#
The 2-3 mixing and mass split: atmospheric neutrinos and magnetized
spectrometers

Abhijit Samantaa,b and A. Yu. Smirnovb
aRamakrishna Mission Vivekananda University, Belur Math, Howrah 711 202, India
bThe Abdus Salam International Centre for Theoretical Physics
Strada Costiera 11, I-34014 Trieste, Italy  email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study dependence of the atmospheric $\nu_{\mu}$ and $\bar{\nu}_{\mu}$
fluxes on the deviations of the 2-3 mixing from maximal,
$|45^{\circ}-\theta_{23}|$, on the $\theta_{23}$-octant and on the neutrino
mass splitting $\Delta m_{32}^{2}$. Analytic expressions for the
$\theta_{23}-$deviation effect and the octant asymmetry are derived. We
present conservative estimations of sensitivities of the iron (magnetized)
calorimeter detectors (ICAL) to these parameters. ICAL can establish the
$\theta_{23}$-deviation at higher than 3$\sigma$ confidence level if
$|45^{\circ}-\theta_{23}|>6^{\circ}$ with the exposure of 1 Mton$\cdot$yr.
Sensitivity to the octant is low for zero or very small 1-3 mixing, but it can
be substantially enhanced for $\theta_{13}>3^{\circ}$. ICAL can measure the
difference of $\Delta m_{32}^{2}$ in $\nu$ and $\bar{\nu}$ channels (the CPT
test) with accuracy $0.8\times 10^{-4}$ eV2 (3$\sigma$) with 1 Mton$\cdot$yr
exposure, and the present MINOS result can be excluded at $>5\sigma$
confidence level. We discuss possible ways to further improve sensitivity of
the magnetized spectrometers.

###### pacs:

14.60.Pq,14.60Lm

##  I Introduction

Determination of the 2-3 mass splitting and leptonic mixing, and in
particular, the deviation of $\theta_{23}$ from the maximal mixing angle,

$\delta_{23}\equiv 45^{\circ}-\theta_{23},$ (1)

is of fundamental importance 111$\delta_{23}$ is related to another deviation
parameter, $D_{23}\equiv 1/2-\sin^{2}\theta_{23}$ used in literature as
$D_{23}=\sin 2\delta_{23}$.. Here we use the standard parameterization of the
PMNS mixing matrix:

$U_{PMNS}=U_{23}(\theta_{23})\Gamma_{\delta}U_{13}(\theta_{13})U_{12}(\theta_{12}),$
(2)

where $U_{ij}$ is the matrix of rotation in the $ij-$plane, and
$\Gamma_{\delta}\equiv{\rm diag}(0,0,e^{i\delta})$. Being maximal or close to
maximal, the 2-3 mixing testifies for existence of certain underlying symmetry
Mohapatra:2006gs . Comparison of the values of $\delta_{23}$ and $\theta_{13}$
as well as the mixing angles in the quarks and lepton sectors can shed some
light on the origins of fermion mass and mixing in general.

The existing results on $\theta_{23}$ and $\Delta m^{2}_{23}$ are summarized
in the Table 1. Note that the global fits of oscillation data
GonzalezGarcia:2010er  (see also Fogli:2008jx ) show some deviation of the 2-3
mixing from maximal: $\delta_{23}=2-3^{\circ}$ (1$\sigma$). Although the data
agree well with maximal mixing, large deviation, $\delta_{23}=\pm 9^{\circ}$,
is still possible.

Concerning the 2-3 mass splitting, the global fit values are in agreement with
the results of SuperKamiokande (SK) Hosaka:2006zd  as well as the MINOS
measurement in the $\nu$ channel minos .

Recently MINOS has reported the values of $\Delta m_{31}^{2}$ and
$\theta_{23}$ in the $\bar{\nu}$ channel minos  which differ from those in the
$\nu$ channel (see tables 1 and 2). If confirmed, this result will testify for
an effective (due to existence of some new interactions Kopp:2010qt ) or
fundamental CPT violation. The analysis of the atmospheric neutrino data does
not confirm MINOS result although the sensitivity of SK to CPT violation is
not high since SK sum up effects of neutrinos and antineutrinos Kopp:2010qt .
Iron calorimeters (ICAL) ino  can perform very sensitive search for the CPT
violation and check MINOS result.

New accelerator experiments T2K Kato:2008zz  and NO$\nu$A Ayres:2004js  will
improve precision of measurements of $\Delta m_{32}^{2}$ by factor 2, but
their accuracy of measurements of $\theta_{23}$ will be only slightly better
than that of the present global fit (see table 1).

There are two aspects of the $\theta_{23}$-measurements:

  * •

determination of the absolute value of the deviation $|\delta_{23}|$, and

  * •

identification of the $\theta_{23}$-octant, i.e. the sign of $\delta_{23}$, or
in other words, resolution of the octant degeneracy.

The problem of determination of $\delta_{23}$ and the octant with atmospheric
neutrinos has been addressed in a number of publications before Kim:1998bv ;
Peres:2003wd ; Peres:2009xe ; GonzalezGarcia:2004cu ; Choubey:2005zy ;
Indumathi:2006gr ; Bandyopadhyay:2007kx . It was realized Peres:2003wd ;
Peres:2009xe ; GonzalezGarcia:2004cu  that at low energies oscillation effects
on the electron neutrino flux are proportional to this deviation, and
therefore searches for an excess (or suppression) of events in the sub-GeV
range would testify for $\delta_{23}$. For water Cherenkov detectors both
aspects of the 2-3 mixing determinations have been explored in Peres:2009xe ;
GonzalezGarcia:2004cu . The study was mainly concentrated on effects in the
electron neutrino flux.

Magnetized calorimeters are mainly aimed at a detection of the muon neutrinos,
but they can distinguish neutrinos and antineutrinos, and this composes their
main advantage. These detectors provide better energy and angular resolution
of the charged leptons, and consequently, neutrinos. A possibility to
disentangle neutrinos and antineutrinos reinforces the following features: i)
The energy and angular resolutions (reconstruction) are different for
neutrinos and antineutrinos. ii) Sensitivities of the neutrino and
antineutrino channels to the oscillation parameters are substantially
different.

Sensitivity of a magnetized calorimeter to the 2-3 mixing and mass splitting
has been explored in Choubey:2005zy  and Indumathi:2006gr . It has been shown
that at nonzero value of the 1-3 mixing the octant discrimination is more
feasible with the magnetized detector than with the water Cherenkov detector
since the former can directly measure the matter effect Choubey:2005zy . In
these studies, however, various simplifications have been made which do not
allow for realistic estimations of potential of the experiments. In the
analysis Samanta:2008af  a possibility of the octant discrimination has been
evaluated for two benchmark values of $\theta_{23}$ and relatively high
$\theta_{13}=7.5^{\circ}$.

Here we present a comprehensive study of sensitivities of the ICAL to the
parameters of 2-3 sector. We assume that by the time of operation of this
detector certain information about $\theta_{13}$ will be obtained.

The paper is organized as follows. In sec. II we study dependence of the
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes on parameters of the 2-3 sector:
$|\delta_{23}|,$ octant and $\Delta m_{23}^{2}$. In sec. III we describe
details of our statistical analysis. We evaluate physics potential of the
magnetized calorimeter to measure these parameters in sec. IV. In sec. V we
consider further improvements of sensitivities of the magnetized calorimeters.
Conclusions are given in sec. VI.

