Octant Degeneracy and Plots of Parameter Degeneracy in Neutrino Oscillations
Revisited

Sho Sugama and Osamu Yasuda

Department of Physics, Tokyo Metropolitan University,

Hachioji, Tokyo 192-0397, Japan

Abstract

The three kinds of parameter degeneracy in neutrino oscillation, the
intrinsic, sign and octant degeneracy, form an eight-fold degeneracy. The
nature of this eight-fold degeneracy can be visualized on the
($\sin^{2}2\theta_{13}$, $1/\sin^{2}\theta_{23}$)-plane, through quadratic
curves defined by $P(\nu_{\mu}\to\nu_{e})=$ const. and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})=$ const., along with a straight line
$P(\nu_{\mu}\to\nu_{\mu})=$ const. After $\theta_{13}$ was determined by
reactor neutrino experiments, the intrinsic degeneracy in $\theta_{13}$
transforms into an alternative octant degeneracy in $\theta_{23}$, which can
potentially be resolved by incorporating the value of
$P(\nu_{\mu}\to\nu_{\mu})$. In this paper, we analytically discuss whether
this octant parameter degeneracy is resolved or persists in the future long
baseline accelerator neutrino experiments, such as T2HK, DUNE, T2HKK and
ESS$\nu$SB. It is found that the energy spectra near the first oscillation
maximum are effective in resolving the octant degeneracy, whereas those near
the second oscillation maximum are not.

##  1 Introduction

In the past twenty-five years, significant progress have been achieved in
measuring neutrino oscillation parameters within the standard three-flavor
framework [1]. The ultimate goal in the study of standard neutrino
oscillations is to determine the CP phase $\delta$. It is expected that future
long baseline experiments such as T2HK [2] and DUNE [3] will precisely measure
the appearance probabilities $P(\nu_{\mu}\to\nu_{e})$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$, enabling the determination of $\delta$.
However, an issue arises in extracting $\delta$ from appearance probabilities
due to the so-called parameter degeneracy. This implies that even with precise
knowledge of $P(\nu_{\mu}\to\nu_{e})$ and $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$
at a fixed energy and baseline length, these probabilities cannot uniquely
determine the values of ($\theta_{23}$, $\theta_{13}$, $\delta$). Since each
degenerate solution gives a different value of $\delta$, resolving the
parameter degeneracy is crucial for the precise determination of $\delta$. In
historical context, the octant degeneracy
$\pi/4-\theta_{23}\leftrightarrow\theta_{23}-\pi/4$ was first identified in
Ref. [4], followed by the recognition of the intrinsic degeneracy in Ref. [5],
and the sign degeneracy $\Delta m^{2}_{\mbox{\rm\scriptsize
atm}}\leftrightarrow-\Delta m^{2}_{\mbox{\rm\scriptsize atm}}$ in Ref. [6].
Ref. [7] highlighted the presence of these as eight-fold degeneracies.
Moreover, a graphical representation was introduced in Ref. [8] where curves
derived from $P(\nu_{\mu}\rightarrow\nu_{e})=$ const. and
$P(\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e})=$ const. formed quadratic patterns
on the ($X\equiv\sin^{2}2\theta_{13}$, $Y\equiv 1/s^{2}_{23}$)-plane, where
$s_{23}\equiv\sin\theta_{23}$. The representation in Ref. [8] visually
demonstrates the existence of eight degenerate solutions, as elaborated in the
subsequent section.

After the determination of $\theta_{13}$ through reactor experiments [9, 10,
11], the primary focus shifts to the sign degeneracy of $\Delta
m^{2}_{\mbox{\rm\scriptsize atm}}$ and the octant degeneracy of $\theta_{23}$.
As far as the sign degeneracy is concerned, nearly all results favor the
normal ordering (NO) [1]. As for the octant degeneracy, on the other hand,
uncertainty remains regarding the correct octant, as the Superkamiokande
atmospheric neutrino results seem to favor the lower octant, while other
datasets indicate a different preference [12].

The purpose of this work is to investigate the behavior of the octant
degeneracy when combining appearance probabilities
$P(\nu_{\mu}\rightarrow\nu_{e})=$ const. and
$P(\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e})=$ const. with fixed values of
$X\equiv\sin^{2}2\theta_{13}=$ const. and $Y\equiv 1/s^{2}_{23}=$ const.,
utilizing the graphical representation from Ref. [8] for the T2HK [2], T2HKK
[13], DUNE [3], and ESS$\nu$SB [14] experiments. Our results demonstrate that,
by utilizing energy bins near the first oscillation maximum, the octant
degeneracy can be resolved by ensuring the intersection of these quadratic
curves with the vertical line $X=$ const. and the two horizontal lines
($Y=1+\cot^{2}\theta_{23}$ or $Y=1+\tan^{2}\theta_{23}$), under the assumption
of small experimental errors.

