde{G}_{\mathclap{f}}^{p}>G_{\mathclap{f}}^{p}.$222$\tilde{G}_{\mathclap{f}}^{p}$
being larger than $G_{\mathclap{f}}^{p}$ may not remain valid if we increase
the order $p$. However, higher-order nonrenormalizable terms are suppressed.
If we obtain a nondegenerate SP at a lower order, higher-order terms cause
only small perturbations to it.

We rewrite (1) as

$\mathcal{V}^{p}=\sum\nolimits_{\alpha=1}^{n}c_{\alpha}\mathcal{I}_{\alpha}+\sum\nolimits_{\alpha=n+1}^{N}c_{\alpha}\mathcal{X}_{\alpha}^{\scriptscriptstyle
T}\mathcal{Y}_{\alpha},$ (16)

with the following three assumptions: i) The first summation consists of $n$
$G_{\mathclap{f}}^{p}$-invariants. We assume that they remain invariant under
the larger group $\tilde{G}_{\mathclap{f}}^{p}$ also. ii) The second summation
involves the rest of the $(N-n)$ $G_{\mathclap{f}}^{p}$-invariants, each of
which is expressed as the dot product
$\mathcal{X}_{\alpha}^{\scriptscriptstyle T}\mathcal{Y}_{\alpha}$ where
$\mathcal{X}_{\alpha}$ and $\mathcal{Y}_{\alpha}$ are multiplets of the same
representation of $G_{\mathclap{f}}^{p}$. We assume that
$\mathcal{X}_{\alpha}$ and $\mathcal{Y}_{\alpha}$ are multiplets of different
representations of $\tilde{G}_{\mathclap{f}}^{p}$. Hence,
$\mathcal{X}_{\alpha}^{\scriptscriptstyle T}\mathcal{Y}_{\alpha}$ is not a
$\tilde{G}_{\mathclap{f}}^{p}$-invariant. iii) Let $\mathcal{C}_{n}$ be the
$n$-dimensional subspace of $\mathcal{C}$ obtained by imposing the constraints
$c_{i}=0$ with $i=n+1,..,N$. Let $\mathcal{V}^{p}_{n}$ and
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}$ be the restrictions of
$\mathcal{V}^{p}$ and $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$
respectively to $\mathcal{C}_{n}$. We assume that

$\mathcal{X}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)_{n}}=0,\,\,\,\mathcal{Y}_{\alpha}\big{|}_{\phi=\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)_{n}}=0\quad\forall
c\in\mathcal{C}_{n}.$ (17)

Since $\mathcal{X}_{\alpha}$ and $\mathcal{Y}_{\alpha}$ are multiplets under
$\tilde{G}_{\mathclap{f}}^{p}$, their vanishing at
$\phi=\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}$ implies

$\partial_{i}(\mathcal{X}_{\alpha}^{\scriptscriptstyle
T}\mathcal{Y}_{\alpha})\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)_{n}}=0\quad\forall\,g\in\tilde{G}_{\mathclap{f}}^{p},\,c\in\mathcal{C}_{n}.$
(18)

Since $g\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}\,\forall
g\in\tilde{G}_{\mathclap{f}}^{p},c\in\mathcal{C}_{n}$ are the SPs of
$\mathcal{V}^{p}_{n}$, and given (18), we obtain

$\begin{split}&\partial_{i}\left(\sum\nolimits_{\alpha=1}^{n}c_{\alpha}\mathcal{I}_{\alpha}+\sum\nolimits_{\alpha=n+1}^{N}c_{\alpha}\mathcal{X}_{\alpha}^{\scriptscriptstyle
T}\mathcal{Y}_{\alpha}\right)\big{|}_{\phi=g\phi^{\mathclap{\,\,\hskip-0.25002pt\circ}\,}(c)_{n}}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=0\quad\forall
g\in\tilde{G}_{\mathclap{f}}^{p},c\in\mathcal{C}_{n},\end{split}$ (19)

i.e., $g\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}\forall
g\in\tilde{G}_{\mathclap{f}}^{p},c\in\mathcal{C}_{n}$ are the SPs of not only
$\mathcal{V}^{p}_{n}$ but also $\mathcal{V}^{p}$. The last $(N-n)$
coefficients of $\mathcal{V}^{p}$ do not play any role in determining these
SPs of $\mathcal{V}^{p}$. Therefore, we obtain
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)=\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{n}$
with $\mathcal{C}=\mathcal{C}_{n}\times\mathbb{R}^{N-n}$.

V. Intrinsic constraints generated from the

stabilizer under $\tilde{G}_{\mathclap{f}}^{p}$

Let $\tilde{H}_{\mathclap{f}}^{p}$ be a subgroup of
$\tilde{G}_{\mathclap{f}}^{p}$ such that it is the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ under
$\tilde{G}_{\mathclap{f}}^{p}$ for a specific value of $c\in\mathcal{C}$.
Following the steps similar to those in the previous section, we can obtain
equations similar to (8)-(13) in which $G_{\mathclap{f}}^{p}$ and
$H_{\mathclap{f}}^{p}$ are replaced with $\tilde{G}_{\mathclap{f}}^{p}$ and
$\tilde{H}_{\mathclap{f}}^{p}$, respectively, leading to the conclusion that
$\tilde{H}_{\mathclap{f}}^{p}$ is the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ for all $c\in\mathcal{C}$.
As a result, we obtain intrinsic constraints of the form
$(h-I)_{ij}\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}_{\mathclap{\,j}}(c)=0\,\forall
h\in\tilde{H}_{\mathclap{f}}^{p},c\in\mathcal{C}$. Thus we conclude that the
stabilizer under not only $G_{\mathclap{f}}^{p}$ but also
$\tilde{G}_{\mathclap{f}}^{p}$ generates intrinsic constraints. We now
demonstrate these concepts with the help of an example.

| $\acute{\phi},\acute{\phi}_{s}$ | $\hat{\phi},\hat{\phi}_{s}$ | $\grave{\phi},\grave{\phi}_{s}$
---|---|---|---
$S_{4}\\!\times\\!\acute{D}_{6}\\!\times\\!\grave{D}_{6}$ | $\boldsymbol{3}\\!\times\\!\acute{\boldsymbol{2}}\\!\times\\!\boldsymbol{1},\boldsymbol{1}$ | $\boldsymbol{1}\\!\times\\!\acute{\boldsymbol{2}}\\!\times\\!\grave{\boldsymbol{2}},\boldsymbol{1}$ | $\boldsymbol{3}\\!\times\\!\boldsymbol{1}\\!\times\\!\grave{\boldsymbol{2}},\boldsymbol{1}$
$Z_{2}\\!\times\\!\acute{Z}_{2}\\!\times\\!\grave{Z}_{2}$ | $1\\!\times\\!-1\\!\times\\!1$ | $-1\\!\times\\!-1\\!\times\\!-1$ | $1\\!\times\\!1\\!\times\\!-1$
Table 1: The action of
$G_{\\!f}=S_{4}\\!\times\\!Z_{2}\\!\times\\!\acute{D}_{6}\\!\times\\!\acute{Z}_{2}\\!\times\\!\grave{D}_{6}\\!\times\\!\grave{Z}_{2}$
on the irreps $\acute{\phi}$, $\acute{\phi}_{s}$, $\hat{\phi}$,
$\hat{\phi}_{s}$, $\grave{\phi}$ and $\grave{\phi}_{s}$.

