# Symplectic geometric flows

Teng Fei and Duong H. Phong

###### Abstract

Several geometric flows on symplectic manifolds are introduced which are
potentially of interest in symplectic geometry and topology. They are
motivated by the Type IIA flow and T-duality between flows in symplectic
geometry and flows in complex geometry. Examples include the Hitchin gradient
flow on symplectic manifolds, and a new flow which is called the dual Ricci
flow.

##  1 Introduction

Geometric flows are now well-recognized as a powerful tool for geometry and
topology. Major successes include the Eells-Sampson theorem on harmonic maps
[ES64], Hamilton’s Ricci flow [Ham82] and Perelman’s proof of the Poincaré
conjecture [Per02, Per03b, Per03a], Donaldson’s heat flow proof of the
Donaldson-Uhlenbeck-Yau theorem on Hermitian-Yang-Mills connections [Don87],
the differentiable sphere theorem of Brendle-Schoen [BS09], and many other
developments. However, these successes were mostly in the settings of
Riemannian and complex geometry, leaving the subject of symplectic geometric
flows underdeveloped. In recent years, a few geometric flows adapted to
symplectic geometry have been introduced, such as Lê-Wang’s anti-complexified
Ricci flow [LW01], Streets-Tian’s symplectic curvature flow [ST14], and He’s
flow of non-degenerate $2$-forms [He21]. Although there has been much progress
on these flows, applications of geometric flows in symplectic geometry and
topology have remained relatively few.

In [FPPZc], in joint work with S. Picard and X.W. Zhang, we introduced the
Type IIA flow for symplectic Calabi-Yau 6-manifolds, motivated by the Type IIA
string equations proposed in e.g. [TY14]. This was further developed in
[FPPZa, FPPZ21, Raf21]. In particular, in [FPPZ21], we successfully applied
the Type IIA flow to prove the stability of the Kähler property for Calabi-Yau
3-folds under symplectic deformations.

The goal of this paper is to introduce some new geometric flows which may
potentially be of interest in symplectic geometry and topology. They are
motivated by duality considerations as well as by the Type IIA flow, to which
several of them are actually closely related. More specifically, in [FP21], S.
Picard and the first author had introduced the dual Anomaly flow as the T-dual
of the Type IIB flow.111This flow coincides with the special case of the
Anomaly flow [PPZ18b, PPZ18a] with slope parameter $\alpha^{\prime}=0$, and
”Type IIB flow” is a more precise name, since there is no longer a gauge field
and no Green-Schwarz Anomaly cancellation mechanism [Pho20, FPPZb]. Both the
Type IIB flow and the dual Anomaly flow are flows in complex geometry, and
they are only dual to each other up to lower order terms. Here we introduce
instead the principle that by applying T-duality, we can derive a flow in
symplectic geometry from a flow in complex geometry. In particular, we show
that the Type IIA flow and the Type IIB flow are indeed related to each other
by T-duality, as suggested by mirror symmetry, and the duality is now exact
without lower order terms. We also derive the T-dual of the Kähler Ricci flow
in the symplectic setting. In Section 3, we propose the study of the gradient
flow of Hitchin’s functional in dimension six. In the presence of a compatible
symplectic form, this flow takes a similar expression to that of the Type IIA
flow. Moreover, it induces a flow of a pair which can be viewed as the anti-
complexified Ricci flow with lower order corrections coupled to a scalar
function. In Section 4 we present a few explicit examples of the gradient flow
of 6D-Hitchin’s functional, showing that the flow can be used to find optimal
almost complex structures on certain locally homogeneous symplectic half-flat
manifolds. These examples naturally provide eternal and convergent solutions
to the anti-complexified Ricci flow.

Acknowledgements The authors dedicate this paper to the beloved Professor
Victor Guillemin. Some ideas behind this paper date back to the many
conversations the first-named author had with Victor when he was a graduate
student at MIT. The second-named author would like to take this opportunity to
express his gratitude for the inspiration he got from Professor Guillemin’s
works, especially in symplectic geometry and partial differential equations,
and for the frequent encouragement he received from him over the years.

##  2 The Type IIA and the Type IIB flows

In this section, we would like to present a general principle by which a flow
in symplectic geometry should arise by T-duality from a flow in complex
geometry. We illustrate this principle by establishing the duality of the Type
IIA and the Type IIB flows in the semi-flat case, which is the case when
T-duality can be implemented explicitly.

Let us first recall the set-up of semi-flat geometry, in the notations from
[FP21]. Let $B$ be a 3-dimensional compact special integral affine manifold.
We can cover $B$ by local coordinates $\\{x^{1},x^{2},x^{3}\\}$ such that the
transition functions are valued in the group
$\textrm{SL}(3,\mathbb{Z})\ltimes\mathbb{R}^{3}$. Let $TB$ be the tangent
bundle of $B$, with $\\{y^{1},y^{2},y^{3}\\}$ the natural coordinates of
tangent directions. Clearly the fiberwise lattice

$\check{\Lambda}=\mathbb{Z}\frac{\partial}{\partial
x^{1}}+\mathbb{Z}\frac{\partial}{\partial
x^{2}}+\mathbb{Z}\frac{\partial}{\partial
x^{3}}=\\{(y^{1},y^{2},y^{3}):y^{j}\in\mathbb{Z}\\}$

is well-defined, and we may form the quotient $\check{X}=TB/\check{\Lambda}$,
which is a smooth $T^{3}$-fibration over $B$. Moreover, $\check{X}$ is
naturally equipped with a complex structure such that
$\\{z^{j}=x^{j}+iy^{j}\\}_{j=1}^{3}$ forms a set of local holomorphic
coordinates. For our purpose, we shall also assume that $B$ is equipped with a
Hessian metric $g$ with local potential $\phi$. In other words, there exists a
local convex function $\phi$ such that

$g_{jk}=\frac{\partial^{2}\phi}{\partial x^{j}\partial x^{k}}.$

We shall also consider the dual affine coordinates
$\left\\{x_{j}=\dfrac{\partial\phi}{\partial x^{j}}\right\\}$ on $B$ under
Legendre transformation. Let $\\{y_{1},y_{2},y_{3}\\}$ be the natural
coordinates for the cotangent directions associated to the local chart
$\\{x^{1},x^{2},x^{3}\\}$. It is not hard to check that the fiberwise lattice

$\hat{\Lambda}=\mathbb{Z}\mathrm{d}x^{1}+\mathbb{Z}\mathrm{d}x^{2}+\mathbb{Z}\mathrm{d}x^{3}=\\{(y_{1},y_{2},y_{3}):y_{k}\in\mathbb{Z}\\}$

is well-defined, and we may form the fiberwise quotient
$\hat{X}=T^{*}B/\hat{\Lambda}$, which is also a smooth $T^{3}$-fibration over
$B$. The natural symplectic form on $T^{*}B$ descends to a symplectic form on
$\hat{X}$, which in local coordinates can be expressed as

$\omega=\mathrm{d}x^{1}\wedge\mathrm{d}y_{1}+\mathrm{d}x^{2}\wedge\mathrm{d}y_{2}+\mathrm{d}x^{3}\wedge\mathrm{d}y_{3}.$

Moreover, $\hat{X}$ is equipped with a holomorphic volume form
$\Omega=\varphi+i\hat{\varphi}=\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\wedge\mathrm{d}z_{3}$
for $z_{j}=x_{j}+iy_{j}$. In local coordinates we have

$\begin{split}\varphi&=\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}-\mathrm{d}x_{1}\wedge\mathrm{d}y_{2}\wedge\mathrm{d}y_{3}-\mathrm{d}y_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}y_{3}-\mathrm{d}y_{1}\wedge\mathrm{d}y_{2}\wedge\mathrm{d}x_{3},\\\
\hat{\varphi}&=\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}y_{3}+\mathrm{d}x_{1}\wedge\mathrm{d}y_{2}\wedge\mathrm{d}x_{3}+\mathrm{d}y_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}-\mathrm{d}y_{1}\wedge\mathrm{d}y_{2}\wedge\mathrm{d}y_{3}.\end{split}$

Now we fix the affine structure on $B$ associated to the local coordinate
$\\{x^{1},x^{2},x^{3}\\}$ and vary the Hessian metric $g$ (or equivalently
vary the potential $\phi$). The symplectic form $\omega$, coming from the
canonical one on $T^{*}B$, does not change. However, the 3-forms $\varphi$ and
$\hat{\varphi}$ are varying since the dual affine structure associated to
$\\{x_{1},x_{2},x_{3}\\}$ is changing.

In particular, if we run the Type IIB flow on $\check{X}$, by the calculation
in [FP21], the flow reduces to the real Monge-Ampère flow of Hessian metrics

$\partial_{t}g_{jk}=\frac{1}{4}\frac{\partial^{2}}{\partial x^{j}\partial
x^{k}}\det g.$ (1)

The goal in this section is to prove the following:

######  Theorem 2.1.

Under the Type IIB flow (1), the associated 3-form $\varphi$ satisfies the
Type IIA flow [FPPZc]

$\partial_{t}\varphi=\frac{1}{16}\mathrm{d}\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi}).$
(2)

###### Proof.

By the definition of Legendre transform, we have

$\frac{\partial x_{j}}{\partial x^{k}}=\frac{\partial^{2}\phi}{\partial
x^{j}\partial x^{k}}=g_{jk},$

hence

$\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}=\det
g\cdot\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}.$

On the other hand, as
$\varphi\wedge\hat{\varphi}=|\varphi|^{2}\dfrac{\omega^{3}}{3!}$, we see that

$|\varphi|^{2}=4\det g.$

Because $\Omega$ is holomorphic, both $\varphi$ and $\hat{\varphi}$ are
closed. It follows that
$\mathrm{d}(|\varphi|^{2}\hat{\varphi})=\mathrm{d}|\varphi|^{2}\wedge\hat{\varphi}$,
and that

$\begin{split}&\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi})=\sum_{j}\frac{\partial|\varphi|^{2}}{\partial
x^{j}}\iota_{\partial y_{j}}\hat{\varphi}\\\
=\quad&\frac{\partial|\varphi|^{2}}{\partial
x^{3}}\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}-\frac{\partial|\varphi|^{2}}{\partial
x^{2}}\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}+\frac{\partial|\varphi|^{2}}{\partial
x^{1}}\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}\\\
&-\frac{\partial|\varphi|^{2}}{\partial
x^{3}}\mathrm{d}y_{1}\wedge\mathrm{d}y_{2}+\frac{\partial|\varphi|^{2}}{\partial
x^{2}}\mathrm{d}y_{1}\wedge\mathrm{d}y_{3}-\frac{\partial|\varphi|^{2}}{\partial
x^{1}}\mathrm{d}y_{2}\wedge\mathrm{d}y_{3}.\end{split}$

Consequently

$\begin{split}&\mathrm{d}\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi})=\left(\frac{\partial^{2}|\varphi|^{2}}{\partial
x_{1}\partial x^{1}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{2}\partial
x^{2}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{3}\partial
x^{3}}\right)\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}\\\
-&\mathrm{d}\left(\frac{\partial|\varphi|^{2}}{\partial
x^{1}}\right)\wedge\mathrm{d}y_{2}\wedge\mathrm{d}y_{3}-\mathrm{d}y_{1}\wedge\mathrm{d}\left(\frac{\partial|\varphi|^{2}}{\partial
x^{2}}\right)\wedge\mathrm{d}y_{3}-\mathrm{d}y_{1}\wedge\mathrm{d}y_{2}\wedge\mathrm{d}\left(\frac{\partial|\varphi|^{2}}{\partial
x^{3}}\right).\end{split}$

Therefore, to prove the theorem, we only need to show that

$\begin{split}\partial_{t}(\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3})=&\frac{1}{16}\left(\frac{\partial^{2}|\varphi|^{2}}{\partial
x_{1}\partial x^{1}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{2}\partial
x^{2}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{3}\partial
x^{3}}\right)\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3},\\\
\partial_{t}(\mathrm{d}x_{j})=&\frac{1}{16}\mathrm{d}\left(\frac{\partial|\varphi|^{2}}{\partial
x^{j}}\right)\textrm{ for }j=1,2,3.\end{split}$