The paper is organized as follows. In Sect. 2, we provide background
information on parameter degeneracy and introduce the plot illustrating the
eight-fold parameter degeneracy. Sect. 3 presents the plots for the four
future long baseline experiments and discusses the distinctive characteristics
of each experiment. Finally, Section 4 summarizes our conclusions.

##  2 The plot of the eight-fold parameter degeneracy

In the second-order approximation with respect to $\theta_{13}$ and $\Delta
m_{21}^{2}$, the appearance probabilities $P(\nu_{\mu}\to\nu_{e},E)$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e},E)$ for a neutrino energy $E$ and a
baseline length $L$ can be expressed as [15, 7]

$\displaystyle P(\nu_{\mu}\to\nu_{e},E)=x^{2}F^{2}+2\,\mbox{\rm sign}(\Delta
m_{31}^{2})\,xyFg\cos\left[\delta+\mbox{\rm sign}(\Delta
m_{31}^{2})\,\Delta\right]+y^{2}g^{2}$ $\displaystyle\hskip 34.1433pt=P(E)$
$\displaystyle
P(\bar{\nu}_{\mu}\to\bar{\nu}_{e},E)=x^{2}\bar{F}^{2}+2\,\mbox{\rm
sign}(\Delta m_{31}^{2})\,xy\bar{F}g\cos\left[\delta-\mbox{\rm sign}(\Delta
m_{31}^{2})\,\Delta\right]+y^{2}g^{2}$ $\displaystyle\hskip
34.1433pt=\bar{P}(E)\,,$ (2)

where

$\displaystyle x\equiv s_{23}\sin 2\theta_{13}\,,$ (5) $\displaystyle
y\equiv\left|\frac{\Delta m^{2}_{21}}{\Delta
m^{2}_{31}}\right|\cos\theta_{23}\sin 2\theta_{12}\,,$
$\displaystyle(F,\bar{F})\equiv\left\\{\begin{array}[]{c}(f,\bar{f})~{}~{}\mbox{\rm
for NO}\\\ (\bar{f},f)~{}~{}\mbox{\rm for IO}\\\ \end{array}\right\\}\,,$
$\displaystyle\left\\{\begin{array}[]{c}f\\\
\bar{f}\end{array}\right\\}\equiv\frac{\sin\left(\Delta\mp
AL/2\right)}{\left(1\mp AL/2\Delta\right)}\,,$ (8) $\displaystyle
g\equiv\frac{\sin\left(AL/2\right)}{AL/2\Delta}\,,$
$\displaystyle\Delta\equiv\frac{|\Delta m_{31}^{2}|L}{4E}\,,$ $\displaystyle
A\equiv\sqrt{2}G_{F}N_{e}\,,$

$G_{F}$ stands for the Fermi coupling constant, and $N_{e}$ represents the
electron density in matter, which we assume to be constant throughout this
paper. The present study focuses on the parameter degeneracy affecting the CP
phase $\delta$, as well as the mixing angles $\theta_{13}$ and $\theta_{23}$,
at different neutrino energies $E$. In the following discussions, we will keep
the values of $\Delta m^{2}_{31}$, $\Delta m^{2}_{21}$ and $\theta_{12}$
fixed, while varying $\delta$, $\theta_{13}$ and $\theta_{23}$ as test
variables around their true values. In Eqs. (2) and (2), $P(E)$ and
$\bar{P}(E)$ are the probabilities evaluated at the neutrino energy $E$ using
the true oscillation parameters $\delta$, $\theta_{13}$ and $\theta_{23}$,
whereas $P(\nu_{\mu}\to\nu_{e},E)$ and $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e},E)$
are calculated with the test value of $\delta$, $\theta_{13}$ and
$\theta_{23}$. By defining

$\displaystyle X\equiv\sin^{2}2\theta_{13}$ $\displaystyle
Y\equiv\displaystyle\frac{1}{s^{2}_{23}}\,,$

and subsequently eliminating $\delta$, we obtain the following expression from
Eqs. (2) and (2) [8]:

$\displaystyle 16CX(Y-1)$ (9)
$\displaystyle=\frac{1}{\cos^{2}\Delta}\left[\left(\frac{P(E)-C}{F}+\frac{\bar{P}(E)-C}{\bar{F}}\right)(Y-1)-(F+\bar{F})X+\frac{P(E)}{F}+\frac{\bar{P}(E)}{\bar{F}}\right]^{2}$
$\displaystyle+\frac{1}{\sin^{2}\Delta}\left[\left(\frac{P(E)-C}{F}-\frac{\bar{P}(E)-C}{\bar{F}}\right)(Y-1)-(F-\bar{F})X+\frac{P(E)}{F}-\frac{\bar{P}(E)}{\bar{F}}\right]^{2},$

where

$\displaystyle C\equiv\left(\frac{\Delta m^{2}_{21}}{\Delta
m^{2}_{31}}\right)^{2}\left[\frac{\sin(AL/2)}{AL/2\Delta}\right]^{2}\sin^{2}2\theta_{12}\,.$