VI. Example

The finite group used in this example consists of the symmetric group $S_{4}$
Pakvasa and Sugawara (1979); Lee and Mohapatra (1994); Ma (2006b); Chakraborty
_et al._ (2020), two dihedral groups of order six (named $\acute{D}_{6}$,
$\grave{D}_{6}$) and three cyclic groups of order two (named $Z_{2}$,
$\acute{Z}_{2}$, $\grave{Z}_{2}$). We denote the irreps of $S_{4}$ by
$\boldsymbol{1}$, $\boldsymbol{\underaccent{\bar}{1}}$, $\boldsymbol{2}$,
$\boldsymbol{3}$, $\boldsymbol{\underaccent{\bar}{3}}$; $\acute{D}_{6}$ by
$\boldsymbol{1}$, $\acute{\boldsymbol{2}}$,
$\acute{\boldsymbol{\underaccent{\bar}{1}}}$ and $\grave{D}_{6}$ by
$\boldsymbol{1}$, $\grave{\boldsymbol{2}}$,
$\grave{\boldsymbol{\underaccent{\bar}{1}}}$. As a convenient basis, we use

$S=\left(\begin{matrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{matrix}\right)\\!,\,T=\left(\begin{matrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{matrix}\right)\\!,\,U=\left(\begin{matrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right)$

and

$s=\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right)\\!,\,t=\left(\begin{matrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{matrix}\right)\\!,\,u=\left(\begin{matrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{1}{2}\end{matrix}\right)$

to generate $\boldsymbol{3}$ and $\boldsymbol{2}$, $\acute{\boldsymbol{2}}$,
$\grave{\boldsymbol{2}}$, respectively. In our analysis, we utilize the tensor
product expansions
$\boldsymbol{3}\times\boldsymbol{3}=\boldsymbol{1}+\boldsymbol{2}+\boldsymbol{3}+\boldsymbol{\underaccent{\bar}{3}}$,
$\acute{\boldsymbol{2}}\times\acute{\boldsymbol{2}}=\boldsymbol{1}+\acute{\boldsymbol{2}}+\acute{\boldsymbol{\underaccent{\bar}{1}}}$
and
$\grave{\boldsymbol{2}}\times\grave{\boldsymbol{2}}=\boldsymbol{1}+\grave{\boldsymbol{2}}+\grave{\boldsymbol{\underaccent{\bar}{1}}}$
where $\boldsymbol{\underaccent{\bar}{3}}$,
$\acute{\boldsymbol{\underaccent{\bar}{1}}}$ and
$\grave{\boldsymbol{\underaccent{\bar}{1}}}$ are antisymmetric products while
the rest are symmetric.

We propose a scalar field
$\phi=(\acute{\phi},\acute{\phi}_{s},\hat{\phi},\hat{\phi}_{s},\grave{\phi},\grave{\phi}_{s})$,
Table 1, which transforms under our flavor symmetry group,
$G_{\\!f}=S_{4}\\!\times\\!Z_{2}\\!\times\\!\acute{D}_{6}\\!\times\\!\acute{Z}_{2}\\!\times\\!\grave{D}_{6}\\!\times\\!\grave{Z}_{2}$.
Since $\acute{\phi}$, $\hat{\phi}$ and $\grave{\phi}$ transform as
$\boldsymbol{3}\\!\times\\!\acute{\boldsymbol{2}}$,
$\acute{\boldsymbol{2}}\\!\times\\!\grave{\boldsymbol{2}}$ and
$\grave{\boldsymbol{2}}\\!\times\\!\boldsymbol{3}$ under
$S_{4}\\!\times\\!\acute{D}_{6}$, $\acute{D}_{6}\\!\times\\!\grave{D}_{6}$ and
$\grave{D}_{6}\\!\times\\!S_{4}$, we express them as $3\\!\times\\!2$,
$2\\!\times\\!2$ and $2\\!\times\\!3$ real matrices, respectively. By taking
their tensor products with themselves, we construct the following quadratic
multiplets. Using $\acute{\phi}$, we obtain

$\displaystyle|\acute{\phi}|^{2}$
$\displaystyle=\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\acute{\phi}],\quad\quad\acute{\phi}^{2}_{\boldsymbol{2}\boldsymbol{1}}=\left(\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{8}\acute{\phi}],\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{3}\acute{\phi}]\right)^{\\!{\scriptscriptstyle T}}\\!\\!,$
$\displaystyle\acute{\phi}^{2}_{\boldsymbol{3}\boldsymbol{1}}$
$\displaystyle=\left(\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{6}\acute{\phi}],\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{4}\acute{\phi}],\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{1}\acute{\phi}]\right)^{\\!{\scriptscriptstyle T}}\\!\\!\\!,$
$\displaystyle\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}$
$\displaystyle=\left(\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\acute{\phi}\sigma_{3}],-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\acute{\phi}\sigma_{1}]\right)\\!,$
$\displaystyle\acute{\phi}^{2}_{\boldsymbol{2}\acute{\boldsymbol{2}}}$
$\displaystyle=\left(\begin{matrix}\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{8}\acute{\phi}\sigma_{3}]&-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{8}\acute{\phi}\sigma_{1}]\\!\\!\\!,\\\
\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{3}\acute{\phi}\sigma_{3}]&-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{3}\acute{\phi}\sigma_{1}]\end{matrix}\right),$
$\displaystyle\acute{\phi}^{2}_{\boldsymbol{3}\acute{\boldsymbol{2}}}$
$\displaystyle=\left(\begin{matrix}\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{6}\acute{\phi}\sigma_{3}]&\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{4}\acute{\phi}\sigma_{3}]&\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{1}\acute{\phi}\sigma_{3}]\\\
-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{6}\acute{\phi}\sigma_{1}]&-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{4}\acute{\phi}\sigma_{1}]&-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{1}\acute{\phi}\sigma_{1}]\end{matrix}\right)^{\\!\\!{\scriptscriptstyle
T}}\\!\\!\\!,$
$\displaystyle\acute{\phi}^{2}_{\boldsymbol{\underaccent{\bar}{3}}\acute{\boldsymbol{\underaccent{\bar}{1}}}}$
$\displaystyle=\left(\text{Tr}[\acute{\phi}^{T}\lambda_{7}\acute{\phi}\sigma_{2}],-\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{5}\acute{\phi}\sigma_{2}],\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\lambda_{2}\acute{\phi}\sigma_{2}]\right)^{\\!{\scriptscriptstyle
T}}\\!\\!,$

where $\sigma_{i}$ and $\lambda_{i}$ are the Pauli matrices and the Gell-Mann
matrices, respectively. The naming of these multiplets is self-explanatory,
e.g., $\acute{\phi}^{2}_{\boldsymbol{2}\acute{\boldsymbol{2}}}$ is the
multiplet quadratic in $\acute{\phi}$ and transforming as $\boldsymbol{2}$ and
$\acute{\boldsymbol{2}}$ under $S_{4}$ and $\acute{D}_{6}$, respectively. We
obtain similar quadratic multiplets using $\grave{\phi}$ also:
$|\grave{\phi}|^{2}$, $\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{2}}$,
$\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{3}}$,
$\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$,
$\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{2}}$,
$\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{3}}$ and
$\hat{\phi}^{2}_{\grave{\boldsymbol{\underaccent{\bar}{1}}}\boldsymbol{\underaccent{\bar}{3}}}$.
The quadratic multiplets obtained using $\hat{\phi}$ are