For the first identity above, we notice that

$\frac{1}{16}\sum_{j}\frac{\partial}{\partial
x_{j}}\left(\frac{\partial|\varphi|^{2}}{\partial
x^{j}}\right)=\frac{1}{4}\sum_{j,k}\frac{\partial x^{k}}{\partial
x_{j}}\frac{\partial^{2}}{\partial x^{k}\partial x^{j}}\det
g=\sum_{j,k}g^{jk}\partial_{t}g_{jk}=\frac{\partial_{t}\det g}{\det g}.$

Therefore

$\begin{split}&\partial_{t}(\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3})=(\partial_{t}\det
g)\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\\\ =~{}&\frac{\det
g}{16}\left(\frac{\partial^{2}|\varphi|^{2}}{\partial x_{1}\partial
x^{1}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{2}\partial
x^{2}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{3}\partial
x^{3}}\right)\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\\\
=~{}&\frac{1}{16}\left(\frac{\partial^{2}|\varphi|^{2}}{\partial x_{1}\partial
x^{1}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{2}\partial
x^{2}}+\frac{\partial^{2}|\varphi|^{2}}{\partial x_{3}\partial
x^{3}}\right)\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}.\end{split}$

From the evolution equation of $g$, we know that there is an affine function
$l$ such that

$\partial_{t}\phi=\frac{1}{4}\det g+l,$

hence

$\partial_{t}(\mathrm{d}x_{j})=\mathrm{d}\left(\partial_{t}\frac{\partial\phi}{\partial
x^{j}}\right)=\mathrm{d}\left(\frac{\partial}{\partial
x^{j}}\left(\partial_{t}\phi\right)\right)=\frac{1}{4}\mathrm{d}\left(\frac{\partial\det
g}{\partial
x^{j}}\right)=\frac{1}{16}\mathrm{d}\left(\frac{\partial|\varphi|^{2}}{\partial
x^{j}}\right).$

∎

######  Remark 2.2.

In [FP21], the dual Anomaly flow was defined as a flow of Hermitian metrics on
the dual manifold. Thus the setting for both the original flow and its dual
were in complex geometry, and the duality only holds modulo lower order terms.
From this point of view, the symplectic and the geometric settings as
described above are more naturally dual, as the duality is now exact.

######  Remark 2.3.

In the above theorem, we made the choice of the phase angle so that $\varphi$
and $\hat{\varphi}$ have the expressions we worked with. In fact, it is well
legitimate to choose $\varphi^{\prime}=\hat{\varphi}$ and
$\hat{\varphi}^{\prime}=-\varphi$ instead. By similar and straightforward
computation, one can show that $\varphi^{\prime}$ also satisfies the Type IIA
flow

$\partial_{t}\varphi^{\prime}=\frac{1}{16}\mathrm{d}\Lambda_{\omega}\mathrm{d}(|\varphi^{\prime}|^{2}\hat{\varphi}^{\prime}),$

therefore Theorem 2.1 holds for arbitrary choice of the phase angle.

In Theorem 2.1, we showed that the Type IIA flow and Type IIB flow are related
to each other by T-duality in the semi-flat limit. Following this idea, once
we have a geometric flow in the complex setting, by applying T-duality, one
may arrive at a natural flow in symplectic geometry. In particular, we can
find the T-dual of the Kähler-Ricci flow in the symplectic world.

In [FP21], the semi-flat reduction of the Kähler-Ricci flow was shown to be
given by

$\partial_{t}g_{jk}=\frac{1}{2}\frac{\partial^{2}}{\partial x^{j}\partial
x^{k}}\log\det g.$ (3)

A similar calculation as in Theorem 2.1 yields the following theorem.

######  Theorem 2.4.

Under the Kähler-Ricci flow (3), the associated 3-form $\varphi$ satisfies the
flow

$\partial_{t}\varphi=\frac{1}{2}\mathrm{d}\Lambda_{\omega}\mathrm{d}(\log|\varphi|^{2}\cdot\hat{\varphi}).$
(4)

By dropping the non-essential factor of $\dfrac{1}{2}$, we shall call the flow

$\begin{cases}&\partial_{t}\varphi=\mathrm{d}\Lambda_{\omega}\mathrm{d}(\log|\varphi|^{2}\cdot\hat{\varphi})\\\
&\varphi_{t=0}=\varphi_{0}\textrm{ is a closed, primitive, and positive
$3$-form}\end{cases}$ (5)

the _dual Ricci flow_.

Due to the presence of the factor $\log|\varphi|^{2}$ in (5), one can imagine
that the short-time existence and uniqueness of (5) would be very challenging
to establish, if there is such a theorem.

##  3 The Hitchin gradient flow on a symplectic manifold

We begin by recalling the functional introduced by Hitchin [Hit00]. Let $M$ be
an oriented compact 6-manifold. Following [Hit00, Fei15], given any positive
3-form $\varphi$ on $M$, or equivalently any reduction of the structure group
of $M$ to $\mathrm{SL}(3,\mathbb{C})$, there is a naturally associated almost
complex structure $J_{\varphi}$, and another 3-form
$\hat{\varphi}=J_{\varphi}\varphi$, such that the form
$\Omega=\varphi+\sqrt{-1}\hat{\varphi}$ is a nowhere vanishing (3,0)-form with
respect to the almost complex structure $J_{\varphi}$.

The Hitchin functional is defined as

$H(\varphi)=\frac{1}{2}\int_{M}\varphi\wedge\hat{\varphi}.$

Assume now that $\varphi$ is closed. Hitchin proposed the variational problem
of finding the critical points of $H(\varphi)$, subject to the constraint of
$\varphi$ being in a given de Rham cohomology class. In particular, he showed
in [Hit00] that

$\delta H=\int_{M}\delta\varphi\wedge\hat{\varphi}.$

Hence the critical points of $H(\varphi)$ are exactly those $\varphi$ such
that $\mathrm{d}\hat{\varphi}=0$, or equivalently, such that $J_{\varphi}$ is
integrable.

We would like to approach this problem by considering a gradient flow for the
Hitchin functional. For this, we need to introduce a metric on $M$. A natural
way to do so is to put a symplectic form $\omega$ on $M$. As shown in [FPPZc],
the almost complex structure $J_{\varphi}$ is compatible with $\omega$ if and
only if $\varphi$ is primitive with respect to $\omega$. In this way, we can
rewrite Hitchin’s variational formula as

$\delta H=\int_{M}(\delta\varphi,\varphi)\frac{\omega^{3}}{3!}.$

Since the cohomology class of $\varphi$ is not changing, we may write
$\delta\varphi=\mathrm{d}\delta\beta$, so

$\delta
H=\int_{M}\langle\mathrm{d}\delta\beta,\varphi\rangle\frac{\omega^{3}}{3!}=\int_{M}\langle\delta\beta,\mathrm{d}^{\dagger}\varphi\rangle\frac{\omega^{3}}{3!}.$

Consequently the gradient flow of Hitchin’s functional is
$\partial_{t}\beta=\mathrm{d}^{\dagger}\varphi$, or

$\partial_{t}\varphi=\partial_{t}\mathrm{d}\beta=\mathrm{d}\mathrm{d}^{\dagger}\varphi.$

In this form, the gradient flow of the Hitchin functional on a compact
$6$-dimensional symplectic manifold can be viewed as the $6$-dimensional
version of Bryant’s Laplacian flow on $7$-manifolds, whose stationary points
are given by manifolds with $G_{2}$-holonomy [Bry06].

Our first observation is that the symplectic version of the gradient flow for
the Hitchin functional is actually a degenerate version of the Type IIA flow
introduced in [FPPZc]:

######  Theorem 3.1.

Let $M$ be a compact $6$-dimensional manifold equipped with a symplectic form
$\omega$. Then the gradient flow of the Hitchin functional with a closed,
primitive, and positive initial $3$-form can be equivalently expressed as

$\partial_{t}\varphi=\mathrm{d}\Lambda_{\omega}\mathrm{d}\hat{\varphi}$ (6)

###### Proof.

Note that, formally, the flow (6) preserves the closedness and primitiveness
of $\varphi$. It suffices to prove the following identity

$\mathrm{d}\mathrm{d}^{\dagger}\varphi=\mathrm{d}\Lambda_{\omega}\mathrm{d}\hat{\varphi}.$

for any closed primitive positive $3$-form $\varphi$. Since
$\mathrm{d}^{\dagger}\varphi=-\\!*\\!\mathrm{d}\\!*\\!\varphi=-\\!*\\!\mathrm{d}\hat{\varphi}$,
we only need to show that
$-\\!*\\!\mathrm{d}\hat{\varphi}=\Lambda_{\omega}\mathrm{d}\hat{\varphi}$. But
it follows from [FPPZc, Lemma 18] that $\mathrm{d}\hat{\varphi}$ is the
product of $\omega$ with a primitive $(1,1)$-form. ∎

We observe that, just as the gradient flow of the $6$-dimensional Hitchin
functional is related to the Type IIA flow by Theorem 3.1, there is also a
7-dimensional version of Hitchin’s functional, whose gradient flow is exactly
Bryant’s Laplacian flow for closed $\mathrm{G}_{2}$-structures, as explained
in [Lot20].

We show next that the symplectic flow (6) is weakly parabolic. For this we
need to determine the eigenvalues of its principal symbol. We follow the
procedure developed in [FPPZc] for the Type IIA flow. We start from the
following formula derived in [FPPZc]

$\delta\hat{\varphi}=-J_{\varphi}(\delta\varphi)-2{\langle\delta\varphi,\hat{\varphi}\rangle\over|\varphi|^{2}}\varphi+2{\langle\delta\varphi,\varphi\rangle\over|\varphi|^{2}}\hat{\varphi}.$
(7)

Since the symbol of the exterior differential is $\xi\wedge{\cdot}$, it
follows that the symbol of the operator
$\varphi\to\mathrm{d}\Lambda_{\omega}\mathrm{d}(\hat{\varphi})$ is given by

$\delta\varphi\ \to\
\xi\wedge\bigg{\\{}\Lambda_{\omega}\left[\xi\wedge(-J_{\varphi}\delta\varphi-2{\langle\delta\varphi,\hat{\varphi}\rangle\over|\varphi|^{2}}\varphi+2{\langle\delta\varphi,\varphi\rangle\over|\varphi|^{2}}\hat{\varphi})\right]\bigg{\\}}.$
(8)

It is no loss of generality to assume that

$\xi=e^{1},\qquad|\varphi|=1,$ (9)

to write $J_{\varphi}=J$, $\Lambda_{\omega}=\Lambda$, and to work on an
adapted frame for $\varphi$, where we have, as shown in [FPPZc]

$\begin{split}&{\omega}=e^{12}+e^{34}+e^{56}\\\ &\varphi={1\over
2}(e^{135}-e^{146}-e^{245}-e^{236}),\quad\hat{\varphi}={1\over
2}(e^{136}+e^{145}+e^{235}-e^{246}).\end{split}$

Note that in the frame $\\{e_{j}\\}$ of vectors, we have $Je_{2k-1}=e_{2k}$,
$J(e_{2k})=-e_{2k-1}$, so in this co-frame, we have $J(e^{2k-1})=-e^{2k}$,
$J(e^{2k})=e^{2k-1}$. We need to identify the eigenvalues of this operator,
restricted to the subspace $W$ of $3$-forms $\delta\varphi$ satisfying the
constraints resulting from $\varphi$ being closed and primitive

$W=\\{\delta\varphi;\quad\xi\wedge\delta\varphi=0,\quad\Lambda(\delta\varphi)=0\\},$
(10)

which can be worked out to be

$W=e^{1}\wedge W^{\prime}$ (11)

where $W^{\prime}$ is the space of $2$-forms $\gamma$ on the vector space
$V^{\prime}$ spanned by $\\{e_{j}\\}_{j=3}^{6}$, which satisfies the
constraint $\Lambda^{\prime}\gamma=0$, where $\Lambda^{\prime}$ is the Hodge
contraction operator on $V^{\prime}$ with respect to the symplectic form
${\omega}^{\prime}=e^{34}+e^{56}$. Next we compute

$\begin{split}&\Lambda[e^{1}\wedge(J\delta\varphi)]=\Lambda[e^{1}\wedge(-e^{2}\wedge
J\gamma)]=-J\gamma\\\ &\Lambda[e^{1}\wedge\hat{\varphi}]={1\over
2}(e^{35}-e^{46})\\\ &\Lambda[e^{1}\wedge\varphi]=-{1\over
2}(e^{45}+e^{36}).\end{split}$