In energy ranges where $|\Delta m_{31}^{2}|L/4E\neq n\pi/2$ ($n=1,2,3,\cdots$)
is satisfied, Eq. (9) represents a quadratic curve in the $(X,Y)$-plane [8].
The rationale behind choosing the variable $Y\equiv 1/s^{2}_{23}$ is its
analytical convenience for demonstrating that there exist two intersections
between the curve (9) and the line $\theta_{23}=$ const. Depending on whether
the test mass ordering is correct or wrong, i.e., whether the test mass
ordering matches the true ordering or not, two potential quadratic curves
emerge, as depicted in Fig. 1 (a). By utilizing the disappearance probability
$P(\nu_{\mu}\to\nu_{\mu})\sim\sin^{2}2\theta_{23}\sin^{2}(\Delta
m_{31}^{2}L/4E)$, we can infer the value of $\sin^{2}2\theta_{23}$, which
yields two possibilities for $Y\equiv
1/s^{2}_{23}=2/(1\pm\sqrt{1-\sin^{2}2\theta_{23}})$. Prior to the measurement
of $\theta_{13}$, therefore, there were potentially eight solutions, as
illustrated in Fig. 1 (a).

Figure 1: Parameter degeneracy for normal ordering and inverted ordering: (a)
Away from the oscillation maximum ($|\Delta m^{2}_{31}|L/4E\neq\pi/2$), (b) At
the oscillation maximum ($|\Delta m^{2}_{31}|L/4E=\pi/2$). In the case of (a),
there are eight solutions with a different value of $\delta$. In the case of
(b), four solutions exist with a two-fold degeneracy.

Conversely, at the oscillation maximum ($|\Delta
m_{31}^{2}|L/4E=\left(n+1/2\right)\pi$; $n=0,1,2,\cdots$), the numerator of
the first term on the right hand side of Eq. (9) must vanish, resulting in a
straight line in the $(X,Y)$-plane:

$\displaystyle\left(\frac{P(E)-C}{F}+\frac{\bar{P}(E)-C}{\bar{F}}\right)(Y-1)-(F+\bar{F})X+\frac{P(E)}{F}+\frac{\bar{P}(E)}{\bar{F}}=0\,.$
(10)

In this case, two potential straight lines also emerge, and they are
illustrated in Fig. 1 (b). When considering the situation at the oscillation
maximum, there exist four feasible solutions, each characterized by a two-fold
degeneracy, as the two-fold intrinsic degeneracy is exact in this case.

Given our primary focus on the intersection between curve (9) and
$\sin^{2}2\theta_{13}=$ const. or $1/s^{2}_{23}=$ const., the classification
of the quadratic curve (9) as hyperbolic or elliptic does not directly
influence the parameter degeneracy characteristics. However, it is feasible to
ascertain whether curve (9) is hyperbolic or elliptic based on the sign of the
following discriminant:

$\displaystyle
D\equiv\left[\frac{1}{\cos^{2}\Delta}\left(\frac{P(E)-C}{F}+\frac{\bar{P}(E)-C}{\bar{F}}\right)(F+\bar{F})\right.$
(11)
$\displaystyle+\left.\frac{1}{\sin^{2}\Delta}\left(\frac{P(E)-C}{F}-\frac{\bar{P}(E)-C}{\bar{F}}\right)(F-\bar{F})+8C\right]^{2}$
$\displaystyle-\left[\frac{1}{\cos^{2}\Delta}\left(\frac{P(E)-C}{F}+\frac{\bar{P}(E)-C}{\bar{F}}\right)^{2}+\frac{1}{\sin^{2}\Delta}\left(\frac{P(E)-C}{F}-\frac{\bar{P}(E)-C}{\bar{F}}\right)^{2}\right]$
$\displaystyle\times\left[\frac{1}{\cos^{2}\Delta}(F+\bar{F})^{2}+\frac{1}{\sin^{2}\Delta}(F-\bar{F})^{2}\right]\,,$

where $D>0$ ($D<0$) corresponds to a hyperbolic (an elliptic) curve. In
general, in the energy region near the first oscillation maximum ($|\Delta
m_{31}^{2}|L/4E\sim\pi/2$), curve (9) exhibits a hyperbolic shape, whereas it
can have an elliptic form in the lower energy region, as explained in the
subsequent discussions for each future experiment.