$\displaystyle|\hat{\phi}|^{2}$
$\displaystyle=\text{Tr}[\hat{\phi}^{\scriptscriptstyle T}\hat{\phi}],$
$\displaystyle\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}$
$\displaystyle=\left(\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\hat{\phi}\sigma_{3}],-\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\hat{\phi}\sigma_{1}]\right),$
$\displaystyle\hat{\phi}^{2}_{\acute{\boldsymbol{\underaccent{\bar}{1}}}\grave{\boldsymbol{\underaccent{\bar}{1}}}}$
$\displaystyle=\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{2}\hat{\phi}\sigma_{2}],$
$\displaystyle\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$
$\displaystyle=\left(\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{3}\hat{\phi}],-\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{1}\hat{\phi}]\right)^{\\!{\scriptscriptstyle T}}\\!\\!,$
$\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\grave{\boldsymbol{2}}}=\left(\begin{matrix}\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{3}\hat{\phi}\sigma_{3}]&-\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{3}\hat{\phi}\sigma_{1}]\\\ -\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{1}\hat{\phi}\sigma_{3}]&\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\sigma_{1}\hat{\phi}\sigma_{1}]\end{matrix}\right).$

Now, we construct the invariants. The quadratic invariants are
$|\acute{\phi}|^{2}$, $|\acute{\phi}_{s}|^{2}$, $|\hat{\phi}|^{2}$,
$|\hat{\phi}_{s}|^{2}$, $|\grave{\phi}|^{2}$ and $|\grave{\phi}_{s}|^{2}$. By
taking the product of any two of these, we obtain $21$ quartic invariants. We
also have another 12 quartic invariants:
$(\acute{\phi}^{2}_{\boldsymbol{2}\boldsymbol{1}})^{\scriptscriptstyle
T}\acute{\phi}^{2}_{\boldsymbol{2}\boldsymbol{1}}$,
$(\acute{\phi}^{2}_{\boldsymbol{3}\boldsymbol{1}})^{\scriptscriptstyle
T}\acute{\phi}^{2}_{\boldsymbol{3}\boldsymbol{1}}$,
$\acute{\phi}_{s}\text{Tr}[\acute{\phi}^{\scriptscriptstyle
T}\acute{\phi}^{2}_{\boldsymbol{3}\acute{\boldsymbol{2}}}]$,
$\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{2}}(\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{2}})^{\scriptscriptstyle
T}$,
$\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{3}}(\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{3}})^{\scriptscriptstyle
T}$, $\grave{\phi}_{s}\text{Tr}[\grave{\phi}^{\scriptscriptstyle
T}\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{3}}]$,
$(\hat{\phi}^{2}_{\acute{\boldsymbol{\underaccent{\bar}{1}}}\grave{\boldsymbol{\underaccent{\bar}{1}}}})^{2}$,
$\hat{\phi}_{s}\text{Tr}[\hat{\phi}^{\scriptscriptstyle
T}\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\grave{\boldsymbol{2}}}]$,
$\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{2}}\acute{\phi}^{2}_{\boldsymbol{2}\boldsymbol{1}}$,
$\grave{\phi}^{2}_{\boldsymbol{1}\boldsymbol{3}}\acute{\phi}^{2}_{\boldsymbol{3}\boldsymbol{1}}$,
$\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$
and
$\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$.
These 39 invariants along with 39 coefficients constitute the most general
potential of order $4$, i.e., for $p=4$, we construct $\mathcal{V}^{4}$ with
$N=39$ terms. Note that in this example, we have
$G_{\mathclap{f}}^{4}=G_{\\!f}$.

Let us replace $\acute{D}_{6}$ in $G_{\mathclap{f}}^{4}$ with two different
dihedral groups $\acute{D}^{a}_{6}$ and $\acute{D}^{b}_{6}$ acting on the rhs
of $\acute{\phi}$ and the lhs of $\hat{\phi}$, respectively. Similarly, we
replace $\grave{D}_{6}$ with $\grave{D}^{a}_{6}$ and $\grave{D}^{b}_{6}$
acting on the lhs of $\grave{\phi}$ and the rhs of $\hat{\phi}$, respectively.
In anticipation of the things to come, let us name the resulting group
$\tilde{G}_{\mathclap{f}}^{4}$, i.e.,
$\tilde{G}_{\mathclap{f}}^{4}=S_{4}\\!\times\\!Z_{2}\\!\times\\!\acute{D}^{a}_{6}\\!\times\\!\acute{D}^{b}_{6}\\!\times\\!\acute{Z}_{2}\\!\times\\!\grave{D}^{a}_{6}\\!\times\\!\grave{D}^{b}_{6}\\!\times\\!\grave{Z}_{2}$.
In relation to $\tilde{G}_{\mathclap{f}}^{4}$, let us verify the three
conditions listed in the previous section.

(i) Every term except
$\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$
and
$\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$
is invariant under $\tilde{G}_{\mathclap{f}}^{4}$, i.e., $n=37$. These $37$
$\tilde{G}_{\mathclap{f}}^{4}$-invariants form $\mathcal{V}^{4}_{37}$. (ii)
Even though
$\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$
and
$\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$
are not $\tilde{G}_{\mathclap{f}}^{4}$-invariants, their constituent terms
$\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}$,
$\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$,
$\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}$ and
$\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$ transform as
multiplets under $\tilde{G}_{\mathclap{f}}^{4}$. (iii) We will show that the
map $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ given by

$\displaystyle\acute{\phi}^{\circ\\!}(c)$
$\displaystyle=\frac{\acute{v}(c)}{\sqrt{3}}\left(\begin{matrix}1&-\frac{1}{2}&-\frac{1}{2}\\\
0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{matrix}\right)^{\\!\\!{\scriptscriptstyle
T}}\\!\\!,$ $\displaystyle\acute{\phi}_{s}^{\circ\\!}(c)$
$\displaystyle=\acute{v}_{s}(c),$ $\displaystyle\hat{\phi}^{\circ\\!}(c)$
$\displaystyle=\hat{v}(c)I/\sqrt{2},$
$\displaystyle\hat{\phi}_{s}^{\circ\\!}(c)$ $\displaystyle=\hat{v}_{s}(c),$
(20) $\displaystyle\grave{\phi}^{\circ\\!}(c)$
$\displaystyle=\frac{\grave{v}(c)}{\sqrt{3}}\left(\begin{matrix}1&-\frac{1}{2}&-\frac{1}{2}\\\
0&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}\end{matrix}\right),$
$\displaystyle\grave{\phi}_{s}^{\circ\\!}(c)$
$\displaystyle=\grave{v}_{s}(c),$

where $\acute{v}(c)$, $\acute{v}_{s}(c)$, etc. are the norms of the irreps, is
a SP of $\mathcal{V}^{4}$ for a domain $\mathcal{C}$ that is an open subset of
$\mathbb{R}^{39}$.