##  II Dependence of the atmospheric neutrino fluxes on 2-3 mixing

The original atmospheric neutrino flux contains both the muon, $F_{\mu}^{0}$,
and electron, $F_{e}^{0}$, neutrino components, so that the muon neutrino flux
at a detector equals

$\displaystyle F_{\mu}$ $\displaystyle=$ $\displaystyle
F_{\mu}^{0}P_{\nu_{\mu}\rightarrow\nu_{\mu}}+F_{e}^{0}P_{\nu_{e}\rightarrow\nu_{\mu}}$
(3) $\displaystyle=$ $\displaystyle
F_{\mu}^{0}\left[P_{\nu_{\mu}\rightarrow\nu_{\mu}}+\frac{1}{r}P_{\nu_{e}\rightarrow\nu_{\mu}}\right].$

Here the flavor ratio

$r(E,\theta_{Z})\equiv\frac{F_{\mu}^{0}(E,\theta_{Z})}{F_{e}^{0}(E,\theta_{Z})}\,$
(4)

is function of the neutrino energy $E$ and zenith angle $\theta_{Z}$.

For the standard parameterization of the mixing matrix dependence of the
oscillation probabilities on the 2-3 mixing $\theta_{23}$ and CP-phase
$\delta$ is explicit for an arbitrary density profile. This follows from the
order of rotation in eq. (2) and the fact that the matrix of matter potentials
has the form $V=diag\\{V_{e},0,0\\}$ in the flavor basis, i.e. it is invariant
under the 2-3 rotations. Indeed, the neutrino evolution can be considered in
the propagation basis,
$\tilde{\nu}\equiv(\nu_{e},\tilde{\nu}_{2},\tilde{\nu}_{3})$, defined via the
following relation with the flavor basis: $\nu_{f}\equiv
U_{23}\Gamma_{\delta}\tilde{\nu}$. Consequently,
$\tilde{\nu}=U_{13}U_{12}\nu_{mass}$, where
$\nu_{mass}\equiv(\nu_{1},\nu_{2},\nu_{3})$ is the basis of mass eigenstates.
In the propagation basis the Hamiltonian, and therefore the amplitudes of
transitions depend on $\theta_{13},\theta_{12},V_{e}$, and mass squared
differences:

$A_{\alpha\beta}=A_{\nu_{\alpha}\rightarrow\nu_{\beta}}(\theta_{13},\theta_{12},V_{e}),~{}~{}~{}~{}\alpha,\beta=e,\tilde{2},\tilde{3},$
(5)

and they do not depend on $\theta_{23}$ and $\delta$. In the flavor basis,
dependence of the amplitudes $A_{f}$ on these parameters appears via
projections of the matrix of amplitudes (5) from the propagation basis to the
flavor basis:

$\hat{A}_{f}=U_{23}\Gamma_{\delta}\hat{A}\Gamma_{-\delta}U_{23}^{T},$ (6)

where $\hat{A}^{f}$ is the matrix of amplitudes in the flavor basis.

In terms of the amplitudes $A_{\alpha\beta}$ using eq. (6) one can rewrite the
expression for the muon neutrino flux, eq. (3), in the following form
Akhmedov:2008qt :

$\displaystyle\frac{F_{\mu}}{F_{\mu}^{0}}$ $\displaystyle\approx$
$\displaystyle 1-\sin^{2}2\theta_{23}\sin^{2}\frac{\phi}{2}$ (7)
$\displaystyle-$
$\displaystyle\frac{1}{2}\sin^{2}2\theta_{23}\cos\phi\left[1-{\rm
Re}(A_{\tilde{2}\tilde{2}}^{*}A_{\tilde{3}\tilde{3}})\right]$ $\displaystyle-$
$\displaystyle\left(s_{23}^{4}-\frac{s_{23}^{2}}{r}\right)\tilde{P}_{A}-\left(c_{23}^{4}-\frac{c_{23}^{2}}{r}\right)\tilde{P}_{S}$
$\displaystyle-$ $\displaystyle\sin 2\theta_{23}P_{\delta},$

where

$\tilde{P}_{A}\equiv|A_{e\tilde{3}}|^{2}~{}~{}{\rm
and}~{}~{}\tilde{P}_{S}\equiv|A_{e\tilde{2}}|^{2}$ (8)

are the probabilities of transitions $\nu_{e}\rightarrow\tilde{\nu}_{2}$ and
$\nu_{e}\rightarrow\tilde{\nu}_{3}$ correspondingly, and $P_{\delta}$ is the
function which depends on the CP-violation phase $\delta$. In eq. (7) $\phi$
is the oscillation phase due to the 2-3 mass splitting: $\phi=\Delta
m_{32}^{2}x/2E$, $c_{23}\equiv\cos^{2}\theta_{23}$, etc.. Notice that
$\tilde{P}_{S}$, $\tilde{P}_{A}$, $P_{\delta}$ and $\phi$ do not depend on
$\theta_{23}$.

In eq. (7) the first two terms are due to vacuum oscillations driven by the
2-3 mixing and mass splitting; the second line describes interference of the
2-3 and 1-2 modes of oscillations. The product of amplitudes in this term can
be approximated as

$|A_{\tilde{2}\tilde{2}}A_{\tilde{3}\tilde{3}}|\approx\sqrt{(1-\tilde{P}_{A})(1-\tilde{P}_{S})}.$
(9)

The terms in the third line describe effects of oscillations due to the 1-2
and 1-3 mixing. The last term in (7) describes the CP-violation. The leading
(second) term as well as the interference and CP-violating terms are symmetric
with respect to change of sign of the deviation:
$\delta_{23}\rightarrow-\delta_{23}$. The octant symmetry (degeneracy) is
broken by the terms in the third line of eq. (7); these terms vanish for the
maximal 2-3 mixing and $r=2$.

For antineutrinos one obtains the same formula as in eq. (7) with substitution
$\tilde{P}_{S}\rightarrow\bar{\tilde{P}}_{S},$
$\tilde{P}_{A}\rightarrow\bar{\tilde{P}}_{A}$ and $r\rightarrow\bar{r}$.

The octant effect can be characterized by the octant asymmetry defined as

$\frac{\Delta^{oct}F_{\mu}}{F^{0}_{\mu}}\equiv\frac{1}{F^{0}_{\mu}}\left[F_{\mu}(45^{\circ}+\delta_{23})-F_{\mu}(45^{\circ}-\delta_{23})\right].$
(10)

For such symmetric deviations from the maximal 2-3 mixing one has

$\displaystyle\Delta(\sin^{2}2\theta_{23})$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle~{}~{}~{}\Delta(\cos^{2}\theta_{23})$ $\displaystyle=$
$\displaystyle\Delta(\cos^{4}\theta_{23})=-\sin 2\delta_{23}.$ (11)

Then according to eq. (7) the octant asymmetry equals

$\frac{\Delta^{oct}F_{\mu}}{F_{\mu}^{0}}=\sin
2\delta_{23}\,\left(1-\frac{1}{r}\right)\,(\tilde{P}_{A}-\tilde{P}_{S}).$ (12)

Notice that $\tilde{P}_{S}$ and $\tilde{P}_{A}$ enter the asymmetry with
opposite signs, and therefore partly cancel each other.