##  3 Behaviors of the plots for the future long baseline experiments

After $\theta_{13}$ was determined, the situation of the eight-fold parameter
degeneracy is changed. The following three pieces of information are
available:

(i)The quadratic curve derived from $\displaystyle
P(\nu_{\mu}\to\nu_{e})=\mbox{\rm
const}.~{}\&~{}P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})=\mbox{\rm const}.$
$\displaystyle\mbox{\rm(ii)}~{}\mbox{\rm The~{}straight~{}line~{}}X=\mbox{\rm
const}.\mbox{\rm~{}obtained~{}from}$ $\displaystyle
P(\bar{\nu}_{e}\to\bar{\nu}_{e})=\mbox{\rm const}.$
$\displaystyle\mbox{\rm(iii)}~{}\mbox{\rm The~{}straight~{}line~{}}Y=\mbox{\rm
const}.\mbox{\rm~{}obtained~{}from}$ $\displaystyle
P(\nu_{\mu}\to\nu_{\mu})=\mbox{\rm
const}.~{}\&~{}P(\bar{\nu}_{\mu}\to\bar{\nu}_{\mu})=\mbox{\rm const}.$ (14)

Our strategy is first to examine the intersection of (i) and (ii), and then to
enforce condition (iii).

As shown in Fig. 2, if we exclusively employ (i) from Eq. (3) and (ii) from
Eq. (3), then generally four solutions arise. In essence, in this case, the
intrinsic degeneracy in $\theta_{13}$ is converted into an additional octant
degeneracy in $\theta_{23}$, as illustrated in Fig.,2.

Figure 2:  Parameter degeneracy after $\theta_{13}$ was determined, assuming
the true mass ordering is normal. The red (blue) curve corresponds to the case
where the test mass ordering is normal (inverted). In case (a), away from the
oscillation maximum, the combination of the quadratic curves through the
appearance probabilities ((i) in Eq. (3)) and the line
$X=\sin^{2}2\theta_{13}=$ const. ((ii) in Eq. (3)) results in four solutions.
The true solution(s) is represented by the black circle, while the fake
solution(s) is indicated by squares. In case (b), at the oscillation maximum,
the merging of the straight lines ((iii) in Eq. (14)) using the appearance
probabilities ((i) in Eq. (3)) and $X=\sin^{2}2\theta_{13}=$ const. ((ii) in
Eq. (3)) leads to two degenerate solutions. The true solution is depicted as a
black circle, coinciding with the fake solution marked by a square.

In the energy region away from the oscillation maximum, as illustrated in Fig.
2 (a), assuming a sufficiently long baseline to differentiate between the mass
orderings, by combining the disappearance channels $\nu_{\mu}\to\nu_{\mu}$ and
$\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ ((iii) in Eq. (14)), we can eliminate the
wrong mass ordering, which is represented by the blue curve in Fig. 2 (a).
This is because the intersections between the blue curve and
$\sin^{2}2\theta_{13}=$ const. yield a value of $1/s_{23}^{2}$ that
contradicts the one suggested from the disappearance channels.

In the energy region at the oscillation maximum, on the other hand, as
depicted in Fig. 2 (b), with a baseline length that may not be sufficiently
long, the two straight lines generated by the appearance probabilities are so
close that it is difficult to distinguish the true and wrong mass orderings.
Thus is the case with T2HK and ESS$\nu$SB, both having relatively shorter
baseline lengths.

Fig. 2 is given for a specific value of the neutrino energy and a specific
baseline length. To observe how this degeneracy is solved or persists, we plot
the values of $1/s_{23}^{2}$ as a function of neutrino energy for the future
long baseline experiments: T2HK, DUNE, T2HKK and ESS$\nu$SB.

To generate these curves, we adopt the following reference values, which
correspond to the arithmetic average of the best fit points in Ref. [1] from
the three groups, except $\delta$ which we assume a value of $-90^{\circ}$:

MO | $\Delta m_{21}^{2}/10^{-5}\mbox{\rm eV}^{2}$ | $|\Delta m_{32}^{2}|/10^{-3}\mbox{\rm eV}^{2}$ | $\theta_{12}$ | $\theta_{13}$ | $\theta_{23}$ | $\delta$ | $\rho/\mbox{\rm g}\cdot\mbox{\rm cm}^{-3}$
---|---|---|---|---|---|---|---
NO | 7.43 | 2.432 | $33.9^{\circ}$ | $8.49^{\circ}$ | $48.1^{\circ}$ | $-90^{\circ}$ | 2.6
IO | 7.43 | 2.496 | $33.9^{\circ}$ | $8.55^{\circ}$ | $48.3^{\circ}$ | $-90^{\circ}$ | 2.6
Table 1:  The reference values of the true oscillation parameters utilized to
generate plots in this study are provided for both normal (NO) and inverted
(IO) mass orderings (MO). We choose a constant density of $\rho$ = 2.6 g/cm3
for all the experiments for the sake of simplicity.