We also work out the inner products $\langle\delta\varphi,\varphi\rangle$ and
$\langle\delta\varphi,\hat{\varphi}\rangle$,

$\begin{split}&\langle\delta\varphi,\varphi\rangle={1\over 2}\langle
e^{1}\wedge\gamma,e^{135}-e^{146}\rangle={1\over
2}\langle\gamma,e^{35}-e^{46}\rangle\\\
&\langle\delta\varphi,\hat{\varphi}\rangle={1\over 2}\langle
e^{1}\wedge\gamma,e^{136}+e^{145}\rangle={1\over
2}\langle\gamma,e^{36}+e^{45}\rangle.\end{split}$

Thus the symbol map on $W^{\prime}$ becomes

$\gamma\ \to\
J\gamma+\langle\delta\varphi,\hat{\varphi}\rangle(e^{45}+e^{36})+\langle\delta\varphi,\varphi\rangle(e^{35}-e^{46})$
(12)

It is convenient to use the following basis for the $5$-dimensional space
$W^{\prime}$,

$\kappa=e^{34}-e^{56},\ \mu_{1}^{\pm}=e^{45}\pm e^{36},\
\mu_{2}^{\pm}=e^{35}\pm e^{46}$ (13)

which are all eigenvectors of $J$,

$\begin{split}&J(e^{34}-e^{56})=(e^{34}-e^{56})\\\
&J(e^{45}+e^{36})=-(e^{45}+e^{36}),\quad J(e^{45}-e^{36})=e^{45}-e^{36}\\\
&J(e^{35}-e^{46})=-(e^{35}-e^{46}),\quad
J(e^{35}+e^{46})=e^{35}+e^{46}.\end{split}$

The symbol becomes the following operator

$\gamma\ \to\ J\gamma+{1\over
2}{\langle}\gamma,\mu_{2}^{-}{\rangle}\mu_{2}^{-}+{1\over
2}{\langle}\gamma,\mu_{1}^{+}{\rangle}\mu_{1}^{+}$ (14)

This implies readily

######  Lemma 3.2.

The above basis turns out to be all eigenvectors of the symbol map

$\begin{split}&\kappa\ \to\kappa,\quad\mu_{1}^{+}\ \to\ 0,\quad\mu_{2}^{+}\
\to\ \mu_{2}^{+}\\\ &\mu_{1}^{-}\ \to\ \mu_{1}^{-},\quad\mu_{2}^{-}\ \to\
0.\end{split}$

Thus the Hitchin gradient flow is more degenerate than the Type IIA flow,
whose principal symbol only has one zero eigenvalue. This additional
degeneracy prevents a proof of short-time existence for general symplectic
manifolds and general data along the lines of [FPPZc]. Nevertheless, as we
shall see in the next section, the Hitchin gradient flow exists on many
interesting manifolds and exhibits a variety of remarkable phenomena. For the
remaining part of this section, we shall just assume that the Hitchin gradient
flow exists on a symplectic manifold, and derive the corresponding evolution
equations for geometric quantities of interest, such as the metric $g_{ij}$
and $|\varphi|^{2}$.

The key point in this derivation is that the Type IIA structure is preserved
under the flow (6), hence we are free to use various identities in Type IIA
geometry developed in [FPPZc, FPPZa]. Otherwise, in [FPPZc] and [FPPZa], two
different methods were given for deriving the evolution equations for the
metric and the term $|\varphi|^{2}$ in the Type IIA flow. Both methods can be
readily adapted to the present case of the Hitchin gradient flow. However,
since the Hitchin gradient flow is given by a Laplacian, and the method of
[FPPZa] gives explicit Bochner-Kodaira formulas for the Laplacian on
$3$-forms, it is easiest to just extract from [FPPZa] what we need.

Recall that the metric $g_{\varphi}$ is defined by
$g_{\varphi}(X,Y)=\omega(X,J_{\varphi}Y)$. We denote it by just $g_{ij}$ for
simplicity, and also consider the metric $\tilde{g}_{ij}$ defined by

$\tilde{g}_{ij}=|\varphi|^{2}g_{ij}=\tilde{g}_{ij}=-\varphi_{jkp}\varphi_{iab}{\omega}^{ka}{\omega}^{pb}.$

We consider first the flow of the metric $\tilde{g}_{ij}$, which is then given
by

$\begin{split}\partial_{t}\tilde{g}_{ij}&=-\partial_{t}\varphi_{jkp}\varphi_{iab}{\omega}^{ka}{\omega}^{pb}+(i\leftrightarrow
j)\\\
&=-(\mathrm{d}\mathrm{d}^{\dagger}\varphi)_{jkp}\varphi_{iab}{\omega}^{ka}{\omega}^{pb}+(i\leftrightarrow
j)\end{split}$

The contribution of $\mathrm{d}\mathrm{d}^{\dagger}$ has been worked out in
[FPPZa], Lemma 14. It is given by

$\begin{split}&-(\mathrm{d}\mathrm{d}^{\dagger}\varphi)_{jkp}\varphi_{iab}{\omega}^{ka}{\omega}^{pb}+(i\leftrightarrow
j)\\\
&\quad=|\varphi|^{2}\big{\\{}Rg_{ij}-2({\mathfrak{D}}_{k}N_{ij}{}^{k}+{\mathfrak{D}}_{k}N_{ji}{}^{k})+(-{\nabla}_{\mu}{\nabla}^{\mu}u+N^{2})g_{ij}+2(N_{i}{}^{k}{}_{j}+N_{j}{}^{k}{}_{i})\partial_{k}u\\\
&\quad\quad\quad\quad-4(N_{-}^{2})_{ij}+8(N_{+}^{2})_{ij}\big{\\}}\end{split}$

where the scalar function $u$ is defined by

$u=\log|\varphi|^{2}$ (15)

and the quadratic expressions $N_{-}^{2}$ and $N_{+}^{2}$ in the Nijenhuis
tensor are defined by

$(N_{+}^{2})_{ij}=N^{pk}{}_{i}N_{pkj},\quad(N_{-}^{2})_{ij}=N^{kp}{}_{i}N_{pkj}.$

We can now make use of the following identities for the Nijenhuis tensor and
the scalar curvature $R$ in Type IIA geometry established in [FPPZc]

$\begin{split}&(N_{-}^{2})_{ij}=2(N_{+}^{2})_{ij}-{1\over 4}|N|^{2}g_{ij},\\\
&R=\Delta u-|N|^{2}={\nabla}_{\mu}{\nabla}^{\mu}u-|N|^{2}\end{split}$

and arrive at the following result:

######  Lemma 3.3.

The flow of the metric $\tilde{g}_{ij}$ is given by

$\partial_{t}\tilde{g}_{ij}=-|\varphi|^{2}\big{\\{}2({\mathfrak{D}}_{k}N_{ij}{}^{k}+{\mathfrak{D}}_{k}N_{ji}{}^{k})-2\partial_{p}u(N_{j}{}^{p}{}_{i}+N_{i}{}^{p}{}_{j})-|N|^{2}g_{ij}\big{\\}}$

Here the covariant derivatives and scalar curvature are with respect to the
metric $g_{ij}$.

Next, we derive the flow of $u=\log|\varphi|^{2}$ and of
$g_{ij}=|\varphi|^{-2}\tilde{g}_{ij}$. We have

$\begin{split}\partial_{t}\log\,{\rm
det}\,\tilde{g}&=\tilde{g}^{ij}\partial_{t}\tilde{g}_{ij}=|\varphi|^{-2}g^{ij}\partial_{t}\tilde{g}_{ij}=-(6R-6{\nabla}_{\mu}{\nabla}^{\mu}u)=-6(R-\Delta
u).\end{split}$

In view of the above formula relating $R$, $\Delta u$ and $|N|^{2}$ in Type
IIA geometry, this reduces to

$\partial_{t}\log\,{\rm det}\,\tilde{g}=6|N|^{2}.$ (16)

Since we also have

$\begin{split}\partial_{t}\log|\varphi|^{2}={1\over 6}\partial_{t}\log{\rm
det}\,\tilde{g}\end{split}$

we obtain the flow for $u=\log|\varphi|^{2}$,

$\partial_{t}\log|\varphi|^{2}=|N|^{2}.$ (17)

The flow of $g_{ij}$ readily follows

$\begin{split}\partial_{t}g_{ij}&=\partial_{t}(|\varphi|^{-2}\tilde{g}_{ij})=|\varphi|^{-2}\partial_{t}\tilde{g}_{ij}-(\partial_{t}\log|\varphi|^{2})|\varphi|^{-2}\tilde{g}_{ij}\\\
&=|\varphi|^{-2}\partial_{t}\tilde{g}_{ij}-(\partial_{t}\log|\varphi|^{2})g_{ij}.\end{split}$

We summarize the formulas in the following lemma:

######  Lemma 3.4.

The flows of the metric $g_{ij}$ and of the scalar function
$u=\log|\varphi|^{2}$ in the Hitchin gradient flow are given by

$\begin{split}&\partial_{t}g_{ij}=-\big{\\{}2({\mathfrak{D}}_{k}N_{ij}{}^{k}+{\mathfrak{D}}_{k}N_{ji}{}^{k})-2\partial_{p}u(N_{j}{}^{p}{}_{i}+N_{i}{}^{p}{}_{j})\big{\\}}\\\
&\partial_{t}u=|N|^{2}.\end{split}$

We note that the Hitchin functional certainly increases along its gradient
flow. But the lemma implies a much stronger property, namely that the
integrand in the functional is pointwise monotone increasing. Now the Ricci
curvature in Type IIA geometry is given by

${\mathfrak{D}}_{k}N_{ij}{}^{k}+{\mathfrak{D}}_{k}N_{ji}{}^{k}=R_{ij}+2(N_{-}^{2})_{ij}-({\nabla}_{i}{\nabla}_{j}u)^{J}$
(18)

so that the above flows can also be rewritten as, with the notation
$({\nabla}_{i}{\nabla}_{j}u)^{J}=\dfrac{1}{2}({\nabla}_{i}{\nabla}_{j}u+J_{i}{}^{p}J_{j}{}^{q}{\nabla}_{p}{\nabla}_{q}u)$
from [FPPZc],

######  Lemma 3.5.

The flows of the metric $g_{ij}$ and of the scalar function
$u=\log|\varphi|^{2}$ in terms of the Ricci curvature are given by

$\begin{split}&\partial_{t}g_{ij}=-\big{\\{}2R_{ij}+4(N_{-}^{2})_{ij}-2({\nabla}_{i}{\nabla}_{j}u)^{J}-2\partial_{p}u(N_{j}{}^{p}{}_{i}+N_{i}{}^{p}{}_{j})\big{\\}}\\\
&\partial_{t}u=|N|^{2}.\end{split}$

There is yet another natural way of rewriting this flow, using the Ricci
curvature formula [FPPZc, Eq. (6.53), Eq. (6.59)]. Thus we get

######  Lemma 3.6.

The flows of the metric $g_{ij}$ and of the scalar function
$u=\log|\varphi|^{2}$ in the Hitchin gradient flow can also be written as

$\begin{split}&\partial_{t}g_{ij}=-R_{ij}+R_{Ji,Jj}+2\partial_{p}u(N_{i}{}^{p}{}_{j}+N_{j}{}^{p}{}_{i})\\\
&\partial_{t}u=|N|^{2}\end{split}$ (19)

######  Remark 3.7.

Formally the coupled system (19) makes sense for any compact symplectic
manifold, therefore it can be viewed as a generalization of the gradient flow
of Hitchin’s functional (6) to arbitrary symplectic manifolds.

######  Remark 3.8.

The system (19) also shows that the gradient flow of the Hitchin functional
can be viewed as a perturbation of the anti-complexified Ricci flow of Lê-Wang
[LW01]. This is the gradient flow of the Blair-Ianus functional [BI86], which
is the $L^{2}$ norm of the Nijenhuis tensor $N$. Explicitly, the gradient flow
can expressed as

$\partial_{t}g_{ij}=-R_{ij}+R_{Ji,Jj}.$

We see that the first equation in (19) only differs from the anti-complexified
Ricci flow by a first order term.