To get an idea about dependences of the probabilities on the neutrino
parameters one can use expressions for the amplitudes in medium with constant
density:

$\displaystyle A_{e\tilde{2}}$ $\displaystyle=$ $\displaystyle c_{13}^{m}\sin
2\theta_{12}^{m}\sin\phi_{21}^{m}~{},$ $\displaystyle A_{e\tilde{3}}$
$\displaystyle=$ $\displaystyle\sin
2\theta_{13}^{m}\left(\sin\phi_{32}^{m}e^{-i\phi_{31}^{m}}+\cos^{2}\theta_{12}^{m}\sin\phi_{21}^{m}\right).$
(13)

The probability $\tilde{P}_{S}\propto\sin^{2}2\theta^{m}_{12}$ decreases with
energy, whereas $\tilde{P}_{A}\propto\sin^{2}2\theta^{m}_{13}$ increases being
resonantly enhanced in the neutrino channel (for the normal mass hierarchy) at
high energies. Here sensitivity of ICAL to the sign of muon charge will play
important role. The two probabilities become comparable at 1 GeV for
$\sin^{2}\theta_{13}\sim 0.01$.

In the limit of zero 1-3 mixing $\tilde{P}_{A}=0$,
$\tilde{P}_{S}\rightarrow{P}_{S}$, and one finds from eq. (7)

$\displaystyle\frac{F_{\mu}}{F_{\mu}^{0}}\approx
1-\sin^{2}2\theta_{23}\sin^{2}\frac{\phi}{2}-\frac{1}{2}\sin^{2}2\theta_{23}\cos\phi\times$
$\displaystyle\times\left(1-\sqrt{1-P_{S}}\right)-\left(c_{23}^{4}-\frac{c_{23}^{2}}{r}\right)P_{S},$
(14)

where ${P}_{S}=\tilde{P}_{S}(\theta_{13}=0)$, which is the $2\nu$ probability
of oscillations driven by $\Delta m^{2}_{21}$ and $\theta_{12}$. For the
octant asymmetry we obtain

$\frac{\Delta^{oct}F_{\mu}}{F_{\mu}^{0}}=-\sin
2\delta_{23}\,\left(1-\frac{1}{r}\right)\,P_{S}.$ (15)

At low energies: $r\approx 2,$ and therefore the asymmetry equals $\sim
1/2\,\sin 2\delta_{23}\,P_{S}$.

In fig. 1 we show the oscillograms for the octant asymmetry, that is, the
lines of equal asymmetry $\Delta^{oct}F_{\mu}/F_{\mu}^{0}$ in the
$E-\cos\theta_{Z}$ plane in neutrino and antineutrino channels. According to
eq. (15) these oscillograms coincide with the oscillograms for $P_{S}$ up to
coefficient which weakly depends on $E$ and $\theta_{Z}$ at $E<1$ GeV. We use
$\delta_{23}=5^{\circ}$, $\theta_{13}=0,$ and other parameters are set at
their best-fit values. The asymmetry increases with decrease of the neutrino
energy. Maximal asymmetry is achieved in the 1-2 resonance ($E\sim 0.1$ GeV):
$\Delta^{oct}F_{\mu}/F_{\mu}^{0}\approx 0.087$. For realistic threshold
$E_{th}=0.3$ GeV and $\delta_{23}=5^{\circ}$ the averaged asymmetry is about
(2 - 3)% and for $E_{th}=0.8$ GeV it is below 1%.

Notice that the octant asymmetry of $\nu_{e}-$flux is about 4 times larger
than the $\nu_{\mu}-$flux asymmetry:

$\frac{\Delta^{oct}F_{e}}{F_{e}^{0}}=-\sin 2\delta_{23}\,r\,P_{S}.$ (16)

Here, however, the original $\nu_{e}-$flux is 2 times smaller. Furthermore,
detection of muons provide better energy and direction resolutions.

Since $P_{S}$ and $\bar{P}_{S}$ are of the same order, separation of the
neutrino and antineutrino signals has no sense here.

Let us consider variations of the $\nu_{\mu}$-flux due to deviation of the 2-3
mixing from maximal. According to (7) the relative change of the flux, which
we will call the $\theta_{23}-$deviation effect, equals

$\displaystyle\frac{\Delta^{dev}F_{\mu}}{F_{\mu}^{0}}$ $\displaystyle\equiv$
$\displaystyle\frac{F_{\mu}(45^{\circ})-F_{\mu}(45^{\circ}-\delta_{23})}{F_{\mu}^{0}}\approx$
(17) $\displaystyle-$
$\displaystyle\Delta(\sin^{2}2\theta_{23})\sin^{2}\frac{\phi}{2}\cong-\frac{1}{2}\sin^{2}2\delta_{23},$

where in the last equality we have averaged the oscillations due to large mass
splitting.

Ratio of the octant asymmetry and the 2-3 deviation effect equals

$\frac{\Delta^{oct}F_{\mu}}{\Delta^{dev}F_{\mu}}=-\frac{1}{\sin
2\delta_{23}}\left(1-\frac{1}{r}\right)(\langle\tilde{P_{A}}\rangle-\langle\tilde{P_{S}}\rangle),$
(18)

where $\langle\tilde{P_{S}}\rangle$ and $\langle\tilde{P_{A}}\rangle$ are the
probabilities averaged over the experimental $E-\cos\theta_{Z}$ ranges.
Although $\Delta^{dev}F_{\mu}^{0}$ is proportional to square of the deviation
parameter, for not very small $\delta_{23}$ ($>5^{\circ}$) and
$\theta_{13}=0$, the integral effect of the deviation from maximal mixing is
stronger than the effect of octant. The reason is that the deviation effect
does not change with energy, whereas the octant asymmetry being proportional
to $P_{S}$ quickly decreases with $E$
($\langle\tilde{P_{S}}\rangle\ll\langle\tilde{P_{A}}\rangle).$ For zero 1-3
mixing and $\delta_{23}=5^{\circ}$ we have
$\Delta^{dev}F_{\mu}/F_{\mu}^{0}=0.015$, and
$\Delta^{oct}F_{\mu}/F_{\mu}^{0}=0.087\langle P_{S}\rangle$.

Figure 1: The oscillograms for the octant asymmetry: The contours of equal
asymmetry $\Delta^{oct}F_{\mu}/F_{\mu}^{0}$ for neutrino (left) and
antineutrino (right) with $\delta_{23}=5^{\circ}$ at $\theta_{13}=0.$ The
other parameters are set at their best-fit values.

For non-zero 1-3 mixing at high energies the ratio of the flux differences is

$\frac{\Delta^{oct}F_{\mu}}{\Delta^{dev}F_{\mu}}\cong-\left(1-\frac{1}{r}\right)\frac{\langle\tilde{P}_{A}\rangle}{\sin
2\delta_{23}},$ (19)

and since $\tilde{P}_{A}$ does not decrease with energy the ratio is not small
in contrast to the previous case.

In our studies of sensitivities to the parameters of 2-3 sector we obtain the
oscillation probabilities solving numerically full three flavor evolution
equation and using the Preliminary Reference Earth Model (PREM)
Dziewonski:1981xy  for the density profile of the Earth. We will use the
consideration presented in this section for interpretation of the numerical
results.

##  III The $\chi^{2}$ analysis for ICAL

To evaluate physics potential of the atmospheric neutrino studies with a
magnetized ICAL detector we generated the atmospheric neutrino events and
considered the muon energy and direction (directly measurable quantities)
using event generator NUANCE-v3 Casper:2002sd . The GEANT geant  simulation of
ICAL detector shows that the energy and angular resolutions of muons are very
high and the corresponding uncertainties are negligible compared to
differences between the angles as well as energies of the neutrino and muon in
the scattering (production) process.