###  3.1 T2HK

The experiment T2HK [2] features a baseline length of 295 km and employs an
off-axis neutrino beam with an approximate energy of 0.6 GeV.

The trajectory of the curve derived from Eq. (9) in the
($\sin^{2}2\theta_{13}$, $1/s^{2}_{23}$)-plane is illustrated in Fig. 3. Since
we assume $\delta=-\pi/2$ as the true value in this paper, if the true mass
ordering is normal, then T2HK alone can rule out the possibility of the wrong
mass ordering (i.e., inverted ordering) [16]. Consequently, the curves
representing the wrong mass ordering, depicted as dotted lines in Fig. 3, are
absent in the left panel of Fig. 3. Conversely, if the true mass ordering is
inverted, then T2HK by itself cannot exclude the possibility of the wrong mass
ordering, leading to the presence of the dotted curves in the right panel of
Fig. 3.

Figure 3:  The trajectory of the curve derived by Eq. (9) for the T2HK
experiment is illustrated. The horizontal thin solid (dotted) straight line
represents the true (fake) value of $1/s^{2}_{23}$, whereas the vertical
dashed straight line indicates the value of $\sin^{2}2\theta_{13}$. The left
(right) panel corresponds to the case where the true mass ordering is normal
(inverted). The solid curves in blue, red and green depict the trajectory of
Eq. (9) for neutrino energies of 0.5 GeV, 0.6 GeV and 0.7 GeV, respectively,
assuming the true mass ordering. On the other hand, the dotted curves
represent trajectories assuming the wrong mass ordering. The energy for the
dotted curves corresponds to that of the solid curve of the same color.

T2HK is conducted at energies near the first oscillation maximum. Therefore,
for the neutrino energy of 0.6 GeV in T2HK, the situation is similar to that
depicted in Fig. 1 (b), where the resolution of the octant degeneracy is
expected to be achieved, provided that the the difference of the true and
wrong values of $1/s^{2}_{23}$, which is obtained from the disappearance
oscillation probabilities $P(\nu_{\mu}\to\nu_{\mu})$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{\mu})$, is larger than the experimental
errors.

Fig. 4 illustrates the values of the true and fake solutions for
$1/s^{2}_{23}$, which are obtained solely from the appearance channels
$\nu_{\mu}\to\nu_{e}$ and $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$ ((i) in Eq. (3))
and the reactor data on $\theta_{13}$ ((ii) in Eq. (3)), plotted against
neutrino energy. This is presented for both the cases of normal ordering (left
panel) and inverted ordering (right panel) as the true mass ordering. The fake
$1/s^{2}_{23}$ value derived from the disappearance oscillation probabilities
$P(\nu_{\mu}\to\nu_{\mu})$ and $P(\bar{\nu}_{\mu}\to\bar{\nu}_{\mu})$ ((iii)
in Eq. (14)) is represented by the horizontal dotted thin line in Fig. 4.
Notice that $P(E)$ and $\bar{P}(E)$ are calculated using the true values of
the oscillation parameters listed in Table 1 for each neutrino energy $E$ and
mass ordering. In the energy range of 0.5 GeV $\lesssim E\lesssim$ 0.7 GeV,
where there is a substantial number of events for the appearance channels, the
discrepancy between the true and fake $1/s^{2}_{23}$ solutions is minimal on
the condition that the correct mass ordering is assumed, and octant degeneracy
is expected to be resolved. We note in passing that, even if the true mass
ordering is assumed to be normal (the left panel of Fig. 4), solutions with
wrong mass ordering can emerge in lower energy ranges, as indicated by the
blue lines. However, the energy in this region is quite low, resulting in a
limited number of events.

Figure 4: Possible values of the two solutions for $1/s^{2}_{23}$ are
displayed for the case of the true mass ordering being normal (left panel) or
inverted (right panel). The horizontal dotted thin straight line represents
the fake value of $1/s^{2}_{23}$ estimated from the disappearance oscillation
probabilities $P(\nu_{\mu}\to\nu_{\mu})$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{\mu})$ ((iii) in Eq. (14)). Additionally, the
expected numbers of events for $\nu_{\mu}\to\nu_{e}$
($\bar{\nu}_{\mu}\to\bar{\nu}_{e}$), extracted from Ref. [17] under the
assumption of $\delta=0$, are included for clarity, enabling a comparison of
statistical errors across different energy regions.  Figure 5: The
discriminant $D$ in Eq. (11) for the quadratic curve (9) in the case of the
T2HK experiment.