Consider the following four group actions under $\tilde{G}_{\mathclap{f}}^{4}$
on $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ which keep it invariant:

$\displaystyle T\,\acute{\phi}^{\circ\\!}(c)\,(t^{2})^{\scriptscriptstyle
T}=\acute{\phi}^{\circ\\!}(c),\quad
t\,\grave{\phi}^{\circ\\!}(c)\,T^{\scriptscriptstyle
T}=\grave{\phi}^{\circ\\!}(c),$ (21) $\displaystyle
t\,\hat{\phi}^{\circ\\!}(c)\,t^{\scriptscriptstyle
T}=\hat{\phi}^{\circ\\!}(c),$ (22) $\displaystyle
U\,\acute{\phi}^{\circ\\!}(c)\,(ut)^{\scriptscriptstyle
T}=\acute{\phi}^{\circ\\!}(c),\,ut\,\grave{\phi}^{\circ\\!}(c)\,U^{\scriptscriptstyle
T}=\grave{\phi}^{\circ\\!}(c),$ (23) $\displaystyle
ut\,\hat{\phi}^{\circ\\!}(c)\,(ut)^{\scriptscriptstyle
T}=\hat{\phi}^{\circ\\!}(c).$ (24)

The group actions (21), (23) generate a dihedral group, say $D^{\prime}_{6}$,
and the group actions (22), (24) generate another dihedral group, say
$D^{\prime\prime}_{6}$. The stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ under
$\tilde{G}_{\mathclap{f}}^{4}$ is nothing but $D^{\prime}_{6}\times
D^{\prime\prime}_{6}$. (23), (24) together form a single group action under
$G_{\mathclap{f}}^{4}$. This action generates a cyclic group, say
$Z^{\prime}_{2}$, which is the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ under
$G_{\mathclap{f}}^{4}$. We have $D^{\prime}_{6}\times
D^{\prime\prime}_{6}<\tilde{G}_{\mathclap{f}}^{4}$, $D^{\prime}_{6}\times
D^{\prime\prime}_{6}\nless G_{\mathclap{f}}^{4}$,
$Z^{\prime}_{2}<G_{\mathclap{f}}^{4}$ and $Z^{\prime}_{2}<D^{\prime}_{6}\times
D^{\prime\prime}_{6}$ as expected.

Orbits with the same conjugacy class of stabilizers are of the same type, and
the union of orbits of the same type forms a stratum Michel and Zhilinskii
(2001). In other words, two points belong to the same stratum if and only if
their stabilizers are conjugate. The action of the group decomposes the space
into strata. An orbit that is isolated in its stratum is called a critical
orbit. The points constituting a critical orbit are SPs of every potential
irrespective of the coefficients Michel (1971); Michel and Zhilinskii (2001).
(21)-(24) fix all degrees of freedom of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ except the norms of the
irreps. If we quotient out the norms Golubitsky  _et al._ (1988); Gaeta
(2006), then in the resulting space, the point corresponding to (20) will be
isolated in its stratum under $\tilde{G}_{\mathclap{f}}^{4}$, and hence it
will be a SP of every potential invariant under
$\tilde{G}_{\mathclap{f}}^{4}$. Let $\mathcal{C}_{37}$ be the subspace of
$\mathcal{C}$ obtained by equating the coefficients of
$\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$
and
$\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$
to zero. Let $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{37}$ and
$\mathcal{V}^{4}_{37}$ be the restrictions of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ (20) and $\mathcal{V}^{4}$
respectively to $\mathcal{C}_{37}$. We may substitute
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{37}$ in
$\mathcal{V}^{4}_{37}$ and equate the derivatives of $\mathcal{V}^{4}_{37}$
with respect to the norms to vanish. Any solution to these equations in which
all the norms are nonvanishing leads to a SP of the form (20), and the
corresponding open subset of $\mathbb{R}^{37}$ is $\mathcal{C}_{37}$. We can
show that the terms $\acute{\phi}^{2}_{\boldsymbol{1}\acute{\boldsymbol{2}}}$,
$\hat{\phi}^{2}_{\acute{\boldsymbol{2}}\boldsymbol{1}}$,
$\hat{\phi}^{2}_{\boldsymbol{1}\grave{\boldsymbol{2}}}$ and
$\grave{\phi}^{2}_{\grave{\boldsymbol{2}}\boldsymbol{1}}$ vanish at (20),
implying that these terms satisfy the third condition (17). Therefore, (20)
provides
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)=\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)_{37}$
with $\mathcal{C}=\mathcal{C}_{37}\times\mathbb{R}^{2}$, and we obtain a set
of SPs of $\mathcal{V}^{4}$ as
$g\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)\,\forall
g\in\tilde{G}_{\mathclap{f}}^{4},c\in\mathcal{C}$.

VII. The framework of the auxiliary group

Almost all flavor symmetry groups discussed in the literature so far have been
in the form of a direct product of a subgroup of $U(3)$ with Abelian groups,
e.g. $A_{4}\\!\times\\!Z_{4}\\!\times\\!Z_{3}$. The apparent reason for this
assumption is that the fermions exist in three families only. However, it was
shown in Babu and Gabriel (2010); Holthausen and Schmidt (2012); Holthausen
_et al._ (2013) that by going beyond the $U(3)$-subgroup paradigm we can
naturally avoid undesirable coefficients in the potential, similar to the
cross-couplings between the two triplets of $A_{4}$ discussed in Altarelli and
Feruglio (2005). References. Babu and Gabriel (2010); Holthausen and Schmidt
(2012); Holthausen  _et al._ (2013) used an enlarged flavor group constructed
as a semidirect product in which the conventional flavor group [the direct
product of a subgroup of $U(3)$ with Abelian groups] appears as the quotient.
In more recent works Krishnan (2020, 2019), a special case of this
construction, in which the semidirect product was replaced with a direct
product, was studied. We named this construction the ‘framework of the
auxiliary group’ and used the notation $G_{\\!f}=G_{r}\\!\times\\!G_{x}$ where
$G_{\\!f}$ is the enlarged flavor group, $G_{r}$ is the conventional flavor
group and $G_{x}$ is the so-called auxiliary group. $G_{x}$ is defined as the
part of the flavor symmetry group under which the fermions remain invariant.
Since the fermions transform nontrivially under $G_{r}$ only, they form an
unfaithful representation of $G_{\\!f}$. The elementary scalar fields, on the
other hand, transform nontrivially under both $G_{r}$ and $G_{x}$. By taking
the tensor products of these elementary fields, we obtain effective fields
that transform nontrivially under $G_{r}$ only so that they can be coupled
with the fermions. In this framework, we obtained novel vacuum alignments for
the irreps of $G_{r}$ Krishnan (2020, 2019). The current paper provides a firm
theoretical foundation for building such alignments. Let us demonstrate this
with our example.

We write our flavor symmetry group as the direct product of
$G_{r}=S_{4}\\!\times\\!Z_{2}$ and
$G_{x}=\acute{D}_{6}\\!\times\\!\acute{Z}_{2}\\!\times\\!\grave{D}_{6}\\!\times\\!\grave{Z}_{2}$.
If coupling with fermions requires the scalar fields to transform as $-1$
under $Z_{2}$, the lowest order of $\phi$ with which the effective fields can
be constructed is cubic. Let us denote such cubic fields with
$\xi=(\xi_{\boldsymbol{1}},\xi_{\boldsymbol{2}},\xi_{\boldsymbol{3}},\xi_{\boldsymbol{\underaccent{\bar}{3}}})$
where the subscripts indicate transformation under $S_{4}$. In the most
general form, they are given by

$\displaystyle\xi_{\boldsymbol{1}}$
$\displaystyle=k_{1}\acute{\phi}_{s}\hat{\phi}_{s}\grave{\phi}_{s}+k^{\prime}_{1}\text{Tr}[\mathcal{S}],$
(25) $\displaystyle\xi_{\boldsymbol{2}}$
$\displaystyle=k_{2}\left(\text{Tr}[\mathcal{S}\lambda_{8}],\text{Tr}[\mathcal{S}\lambda_{3}]\right),$
(26) $\displaystyle\xi_{\boldsymbol{3}}$
$\displaystyle=k_{3}\left(\mathcal{S}_{23},\mathcal{S}_{31},\mathcal{S}_{12}\right),$
(27) $\displaystyle\xi_{\boldsymbol{\underaccent{\bar}{3}}}$
$\displaystyle=k_{\underaccent{\bar}{3}}\left(\mathcal{A}_{23},\mathcal{A}_{31},\mathcal{A}_{12}\right),$
(28)

where $k$’s are arbitrary constants, and $\mathcal{S}$ and $\mathcal{A}$ are
the symmetric and the antisymmetric matrices