$\chi^{2}$ is calculated according to the Poisson distribution. The term due
to contribution of prior information on the oscillation parameters measured by
other experiments is not added to $\chi^{2}$ here for conservative estimation;
the effect of priors will be considered in sec. V. The data have been binned
in cells of equal size in the ($\log_{10}E$ \- $L^{0.4}$) plane, where
$L=2R\cos\theta_{Z}$ is the length of neutrino trajectory. Choice of this
binning is motivated by pattern of the oscillation probability
$P(\nu_{\mu}\rightarrow\nu_{\mu})$ in the $L-E$ plane Samanta:2008ag . The
distance between two consecutive oscillation peaks in this plane increases
(decreases) as one goes to lower $L$ ($E$) values for a given $E$ ($L$). The
binning of $L$ has been optimized to get better sensitivity to the oscillation
parameters. To maintain $\chi^{2}/d.o.f.\approx 1$ in Monte Carlo simulation
study, number of events should be $>4$ per cell Samanta:2008af . If the number
is smaller than 4 (which happens in the high energy bins), we combine results
from the nearest cells.

For each set of oscillation parameters we integrate the atmospheric neutrino
flux at the detector over the energy and zenith angle folding it with the
cross-section, the exposure time, the target mass, the efficiency of detection
and the two dimensional energy-angle correlated resolution functions to obtain
data for the $\chi^{2}$ analysis. We use the charge current cross section of
NUANCE-v3 Casper:2002sd  and the neutrino flux of the 3-dimensional scheme
Honda:2006qj .

The systematic uncertainties of the atmospheric neutrino flux are crucial for
determination of the oscillation parameters. We have divided them into two
categories: (i) the overall flux normalization uncertainties which are
independent of the neutrino energy and zenith angle, and (ii) the spectral
tilt uncertainties which depend on $E$ and $\theta_{Z}$. The flux with
uncertainties included can be written as

$\displaystyle\Phi(E,\theta_{Z})=\Phi_{0}(E)\left[1+\delta_{E}\log_{10}\frac{E}{E_{0}}\right]$
$\displaystyle\times\left[1+\delta_{Z}(|\cos\theta_{Z}|-0.5)\right]\times\left[1+\delta_{f_{N}}\right].$
(20)

For $E<1$ GeV we take the energy-dependent uncertainty: $\delta_{E}=15\%$ and
$E_{0}=1$ GeV, and for $E>10$ GeV correspondingly, $\delta_{E}=5\%$ and
$E_{0}=10$ GeV. The uncertainty is $\delta_{E}\sim 7\%$ in the range $E=1-10$
GeV. The overall flux uncertainty as function of the zenith angle is
parameterized by $\delta_{Z}$. According to Honda:2006qj  we use
$\delta_{Z}=4\%$ which leads to 2% vertical/horizontal flux uncertainty. We
take for the overall flux normalization uncertainty $\delta_{f_{N}}=10\%$ and
for the neutrino cross-section uncertainty: $\delta_{\sigma}=10\%$.

In our $\chi^{2}$ analysis the numbers of events have been computed for the
theoretical (fit) values and experimental (true) values of parameters in the
same way by migrating the number of events from the neutrino to muon energy
and zenith angle bins. The resolution functions have been taken from the
previous work Samanta:2009qw .

In studies of sensitivity to the 2-3 mixing we marginalize $\chi^{2}$ over
$\Delta m_{32}^{2},~{}\theta_{13}$, $\delta_{CP}$ for $\nu$’s and
$\bar{\nu}$’s separately. Then we find the total $\chi^{2}$ as
$\chi^{2}=\chi^{2}_{\nu}+\chi^{2}_{\bar{\nu}}$. We have chosen the following
benchmark values of the neutrino parameters: $\Delta m_{32}^{2}=2.5\times
10^{-3}$ eV2, $\delta_{CP}=0$, $\Delta m_{21}^{2}=7.9\times 10^{-5}$ eV2 and
$\theta_{12}=34.24^{\circ}$. In marginalization we use, first, flat
distributions of values of the parameters in the following ranges: $\Delta
m_{32}^{2}=(2.3-2.7)\times 10^{-3}$eV2, $\theta_{23}=36^{\circ}-54^{\circ}$
and $\theta_{13}=0^{\circ}-10.5^{\circ}$. The range of $\theta_{13}$ is
changed for some particular analyses. (The non-flat distributions of
parameters will be considered in sec. V.)

The parameters $\Delta m_{21}^{2}$ and $\theta_{12}$ produce subleading
effects on the atmospheric neutrino fluxes for $E>1$ GeV. Moreover, effect of
these parameters in marginalization is very small due to their narrow allowed
regions. Therefore we have taken fixed values of $\Delta m_{21}^{2}$ and
$\theta_{12}$ in our analysis.

Figure 2: Dependence of $\Delta\chi^{2}$ on fit value of $\theta_{23}$ for
fixed input (true) values $\theta_{23}=37^{\circ}$ (left) and $40^{\circ}$
(right). We used $\theta_{13}=0$, ${\mathcal{E}=}$ 1, 2, and 4 Mton$\cdot$yr,
and the energy threshold 0.141 GeV. The $\chi^{2}$ is marginalized with
respect to all the oscillation parameters except $\theta_{23}$. The range of
marginalization for $\theta_{13}$ is $0-10.5^{\circ}$.  Figure 3: Dependence
of the ICAL sensitivity to the deviation from maximal mixing on the input
value of $\theta_{23}$ for different exposure times: ${\mathcal{E}=}$ 1, 2,
and 4 Mton$\cdot$yr. We use the threshold 0.141 GeV. Here
$\Delta^{dev}\chi^{2}=\chi^{2}(45^{\circ})-\chi^{2}(\theta_{23}^{true})$.
$\chi^{2}$ is marginalized with respect to all the oscillation parameters
except $\theta_{23}$. The range of marginalization for $\theta_{13}$ is
$0-10.5^{\circ}$.  Figure 4: Dependence of the ICAL sensitivity to the
$\theta_{23}$-octant on the input (true) value of $\theta_{23}$ for different
values of ${\mathcal{E}}$, and threshold 0.141 GeV. Here
$\Delta^{oct}\chi^{2}=\chi^{2}(90^{\circ}-\theta_{23})-\chi^{2}(\theta_{23}).$
$\chi^{2}$ is marginalized with respect to all the oscillation parameters
except $\theta_{23}$. The range of marginalization for $\theta_{13}$ is
$0-10.5^{\circ}$.  Figure 5: Dependence of the ICAL sensitivity to the
deviation from maximal 2-3 mixing on the input value of $\theta_{23}$ for
different marginalization ranges of $\theta_{13}$. We use ${\mathcal{E}}=$ 1
Mton$\cdot$yr and threshold 0.806 GeV. Here
$\Delta^{dev}\chi^{2}\equiv\chi^{2}(45^{\circ})-\chi^{2}(\theta_{23}^{true}).$
$\chi^{2}$ is marginalized with respect to all oscillation parameters except
$\theta_{23}$.  Figure 6: Dependence of the ICAL sensitivity to the
$\theta_{23}$-octant on the input value of $\theta_{23}$ with ${\mathcal{E}}=$
1 Mton$\cdot$yr and threshold of 0.806 GeV. Here
$\Delta^{oct}\chi^{2}\equiv\chi^{2}(90^{\circ}-\theta_{23})-\chi^{2}(\theta_{23}).$
$\chi^{2}$ is marginalized with respect to all oscillation parameters except
$\theta_{23}$.