Fig. 5 illustrates the sign of discriminant (11) of the quadratic curve (9)
plotted against the neutrino energy in the case of the T2HK experiment. In
most of the energy region where the numbers of events of the appearance
channels are large, the discriminant is positive, indicating a hyperbolic
quadratic curve. Conversely, and at low energies, there are regions where the
discriminant becomes negative, resulting in an elliptic quadratic curve.

###  3.2 DUNE

The DUNE experiment [3] employs a baseline length of 1300 km and utilizes a
wideband beam with an average neutrino energy around 2.5 GeV.

The trajectory of the curve described by Eq. (9) for the DUNE experiment is
illustrated in Fig. 6. Due to the considerable baseline length, the matter
effect at DUNE is significant, which leads to the exclusion of the possibility
of a wrong mass ordering by DUNE alone, regardless of the true mass ordering.
Although DUNE employs a wideband beam, the situation for the energy range 2.5
GeV $\lesssim E\lesssim$ 3.5 GeV is similar to that at the first oscillation
maximum. Fig. 7 presents the values of $1/s^{2}_{23}$ for both true and fake
solutions as functions of neutrino energy, in the case of normal ordering
(left panel) and inverted ordering (right panel) as the true mass ordering. In
the energy range of 2.5 GeV $\lesssim E\lesssim$ 3.5 GeV, where the number of
events of the appearance channels is significant, difference between the true
and fake solutions for $1/s^{2}_{23}$ is minor, suggesting a potential
resolution of the octant degeneracy, as long as the difference of the true and
wrong values of $1/s^{2}_{23}$, derived from the disappearance oscillation
probabilities ((iii) in Eq. (14)), exceeds the experimental errors. Similar to
the case of T2HK, for DUNE as well, even if the true mass ordering is assumed
to be normal, there can be fake solutions in lower energy ranges, as depicted
in blue (shown in the left panel of Fig. 7), although the energy in this
region is too low to yield a substantial number of events.

Figure 6:  The trajectory of the curve described by Eq. (9) for the DUNE
experiment is depicted. The meaning of the straight thin lines is the same as
in Fig. 3. The left (right) panel corresponds to the case of true normal
(inverted) mass ordering. The solid curves in blue, red and green represent
the trajectory of Eq. (9) for neutrino energy of 2.5 GeV, 3.0 GeV and 3.5 GeV,
respectively, assuming the true mass ordering (normal (inverted) ordering in
the left (right) panel).

Figure 7:  The potential values of the two solutions for $1/s^{2}_{23}$ are
depicted for both the true normal (left panel) or inverted (right panel) mass
ordering cases. Similar to Fig. 4, the horizontal dotted thin line represents
the fake value of $1/s^{2}_{23}$ estimated from the disappearance oscillation
probabilities ((iii) in Eq. (14)). Additionally, the expected numbers of
events for $\nu_{\mu}\to\nu_{e}$ ($\bar{\nu}_{\mu}\to\bar{\nu}_{e}$), obtained
from Ref. [18], are provided for reference.

Fig. 8 displays the sign of discriminant (11) for the quadratic curve (9)
plotted against neutrino energy for the DUNE case. Similarly, in this case,
the discriminant is positive for most of the energy region where the number of
events of the appearance channels is significant. However, at lower energies,
there are regions where the discriminant transitions to negative values.

Figure 8: The discriminant (11) for the quadratic curve (9) in the case of
DUNE.

###  3.3 T2HKK

The T2HKK experiment [13] employs a baseline length of 1100 km and utilizes a
wideband beam with an average neutrino energy around 1 GeV. Unlike the T2HK
and DUNE experiments, the T2HKK experiment does not cover the energy range at
the first oscillation maximum, which occurs at $E=|\Delta
m_{31}^{2}|L/(4\times\pi/2)$=2.2 GeV, but rather covers the range at the
second oscillation maximum, occurring at $E=|\Delta m_{31}^{2}|L/(4\times
3\pi/2)$=0.75 GeV.

Fig. 9 illustrates the trajectory of the curve described by Eq. (9). Due to
the relatively low neutrino energy, the matter effect at T2HKK is not so large
and the curves corresponding to the wrong mass ordering appear in both true
mass ordering cases.

Figure 9:  The trajectory traced by the curve described by Eq. (9) for the
T2HKK case is showcased. The meaning of the straight thin lines is the same as
in Fig. 3. The left (right) panel corresponds to the true normal (inverted)
mass ordering. The solid curves colored in blue, red, green and purple denote
the trajectory of Eq. (9) for neutrino energy of 0.5 GeV, 0.6 GeV, 0.7 GeV and
0.8 GeV, respectively, assuming the true mass ordering (normal (inverted)
ordering in the left (right) panel). Conversely, the dotted curves represent
trajectories assuming the wrong mass ordering. The energy value for the dotted
curves is the same as that of the solid curve of the corresponding color.