######  Remark 3.9.

If we run the flow (6) on locally homogeneous half-flat symplectic manifolds
(as in the case of [FPPZc, Section 9.3.2]), the induced flow of $g$ is exactly
Lê-Wang’s anti-complexified Ricci flow because that $u$ is a constant in
space. In particular, this flow can be used to find optimal compatible almost
complex structures.

##  4 Examples

In this section, we present a few explicit examples of the gradient flow of
6D-Hitchin’s functional on locally homogeneous symplectic half-flat manifolds.
These manifolds have been studied in [FPPZc] from the perspective of the Type
IIA flow.

######  Example 4.1.

Let $M$ be the nilmanifold constructed by de Bartolomeis-Tomassini [dBT06,
Example 5.2]. The Lie algebra of the nilpotent Lie group is characterized by
invariant 1-forms $\\{e^{1},\dots,e^{6}\\}$ satisfying

$\begin{split}&\mathrm{d}e^{1}=\mathrm{d}e^{2}=\mathrm{d}e^{3}=\mathrm{d}e^{5}=0,\\\
&\mathrm{d}e^{4}=e^{15},\qquad\mathrm{d}e^{6}=e^{13}.\end{split}$

Clearly $\omega=e^{12}+e^{34}+e^{56}$ defines an invariant symplectic
structure. Moreover, this nilpotent Lie group admits co-compact lattices so
all the constructions descend to compact nilmanifolds. Consider the ansatze

$\varphi=\varphi_{a,b}=(1+a)e^{135}-e^{146}-e^{245}-e^{236}+b(e^{134}-e^{156}),$
(20)

it is easy to check that $\varphi_{a,b}$ is primitive and closed for any
$a,b$. The positivity condition for $\varphi_{a,b}$ is that
$\dfrac{1}{16}|\varphi|^{4}=1+a-b^{2}>0$. By straightforward calculations, we
get

$|\varphi|^{2}\hat{\varphi}=4((1+a-b^{2})e^{1}\\!\wedge\\!(e^{36}+e^{45})+e^{2}\wedge(be^{34}+(1+a)e^{35}\\!-\\!e^{46}\\!-\\!be^{56})).$

It follows that

$\begin{split}&\mathrm{d}(|\varphi|^{2}\hat{\varphi})=4e^{12}(e^{34}+2be^{35}-e^{56}),\\\
&\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi})=4(e^{34}+2be^{35}-e^{56}),\\\
&\mathrm{d}\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi})=8e^{135}.\end{split}$

So the gradient flow of 6D-Hitchin’s functional reduces to the ODE system

$\begin{cases}&\dot{a}=\dfrac{2}{\sqrt{1+a-b^{2}}}\\\ &\dot{b}=0\end{cases},$

which can be solved explicitly

$\begin{cases}&(1+a-b_{0}^{2})^{3/2}=(1+a_{0}-b_{0}^{2})^{3/2}+3t,\\\
&b=b_{0}\end{cases}.$

In particular the flow exists for all time and

$|N|^{2}=(1+a-b^{2})^{-3/2}=\frac{1}{(1+a_{0}-b_{0}^{2})^{3/2}+3t}\to 0,$

as $t$ goes to infinity.

######  Example 4.2.

Consider the symplectic half-flat structure on the solvmanifold $M$
constructed by Tomassini and Vezzoni in [TV08, Theorem 3.5]. The geometry of
this solvmanifold is characterized by invariant 1-forms
$\\{e^{j}\\}_{j=1}^{6}$ satisfying

$\begin{split}&\mathrm{d}e^{1}=-\lambda e^{15},\quad\mathrm{d}e^{2}=\lambda
e^{25},\quad\mathrm{d}e^{3}=-\lambda e^{36},\\\ &\mathrm{d}e^{4}=\lambda
e^{46},\quad\mathrm{d}e^{5}=0,\quad\mathrm{d}e^{6}=0,\end{split}$

where $\lambda=\log\dfrac{3+\sqrt{5}}{2}$. One can easily check that
$\omega=e^{12}+e^{34}+e^{56}$ is an invariant symplectic form on $M$. Consider
the ansatze

$\varphi=\alpha(e^{135}+e^{136})+\beta(e^{145}-e^{146})+\gamma(e^{235}-e^{236})-\delta(e^{245}+e^{246}).$
(21)

A direct calculation gives

$\begin{split}|\varphi|^{2}\hat{\varphi}&=8(-\alpha\beta\gamma(e^{135}-e^{136})+\alpha\beta\delta(e^{145}+e^{146})\\\
&+\alpha\gamma\delta(e^{235}+e^{236})+\beta\gamma\delta(e^{245}-e^{246})).\end{split}$

The nondegenerate condition is that
$|\varphi|^{4}=64\alpha\beta\gamma\delta>0$. It follows that

$\begin{split}\mathrm{d}(|\varphi|^{2}\hat{\varphi})&=16\lambda(\alpha\beta\gamma
e^{1356}+\alpha\beta\delta e^{1456}-\alpha\gamma\delta
e^{2356}+\beta\gamma\delta e^{2456}),\\\
\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi})&=16\lambda(\alpha\beta\gamma
e^{13}+\alpha\beta\delta e^{14}-\alpha\gamma\delta e^{23}+\beta\gamma\delta
e^{24}),\\\
\mathrm{d}\Lambda_{\omega}\mathrm{d}(|\varphi|^{2}\hat{\varphi})&=16\lambda^{2}(\alpha\beta\gamma(e^{135}+e^{136})+\alpha\beta\delta(e^{145}-e^{146})\\\
&+\alpha\gamma\delta(e^{235}-e^{236})-\beta\gamma\delta(e^{245}+e^{246})).\end{split}$

After linear time rescaling, the gradient flow of Hitchin’s functional under
our ansatze reduces to

$\begin{split}&\dot{\alpha}=\frac{\alpha\beta\gamma}{\sqrt{\alpha\beta\gamma\delta}},\qquad\dot{\beta}=\frac{\alpha\beta\delta}{\sqrt{\alpha\beta\gamma\delta}},\\\
&\dot{\gamma}=\frac{\alpha\gamma\delta}{\sqrt{\alpha\beta\gamma\delta}},\qquad\dot{\delta}=\frac{\beta\gamma\delta}{\sqrt{\alpha\beta\gamma\delta}}.\end{split}$

For simplicity, let us assume that all of $\alpha,\beta,\gamma,\delta$ are
positive. It is easy to see that there exist positive constants $C_{1}$ and
$C_{2}$ such that $\alpha(t)=C_{1}\delta(t)$ and $\beta(t)=C_{2}\gamma(t)$.
The ODE system simplifies to

$\dot{\gamma}=\sqrt{\frac{C_{1}}{C_{2}}}\delta,\qquad\dot{\delta}=\sqrt{\frac{C_{2}}{C_{1}}}\gamma.$

Again this system can be solve explicitly as

$\begin{split}&\alpha=\sqrt{C_{1}}(Ae^{t}+Be^{-t}),\quad\beta=\sqrt{C_{2}}(Ae^{t}-Be^{-t}),\\\
&\gamma=\frac{1}{\sqrt{C_{2}}}(Ae^{t}-Be^{-t}),\quad\delta=\frac{1}{\sqrt{C_{1}}}(Ae^{t}+Be^{-t}),\end{split}$

where $A>0$, $B$ are constants determined by initial data. In particular the
flow exists for all time and $\lim_{t\to\infty}|\varphi|^{2}=\infty$. However,
the limit $\lim_{t\to\infty}J_{t}=J_{\infty}$ does exist. This is because

$\begin{split}\varphi_{\infty}:=\lim_{t\to\infty}\frac{\varphi}{|\varphi|}=~{}&\sqrt{\frac{C_{1}}{8}}(e^{135}+e^{136})+\sqrt{\frac{C_{2}}{8}}(e^{145}-e^{146})\\\
&+\sqrt{\frac{1}{8C_{2}}}(e^{235}-e^{236})-\sqrt{\frac{1}{8C_{1}}}(e^{245}+e^{246})\end{split}$

exists. In fact, $J_{\infty}$ is a harmonic almost complex structure in the
sense of [LW01], namely the Ricci curvature is $J_{\infty}$-invariant. In
addition, $\varphi_{\infty}$ satisfies

$\mathrm{d}\Lambda_{\omega}\mathrm{d}\hat{\varphi}_{\infty}=2\lambda^{2}\varphi_{\infty}.$

This case also provides an example of convergence of anti-complexified Ricci
flow to a non-Kähler metric.

######  Remark 4.3.

When $|\varphi|$ is a constant over space, the Hitchin gradient flow is simply
a time-rescaled version of the Type IIA flow. Therefore we can use Raffero’s
technique [Raf21] to produce many special solutions to the Hitchin gradient
flow (6) as well.

##  5 Additional remarks

It may be worth considering the following $\epsilon$-regularization of the
Hitchin gradient flow

$\partial_{t}\varphi=\mathrm{d}\Lambda_{\omega}\mathrm{d}(|\varphi|^{\epsilon}\hat{\varphi})$
(22)

for each $\epsilon>0$. The same arguments for the Type IIA flow show that the
flow preserves primitiveness and should be a well-defined flow of Type IIA
geometries. While we do not expect that the corresponding solutions will have
a limit as $\epsilon\to 0$, it is conceivable that certain important notions
may have a limit. A model situation may be Landau-Ginzburg models and
renormalized energies. It is also intriguing that the dual Ricci flow can be
interpreted as another limit of this regularization.

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section [0pt] *[0pc]

# Synergy between DUNE and T2HKK to probe Invisible Neutrino Decay

Zannatun Firdowzy Dey<EMAIL_ADDRESS>Assam Don Bosco University,
Tapesia Campus, Sonapur, Assam, 782402, India  Debajyoti Dutta
<EMAIL_ADDRESS>Assam Don Bosco University, Tapesia Campus,
Sonapur, Assam, 782402, India  Bhattadev University, Pathsala Town, Barpeta,
Assam 781325

###### Abstract

We address the consequence of invisible neutrino decay in the context of two
long base-line neutrino experiments; T2HKK (Tokai-to-Hyper-Kamiokande-to-
Korea) and DUNE (Deep Underground Neutrino experiment). The main purpose of
this work is to bring out the aspects of CC (charged current) and NC (neutral
current) measurement at DUNE in the context of invisible neutrino decay. We
find that the inclusion of NC measurements (NC) at DUNE with the charged
current measurements (CC) enhances its ability to constrain invisible neutrino
decay. This enhancement further increases if the total run-time changes from 7
years to 10 years at DUNE. The synergy between DUNE and T2HKK improves the
constraints on invisible neutrino decay. The derived constraints are
$\tau_{3}/m_{3}\leq 5.51\times 10^{-11}$ s/eV at $3\sigma$ CL. This constraint
on the decay parameter increases further if we assume that all the oscillation
parameters are measured precisely. In that case, the 3$\sigma$ constraint
obtained by using the synergy between DUNE and T2HKK is $\tau_{3}/m_{3}\leq
5.69\times 10^{-11}$ s/eV. Again if nature prefers $\nu_{3}$ to be unstable
and the decay width is $\tau_{3}/m_{3}=2.2\times 10^{-11}$ s/eV then this
combination can exclude the no decay scenario at more than $5\sigma$ C. L. .
Although the CP sensitivity is not much hindered in the presence of invisible
neutrino decay, the measurements of $\theta_{23}$, as well as the octant
sensitivity, are very much affected. Octant sensitivity improves in the lower
octant in the presence of the invisible neutrino decay. However, the synergy
between DUNE and T2HKK improves the overall octant sensitivity in the presence
of invisible neutrino decay.