##  IV Sensitivities of ICAL

In our computations we explored the neutrino energy range (0.141 -15) GeV, we
used different energy thresholds and different exposures, ${\mathcal{E}}$, of
0.25, 1, 2, and 4 Mton$\cdot$yr.

###  IV.1 Determination of 2-3 mixing for $\theta_{13}=0$

The sensitivity of ICAL experiment to $\theta_{23}$ is shown in fig. 2. We
plot

$\Delta\chi^{2}\equiv\chi^{2}(\theta_{23})-\chi^{2}(\theta^{true}_{23})$ (21)

as function of the fit value for fixed input values
$\theta_{23}^{true}=37^{\circ}$, (left) and $40^{\circ}$ (right) with
${\mathcal{E}}=$ 1, 2, and 4 Mton$\cdot$yr. We have marginalized $\chi^{2}$
with respect to all the oscillation parameters except $\theta_{23}$. The
figure shows high sensitivity to the deviation $\delta_{23}$: it would be
possible to discriminate between a given $\theta_{23}$ and maximal mixing at
99% C.L., if $|\delta_{23}|>5^{\circ}$. For instance, after 1 Mton$\cdot$yr
the angle $\theta_{23}=37^{\circ}$ can be distinguished from $45^{\circ}$ at
$8\sigma$ level.

The figure shows also low sensitivity of the experiment to the octant. Indeed,
$\Delta\chi^{2}$ is higher in the right minima which correspond to the wrong
octant. After 1 Mton$\cdot$yr exposure the difference of $\Delta\chi^{2}$ in
the true and wrong octants is smaller than 1. The difference becomes more than
2 (90% CL) only after 4 Mton$\cdot$yrs. Identification of the octant becomes
even more difficult for smaller $\delta_{23}$. If $\delta_{23}=5^{\circ},$ we
find $\Delta\chi^{2}=1.2$ for ${\mathcal{E}}=4$ Mton$\cdot$yr. This result can
be readily seen from our analytical consideration in sec. II. The probability
$P_{S}$ averaged over the energy interval $(0.14-15)$ GeV equals $\langle
P_{S}\rangle\sim 0.02$, so that for $\delta_{23}=8^{\circ}$:
$\Delta^{oct}F_{\mu}/F^{0}_{\mu}\sim 4\cdot 10^{-3}$, whereas
$\Delta^{dev}F_{\mu}/F^{0}_{\mu}\sim 0.04$ \- an order of magnitude larger. As
we mentioned before, this big difference of sensitivities to the deviation and
octant (degeneracy) is because the octant asymmetry is collected only at very
low energies where $P_{S}$ is unsuppressed, whereas whole energy range
contributes to the sensitivity to the deviation $\delta_{23}$.

The sensitivity to $\delta_{23}$ drops down substantially with decrease of
$\delta_{23}$. Reducing $\delta_{23}$ from $8^{\circ}$ to $5^{\circ}$ (compare
the left and right panel of fig. 2) leads to decrease of the flux difference
of eq. (17) by factor 2.5, and correspondingly, significance of discrimination
from maximal mixing becomes $2\sigma$ (for ${\mathcal{E}}=1$ Mton$\cdot$yr).
Sensitivity to the octant at the $1\sigma$ level appears only if
${\mathcal{E}}>4$ Mton$\cdot$yr.

In fig. 3 we show the marginalized $\Delta\chi^{2}$, calculated at the maximal
mixing (fit value) for different input (true) values of $\theta_{23}$:

$\Delta^{dev}\chi^{2}\equiv\chi^{2}(45^{\circ})-\chi^{2}(\theta_{23}^{true}).$
(22)

The picture is complementary to that in fig. 2, and there is an approximate
symmetry with respect to $\theta_{23}=45^{\circ}$. The reason of sharp
increase of $\Delta\chi^{2}$ at 42 and $48^{\circ}$ is related to weak
dependence of the oscillation probability on $\delta_{23}=0$ around
$\delta_{23}=0$ and to overall flux uncertainty.

In fig. 4 we illustrate dependence of the sensitivity to the octant on
$\theta_{23}$. For different input (true) values of $\theta_{23}$ we plotted

$\Delta^{oct}\chi^{2}\equiv\chi^{2}(90^{\circ}-\theta_{23})-\chi^{2}(\theta_{23}).$
(23)

According to fig. 4, it will be possible to discriminate the octant at 90% CL
if $\theta_{23}\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}38^{\circ}$ or $\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}52^{\circ}$ after
${\mathcal{E}}=4$ Mton$\cdot$yr. Notice that the curves are nearly symmetric
with respect to $\theta_{23}=45^{\circ}$.

Due to fast decrease of $P_{S}$ with increase of energy (see fig. 1) the
sensitivity to the octant disappears for high values of the threshold.

###  IV.2 Determination of $\theta_{23}$ in the presence of non-zero 1-3
mixing.

Analysis of the oscillation data testifies for the non-zero 1-3 mixing,
although significance of this result is not high and zero value of
$\theta_{13}$ is not yet excluded. The global fit at 1$\sigma$ C.L. gives
$\theta_{13}={7.3^{\circ}}\,^{+2.1^{\circ}}_{-3.2^{\circ}}$, with 3$\sigma$
upper bound of $\theta_{13}<13^{\circ}$, and $\delta_{\text{CP}}\in[0,\,360]$
GonzalezGarcia:2010er . Analysis of the solar and KAMLAND data by SNO
collaboration leads to
$\theta_{13}={8.13^{\circ}}\,^{+3.53^{\circ}}_{-7.03^{\circ}}$ at 95% CL
Aharmim:2009gd . New and forthcoming experiments Double Chooz, Daya Bay, RENO,
T2K, NO$\nu$A can confirm this result with higher confidence level or put new
stringent upper bound Mezzetto:2010zi  which would correspond approximately to
a situation with zero 1-3 mixing considered in the previous section.

By the time when ICAL will collect significant statistics the angle
$\theta_{13}$ will be known with relatively good accuracy. To clarify an
impact of this information on the determination of parameters of the 2-3
sector we have performed analysis for non-zero $\theta_{13}$. For illustration
purpose we use $\theta_{13}=5^{\circ}$ as the true value and different fit
intervals with flat distribution (which could reflect errors in measurements
of $\theta_{13}$).

In fig. 5 we show dependence of $\Delta^{dev}\chi^{2}$ defined in eq. (22) on
the true (input) value of $\theta_{23}$ for the fit value
$\theta_{23}=45^{\circ}$ and ${\mathcal{E}}=$ 1 Mton$\cdot$yr. Different
curves correspond to different marginalization intervals of $\theta_{13}$.
Comparing these results with the results of fig. 3 we find that inclusion of
the 1-3 mixing does not change significantly the sensitivity to $\delta_{23}$.
The reason is that this sensitivity follows from the main mode of the
$\nu_{\mu}-$oscillations; large probability for this mode extends to higher
energies and 1-3 mixing produces just additional distortion of the oscillatory
pattern. However, inclusion of the 1-3 mixing makes the curve less symmetric
with respect to $45^{\circ}$ which reflects an increase of sensitivity to the
octant. Also sensitivity to the deviation weakly depends on the
marginalization interval for $\theta_{13}$.