Fig. 10 presents the values of $1/s^{2}_{23}$ for both true and fake solutions
as functions of neutrino energy, for the case of normal ordering (left panel)
and inverted ordering (right panel) as the true mass ordering. The behavior of
the curves is sensitive to the neutrino energy in the energy range below the
first oscillation maximum (i.e., 2.2 GeV). Even in the energy range 0.5 GeV
$\lesssim E\lesssim$ 0.8 GeV, where the number of events of the appearance
channels is significant, the difference between the values of $1/s^{2}_{23}$
for true and fake solutions is substantial compared to the difference between
the values of the lower and higher octant solutions derived from the
disappearance channels $\nu_{\mu}\to\nu_{\mu}$ and
$\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$. Furthermore, it is evident that the
energy dependence of the difference between the values of $1/s^{2}_{23}$ for
the true and fake solutions is significant. In other words, a slight change in
neutrino energy leads to a considerable change in the $1/s^{2}_{23}$ value for
both true and fake solutions, causing it to shift from below the correct value
to above it. Consequently, integrating this difference over a certain neutrino
energy interval yields results that depend on the size of the energy interval,
rendering a reliable conclusion difficult to obtain. Hence, it is not expected
that the T2HKK experiment alone will resolve the octant degeneracy. It has
been previously reported [20] that the detector located at 1100 km in the
T2HKK experiment exhibits limited sensitivity to octant degeneracy. The
aforementioned discussion regarding the behavior near the second oscillation
maximum provides a rationale for the restricted sensitivity observed in T2HKK.
However, it is expected that the T2HKK experiment will be combined with the
T2HK experiment, which is likely to result in the resolution of the octant
degeneracy.

Figure 10:  The potential values of the two solutions for $1/s^{2}_{23}$ are
displayed, with the left panel representing the case of true normal mass
ordering and the right panel depicting the true inverted mass ordering case.
Similar to Fig. 4, the horizontal dotted thin line indicates the fake value of
$1/s^{2}_{23}$ derived from the disappearance oscillation probabilities ((iii)
in Eq. (14)). For the sake of clarity and to facilitate comparison, the
expected numbers of events for $\nu_{\mu}\to\nu_{e}$
($\bar{\nu}_{\mu}\to\bar{\nu}_{e}$) are provided, sourced from Ref. [13].

Fig. 11 visualizes the sign of the discriminant (11) associated with the
quadratic curve (9), portrayed as a function of the neutrino energy for the
T2HKK case. Unlike the cases of T2HK and DUNE, even in regions where the
number of events in the appearance channels is substantial, the discriminant
takes on negative values.

Figure 11: The discriminant (11) for the quadratic curve (9) in the case of
T2HKK.

###  3.4 ESS$\nu$SB

Figure 12: Trajectory of the curve described by Eq. (9) for the ESS$\nu$SB
case. The meaning of the straight thin lines is the same as in Fig. 3. The
left (right) panel corresponds to the case of true normal (inverted) mass
ordering. The solid curves in blue, red and green illustrate the trajectory of
Eq. (9) for neutrino energy of 0.2 GeV, 0.3 GeV and 0.6 GeV, respectively,
assuming the true mass ordering (normal (inverted) ordering in the left
(right) panel).

The ESS$\nu$SB experiment,[14] utilizes a baseline length of 360 km111During
the earlier stages of the ESS$\nu$SB plan, alternative baseline lengths were
considered, but the baseline length is now established at 360 km. and employs
a wideband beam with an average neutrino energy of approximately 0.4 GeV. The
ESS$\nu$SB experiment aims to cover energy ranges corresponding to both the
first oscillation maximum ($E\sim$ 0.73 GeV) and the second oscillation
maximum ($E\sim$ 0.24 GeV).

Fig. 12 illustrates the trajectory of the curve described by Eq. (9). Due to
its relatively short baseline length, the matter effect at ESS$\nu$SB is
small, and curves corresponding to the wrong mass ordering appear in both true
mass orderings. An exception is observed at 0.6 GeV for the true normal mass
ordering, where the wrong mass ordering is rejected for the same reason as
T2HK.