###### Contents

  1. 1 Introduction
  2. 2 Theory of invisible neutrino decay
  3. 3  Experimental and simulation details
    1. 3.1 DUNE (Deep Underground Neutrino Experiment)
    2. 3.2 T2HKK(Tokai-to-Hyper-Kamiokande-to-Korea)
  4. 4 Results
    1. 4.1 Probability at DUNE and T2HKK
    2. 4.2 $\chi^{2}$ analysis
      1. 4.2.1 Constraints on Invisible Neutrino Decay
      2. 4.2.2 Measurements of $\theta_{23}$ in presence of neutrino decay
      3. 4.2.3 Octant and CP violation Sensitivity in the presence of Invisible Neutrino Decay
      4. 4.2.4 Correlation plots
  5. 5 Summary and Conclusion

##  1 Introduction

The discovery of neutrino oscillations has led to a new area of research
beyond the Standard Model (BSM), explaining the existence of neutrino mass.
Out of the six oscillation parameters, $\theta_{12},\theta_{13}$ and
$\Delta{m}^{2}_{21}$ are measured precisely, but the sign of
$\Delta{m}^{2}_{31}$, octant and absolute value of $\theta_{23}$ and the CP
violating phase $\delta_{cp}$ are yet to be measured precisely. There are a
few indications regarding these unknown neutrino oscillation parameters from
the ongoing neutrino oscillation experiments T2K [1] and NO$\nu$A [2]. Both
the experiments predict the same best-fit value for $\sin^{2}\theta_{23}$ at
1$\sigma$ yet they have strong disagreements on the measured best-fit value of
$\delta_{cp}$. With more data from these experiments or from a combined
analysis of the present data, better constraints can be put on these
oscillation parameters. In these contests, the upcoming experiments like
DUNE[3], T2HK/T2HKK[4; 5], ESS$\nu$SB[6], JUNO[7], INO[8], PINGU[9], KM3Net-
ORCA[10] etc. are aiming to determine these unknown parameters with more
precision. Since the discovery of neutrino oscillations has made huge progress
and this phenomenon is now well established, a complete understanding of
neutrino oscillations will provide an opportunity to explore new physics
beyond the standard model. In the present scenario, BSM physics like the
existence of sterile neutrinos, vector and scalar NSIs, invisible neutrino
decay, etc are some interesting topics to be explored further, as the
existence of such new physics will hamper the measurements of the standard
neutrino oscillation parameters in these upcoming experiments. Upcoming long
and short-baseline neutrino experiments will be able to address some of these
questions.

Invisible neutrino decay is one such new topic of interest, which was first
proposed by [11] to explain the solar neutrino problem. There are two possible
modes of neutrino decay: visible and invisible decay modes. Both Dirac and
Majorana neutrinos can undergo invisible decay. For Dirac neutrinos, the
coupling between the neutrinos and the light scalar boson (s) leads to the
decay mode: $\nu_{j}\rightarrow\bar{\nu}_{iR}+\chi$, where $\bar{\nu}_{iR}$ is
the right-handed singlet and $\chi$ is the iso-singlet scalar. If neutrinos
are Majorana particles, then the decay mode is $\nu_{j}\rightarrow\nu_{s}+J$,
where $\nu_{s}$ is the sterile neutrino and J is a Majoron coupled by a
pseudo-scalar coupling. This Majoron has to be singlet as constrained by the
LEP data. In both scenarios, the decay products are sterile and are invisible
to the detectors. If the neutrino decays via $\nu_{j}\rightarrow\nu_{i}+J$ and
$\nu_{j}\rightarrow\bar{\nu}_{i}+J$ channels, then the final states can be
detected, and these modes of decay are known as visible neutrino decay. In
visible neutrino decay, the final state fermion is a lighter neutrino, which
is visible to the detectors, while in the other case, the final state
particle, being a sterile neutrino, cannot be detected. Assuming $\nu_{2}$ to
be unstable, neutrino decay with oscillation has been studied in [12; 13; 14;
15; 16]. These studies put constraints on the lifetime of $\nu_{2}$. Solar
neutrino data constraints: $\nu_{2}/m_{2}>8.5\times 10^{-7}$ s/eV. Low-energy
data from solar neutrinos as well as data from supernova neutrinos put bounds
on both $\nu_{2}$ and $\nu_{1}$. Data from the high-resolution multi-ton Xenon
detector gives very stringent bounds on $\nu_{2}$ and $\nu_{1}$ and at
$2\sigma$, solar neutrino data from this detector gives $\nu_{2}/m_{2}>8\times
10^{-3}$ s/eV and $\nu_{1}/m_{1}>3\times 10^{-2}$ s/eV respectively. $\nu_{e}$
lifetime is also strongly constrained by SN1987A data. The bound on $\nu_{3}$
comes from atmospheric and long-baseline neutrino experiments. Earlier, to
overcome the atmospheric neutrino problem, the possibility of neutrino decay
was proposed in [17]. But the fit was very poor. In [18; 19], authors studied
neutrino decay with mixing to fit SK data. Later, the SK collaboration showed
that its data could be better fit with neutrino oscillation only [20].
However, in a scenario where an unstable neutrino decays to a sterile neutrino
with $\Delta m^{2}\sim 0.003$ $eV^{2}$, the presence of small non-zero decay
gives a better fit to SK data [21]. So decay with oscillation can still give a
better fit to SK data. The global analysis of MINOS data with the atmospheric
data constrained $\nu_{3}/m_{3}\geq 2.9\times 10^{-10}$ s/eV [22] at 90$\%$
CL. The atmospheric events at INO in India can also constrain the lifetime of
$\nu_{3}$ [23]. Visible neutrino decay is very tightly constrained and some
studies have been done in the context of long-baseline experiments in [24; 25]

In the context of invisible neutrino decay, the performance of DUNE has been
studied in ref [26]. A run time of 5+5 years at DUNE can constrain
${\tau_{3}}/{m_{3}}>4.50\times 10^{-11}$ s/eV at $90\%$ CL. The inclusion of
neutral current (NC) measurements with the charge current (CC) measurements at
DUNE can improve the bounds further [27]. The bounds obtained from the MOMENT
experiment on ${\tau_{3}}/m_{3}$ is $>1.6\times 10^{-11}$ s/eV at $3\sigma$
[28]. In ref. [29] authors have shown a $3\sigma$ bound on the decay parameter
in the ESSnuSB experiment in the context of invisible neutrino decay for the
two baselines. The constraint obtained for 540 km is found to be poorer than
the constraint obtained from the 360 km baseline. The medium baseline reactor
experiment JUNO gives can constrain ${\tau_{3}}/{m_{3}}>7.5\times{10}^{-11}$
s/eV at $95\%$ by 100 kt.years exposure [7]. Considering two generations of
neutrinos, bounds on ${\tau_{3}}/{m_{3}}$ were calculated in ref. [30] using
MINOS and T2K data. In ref. [31], authors have considered a proper three-
generation scenario. Using the preliminary data from the T2K and NO$\nu$A
experiments, they measured the bound on the lifetime of ${\tau_{3}}/{m_{3}}$
considering neutrino decay with oscillation. The measured bound is
${\tau_{3}}/{m_{3}}>1.5\times{10}^{-12}$ s/eV at $3\sigma$. In ref. [32],
authors have recently shown that the presence of invisible neutrino decay can
solve the tension between track and cascade at IceCUBE.

Neutrino physics is in its precision era and upcoming neutrino experiments
strive to achieve precise measurements of neutrino oscillation parameters.
However, the existence of parameter degeneracy [33] among the mixing
parameters introduces challenges in accurately determining these parameters.
Amalgamating different neutrino experiments helps in resolving parameter
degeneracies. In the context of invisible neutrino decay, a combined analysis
of T2HK plus ESS$\nu$EB and T2HKK plus ESS$\nu$EB has been performed in [34]
and these combinations can constrain ${\tau_{3}}/{m_{3}}$ as
${\tau_{3}}/{m_{3}}\leq 4.36\times{10}^{-11}$ s/eV and ${\tau_{3}}/{m_{3}}\leq
5.53\times{10}^{-11}$ s/eV respectively, at 3$\sigma$ CL. The bounds obtained
for these combinations are better than the individual bounds of these
experiments calculated in the paper.

In this paper, we perform a synergistic study of DUNE and T2HKK in the context
of invisible neutrino decay. Both DUNE and T2HKK are two promising upcoming
long-baseline neutrino experiments. DUNE is a beam-based high-statistics
experiment with a large detector mass, a long baseline, and a sufficient
matter effect. On the other hand, T2HKK is also a very special experiment
where it has two detector sites, and the detectors are placed close to their
1st and 2nd oscillation maxima. It also has a huge detector mass, huge
statistics, and sufficient sensitivity to the matter’s effect, like DUNE. So
combining such long-baseline experiments enhances the overall sensitivity and
helps in finding potential synergies among the experiments.

Although separate studies have been performed in the presence of invisible
neutrino decay at DUNE and T2HKK, we have performed a combined study of these
two experiments to explore possible synergies among them. In the case of DUNE,
we have separately studied the sensitivity to ${\tau_{3}}/{m_{3}}$ both for
charge current (CC) and neutral current (NC) measurements and finally combined
both of them to obtain improved sensitivity. Then, with the combined CC and NC
measurements at DUNE, we combined the T2HKK experiment to observe further
improved sensitivity to ${\tau_{3}}/{m_{3}}$. The analysis is performed
considering three-flavor neutrino oscillation scenarios in the presence of
matter effect. We have also considered two different run times at DUNE and
found that with a higher run time, it is possible to improve the constraints
on ${\tau_{3}}/{m_{3}}$. The correlations between $\theta_{23}$ and
${\tau_{3}}/{m_{3}}$ have been studied extensively in this work. If invisible
neutrino decay is a natural phenomenon, then how its presence will hamper the
octant and CP violation sensitivities at the far detectors of DUNE and T2HKK
is studied in detail in this work. The same is also explored for the
combination of the two experiments.

This paper is structured as follows: We address the theory of invisible
neutrino decay in Section 2. In Section 3 we give a brief description of the
two experiments and the simulation details. We present our results in Section
4 and we conclude in Section 5.

##  2 Theory of invisible neutrino decay

We suppose that the heaviest mass eigenstate $\nu_{3}$ is not stable and
undergoes decay into a sterile state, $\nu_{4}$, with lifetime $\tau_{3}$ into
a new sterile state and a scalar singlet $J$;
$(\nu_{3}\rightarrow\bar{\nu_{4}}+J)$. Additionally, we assume that there is
no mixing of the three active neutrinos with the sterile neutrino. Taking this
into account, the neutrino mixing matrix can be described using the standard
three-family mixing matrix U in the following manner

$\begin{pmatrix}\nu_{\alpha}\\\ \nu_{s}\end{pmatrix}=\begin{pmatrix}U&0\\\
0&1\end{pmatrix}\begin{pmatrix}\nu_{i}\\\ \nu_{4}\end{pmatrix}$ (1)

Where U is the standard PMNS (Pontecorvo–Maki–Nakagawa–Sakata matrix) matrix
describing the standard three neutrino oscillation,
$\alpha\rightarrow\nu_{e},\nu_{\mu},\nu_{\tau}$ indicates the flavor
eigenstate and $i=1,2,3$ the mass eigenstates. We assume normal mass ordering
i.e., $m_{1}>m_{2}>m_{3}$. Furthermore, we assume that the mass and decay
eigenstates are identical. Based on these presumptions, the evolution of
neutrinos in the presence of matter can be expressed as follows:

${i\frac{d}{dx}}{\begin{pmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{pmatrix}}=\left[U\left[\frac{1}{2E}{\begin{pmatrix}0&0&0\\\
0&\Delta{m}^{2}_{21}&0\\\
0&0&\Delta{m}^{2}_{31}\end{pmatrix}}-{i\frac{m_{3}}{2E{\tau_{3}}}}{\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&1\end{pmatrix}}\right]{U^{\dagger}}+{\begin{pmatrix}A_{cc}&0&0\\\
0&0&0\\\ 0&0&0\end{pmatrix}}\right]{\begin{pmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{pmatrix}}$ (2)

Simplifying the equation 2 we can write,

$i\frac{d}{dx}v_{f}=\frac{1}{2E}\left[U\widetilde{H}{U^{\dagger}}+A\right]{v_{f}}.$
(3)

Where,

$\widetilde{H}=\begin{pmatrix}0&0&0\\\ 0&\Delta{m}^{2}_{21}&0\\\
0&0&\Delta{m}^{2}_{31}-\frac{{i}m_{3}}{\tau_{3}}\\\
\end{pmatrix},A=\begin{pmatrix}A_{cc}&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}$
(4)

Here $A=2\sqrt{2}G_{F}n_{e}E$ represents the matter potential due to neutrino
electron scattering, $G_{F}$ is the Fermi coupling constant, E is the energy
and $n_{e}$ is the density of the electron. The probability of getting a
neutrino from an initial state $\nu_{a}$ to a final state $\nu_{b}$ can be
expressed as

$P_{ab}={|<\nu_{b}|\nu_{a}>|}^{2}.$ (5)

Where, a, and b correspond the flavor states $e,\mu,$ and $\tau$. The decay
rate of the unstabe state ${\nu_{3}}$ is defined as ${\tau_{3}}/{m_{3}}$.
Therefore, in the probability equation, the effect of the decay appears as exp
$[-({m_{3}}/{\tau_{3}})\frac{L}{E}]$. Hence, a lower value of the decay
parameter ${\tau_{3}}/{m_{3}}$ will affect an experiment with a longer
baseline or lower energy.