$\mathcal{S}=\frac{1}{2}(\acute{\phi}\hat{\phi}\grave{\phi}+(\acute{\phi}\hat{\phi}\grave{\phi})^{\scriptscriptstyle
T}),\quad\mathcal{A}=\frac{1}{2}(\acute{\phi}\hat{\phi}\grave{\phi}-(\acute{\phi}\hat{\phi}\grave{\phi})^{\scriptscriptstyle
T}).$ (29)

Let
$\xi^{\circ\\!}(c)=(\xi_{\boldsymbol{1}}^{\circ\\!}(c),\xi_{\boldsymbol{2}}^{\circ\\!}(c),\xi_{\boldsymbol{3}}^{\circ\\!}(c),\xi_{\boldsymbol{\underaccent{\bar}{3}}}^{\circ\\!}(c))$
denote the values of the effective fields at the SP (20). We obtain

$\displaystyle\xi_{\boldsymbol{1}}^{\circ\\!}(c)$
$\displaystyle=k_{1}\acute{v}_{s}(c)\hat{v}_{s}(c)\grave{v}_{s}(c),$ (30)
$\displaystyle\xi_{\boldsymbol{2}}^{\circ\\!}(c)$
$\displaystyle=-k_{2}\acute{v}(c)\hat{v}(c)\grave{v}(c)\frac{1}{\sqrt{6}}(-\frac{1}{2},-\frac{\sqrt{3}}{2}),$
(31) $\displaystyle\xi_{\boldsymbol{3}}^{\circ\\!}(c)$
$\displaystyle=k_{3}\acute{v}(c)\hat{v}(c)\grave{v}(c)\frac{1}{6\sqrt{2}}(2,-1,-1),$
(32) $\displaystyle\xi_{\boldsymbol{\underaccent{\bar}{3}}}^{\circ\\!}(c)$
$\displaystyle=0$ (33)

The nonvanishing SPs, i.e., $\xi_{\boldsymbol{1}}^{\circ\\!}(c)$,
$\xi_{\boldsymbol{2}}^{\circ\\!}(c)$ and $\xi_{\boldsymbol{3}}^{\circ\\!}(c)$
contribute towards the construction of the fermion mass matrix.

Let us focus on $\xi_{\boldsymbol{3}}^{\circ\\!}(c)\propto(2,-1,-1)$. We have
$U\xi_{\boldsymbol{3}}^{\circ\\!}(c)^{\scriptscriptstyle
T}=\xi_{\boldsymbol{3}}^{\circ\\!}(c)^{\scriptscriptstyle T}$. The stabilizer
of $\xi_{\boldsymbol{3}}^{\circ\\!}(c)$ under $G_{r}=S_{4}\times Z_{2}$ is the
cyclic group $Z^{\prime}_{2}$ generated by $U$, which is the same as the
stabilizer of $\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ under
$G_{\mathclap{f}}^{4}$ generated by the group action (23), (24). This
stabilizer produces the intrinsic constraint
$\xi_{\boldsymbol{3}_{2}\\!}^{\circ\\!}(c)-\xi_{\boldsymbol{3}_{3}\\!}^{\circ\\!}(c)=0$.
We also have a second intrinsic constraint
$\xi_{\boldsymbol{3}_{1}\\!}^{\circ\\!}(c)+2\xi_{\boldsymbol{3}_{2}\\!}^{\circ\\!}(c)=0$.
It is generated as a consequence of $D^{\prime}_{6}\times
D^{\prime\prime}_{6}$, which is the stabilizer of
$\phi^{\mathclap{\,\,\hskip-0.35002pt\circ}\,}(c)$ under
$\tilde{G}_{\mathclap{f}}^{4}$. If $\xi_{\boldsymbol{3}}$ were an elementary
triplet, we would not have been able to obtain this constraint intrinsically.
A triplet under $S_{4}\times Z_{2}$ represents the symmetry group of a cube.
Its normalized space has three critical orbits only Michel and Zhilinskii
(2001): the orbit of $\frac{1}{\sqrt{3}}(1,1,1)$ (vertices), the orbit of
$\frac{1}{\sqrt{2}}(1,1,0)$ (edge centers) and the orbit of $(1,0,0)$ (face
centers). Therefore, the only way to obtain the SP $\propto(2,-1,-1)$ for an
elementary triplet is by constraining the coefficients in the potential. On
the other hand, by utilizing the concepts introduced in this work, we obtained
the SP $\propto(2,-1,-1)$ for an effective triplet without imposing
constraints on the coefficients. In a recent flavor model Krishnan (2019), an
effective $S_{4}$ triplet with a similar SP [belonging to the orbit of the SP
$\propto(2\sqrt{6},-1,-1)$] was proposed resulting in $\text{TM}_{1}$ neutrino
mixing with $\sin^{2}\theta_{13}=\frac{1}{3}\sin^{2}\frac{\pi}{12}$. We hope
that the present work will form the theoretical basis for such flavor models
that aspire to obtain phenomenologically interesting predictions entirely from
discrete symmetries without imposing constraints on the coefficients.

VIII. Summary

We consider the most general potential of order $p$ invariant under a finite
symmetry group $G_{\mathclap{f}}^{p}$. We classify the constraints on a
stationary point (SP) of the potential into (a) intrinsic: that originate from
the group properties and (b) extrinsic: that are obtained by imposing
constraints on the coefficients in the potential. Given a SP, we consider the
largest finite group such that its action on the SP produces a set of SPs,
independently of the coefficients. We name this group
$\tilde{G}_{\mathclap{f}}^{p}$ and discover that it can be larger than
$G_{\mathclap{f}}^{p}$. We show that the stabilizer of a SP under not only
$G_{\mathclap{f}}^{p}$ but also the larger group
$\tilde{G}_{\mathclap{f}}^{p}$ generates intrinsic constraints. We briefly
review the framework of the auxiliary group where several elementary fields
are coupled together to obtain an effective field. Intrinsic constraints on
the SPs of the elementary fields (generated by the stabilizer under
$\tilde{G}_{\mathclap{f}}^{p}$) lead to intrinsic constraints on the SP of the
effective field. Using an example, we show that these constraints on the SP of
the effective field cannot always be obtained intrinsically if the field were
elementary. These results have a direct application to flavor models in
particle physics. We hope that they find applications in other areas of
physics that involve discrete symmetries as well, besides being of general
interest in mathematical physics.

Acknowledgments

The author would like to gratefully acknowledge the many stimulating
discussions with Sujatha Ramakrishnan. The author would also like to thank the
referee of Krishnan (2020), whose comments served as an inspiration for this
work.