Fig. 13 illustrates the value of $1/s^{2}_{23}$ for both true and fake
solutions, presented as functions of neutrino energy in the case of normal
ordering (left panel) or inverted one (right panel) as the true mass ordering.
The behavior of the curves around the energy region of the first oscillation
maximum is analogous to that observed in T2HK and DUNE. In this energy range,
resolving the octant degeneracy becomes potentially feasible if a substantial
number of appearance events are available for both neutrinos and
antineutrinos. However, practical observations from Fig. 13 reveal that the
numbers of events $N(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ are relatively low
around the energy range corresponding to the first oscillation maximum.
Consequently, the resulting larger experimental errors make it challenging to
effectively resolve the octant degeneracy. Furthermore, the behavior of the
curves around the energy region of the second oscillation maximum resembles
that of T2HKK. Octant degeneracy is not likely to be resolved solely using the
energy bins around this range. Prior studies [21, 22, 23] have highlighted the
limited sensitivity of ESS$\nu$SB to octant degeneracy. The explanations
provided above regarding behaviors near the first and second oscillation
maxima clarify the reasons for the observed limitations in ability of
ESS$\nu$SB to address octant degeneracy.

Figure 13: Potential values of $1/s^{2}_{23}$ for the true and fake solutions
are shown for the case where the true mass ordering is normal (left panel) or
inverted (right panel). Similar to Fig. 4, the horizontal dotted thin line
represents the fake value of $1/s^{2}_{23}$ derived from the disappearance
oscillation probabilities ((iii) in Eq. (14)). Additionally, the expected
numbers of events for $\nu_{\mu}\to\nu_{e}$
($\bar{\nu}_{\mu}\to\bar{\nu}_{e}$), taken from Ref.,[19], are provided for
reference.

Fig. 14 illustrates the sign of the discriminant (11) for the quadratic curve
(9) plotted as a function of the neutrino energy in the case of ESS$\nu$SB.
Similar to the cases of T2HK and DUNE, the discriminant remains positive for
the energy range around the first oscillation maximum ($E\sim$ 0.73 GeV),
while it transitions to negative values around $E\sim$ 0.3 GeV.

Figure 14: The discriminant (11) of the quadratic curve (9) in the case of
ESS$\nu$SB.

From these plots, a general trend emerges suggesting that octant degeneracy
can potentially be resolved using energy bins around the first oscillation
maximum, assuming sufficiently large numbers of appearance events in this
energy range for both neutrinos and antineutrinos. However, it is anticipated
to be challenging to effectively address the degeneracy using energy bins
centered around the second oscillation maximum.

##  4 Conclusions

In this paper we conducted an analytical investigation of octant degeneracy
after the determination of $\theta_{13}$, utilizing the trajectory defined by
the appearance probabilities in the ($\sin^{2}2\theta_{13}$,
$1/s^{2}_{23}$)-plane. By considering the appearance channels
$\nu_{\mu}\to\nu_{e}$ and $\bar{\nu}_{\mu}\to\bar{\nu}_{e}$, along with the
reactor data on $\theta_{13}$, we identified four potential solutions for
$1/s^{2}_{23}$. Incorporating the disappearance channels
$\nu_{\mu}\to\nu_{\mu}$ and $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$, it is
theoretically feasible to resolve octant degeneracy, assuming that
experimental errors are sufficiently small compared to the difference between
the true and fake solutions.

We examined the values of $1/s^{2}_{23}$ for the true and fake solutions in
future long baseline experiments, including T2HK, DUNE, T2HKK, and ESS$\nu$SB.
T2HK and DUNE have adequate numbers of appearance events for both neutrinos
and antineutrinos in the energy region near the first oscillation maximum,
where the difference between the values of $1/s^{2}_{23}$ for the true and
fake solutions is so minor that the fake octant solution inferred from the
disappearance channels can be excluded. As a result, these experiments hold
the potential to effectively resolve the octant degeneracy. On the other hand,
T2HKK and ESS$\nu$SB encounter challenges due to insufficient numbers of
appearance events for either neutrinos or antineutrinos around the first
oscillation maximum. Moreover, the significant energy-dependent variation
between the true and fake solutions around the second oscillation maximum in
these experiments makes it difficult to solely overcome the degeneracy through
experimental data.

Additionally, we calculated the discriminant of the quadratic curves as a
function of energy, illustrating the energy regions where they exhibit
hyperbolic or elliptic behavior for each experiment. This aspect was not
discussed in Ref. [8].

It is important to note that our study relies on discussions centered around
the best-fit points of oscillation parameters. In practice, however, due to
experimental errors, conclusions of octant degeneracy resolution should be
interpreted with a certain level of confidence in their potential. Our primary
aim in this study is to provide analytical insights into which experiments and
energy ranges show promise for resolving the octant degeneracy. We hope that
our work contributes valuable insights into how octant degeneracy might be
addressed in future long baseline experiments.

## Acknowledgments

One of the authors (O.Y.) would like to thank Monojit Ghosh for discussions.
This research was partly supported by a Grant-in-Aid for Scientific Research
of the Ministry of Education, Science and Culture, under Grant No. 21K03578.

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