##  3  Experimental and simulation details

DUNE [3] and T2HKK are two long baseline experiments. In this section, we give
a brief description of these two experiments.

###  3.1 DUNE (Deep Underground Neutrino Experiment)

DUNE [3] is a proposed experiment aiming to find out if there’s CP violation
in the lepton world. Besides that big question, they’re also looking into
figuring out the mass hierarchy and a thing called the octant of
$\theta_{23}$. They plan to shoot a special beam of neutrinos (and their anti-
particles) from Fermilab in Illinois. They’re using a powerful
$1.07MW\textendash 80$ GeV proton beam, expecting around $1.47\times 10^{21}$
protons to hit the target each year. A $40$ kt Liquid Argon (LAr) detector
will be placed in a mine in South Dakota, about $1300$ km away. The whole
experiment is set to run for 10 years, spending equal time on neutrinos and
anti-neutrinos. That adds up to a total exposure of $4.12\times 10^{23}$ kt-
POT-yr. All the technical details about how they define signals and
background, plus their assumptions about detector efficiency for certain
events are given in ref. [35]. The information on anticipated NC events at
DUNE was sourced from [36]. We assumed a detection efficiency of $90\%$ for NC
events and employed migration matrices to faithfully reproduce their spectra.
In NC events, the outgoing (anti-)neutrino takes away a portion of the
incoming energy. Consequently, the reconstructed visible energy is less than
the total initial energy carried by the neutrino. In our analysis, we adopted
50 MeV energy bins for NC events. For the analysis of CC events, we followed
the 125 MeV energy binning approach used in [35]. The background for NC events
includes misidentified CC events, such as electron events (from CC signals
like $\nu_{\mu}\rightarrow\nu_{e}$ or intrinsic beam
$\nu_{e}\rightarrow\nu_{e}$), muon events $(\nu_{\mu}\rightarrow\nu_{\mu})$,
tau events $(\nu_{\mu}\rightarrow\nu_{\tau})$, and their corresponding CP-
reversed channels due to anti-neutrino/neutrino contaminations in the beam.

###  3.2 T2HKK(Tokai-to-Hyper-Kamiokande-to-Korea)

The next Hyper-K experiment is planned to construct two identical water-
Cherenkov detectors, each 187 kt in size. The first will be located at the
Tochibora mine in Japan, with a baseline of 295 km from the J-PARC neutrino
target and an off-axis angle of 2.5∘. The other detector may be located in
Korea at a baseline of 1100 km. The flux at 295 km will cover the first
oscillation maximum and the flux at 1100 km will cover the second oscillation
maximum. The first and second oscillation maxima of the PMNS neutrino
appearance probability can be reached due to Korea’s longer baseline. The
longer baseline of the Korean sites improves the oscillation probability’s CP-
violating component and resolves parameter combinations between the CP-
violating phase and neutrino mass ordering that would be almost degenerate if
measurements were limited to the Japanese site.T2HKK offers a baseline that is
nearly as long as the proposed DUNE experiment, but it is in an energy band
that is similar to that of the current T2K experiment. This enables it to
probe oscillations at the second oscillation maximum, a capability that is
only shared by the proposed ESS neutrino beam and not available to any other
experiment[37].

Channels |  DUNE (1300 km) |  T2HKK (295 km, 1100 km)
---|---|---
$\nu_{e}$ appearance |  2 $\%$ (5)$\%$ |  3.2 $\%$ (5)$\%$
$\nu_{\bar{e}}$ appearance |  2 $\%$ (5)$\%$ |  3.9 $\%$ (5)$\%$
$\nu_{\mu}$ disappearance |  2 $\%$ (5)$\%$ |  3.6 $\%$ (5)$\%$
$\nu_{\bar{\mu}}$ disappearance |  2 $\%$ (5)$\%$ |  3.6 $\%$ (5)$\%$
Table 1: The signal (background) normalization uncertainties of DUNE and T2HKK experiments for various channels. Oscillation parameters |  True Values |  Marginalization Range
---|---|---
$\theta_{13}$ |  $8.6^{\circ}$ |  [$8.1^{\circ},8.9^{\circ}$]
$\theta_{12}$ |  $33.82^{\circ}$ |  Fixed
$\sin^{2}\theta_{23}$ |  $41.95^{\circ}$ (LO), $48.44^{\circ}$ (HO) |  [$38^{\circ},52^{\circ}$]
$\Delta m^{2}_{21}(eV^{2})$ |  $7.39\times 10^{-5}$ |  Fixed
$\Delta m^{2}_{31}(eV^{2})$ |  $2.52\times 10^{-3}$ |  [$2.3\times 10^{-3},2.6\times 10^{-3}$]
$\delta$ |  $-90^{\circ}$ |  [${-180}^{\circ},180^{\circ}$]
Table 2: The values of three neutrino oscillation parameters used in the
present analysis.

In our analysis, we have used GLoBES (General Long Baseline Neutrino
Experiment simulator) [38] software package to simulate the DUNE and T2HKK
experiments. We have simulated DUNE for a run time of 10 years and 7 years,
which are equally divided between neutrinos and anti-neutrinos. Similarly, for
T2HKK we have considered a run time of 2.5 years for neutrinos and 7.5 years
for anti-neutrinos, respectively. Table 2 provides the values of the three
neutrino oscillation parameters that we used in our simulation. The values
provided in the table align with the current global fit ranges [39]. Unless
mentioned, we have marginalized over $\theta_{23}$, $\theta_{13}$,
$\Delta{m}^{2}_{31}$ and $\delta_{cp}$ in their 3$\sigma$ range given in Table
2. The calculations have been done considering only normal mass ordering. The
signal (background) normalization uncertainties of DUNE and T2HKK experiments
for various channels used in this analysis is given in Table 1.

##  4 Results

In this section, we present the results of our analysis. We present the
probability of neutrino oscillations for the DUNE and T2HKK experiments as a
function of energy, both with and without invisible neutrino decay.

###  4.1 Probability at DUNE and T2HKK

Figure 1: The appearance (left panels) and disappearance (right panels)
channels of neutrino oscillation probabilities for DUNE and T2HKK experiments
as a function of neutrino energy. The top panel corresponds to DUNE, and the
bottom panel corresponds to T2HKK.

Figure 1 shows the probabilities of appearance (left) and disappearance
(right) for the DUNE and T2HKK experiments as a function of energy for the
baseline of 1300km and 1100 km respectively. In the top panel, we present the
probability plots for the DUNE and the bottom panel shows the probability
plots for T2HKK. We present the plots both for decay and no decay cases. For
all cases, the dashed lines are for ${\tau_{3}}/{m_{3}}=0$ and the solid lines
are for ${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV. We show the results both
for the lower octant (sin${}^{2}\theta_{23}=41.95$) and the higher octant
(sin${}^{2}\theta_{23}=48.44$). The red (cyan) dashed lines represent the
probability of no decay in HO (LO), while the green (blue) solid lines
represent probability plots in the presence of invisible neutrino decay in HO
(LO).

From figure 1, we observe that $P_{\mu e}$ decreases in the presence of
neutrino decay both in the lower and the higher octants. Around the
oscillation maxima, the differences between the no-decay and the with-decay
cases are significant. For a fixed value of ${\tau_{3}}/{m_{3}}=2.2\times
10^{-11}$ s/eV the probability in the lower octant is smaller than the
probability in the higher octant. Also around the oscillation maxima, the
effect of decay is more prominent in T2HKK than DUNE. The disappearance
probabilities ($\nu_{\mu}\rightarrow\nu_{\mu}$) are shown in the right panel
of figure 1. The disappearance probabilities for the no decay case overlap in
both the lower and the higher octants and this behavior is the same in both
DUNE and T2HKK. But in the presence of neutrino decay, the disappearance
probability decreases. Around the oscillation peaks, the difference is
significant, especially in T2HKK. This is clear from the following two-
generation survival probability in a vacuum [34]:

$P_{\mu\mu}=[1-\sin^{2}\theta_{23}(1-e^{-\frac{m_{3}L}{\tau_{3}E}})]^{2}-\sin^{2}{2\theta_{23}}e^{-\frac{m_{3}L}{\tau_{3}2E}}\sin^{2}(\frac{\Delta
m^{2}_{31}L}{4E})$ (6)

In the case when $\tau_{3}/m_{3}=0$, the probability is determined by
${\sin^{2}}2\theta_{23}$. However, if $\tau_{3}/m_{3}$ is not zero, or when
decay is present, the $e^{-\frac{m_{3}L}{\tau_{3}E}}$ factor becomes
significant and causes octant sensitivity.

Figure 2: Probability as a function of sin${}^{2}\theta_{23}$ . The left panel
is for DUNE and the right panel is for T2HKK. The dashed red lines are for
standard oscillation case and the solid lines are for
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV in data. The red lines represent
the $P_{\mu\mu}$ and the green lines represent the $P_{\mu e}$.

Figure 2 shows the probability of DUNE on the left and T2HKK on the right as a
function of $\theta_{23}$, both in the presence and absence of invisible
neutrino decay. Plots are generated at a fixed energy E$\sim 0.7$ GeV for
T2HKK and E$\sim 2.5$ GeV for DUNE. In both experiments, the probability of
$P_{\mu e}$ rises as $\theta_{23}$ increases. But the probability decreases
when neutrino decay is present. On the other hand, we observe that
P${}_{\mu\mu}\sim 0$ at $46^{\circ}$ but not at $45^{\circ}$ for standard
oscillation. This is because we choose the non-zero value of $\theta_{13}$,
which is discussed in ref. [40]. It can be seen that the decay probability is
smaller in the higher octant but larger in the lower octant compared to the no
decay probability. We shall examine $\theta_{23}$ measurement in the context
of invisible neutrino decay in brief in our later analysis.

###  4.2 $\chi^{2}$ analysis

####  4.2.1 Constraints on Invisible Neutrino Decay

Figure 3: The sensitivity $\chi^{2}$ as a function of ${\tau_{3}}/{m_{3}}$ for
DUNE (3.5+3.5), DUNE (5+5) and T2HKK for different marginalization effect. The
left panel is for DUNE (3.5+3.5), the middle panel is for DUNE (5+5) and the
right panel is for T2HKK. The data have been generated for
${\tau_{3}}/m_{3}=0$.

In this section, we have performed a $\chi^{2}$ analysis to show how these
experiments and their combinations can constrain invisible neutrino decay. We
have estimated the statistical $\chi^{2}$ function using the following
relation:

$\chi^{2}_{stat}=2\sum_{i}\left[N^{test}_{i}-N^{true}_{i}+N^{true}_{i}log\frac{N^{true}_{i}}{N^{test}_{i}}\right]$
(7)

Where N${}^{true}_{i}$ represents the number of true events and
N${}^{test}_{i}$ represents the number of test events. For DUNE, we have
considered 7 years and 10 years of run-time which are equally divided between
neutrinos and anti-neutrinos. The effect of both Charged current (CC) and
neutral current (NC) measurements at DUNE has been considered and studied
separately (as DUNE-NC and DUNE-CC) as well as in combination (As
DUNE-(CC+NC)). We then combined both DUNE-(CC+NC) with T2HKK to constrain
invisible neutrino decay.