Fig. 6 illustrates the sensitivity of ICAL to the octant in the presence of
non-zero 1-3 mixing. We show dependence of $\Delta^{oct}\chi^{2}$ defined in
eq. (23) on the true value of $\theta_{23}^{true}$ for the fit value
$\theta_{23}^{fit}=90^{\circ}-\theta_{23}$. Different curves correspond to
different marginalization intervals for $\theta_{13}$. There are two important
features of the fig. 6. First, the sensitivity to the octant is substantially
better for non-zero value of $\theta_{13}$ than for vanishing 1-3 mixing (fig.
6) as was also shown in previous publications GonzalezGarcia:2004cu ;
Hiraide:2006vh ; Kajita:2006bt ; Meloni:2008it . This is related to the fact
that the octant asymmetry of the flux is determined now by

$\frac{\Delta^{oct}F}{F_{\mu}^{0}}\approx\sin
2\delta_{23}\left(1-\frac{1}{r}\right)\langle P_{A}\rangle$ (24)

and $\langle P_{A}\rangle\sim 0.1$ in the interval $E=(0.14-15)$ GeV; it is
much larger than $\langle P_{S}\rangle$ being enhanced in the energy range E =
(3 - 10) GeV (in the resonance channel). Furthermore, at high energies
$r\sim(3-4)$ and value of the coefficient in eq. (24) becomes larger. As a
result, for $\theta_{23}=51^{\circ}$ the octant can be identified at $2\sigma$
level ($\Delta^{oct}\chi^{2}=4$) with ${\mathcal{E}}=$ 1 Mton$\cdot$yr, as
compared to $\Delta^{oct}\chi^{2}=0.3$ for $\theta_{13}=0$.

The second feature is significant asymmetry of the curves with respect to
$\theta_{23}=45^{\circ}$. The asymmetry is practically absent for fixed input
value of $\theta_{13}$ but it increases with broadening of the marginalization
interval, and more importantly, with increase of the lower border of this
interval. For $\theta_{23}>45^{\circ}$ the curves are practically not changed
with change of the interval, whereas for $\theta_{23}<45^{\circ}$ the
sensitivity substantially decreases. For instance, taking
$\theta_{23}=40^{\circ}$ we obtain $\Delta\chi^{2}=1$ for the interval
$\theta_{13}=(3^{\circ}-12.5^{\circ})$ instead of $\Delta^{oct}\chi^{2}=3$ for
fixed value $\theta_{13}=5^{\circ}$.

This asymmetry can be readily understood from the analytic consideration of
sec. II. Neglecting the effect of 1-2 mixing the $\nu_{\mu}-$flux can be
presented as

$\frac{F_{\mu}}{F_{\mu}^{0}}\approx K(\sin
2\theta_{23})-f(\theta_{23})\left(1-\frac{1}{r}\right)P_{A}(\theta_{13}),$
(25)

where $K(\sin 2\theta_{23})$ is an even function of the deviation (symmetric
with respect to change of the octant), and

$f(\theta_{23})\equiv\left(s_{23}^{4}-\frac{s^{2}_{23}}{r}\right)$ (26)

quickly increases with $\theta_{23}$, so that for $r=3-4$ one has
$f(\theta_{23}=40^{\circ})\ll f(\theta_{23}>50^{\circ})$. Therefore for
$\theta_{23}<45^{\circ}$ the flux $F_{\mu}$ has much weaker dependence on
$\theta_{13}$ than for $\theta_{23}>45^{\circ}$. In the process of
marginalization over $\theta_{13}$ we compare the true value of the flux,
$F_{\mu}^{true}$, with the fit value, $F_{\mu}^{fit}$, and
$\Delta^{oct}\chi^{2}$ is proportional to their difference:

$\displaystyle\frac{1}{F_{\mu}^{0}}\left|F_{\mu}^{true}-F_{\mu}^{fit}\right|=\left(1-\frac{1}{r}\right)\times$
$\displaystyle\left|f(\theta_{23})\langle
P_{A}(\theta_{13}^{true})\rangle-f(90^{\circ}-\theta_{23})\langle
P_{A}(\theta_{13})\rangle\right|.$ (27)

If the fit value of 1-3 mixing is fixed:
$\theta_{13}^{fit}=\theta_{13}^{true}$, the curve is approximately symmetric
with respect to change $\theta_{23}\leftrightarrow(90^{\circ}-\theta_{23})$.
Indeed, in this case we have from eq. (27)

$\displaystyle\frac{1}{F_{\mu}^{0}}\left|F_{\mu}^{true}-F_{\mu}^{fit}\right|=\left(1-\frac{1}{r}\right)$
$\displaystyle\times\langle
P_{A}(\theta_{13}^{true})\rangle\left|f(\theta_{23})-f(90^{\circ}-\theta_{23})\right|.$
(28)

The situation is different if $\theta_{13}$ varies in certain interval
$\theta_{13}=[\theta_{13}^{min}-\theta_{13}^{max}]$, and we perform
marginalization over $\theta_{13}$ in this interval. Marginalization minimizes
the difference of fluxes (eq. (27)) over $\theta_{13}$ for a given value of
$\theta_{23}$. If $\theta_{23}<45^{\circ}$, then $f(\theta_{23})\ll
f(90^{\circ}-\theta_{23})$. In this case the difference of fluxes is minimal
if $\theta_{13}\sim\theta_{13}^{min}$. Indeed, since $\langle P_{A}\rangle$
decreases with $\theta_{13}$, a small value of $\langle P_{A}\rangle$
partially compensates large value of $f(90^{\circ}-\theta_{23})$ in the second
term on the right hand side of eq. (27). Furthermore, the smaller
$\theta_{13}^{min}$ the stronger compensation, and therefore the smaller
$\Delta\chi^{2}$ can be obtained. If $\theta_{23}>45^{\circ}$, then
$f(\theta_{23})\gg f(90^{\circ}-\theta_{23})$. Now to compensate the first
term in eq. (27) one should take $\langle
P_{A}(\theta_{13}^{fit})\rangle\gg\langle P_{A}(\theta_{13})\rangle$. This is,
however, not possible for the considered values of $\theta_{13}$. Thus, in the
case of unprecise determination of $\theta_{13}$ sensitivity to the octant is
higher for $\theta_{23}>45^{\circ}$.

Figure 7: Dependence of $\Delta\chi^{2}$ on the fit value of $\Delta
m_{32}^{2}$ for the true value $\Delta m_{32}^{2}=2.35\times 10^{-3}$ eV2 in
the neutrino and antineutrino channels. We use ${\mathcal{E}}=$ 0.25 and 1
Mton$\cdot$yr and the threshold 0.8 GeV. The marginalization is done over all
the oscillation parameters except $\Delta m_{32}^{2}$. The $\chi^{2}$ values
1, 4, 9 correspond to 1$\sigma$ (68.3%), 2$\sigma$ (95.4%), and 3$\sigma$
(99.73%), respectively.

Figure 8: The iso-$\chi^{2}$ contours in the $\Delta m^{2}_{32}-\theta_{23}$
plane for neutrinos (left) and antineutrinos (right) obtained with
${\mathcal{E}}=$ 0.25 Mton$\cdot$yr. The inner contour corresponds to
$\chi^{2}=1$ and others with increment 1. We use the input value $\Delta
m^{2}_{32}=-2.35\times 10^{-3}$eV2, $\theta_{23}=45^{\circ}$,
$\theta_{13}=5^{\circ}$ and the threshold 0.8 GeV. Here $\Delta\chi^{2}$
values 2.3, 4.6, 9.2 correspond to 1$\sigma$ (68.3%), 2$\sigma$ (90%), and
3$\sigma$ (99%), respectively. Intensity of color reflects value of
$\Delta\chi^{2}$ (see column on the right hand sides of the panels for
identification).