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Vol. 000 No.0, 000–000

11institutetext: School of Physics, Peking University, Beijing 100871, China
22institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China
33institutetext: College of Mathematics and Physics, Wenzhou university,
Wenzhou Zhejiang 325035, China<EMAIL_ADDRESS>
44institutetext: Department of Physics and Electronic Engineering, Nanyang
Normal University, Nanyang Henan 473061, China

# To investigate the magnetic-field-induced distortion of NSs through GRB
X-ray plateaus

Tingting Lin 1122 Shuang Du 3311 Shujin Hou 44 Renxin Xu 1122

(Received 2020 xx xx; accepted 20xx xx xx)

###### Abstract

Magnetic field may distort neutron stars (NSs), but the effect has not been
robustly tested through gravitational-wave observation yet due to the absence
of a fast rotating Galactic magnetar. The central objects of Gamma-ray bursts
(GRBs) could be millisecond magnetars. Under the magnetar scenario on the
X-ray plateaus of GRB afterglows, the spindown evolution modulated by the
gravitational-wave radiation may be inferred from some special samples, so
that the magnetically-induced distorting can be further estimated. According
to two samples, GRB 060807 and GRB 070521, we found that the correlation
between the effective ellipticity, $\varepsilon_{\rm B,eff}$, and effective
dipole magnetic field strength on a neutron star (NS) surface, $B_{\rm eff}$,
is $\varepsilon_{\rm B,eff}\sim 10^{-4}(\frac{B_{\rm eff}}{10^{14}\;\rm
G})^{2}$. This result demands that $B_{\rm eff}\sim 0.01B_{\rm t}$ with
$B_{\rm t}$ being the internal toroidal magnetic field strength of NSs. We
suggested that the torque generated during few unsymmetrical massive-star
collapses may induce differential rotations in proto-NSs to amplify the
internal toroidal fields.

###### keywords:

gravitational waves – gamma-ray burst: individual(GRB 060807, GRB 070521) –
stars: magnetars

##  1 Introduction

Rapidly rotating, distorted neutron stars (NSs) are potential gravitational-
wave sources. Many authors consider the distortion could be induced by the
anisotropic pressure from the internal magnetic field in an NS, especially the
internal toroidal magnetic field (Ostriker & Gunn, 1969; Bonazzola &
Gourgoulhon, 1996; Konno, Obata, & Kojima, 2000; Ioka, 2001; Palomba, 2001;
Cutler, 2002; Ioka & Sasaki, 2004; Stella et al., 2005; Tomimura & Eriguchi,
2005; Haskell et al., 2008; Dall’Osso, Shore, & Stella, 2009; Mastrano et al.,
2011). However, this gravitational-wave emission does not show obvious effect
in Galactic NSs (Aasi et al., 2014; Abbott et al., 2017a; Gao, Cao, & Zhang,
2017) since these NSs do not maintain strong magnetic field ($>10^{14}\;\rm
G$) and fast spin (in LIGO’s range), which are needed to produced a strong
gravitational-wave radiation, simultaneously. Soft gamma repeaters and
anomalous X-ray pulsars should be corresponding to magnetars (with inferred
effective surface magnetic dipole field strength $\sim 10^{14}-10^{15}\;\rm
G$, see Rea & Esposito 2011 for a review), but their spin periods are usually
longer than $1\;\rm s$ due to the effective brake by magnetic dipole
radiation. Besides, the decay of internal toroidal magnetic field (Li et al.,
2016; Beloborodov & Li, 2016; Gao et al., 2019) may also weaken the
distortion. Accretion may accelerate the spin of an NS, but it will induce the
decay of the magnetic field in turn (the magnetic field strengths of
millisecond pulsars are usually weaker than that of normal pulsars; e.g.,
Manchester et al. 2005). To test the effect of magnetically-induced distortion
through gravitational-wave channel, a fast rotating, strong magnetized NS is
needed.

According to the dynamo theory, Duncan & Thompson (1992) shown that a nascent
NS with millisecond period would generate an internal magnetic field up to
$10^{16}\;\rm G$ as long as the NS experiences a phase of neutrino-driven
turbulent convection111An alternative ‘powerful contender ’ is the fossil
field scenario (Ferrario & Wickramasinghe 2006, 2008; see Turolla et al. 2015
for a review).. Then the problem is which event can provide such a turbulent
condition and tell us there is a millisecond magnetar. Since gamma-ray bursts
(GRBs) originate from massive star collapses (Woosley, 1993; Paczyński, 1998;
Galama et al., 1998; MacFadyen & Woosley, 1999; Hjorth et al., 2003; Stanek et
al., 2003) and NS binary mergers (Paczynski, 1986; Goodman, Dar, & Nussinov,
1987; Eichler et al., 1989; Abbott et al., 2017b), the catastrophes of GRB
progenitors may provide such a condition to form millisecond magnetars, i.e.,
GRB prompt emissions/GRB afterglows are potential events that powered by
nascent millisecond magnetars (Usov, 1992; Dai & Lu, 1998a, b; Kluzniak &
Ruderman, 1998; Zhang & Mészáros, 2001; Dai et al., 2006; Metzger et al.,
2011; Rowlinson et al., 2013). Especially, several GRB X-ray plateaus seem to
be only explained reasonably under magnetar scenario (Du, 2020).

Although, so far, it is hard to detect the gravitational-wave radiation from a
millisecond magnetar directly (Abbott et al., 2019), under the GRB magnetar
scenario, the spindown of the magnetar will record the information of
gravitational-wave radiation which may be extracted from the electromagnetic
channel (some GRB X-ray plateaus). Therefore, these events may be used to test
the effect of magnetically-induced distortion. In this paper, we show the
method to infer the magnetically-induced distortion under two assumptions, (i)
GRB X-ray plateaus correspond to energy releases of magnetar winds solely
behind GRB jets (Du, 2020); (ii) gravitational-wave radiations of magnetars
are stimulated by magnetically-induced distortion. The merit of this method is
that once the gravitational-wave radiation from the central magnetar of a GRB
is detected, the results obtained from gravitational-wave channel and
electromagnetic channel can be double checked.

The remaining part of this work is organized as follows. We show our method to
estimate the magnetically-induced ellipticity under GRB magnetar scenario in
Section 2. We present two-case study of _Swift_ GRB samples, GRB 060807 and
GRB 070521, in Section 3. The summary is given in Section 4.

##  2 The method

The gravitational radiation produced by magnetic-field-induced distortion
depends on the tilt angle between the magnetic and spin axes (Cutler & Jones,
2001). Since the tilt angle can not be determined, as well as the realistic
magnetically-induced ellipticity, we introduce an effective magnetically-
induced ellipticity, $\varepsilon_{\rm B,eff}$, so that the luminosity of
gravitational radiation is (e.g., Maggiore 2008)

$\displaystyle L_{\rm gw}(t)=\frac{32GI^{2}\varepsilon_{\rm
B,eff}^{2}\Omega(t)^{6}}{5c^{5}}.$ (1)

The other parameters are the gravitational constant $G$, moment of inertia
$I$, speed of light $c$ and angular velocity $\Omega(t)$. Similarly, we use an
effective dipole magnetic field strength on the NS surface, $B_{\rm eff}$, to
include the deviation from the actual electromagnetic radiation, then the
power of spin-down wind is (e.g., Landau & Lifshitz 1979; Shapiro & Teukolsky
1983)

$\displaystyle L_{\rm em}(t)=\frac{B_{\rm
eff}^{2}R^{6}\Omega(t)^{4}}{6c^{3}},$ (2)

where $R$ is the equatorial radius. To connect $\varepsilon_{\rm B,eff}$ and
$B_{\rm eff}$ and facilitate comparison with theoretic results, we introduce a
parameter, $\xi_{\rm eff}$, that $\varepsilon_{\rm B,eff}=\xi_{\rm
eff}\left(\frac{B_{\rm eff}}{10^{14}\;\rm G}\right)^{2}$. Therefore, if the
brake is dominated by the gravitational radiation, there is

$\displaystyle\frac{\xi_{\rm eff}}{(10^{14}\;\rm
G)^{2}}>\sqrt{\frac{5}{192G}}\frac{R^{3}c}{I\Omega_{0}B_{\rm eff}},$ (3)

where $\Omega_{0}\equiv\Omega(t=0)$ (similarly hereinafter for the parameter
with the subscript $0$) . Otherwise, the left side of equation (3) should be
less than the term on the right.