Here we have shown the $\chi^{2}$ sensitivity as a function of the decay
parameter. We have also explored the effect of different marginalization while
constraining invisible neutrino decay. The results are shown in figure 3. The
data in all the cases shown in figure 3 are generated with the standard
oscillation parameters given in Table 2. The true value of $\theta_{23}$ is
considered in the higher octant i.e. $\theta_{23}=48.44$. The blue dashed
lines represent the scenario where we have not marginalized any parameters.
The dashed red, yellow, and magenta lines represent the marginalization of
$\theta_{23}$, $\Delta m^{2}_{31}$ and $\delta_{cp}$ respectively in their
3$\sigma$ allowed range. The solid brown, cyan and green lines represent the
cases where the marginalization is over $\theta_{23}$ and
$\Delta{m}^{2}_{31}$, $\Delta{m}^{2}_{31}$ and $\delta_{cp}$ and
$\Delta{m}^{2}_{31}$ and $\delta_{cp}$ respectively. Finally, the black line
shows the effect of marginalization over $\theta_{23}$, $\Delta{m}^{2}_{31}$
and $\delta_{cp}$.

Figure 4: $\chi^{2}$ as a function of ${\tau_{3}}/{m_{3}}$ for standalone and
combination of experiments. The left panel is for the case where
${\tau_{3}}/{m_{3}}$=0 and the right panel is for
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV in data. The top panel is for the
combination of DUNE (5+5) and T2HKK experiments and the bottom panel is for
the combination of DUNE (3.5+3.5) and T2HKK experiments.

We have observed from figure 3 that at DUNE, the dashed blue line overlaps
with the dashed magenta line, the dashed yellow line overlaps with the solid
green line, and the dashed red line overlaps with the solid cyan line
respectively. It clearly shows that the marginalization over $\delta_{cp}$ has
minimal effect while constraining invisible neutrino decay. Even when it is
combined with $\theta_{23}$ and $\Delta m^{2}_{31}$ individually, only the
effects of $\theta_{23}$ and $\Delta m^{2}_{31}$ are observed. But, the effect
of marginalizing over $\theta_{23}$ and $\Delta m^{2}_{31}$ is significant and
adding $\delta_{cp}$ can affect the constrain further. In the case of T2HKK,
there is no effect of $\Delta{m}^{2}_{31}$ and $\delta_{cp}$ marginalization.
However, the marginalization effect is seen only for $\theta_{23}$. Here the
constraint obtained by marginalizing all three parameters and the constraint
obtained by only marginalizing $\theta_{23}$ are identical. So unless stated,
in the remaining part of this work, we marginalize over $\theta_{23}$, $\Delta
m^{2}_{31}$, and $\delta_{cp}$ for both the experiment.

In figure 4, we show the sensitivities of DUNE, T2HKK, and their combinations
to invisible neutrino decay. The results are shown for DUNE-NC, DUNE-CC
DUNE-(CC+NC), T2HKK, and DUNE-(CC+NC)+T2HKK. The top panel is for DUNE (5+5)
years while the bottom panel is for DUNE (3.5+3.5) years. We consider the true
value of $\theta_{23}$ in the higher octant (HO) and the value is 48.44∘. The
red dashed lines in figure 4 are for DUNE-CC, blue solid lines are for DUNE-
NC, magenta dashed lines are for T2HKK, green dashed lines are for
DUNE-(CC+NC) and the dashed black lines are for a combination of DUNE-(CC+NC)
and T2HKK. On the top-left and the bottom-left panel, the data is generated
with the true value of the oscillation parameters as given in 2, keeping
${\tau_{3}}/m_{3}=0$. These plots show how well DUNE, T2HKK, and their
combinations can constrain invisible neutrino decay. In the top-left panel, we
see that the inclusion of the neutral current with the charged current
improves the limits slightly on the decay parameter. Also the combination of
DUNE-(CC+NC)+T2HKK gives the highest precision. After 7 years (3.5 years of
$\nu$ and 3.5 years of $\bar{\nu}$) of running of DUNE, the constraint on
${\tau_{3}}/m_{3}$, that can be obtained from DUNE-NC, DUNE-CC, DUNE-(CC+NC)
and DUNE-(CC+NC)+T2HKK are $2.5\times 10^{-12}s/eV$ , $2.68\times
10^{-11}s/eV$, $3.04\times 10^{-11}s/eV$ and $5.09\times 10^{-11}s/eV$
respectively at 3$\sigma$ CL. These constraints improve further if DUNE runs
for 10 years. After 10 years (5 years of $\nu$ and 5 years of $\bar{\nu}$) of
running of DUNE, the limits obtained by DUNE-CC, DUNE-(CC+NC) and
DUNE-(CC+NC)+T2HKK are $2.72\times 10^{-11}s/eV$, $3.17\times 10^{-11}s/eV$
and $5.51\times 10^{-11}s/eV$ respectively at $3\sigma$ CL. The constraint
obtained from T2HKK is $4.39\times 10^{-11}s/eV$ at $3\sigma$ CL which is
better than DUNE-CC and DUNE-(CC+NC).

On the top-right and bottom-right panels of figure 4, we have shown how these
experiments and their combinations can exclude the present standard
oscillation scenario if invisible neutrino decay is a natural phenomenon with
${\tau_{3}}/m_{3}=2.2\times 10^{-11}$ s/eV. The true value of
${\tau_{3}}/m_{3}$, considered in this work, is within $1.5\sigma$ allowed
region of T2K and NO$\nu$A data analysis [31]. So if $\nu_{3}$ is unstable in
nature with decay width ${\tau_{3}}/m_{3}=2.2\times 10^{-11}$ s/eV, then DUNE,
after running for 10 years (5 years of $\nu$ and 5 years of $\bar{\nu}$) can
exclude no decay scenario at 3$\sigma$ CL while T2HKK alone can exclude the
same at more than 5$\sigma$ CL. The synergy between DUNE and T2HKK can further
improve the exclusion limit at 5$\sigma$ CL. It can be seen from the figure 4
that, T2HKK has better precision than DUNE and for a given true value of
${\tau_{3}}/m_{3}=2.2\times 10^{-11}$ s/eV, it can measure the decay parameter
in the range [$5.23\times 10^{-11}>\tau_{3}/m_{3}>1.37\times 10^{-11}$] s/eV
at $3\sigma$ C. L. . DUNE-CC, when runs for 10 years, can precisely measure
the decay parameter in the range [$1.26\times
10^{-10}>\tau_{3}/m_{3}>1.21\times 10^{-11}$] s/eV at 3$\sigma$ C. L. for the
given true ${\tau_{3}}/m_{3}=2.2\times 10^{-11}$ s/eV. Although, with 7 years
of running, DUNE-CC alone can not exclude the no-decay scenario at 3$\sigma$
CL yet, the synergy between DUNE-(CC+NC) and T2HKK can exclude no decay
scenario at 5$\sigma$ ($3\sigma$) CL and they can measure the value of the
decay parameter in the range [$1.19\times 10^{-10}>\tau_{3}/m_{3}>1.17\times
10^{-11}$] s/eV ([$4.45\times 10^{-11}>\tau_{3}/m_{3}>1.47\times 10^{-11}$]
s/eV). After 10 years of data collection at DUNE, this synergy improves the
precision further and can measure the decay parameter in the range [$9.6\times
10^{-11}>\tau_{3}/m_{3}>1.22\times 10^{-11}$] s/eV ([$4.19\times
10^{-11}>\tau_{3}/m_{3}>1.51\times 10^{-11}$] s/eV) at 5$\sigma$ ($3\sigma$)
CL.

Figure 5: $\chi^{2}$ as a function of ${\tau_{3}}/{m_{3}}$ for standalone and
combination of experiments. The left panel is for the case where
${\tau_{3}}/{m_{3}}$=0 and the right panel is for
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV in data.

Neutrino physics is in the precision era and in the next decade the
oscillation parameters will be measured more precisely. By the time DUNE takes
data for 10 years, the six oscillation parameters will be measured more
precisely. In such a scenario, assuming that we measure all the six
oscillation parameters precisely, we have generated figure 5 where we have not
marginalized over any parameters. The colour code of the plots are same as
figure 4. Under this scenario, the constraints improve significantly and the
measured constraints on invisible neutrino decay at DUNE-(CC+NC) and T2HKK at
$3\sigma$ CL are $3.44\times 10^{-11}$ s/eV and $4.49\times 10^{-11}$ s/eV
respectively (See left panel of figure 5 ). The synergy between the two
experiments can improve this constraint and at $3\sigma$, the measured bound
is $5.69\times 10^{-11}$ s/eV. Again, for the assumed true value of the decay
parameter ${\tau_{3}}/m_{3}=2.2\times 10^{-11}$ s/eV, the exclusion limit is
enhanced in both experiments. As a result, $3\sigma$ CL, precise measurement
of the decay parameter can be done at DUNE-(CC+NC) and T2HKK in the range
[$7.22\times 10^{-11}>\tau_{3}/m_{3}>1.26\times 10^{-11}$] s/eV, [$5.02\times
10^{-11}>\tau_{3}/m_{3}>1.37\times 10^{-11}$] s/eV respectively. Using the
synergy between DUNE-(CC+NC) and T2HKK, we can exclude the no-decay scenario
at 5$\sigma$ CL and hence can measure the decay parameter in the range
[$7.74\times 10^{-11}>\tau_{3}/m_{3}>1.24\times 10^{-11}$ ] s/eV.

####  4.2.2 Measurements of $\theta_{23}$ in presence of neutrino decay

Here in this section, we show the effect of invisible neutrino decay in the
measurements of $\theta_{23}$. We show the results considering the HO i.e.
$\theta_{23}=48.44^{\circ}$ as the true octant. The value of the decay
parameter considered is ${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV. We have
considered two scenarios: (i) We assume ${\tau_{3}}/{m_{3}}=2.2\times
10^{-11}s/eV$ in data and in the fit, we keep ${\tau_{3}}/{m_{3}}=0$.(ii) We
assume decay in fit assuming ${\tau_{3}}/{m_{3}}=0$ in data. In figure 6, the
dashed black lines represent the case when we assume decay in the data, the
dashed blue lines represent the case when we assume decay in the fit, and the
dashed red lines represent the standard scenarios for each case. We have
marginalized over $\Delta m^{2}_{31}$ and $\delta_{cp}$ over their $3\sigma$
allowed ranges given in Table 2 in the fit.

Figure 6: ${\chi}^{2}$ as a function of sin${}^{2}\theta_{23}$ (test). The top
left panel corresponds to DUNE-(CC+NC), the top right panel corresponds to
T2HKK and the bottom panel corresponds to DUNE-(CC+NC)+T2HKK experiments. The
dashed red lines are for standard oscillation case, dashed black lines are for
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV in data and the dashed blue lines
are for ${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV in test.

As seen in figure 6, the global true minima for the stable neutrino case is at
the HO as the true value of $\theta_{23}=48.44^{0}$ lies in the HO. Hence the
fake minima appear in the LO at $\theta_{23}=43.65^{0}$. However, the
inclusion of neutrino decay in the data (fit) shifts the point of minima in
both the octants. Here in the HO of DUNE, the minima of the $\chi^{2}$ appear
at $\theta_{23}=47.35^{0}(49.42^{0})$ while in the LO, fake minima appear at
$\theta_{23}=44.25^{0}(44.30^{0})$. This behavior is consistent in both DUNE
and T2HKK and also in the combinations of both. In T2HKK, minima in the HO
appears at $\theta_{23}=47.8^{0}(48.91^{0})$ and in the LO, fake minima
appears at $\theta_{23}=43.62^{0}(43.51^{0})$. Even after combining DUNE and
T2HKK, the minima in the HO appears at $\theta_{23}=47.7^{0}(48.99^{0})$ and
in the LO, it appears at $\theta_{23}=44.3^{0}(43.76^{0})$. We see that for
both scenarios, the global minima gets shifted from the minima position
corresponding to the stable neutrino case. In the first scenario, the true
minima get shifted towards lower $\theta_{23}$ while in the second scenario,
true minima get shifted towards higher $\theta_{23}$. The reason for this can
be understood from figure 2 and is well explained in [26; 34].

####  4.2.3 Octant and CP violation Sensitivity in the presence of Invisible
Neutrino Decay

Figure 7: Expected octant-sensitivity at DUNE-(CC+NC) top left, T2HKK top
right , and DUNE-(CC+NC)+T2HKK bottom panel. The red dashed lines are for
standard oscillation case and the black dashed lines are in the presence of
invisible neutrino decay.