Figure 9: The same as in fig. 8 but for ${\mathcal{E}}=$ 1 Mton$\cdot$yr. The
inner contour corresponds to $\chi^{2}=2$ and for other contours the increment
$\Delta\chi^{2}=2$.

###  IV.3 Determination of the 2-3 mass split. CPT test

Important advantage of a magnetized calorimeter is that it allows one to
measure the neutrino mass differences and mixing angles in the neutrino and
antineutrino channel separately. A difference of results can be related to
some effective or fundamental violation of the CPT symmetry.

In fig. 7 we show dependence of $\Delta\chi^{2}$ on the fit value of $\Delta
m^{2}_{32}$ for the true value $\Delta m_{32}^{2}=2.35\cdot 10^{-3}$ eV2 in
the neutrino and antineutrino channels. We take $\theta_{13}=5^{\circ}$ and
$\theta_{23}=45^{\circ}$. According to this figure with ${\mathcal{E}}=0.25$
Mton$\cdot$yr the value $\Delta m_{32}^{2}=3.3\cdot 10^{-3}$ eV2 can be
discriminated from the true value at about 2$\sigma$ level.

The accuracy of measurement of $\Delta m_{32}^{2}$ is better in the
$\nu-$channel. For ${\mathcal{E}}=0.25$ Mton$\cdot$yr the error in
$\nu$-channel is about two times smaller than that in $\bar{\nu}$ channel. The
difference of accuracies decreases with increase of exposure and e.g. for 1
Mton$\cdot$yr it becomes about 25%. The curves $\Delta\chi^{2}$ are asymmetric
with respect to ${\Delta m_{32}^{2}}^{true}$, which is related to the
dependence of oscillation probability on $\Delta m_{32}^{2}$.

The $1\sigma$ error for $\Delta m^{2}_{23}$ could be $0.15\cdot 10^{-3}$ eV2
and $0.04\cdot 10^{-3}$ eV2 after 0.25 Mton$\cdot$yr and 1 Mton$\cdot$yr
exposures correspondingly. With ${\mathcal{E}}=1$ Mton$\cdot$yr the error
$0.15\cdot 10^{-3}$ eV2 can be achieved at $3\sigma$ level. With
${\mathcal{E}}=1$ Mton$\cdot$yr one can obtain an accuracy $0.08\cdot 10^{-3}$
eV2 ($90\%$ C.L.) which is better than the present MINOS accuracy.

In figs. 8 and 9 we show $\Delta\chi^{2}$ as function of $\Delta m_{32}^{2}$
and $\theta_{23}$ in the neutrino and antineutrino channels for
${\mathcal{E}}=$ 0.25 and 1 Mton$\cdot$yr. As true values we take
$\theta_{23}=45^{\circ}$ and $\Delta m_{32}^{2}=2.35\cdot 10^{-3}$ eV2.
According to fig. 9 the present MINOS result for $\bar{\nu}$ ($|\Delta
m_{31}^{2}|=3.36\cdot 10^{-3}~{}{\rm eV}^{2},$ $\theta_{23}=34^{\circ}$) can
be excluded by ICAL at about $6\sigma$ level with ${\mathcal{E}}=$ 1
Mton$\cdot$yr.

##  V Further improvements of sensitivities

There are several directions in which sensitivity of ICAL can be further
improved.

###  V.1 Adding information about hadrons

Figure 10: Effect of inclusion in the analysis the information on hadrons. The
iso-$\chi^{2}$ contours (from inner side $\chi^{2}=2$ with increment 2) in the
$\Delta m^{2}_{32}-\theta_{23}$ plane without hadrons (left) and with hadrons
(right). We take 1 Mton$\cdot$yr, the threshold 0.8 GeV for both muons and
hadrons. and sum up the signals from neutrinos and antineutrinos.

Measurements of the hadron energy in ICAL in addition to the muon energy is
expected to improve reconstruction of the neutrino energy for $E\mathrel{\hbox
to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}2$ GeV.
However, the total hadron energy in an event is carried out by multiple low
energy hadrons. The average energy per hadron per event is $\mathrel{\hbox
to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1$GeV
and the average number of hadrons per event is $\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}2$ At $E\mathrel{\hbox
to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1$GeV
the energy resolution is very poor, roughly $80\%$, and the number of hits
(number of active detector layers in which signal is detected) increases only
logarithmically with $E$.

The resolution of hadron energy (for all pions and kaons) at ICAL has been
obtained from GEANT4 simulation and parametrized as

$\sigma_{had}/E_{had}=a/\sqrt{E_{had}}+b,$ (29)

where, $a\approx 0.60$ and $b\approx 0.1$ for the thickness of iron layer 5.6
cm and after averaging over all directions.

For each hadron in an event we find the reconstructed hadron energy
($E_{had}^{rec}$) by a random number method using the value of $\sigma_{had}$
from eq. 29. Then we find the final resolution as a function of
$[E_{\nu}-(E_{\mu}+E_{had}^{recS})]/E_{\nu}$; where, $E_{had}^{recS}$ is the
sum of the energies of all reconstructed hadrons in an event. We use the
atmospheric neutrino events without oscillations for an exposure of $50\times
1000$ kTon$\cdot$yr generated by Nuance to reconstruct the resolution
functions for each energy and zenith angle bins.

In fig. 10 we show the iso-$\chi^{2}$ contours with and without inclusion of
information about hadrons. We find that improvement of the sensitivity to
$\theta_{23}$, and therefore $\delta_{23}$, is marginal. Also there is a very
small change in the sensitivity to $\Delta m_{32}^{2}$: $\Delta(\Delta
m_{32}^{2})=0.02\cdot 10^{-3}$ eV2.

###  V.2 Cross-sections and fluxes

Figs. 11 and 12 illustrate improvements of the sensitivities with reduction of
different systematic uncertainties. Decrease of uncertainties of the overall
flux normalization, the ratio of horizontal/vertical flux, the neutrino cross-
section, and the tilt (below 1 GeV) from 10%, 10%, 2%, and 15% to 2%, 2%, 2%,
and 3%, respectively, do not lead to significant improvement of the
sensitivities. The reason is that the up-going neutrinos oscillate and down-
going neutrinos remain practically unchanged. In $\chi^{2}$ analysis these
down-going neutrinos allow to reduce the effect of systematic uncertainties in
fluxes. Significant improvement occurs only when all systematic uncertainties
are zero.

Figure 11: The same as in fig. 5, but with reduced systematic uncertainties.
Figure 12: The same as in fig. 6, but with reduced systematic uncertainties.

###  V.3 Adding priors to $\chi^{2}$

In $\chi^{2}$ analysis in sec. 4 for simplicity we used flat distributions of
uncertainties of the oscillation parameters in marginalization. The prior
contribution of the parameters to $\chi^{2}$. should improve the sensitivity.
Proper procedure would require the use of the best-fit values and variations
of parameters (especially $\theta_{13}$) which will be possible after results
of forthcoming experiments will be known.

Figure 13: The same as in fig. 5, but with the contributions from priors for
the oscillation parameters added to $\chi^{2}$ (see text). The marginalization
range for $\theta_{13}$ is $3^{\circ}-12.5^{\circ}(0^{\circ}-12.5^{\circ})$ in
absence (presence) of prior contribution.  Figure 14: The same as in fig. 6,
but with the contribution from priors for the oscillation parameters added to
the $\chi^{2}$. The marginalization range for $\theta_{13}$ is
$3^{\circ}-12.5^{\circ}(0^{\circ}-12.5^{\circ})$ in absence (presence) of
prior contribution.