The next step is to determine what dominates the spindown (see Appendix for
more general discussion). As mentioned in the Introduction, GRB X-ray plateaus
may be powered by the spin-down winds from their central magnetars. When the
magnetic dipole radiation dominates the spindown, i.e.,
$I\Omega\dot{\Omega}\approx-L_{\rm em}$, the power of the spin-down wind is

$\displaystyle L_{\rm em}(t)=L_{\rm em,0}\left(1+\frac{t}{\tau_{\rm
em}}{}\right)^{-2},$ (4)

where

$\displaystyle\tau_{\rm em}=\frac{3c^{3}I}{B_{\rm
eff}^{2}R^{6}\Omega_{0}^{2}};$ (5)

When the gravitational radiation dominates the spindown, i.e.,
$I\Omega\dot{\Omega}\approx-L_{\rm gw}$, the power of the spin-down wind is

$\displaystyle L_{\rm em}(t)=L_{\rm em,0}\left(1+\frac{t}{\tau_{\rm
gw}}{}\right)^{-1},$ (6)

where

$\displaystyle\tau_{\rm gw}=\frac{5c^{5}}{128GI\varepsilon_{\rm
B,eff}^{2}\Omega_{0}^{4}}.$ (7)

Under the magnetar scenario, GRB X-ray plateaus must track the time evolution
of the spin of central NSs (Du, 2020), so that one can derive which mechanism
dominates the spindown through the slope of the X-ray light curve following
the plateau. However, at this point, we can’t use equation (3) to constrain
$\xi_{\rm eff}$ since the parameters on the right side of this equation are
uncertain.

Note that, under the magnetic-dipole-radiation-dominated case, the luminosity
of the GRB X-ray afterglow that powered by the spin-down wind should be
smaller than the spin-down power itself, i.e.,

$\displaystyle L_{\rm x}(t)\leq L_{\rm em}(t)$ (8)

and the total energy of the GRB X-ray afterglow that powered by the spin-down
wind should be smaller than the rotational energy of the central NS, i.e.,

$\displaystyle\int_{t_{0}}^{t_{\rm e}}L_{\rm
x}(t)dt\leq\frac{1}{2}I\Omega_{0}^{2}.$ (9)

As shown in Du (2020), to power an X-ray plateau, the dissipation of the spin-
down wind must track the spin-down rate, i.e., the magnetic energy of the
spin-down wind should be almost released in X-ray emission totally, so we can
approximately equal the left and right sides of equations (8) and (9),
respectively. Therefore, the uncertain parameters on the right side of
equation (3) can be connected to observable quantities (Du et al., 2019a),
i.e.,

$\displaystyle\Omega_{0}\approx\left(\frac{2L_{\rm em,0}\tau_{\rm
em}}{I}\right)^{1/2},$ (10)

and

$\displaystyle B_{\rm eff}\approx\left(\frac{3c^{3}I^{2}}{2L_{\rm
em,0}R^{6}\tau_{\rm em}^{2}}\right)^{1/2}.$ (11)

In addition, the power of gravitational radiation decays faster than that of
the magnetic dipole radiation (see equations (1) and (2)). When the spindown
is dominated by gravitational radiation initially, there will be a moment
$\tau_{\ast}$ that the spindown dominated by gravitational radiation will
transform into the case of magnetic-dipole-radiation domination (Zhang &
Mészáros, 2001; Lasky & Glampedakis, 2016), i.e.,

$\displaystyle\tau_{\ast}=\frac{\tau_{\rm em}}{\tau_{\rm gw}}\left(\tau_{\rm
em}-2\tau_{\rm gw}\right).$ (12)

From equation (12), $\tau_{\rm em}$ can be read as

$\displaystyle\tau_{\rm em}=\tau_{\rm gw}+\sqrt{\tau_{\rm
gw}^{2}+\tau_{\ast}\tau_{\rm gw}}.$ (13)

The extreme sample of this case is that the slope of the X-ray light curve
decays as $0\rightarrow\sim-1\rightarrow\sim-2$ (see equations (4) and (6)).
If there is such a sample, one can obtain $\tau_{\rm gw}$ and $\tau_{\ast}$,
as well as $\tau_{\rm em}$, through data fitting, so that equations (10) and
(11) are available and $\varepsilon_{\rm B,eff}$ can be directly connected to
observable quantities (Du et al., 2019b). Dividing equation (5) by equation
(7), there is

$\displaystyle\frac{\xi_{\rm eff}}{(10^{14}\;\rm
G)^{2}}=\sqrt{\frac{5\tau_{\rm em}}{384\tau_{\rm
gw}G}}\frac{R^{3}c}{I\Omega_{0}B_{\rm eff}}.$ (14)

Substituting equations (10) and (11) into equation (14), there is

$\displaystyle\frac{\xi_{\rm eff}}{(10^{14}\;\rm
G)^{2}}=\sqrt{\frac{5\tau_{\rm em}^{2}}{1152G\tau_{\rm
gw}c}}\frac{R^{6}}{I\sqrt{I}}.$ (15)

Once $R$ and $I$ are obtained from the equation of state (EoS) of NSs,
$\xi_{\rm eff}$ could be estimated through equation (15).

##  3 two-case study

Equation (15) shows $\xi_{\rm eff}$ depends on the equatorial radius and
moment of inertia of the NS. At present, the EoS of NSs is still uncertain, as
well as $R$ and $I$ of an NS with known mass. However, observations have
provided substantial information to help us choosing appropriate value of
parameters, so that the result given by equation (15) keeps rationality.

GRB/type-Ic supernova associations (Galama et al., 1998; Hjorth et al., 2003;
Stanek et al., 2003) indicate that long GRBs (duration $>2\;\rm s$) originate
from massive star collapses. According to the mass distribution of the pulsars
in Milky Way (see Özel & Freire 2016 for a review), the mass of the
millisecond NS associated with a long GRB could be $\sim 1.4\;\rm M_{\odot}$.
The direct measurement of NS masses (Demorest et al., 2010; Antoniadis et al.,
2013; Cromartie et al., 2019) and the observations/theoretical studies to the
binary NS merger (Abbott et al., 2017b; Margalit & Metzger, 2017; Annala et
al., 2018; Hajela et al., 2021) show that the equatorial radius and moment of
inertia of a millisecond NS with mass $\sim 1.4\;\rm M_{\odot}$ could be
$R\sim 17\;\rm km$ and $I=3\times 10^{45}\;\rm g\cdot cm^{2}$ (Cipolletta et
al., 2015; Riahi et al., 2019).