It has been observed in the previous subsection that in the presence of
invisible neutrino decay, measurements of $\theta_{23}$ get hampered as the
position of the global minima gets shifted from the global minima
corresponding to the standard three-flavor neutrino oscillation. In this
section, we study the effect of neutrino decay in the measurements of octant
and CP violation sensitivity at the far detector of these experiments. Figure
7 and 8 show the octant sensitivity and CP violation sensitivity for
DUNE-(CC+NC), T2HKK, and the combination of DUNE-(CC+NC) and T2HKK
respectively. In figure 7, the dashed red lines represent the standard
oscillation scenario while the dashed black lines represent the decay
scenario. To study the effect of invisible neutrino decay in octant
sensitivity and CP violation sensitivity we consider the presence of decay in
both data and theory. We assume the value of decay parameter
$\tau_{3}/m_{3}=2.2\times 10^{-11}$ s/eV. We generate the data at the values
of oscillation parameters given in Table 2. While analyzing the octant
sensitivity in the lower octant (LO), for every true value of $\theta_{23}$ in
the lower octant, we vary the test $\theta_{23}$ in the higher octant in the
range [$45^{\circ},52^{\circ}$]. Similarly for any true value of $\theta_{23}$
in the higher octant (HO), the test $\theta_{23}$ is varied in LO in the range
[$38^{\circ},45^{\circ}$]. In the fit we marginalized over
$\theta_{13},\Delta{m}^{2}_{31}$ and $\delta_{cp}$ in their allowed $3\sigma$
range given in Table 2.

We observe that for all the cases in the presence of invisible neutrino decay
the octant sensitivity increases in the lower octant and decreases in the
higher octant. This is consistent with the probability case ($P_{\mu\mu}$)
shown in figure 2 where we already found that the decay probability for LO is
higher than the decay probability in HO and hence in the presence of invisible
neutrino decay the sensitivity increases in LO and deteriorates in HO.
Furthermore, we note that T2HKK has an overall higher octant sensitivity than
DUNE-(CC+NC). Again when we combine DUNE-(CC+NC) with T2HKK, octant
sensitivity enhances.

Figure 8: CP violation sensitivity for DUNE-(CC+NC) (red), T2HKK (magenta),
and DUNE-(CC+NC)+T2HKK (black) experiments. The dashed lines are for the
standard oscillation case and the solid lines are for the decay case.

In figure 8 we show the CP violation sensitivity plots for DUNE-(CC+NC),
T2HKK, and the combination of DUNE-(CC+NC) and T2HKK. The true value of
$\theta_{23}$ considered in this case is 48.44∘. Here, all the solid lines
represent the decay scenario while the dashed lines represent the stable
neutrino case. In figure 8 the red lines are for DUNE-(CC+NC), magenta lines
are for T2HKK, and the black lines are for DUNE-(CC+NC)+T2HKK. CP violation
sensitivity of an experiment is the measure of excluding the CP-conserving
values. To measure the CP violation sensitivity, we vary the true
$\delta_{cp}$ within the range $[-\pi,\pi]$ in data while in the fit, we fix
$\delta_{cp}$ at $0$ and $\pm\pi$. The oscillation parameters $\theta_{23}$
and $\Delta{m}^{2}_{31}$ are marginalized in their $3\sigma$ range keeping the
other parameter fixed as given in Table 2. We observe that the invisible
neutrino decay brings almost no change in the plots.

####  4.2.4 Correlation plots

Figure 9: The $3\sigma$ confidence contours in the
${\tau_{3}}/{m_{3}}-\theta_{23}$ plane for DUNE (CC-red line, CC+NC-green
line), T2HKK ( blue line) and DUNE-(CC+NC)+T2HKK (black line). The left panel
is for ${\tau_{3}}/{m_{3}}=0$ in data and the right panel is for
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV in data.

In figure 9, we have shown the correlation between ${\tau_{3}}/{m_{3}}$ and
$\theta_{23}$ at $3\sigma$ CL in ${\tau_{3}}/{m_{3}}-\theta_{23}$ plane. The
results are generated for DUNE, T2HKK and for their combinations. For DUNE, 10
years of runtime is considered in this case. While generating the data in the
left panel, we consider no-decay scenario and in the right panel, we assume
that neutrino decay is a natural phenomenon in addition to oscillations. In
the analysis, we use the true value of the atmospheric mixing angle
$\theta_{23}=48.44^{\circ}$ and the true value of the decay parameter
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV. This plot complements the
results shown in figure 5 but in ${\tau_{3}}/{m_{3}}-\theta_{23}$ plane. The
smaller the parameter space, the more precise the measurements are. For DUNE,
parameter space shrinks in both the panels, when we add neutral current
measurements to the charged current measurements. But yet, T2HKK performs
better than DUNE in both scenarios. If neutrino decays in addition to
oscillations, then T2HKK can significantly constrain the parameter space and
hence can make more precise measurements in the 3$\sigma$ allowed region. The
combinations of DUNE and T2HKK enhance the precision. Here also, like in the
previous cases, the synergy between DUNE and T2HKK helps in all aspects to
probe invisible neutrino decay.

Figure 10: The $3\sigma$ confidence contours in the
$\theta_{23}-\Delta{m}^{2}_{31}$ plane for DUNE-(CC+NC) (top left), T2HKK (top
right), and DUNE-(CC+NC)+T2HKK (bottom right) experiments. The red dashed
contours are for ${\tau_{3}}/{m_{3}}=0$ and solid black contours are for
${\tau_{3}}/{m_{3}}=2.2\times 10^{-11}$ s/eV.

In figure 3, we have already observed the effects of $\theta_{23}$ and
$\Delta{m}^{2}_{31}$ on constraining invisible neutrino decay. Both parameters
play an important role in the presence of invisible neutrino decay. Here in
figure 10, we show the $3\sigma$ allowed region in
$\theta_{23}-\Delta{m}^{2}_{31}$ plane for DUNE, T2HKK and for the combination
of both the experiments, with and without neutrino decay. For DUNE, we have
considered 10 years of runtime. For all the cases in figure 10 the red dashed
contours represent the stable neutrino case while the solid black contours
represent the decay case. For the decay case, we have considered decay both in
data and fit. In the fit, we have varied the decay parameter in the range
$10^{-12}$ to $10^{-9}$. $\delta_{cp}$ is marginalized in its 3$\sigma$
allowed range in both the cases. We observe that in the presence of invisible
neutrino decay with oscillation, the allowed region increases more than the
standard case, in all three panels. The precision of measurement at T2HKK is
higher than that of DUNE as the $\theta_{23}-\Delta{m}^{2}_{31}$ parameter
space shrinks in the case of T2HKK. The precision of measurements increases
when we combine both experiments. In table 3, we show a comparison of the
limit on decay parameters obtained from various experiments.

Experiment |  $90\%$ CL $(3\sigma)$ bound on $\tau_{3}/m_{3}$ (s/eV) |  Ref.
---|---|---
DUNE - CC |  $4.50(2.38)\times 10^{-11}$ |  Ref. [26]
DUNE - (CC+NC) (5+5) |  $5.1(2.7)\times 10^{-11}$ |  Ref. [41]
DUNE - (CC+NC) (5+5) |  $5.77(3.17)\times 10^{-11}$ |  This work
DUNE - (CC+NC) (5+5)+T2HKK (With marg) |  $1.06\times 10^{-10}(5.51\times 10^{-11})$ |  This work
DUNE - (CC+NC) (5+5)+T2HKK (W/O marg) |  $1.1\times 10^{-10}(5.69\times 10^{-11})$ |  This work
ESSnuSB (540 km) |  $4.22(1.68)\times 10^{-11}$ |  Ref. [29]
ESSnuB (360 km) |  $4.95(2.64)\times 10^{-11}$ |  Ref. [29]
JUNO |  $9.3(4.7)\times 10^{-11}$ |  Ref. [42]
INO |  $1.51(0.566)\times 10^{-10}$ |  Ref. [23]
KM3NeT-ORCA |  $2.5(1.4)\times 10^{-10}$ |  Ref. [10]
T2HK |  $4.43(2.72)\times 10^{-11}$ |  Ref. [34]
T2HKK |  $1.01\times 10^{-10}(4.36\times 10^{-11})$ |  Ref. [34]
T2HKK+ESSnuSB |  $1.064\times 10^{-10}(5.53\times 10^{-11})$ |  Ref. [34]
T2K + NO$\nu$A |  $2.3(1.5)\times 10^{-11}$ |  Ref. [31]
ESSnuSB |  $3.69(2.43)\times 10^{-11}$ |  Ref. [34]
T2HK+ESSnuSB |  $1.01\times 10^{-10}(4.36\times 10^{-11})$ |  Ref. [34]
Table 3: Comparision of bound on decay parameter ${\tau_{3}}/{m_{3}}$ from
various experiments.

##  5 Summary and Conclusion

We have explored the impact of invisible neutrino decay in two upcoming long
baseline experiments DUNE and T2HKK. We assume that the mass eigenstate
$m_{3}$ is heavier than $m_{4}$ and hence $\nu_{3}$ being unstable can decay
to a lighter sterile state. Throughout this analysis, we have assumed normal
mass hierarchy as the true hierarchy. In this work, we have combined the
neutral current measurements and charged current measurements at DUNE to
constrain invisible neutrino decay. We found that this combination of CC and
NC enhances DUNE’s ability to constrain ${\tau_{3}}/{m_{3}}$. We found that
the synergy between DUNE and T2HKK can provide tight constraints on
${\tau_{3}}/{m_{3}}$.

First, we have checked the effect of marginalization on constraining invisible
neutrino decay. It is observed that the marginalization of $\delta_{cp}$ has a
minimal effect while that of $\theta_{23}$ and $\Delta{m}^{2}_{31}$ have a
significant effect on invisible neutrino decay, especially at DUNE. Based on
this analysis, we have considered the marginalization of $\theta_{23}$,
$\Delta{m}^{2}_{31}$ and $\delta_{cp}$ while deriving the constraint on
invisible neutrino decay. We found that after taking data for 10 years, DUNE
(with both CC and NC measurements) can constrain ${\tau_{3}}/{m_{3}}$ at
3$\sigma$ and the measured value is $3.17\times 10^{-11}s/eV$. We have
observed that with T2HKK experiment has a better ability to constrain
invisible neutrino decay than DUNE. This constraint on invisible decay
improves further when combined with the T2HKK experiment and at 3$\sigma$, the
limit is $5.5\times 10^{-11}s/eV$. If nature has invisible neutrino decay in
addition to oscillations and the true value of ${\tau_{3}}/m_{3}=2.2\times
10^{-11}$ s/eV, the synergy between these two experiments can rule out a
stable neutrino scenario at 5$\sigma$ CL. In that case, the combination of
DUNE and T2HKK can measure the decay parameter in the range [$9.6\times
10^{-11}>\tau_{3}/m_{3}>1.22\times 10^{-11}$] s/eV. We have observed that the
synergy between DUNE( with both NC and CC measurements) and T2HKK can give the
best possible constrain of $\tau_{3}/m_{3}\leq 5.69\times 10^{-11}$ s/eV at
3$\sigma$ if all the six oscillation parameters measured precisely by the time
DUNE take data for 10 years.

The impact of decay in $\theta_{23}$ measurement has also been studied. We
have found that the $\theta_{23}$ measurements are hampered in the presence of
invisible neutrino decay. If the data is generated with neutrino decay, both
the true and fake minima deviate from the minima corresponding to the stable
neutrino. Similar behaviour is observed when stable neutrino data is fitted
with a neutrino decay scenario. If neutrino decay is a natural process, then
the measurements of $\theta_{23}$ get affected as also seen from the
correlation plots in the ${\tau_{3}}/{m_{3}}-\theta_{23}$ plane.

This study also reveals that in the presence of invisible neutrino decay
octant sensitivity increases in the lower octant but decreases in the higher
octant. The combination of DUNE with T2HKK enhances the overall octant
sensitivity. The effect of invisible neutrino decay in the measurement of CP
violation sensitivity is nominal.

While the neutral current (NC) measurements at DUNE have no significant effect
on invisible neutrino decay, combining it with the charged current
measurements provides a higher precision. Furthermore, combining DUNE (CC+NC)
with T2HKK improves overall sensitivity and constraints on the decay
parameter.

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