In figs. 13 and 14 we show improvements of the sensitivities to the
$\theta_{23}$ deviation and the octant due to inclusion of the prior
contribution. We have assumed the Gaussian distribution of the uncertainties
around the best fit with width $\sigma(\sin^{2}2\theta_{13})=0.01$,
$\sigma(\sin^{2}2\theta_{23})=0.015$ and $\sigma(\Delta m_{32}^{2})/\Delta
m_{32}^{2}=0.015$ following Huber:2009cw ; Huber:2004ug . The $2\sigma$ errors
in measurements of $\Delta m_{32}^{2}$ are $\pm 0.09$ with flat uncertainties
of oscillation parameters, $\pm 0.07$ with prior information from present
global-fit, and $\pm 0.018$ with prior information from possible T2K result.

The asymmetry in the sensitivity to the octant (which was due to the
uncertainty of $\theta_{13}$ in absence of prior contribution) now disappears,
and the result does not depend on marginalization range.

Of course, larger values of $\theta_{13}$ can substantially enhance the
sensitivity to the octant since $P_{A}\sim\sin^{2}\theta_{13}$ at low
energies.

Table 1: Results of determination of $\theta_{23}$ $\theta_{23}$ | CL | Source
---|---|---
${42.9^{\circ}}^{+4.1^{\circ}}_{-2.8^{\circ}}$ | 1$\sigma$ | global-fitGonzalezGarcia:2010er
$35.7^{\circ}-54^{\circ}$ | 3$\sigma$ | global-fitGonzalezGarcia:2010er
${45^{\circ}}^{+10^{\circ}}_{-7.8^{\circ}}$ | 99% | SK Hosaka:2006zd
$45^{\circ}\pm 9^{\circ}$ | 90% | MINOS ($\nu$) Hosaka:2006zd
${34^{\circ}}^{+6^{\circ}}_{-4^{\circ}}$ or ${56^{\circ}}^{+4^{\circ}}_{-6^{\circ}}$ | 90% | MINOS ($\bar{\nu}$) minos
$39^{\circ}-51^{\circ}$ | 2$\sigma$ | T2K Huber:2004dv
$36^{\circ}-54^{\circ}$ | 2$\sigma$ | NO$\nu$A Huber:2004dv
$40^{\circ}-50^{\circ}$ | 2$\sigma$ | INO (1 Mton$\cdot$yr)
Table 2: Results of determination of $\Delta m^{2}_{31}$ $\Delta m_{32}^{2}(10^{-3}$eV2) | CL | Source
---|---|---
$-2.36\pm 0.07(\pm 0.36)$ | 1 (3)$\sigma$ | global-fit GonzalezGarcia:2010er
$+2.47\pm 0.12(\pm 0.37)$ | 1 (3)$\sigma$ | global-fit GonzalezGarcia:2010er
$2.5^{+0.52}_{-0.60}$ | 99% | SK 3$\nu$ Hosaka:2006zd
$2.35^{+0.11}_{-0.08}$ | 90% | MINOS $\nu$ minos
$3.36^{+0.45}_{-0.40}$ | 90% | MINOS $\bar{\nu}$ minos
$2.5\pm 0.04$ | 2$\sigma$ | T2K Huber:2004dv
$2.5_{-0.04}^{+0.07}$ | 2$\sigma$ | NO$\nu$A Huber:2004dv
$2.5\pm 0.07$ | 2$\sigma$ | INO (1 Mton$\cdot$yr)

##  VI Conclusion

We have studied analytically the dependence of the $\theta_{23}-$deviation
effect and octant asymmetry of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes on the
neutrino parameters $\theta_{23}$ and $\theta_{13}$. We explored numerically a
sensitivities of a magnetized calorimeter to the $\theta_{23}$-deviation, to
the octant and to $\Delta m_{32}^{2}$.

We show that for $\theta_{13}=0$ the sensitivity of ICAL to the octant is low
even for maximally allowed values of the deviation of the 2-3 mixing from
maximal. This is related to the fact that the octant asymmetry is proportional
to the “solar” probability $P_{S}$ which is large: $O(1)$ at $E\sim 0.1$ GeV
but quickly, as $\propto E^{-2}$, decreases with energy. The situation can be
improved by lowering the threshold, increasing exposure and reducing
systematic errors (especially in spectral index). We find that
$sign(\delta_{23})$ (octant) can be established at $90\%$ C.L. if
$|\delta_{23}|=7^{\circ}$, ${\mathcal{E}}=$ 4 Mton$\cdot$yr and $E_{th}=0.141$
GeV.

ICAL has good sensitivity to the $\theta_{23}$-deviation from maximal 2-3
mixing: the effect is proportional to the probability of the main channel of
oscillations, $\nu_{\mu}-\nu_{\tau}$, which is unsuppressed in whole
considered neutrino energy range. As a result, dependence of the sensitivity
on the energy threshold is weak and it does not change substantially when the
effect of 1-3 mixing is included. We find that with the 1 Mton$\cdot$yr
exposure the $3\sigma$ accuracy of determination of the deviation will be
$|\delta_{23}|\approx 6^{\circ}$, which is better than the present global fit
result and slightly better than expected sensitivity of T2K ($\approx
9^{\circ}$).

The oscillations driven by non-zero 1-3 mixing substantially improve the
sensitivity to the octant. One can determine the octant for
$\delta_{23}=5^{\circ}$ and $\theta_{13}=5^{\circ}$ at 90% C.L. with 1
Mton$\cdot$yr exposure. We find that this sensitivity depends crucially on the
uncertainty range of $\theta_{13}$. For a given nonzero $\theta_{13}$, the
sensitivity to octant discrimination is symmetric in $\theta_{23}$ with
respect to $\theta_{23}=45^{\circ}.$ However, the asymmetry arises (smaller
sensitivity for $\theta_{23}<45^{\circ}$) if value of $\theta_{13}$ can vary
in large range. The symmetry is restored if prior for the 1-3 mixing is added.

The accuracy of measurements of $\Delta m_{23}^{2}$ by ICAL, $\Delta(\Delta
m_{23}^{2})=0.15\cdot 10^{-3}$ eV2 ($3\sigma$, 1 Mton$\cdot$yr exposure), is
two times better than the accuracy of the present global fit and it is
worthier than the expected sensitivity of T2K.

ICAL can measure the difference of $\Delta m_{32}^{2}$ in $\nu$ and
$\bar{\nu}$ channels (the CPT test) with accuracy $0.8\times 10^{-4}$ eV2 at
3$\sigma$ confidence level with 1 Mton$\cdot$yr exposure and the present MINOS
result can be excluded at $>5\sigma$ confidence level.

We find that inclusion of information about hadrons from neutrino interactions
does not change the sensitivity to the oscillation parameters substantially.
Also improvements of the sensitivity due to better determinations of the
cross-section and neutrino flux are rather modest. However, the sensitivity
improves substantially with adding priors, especially for the 1 - 3 mixing,
and of course, with increase of the absolute value of 1-3 mixing.

Acknowledgments: The use of general cluster facility of Harish-Chandra
Research Institute for a part of this work is gratefully acknowledged. A. S.
wants to acknowledge Moon Moon Devi and Amol Dighe for providing the hadron
resolution function.

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