Figure 1: The fitting result of two _Swift_ GRB samples.  Table 1: The fitting result Sample | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\tau_{\rm gw}({\rm s})$ | $\tau_{\ast}({\rm s})$
---|---|---|---|---|---
GRB 060807 | -0.05 | 1.20 | 2.20 | 5132 | 36721
GRB 070521 | -0.17 | 0.85 | 2.34 | 362 | 12894

According to the discussion shown in Section 2, we select two _Swift_ samples
(see Figure 1; Evans et al. 2007, 2009), GRB 060807 (with duration
$T_{90}\approx 34\;\rm s$; Barthelmy et al. 2006) and GRB 070521
($T_{90}\approx 37.9\;\rm s$; Palmer et al. 2007), to estimate $\xi_{\rm
eff}$. We use the following function (e.g., Yu et al. 2010) to fit the data,

$\displaystyle Flux=F\left[\left(\frac{t}{\tau_{\rm
gw}}\right)^{\omega\alpha_{1}}+\left(\frac{t}{\tau_{\rm
gw}}\right)^{\omega\alpha_{2}}+\left(\frac{\tau_{*}}{\tau_{\rm
gw}}\right)^{\omega\alpha_{2}}\left(\frac{t}{\tau_{*}}\right)^{\omega\alpha_{3}}\right]^{-1/\omega}.$
(16)

The fitting result is shown in Table 1. According to equations (13) and (15)
and Table 1, we have $\xi_{\rm eff}({\rm GRB\;060807})=5.09\times 10^{-4}$ and
$\xi_{\rm eff}({\rm GRB\;070521})=2.47\times 10^{-4}$. In addition, according
to the measured effective surface magnetic dipole fields of Galactic magnetars
($10^{14}-10^{15}\;\rm G$; Rea & Esposito 2011; see Du et al. 2019a for the
GRB situation), one can estimate that $\varepsilon_{\rm B,eff}\sim
10^{-4}-10^{-2}$.

Our result shows that $\varepsilon_{\rm B,eff}\sim 10^{-4}(\frac{B_{\rm
eff}}{10^{14}\;\rm G})^{2}$. As a contrast, theoretical work (e.g, Cutler
2002) predicts that the ellipticity, $\varepsilon_{\rm B}$, induced by the
internal toroidal field, $B_{\rm t}$, satisfies $\varepsilon_{\rm B}\sim
10^{-4}\left(\frac{B_{\rm t}}{10^{16}\;\rm G}\right)^{2}$. If the two
conclusions are compatible, there should be a relation that $B_{\rm eff}\sim
0.01B_{\rm t}$. This is consistent with the requirement in modeling AXPs/SGRs
(Thompson & Duncan 1993, 1995) and the implication of the phase modulation
observed for two magnetars (Makishima et al., 2014, 2016) that the internal
toroidal fields of magetars can be up to $10^{16}\;\rm G$.

##  4 Summary

In this paper, we aim at investigating how strong an NS can be distorted by
magnetic field. We propose a method to test magnetic-induced distortion of NSs
through GRB X-ray plateaus under GRB magnetar scenario222The afterglows like
that of GRB 060807 and GRB 070521 may also be explained as that the transition
of the decay slope from $-1$ to $-2$ is just the jet break of the GRB jet with
a smaller opening angle (Rhoads, 1999) and the plateau is caused by the jet
viewed slightly off-axis (Beniamini et al., 2019). . If the distortion is
strong enough, the corresponding gravitational radiation can dominate the
initial NS spindown and affect the time evolution of the NS electromagnetic
radiation, i.e., the decay of GRB X-ray luminosity (with decay index from
$\sim 0\rightarrow\sim 1\rightarrow\sim 2$). Thus, the ellipticity may be
connected to some measurable and inferred parameters. According to the two-
case study on GRB 060807 and GRB 070521, we find that $\varepsilon_{\rm
B,eff}\sim 10^{-4}(\frac{B_{\rm eff}}{10^{14}\;\rm G})^{2}$. To be consistent
with the theoretical result, e.g., $\varepsilon_{\rm B}\sim
10^{-4}(\frac{B_{\rm t}}{10^{16}\;\rm G})^{2}$, our result shows that there
should be $B_{\rm eff}\sim 0.01B_{\rm t}$. This deduction could be used as the
initial internal magnetic field structure to model the external magnetosphere
of an NS.

Large toroidal fields may induce a certain instability for given magnetic
configuration (Akgün et al, 2013) and lead to a short-duration magnetar wind
under another relation between $B_{\rm t}$ and $B_{\rm eff}$, e.g., $B_{\rm
t}=10B_{\rm eff}$ adopt in Beniamini & Lu (2021) (so that if $B_{\rm t}>\sim
10^{16}\rm\;G$, there should be $B_{\rm eff}>\sim 10^{15}\;\rm G$ which leads
to strong brake through magnetic dipole radiation) . According to the
observation of Galactic magnetars (Rea & Esposito, 2011) and the model-
independent constraint on the magnetic field for GRB magnetar scenario (Du et
al., 2019a), we tend to use $B_{\rm eff}<\sim 10^{15}\;\rm G$ as shown above.
The larger disparity between $B_{\rm eff}$ and $B_{\rm t}$ shown in this paper
depends on the premise that the early-time evolution of afterglows of GRB
060807 and GRB 070521 are dominated by gravitational-wave radiation and the
requirement that our result is consistent with the theoretic predict, i.e.,
$\varepsilon_{\rm B}\sim 10^{-4}(\frac{B_{\rm t}}{10^{16}\;\rm G})^{2}$.
Despite the above uncertainties, strong toroidal fields seem to be inevitable
(Ciolfi & Rezzolla, 2013). Under GRB situation, the possible unsymmetrical
collapses and explosions of long GRB progenitors (for the rare cases, e.g.,
GRB 060807 and GRB 070521) may induce an extra torque which results in
differential rotations between the interiors and exteriors of proto-NSs to
amplify the internal toroidal fields. In the future, several supermassive star
collapses and rare binary NS mergers in $\sim 100\;\rm Mpc$ may provide
changes to test the result shown in this paper (Abbott et al., 2017).

##  5 Acknowledgement

We thank the referee for useful comments. We acknowledge the use of the public
data from the Swift data archives. This work was supported by the National Key
R&D Program of China (Grant No. 2017YFA0402602), the National Natural Science
Foundation of China (Grant Nos. 11673002, and U1531243), and the Strategic
Priority Research Program of Chinese Academy Sciences (Grant No. XDB23010200).

## Appendix

Figure 2: The time evolution of $L_{\rm em}$ and $L_{\rm gw}$. The thicker
lines correspond to the parameters that $\varepsilon_{\rm B,eff}=2\times
10^{-3}$, $B_{\rm eff}=2\times 10^{14}\;\rm G$, $I=3\times 10^{45}\;\rm g\cdot
cm^{2}$ and $\Omega_{0}=6280\;\rm rad\cdot s^{-1}$. The thinner lines
correspond to the same parameters as that of thicker lines except
$\varepsilon_{\rm B,eff}=5\times 10^{-4}$.

To show the general evolution of the spindown, we analytically solve the
equation, $I\Omega\dot{\Omega}=-L_{\rm em}-L_{\rm gw}$, and get333Ho (2016)
missed $\gamma_{\rm e}$ in the logarithmic term in his equation (17).

$\displaystyle t=\left[\frac{\gamma_{\rm
e}}{2\beta^{2}}\ln\left(\frac{\beta+\gamma_{\rm e}\Omega(t)^{2}}{\gamma_{\rm
e}\Omega(t)^{2}}\right)-\frac{1}{2\beta\Omega(t)^{2}}\right]\bigg{|}_{\Omega}^{\Omega_{0}},$
(17)

Where

$\displaystyle\beta=\frac{B_{\rm eff}^{2}R^{6}}{6Ic^{3}},$ (18)
$\displaystyle\gamma_{\rm e}=\frac{32GI\varepsilon_{\rm B,eff}^{2}}{5c^{5}}.$
(19)

For comparison, we show two different situations that gravitational-radiation
domination initially (thicker lines) and magnetic-dipole-radiation domination
initially (thinner lines) in Fig. 2.

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