Abstract

INO/ICAL/PHY/NOTE/2015-01

ArXiv:1505.07380 [physics.ins-det]


Pramana - J Phys (2017) 88 : 79

doi:10.1007/s12043-017-1373-4

Physics Potential of the ICAL detector at

[0.2cm] the India-based Neutrino Observatory (INO)

[1.5cm]


The ICAL Collaboration






[The ICAL Collaboration]




Shakeel Ahmed, M. Sajjad Athar, Rashid Hasan, Mohammad Salim, S. K. Singh

Aligarh Muslim University, Aligarh 202001, India



S. S. R. Inbanathan

The American College, Madurai 625002, India


Venktesh Singh, V. S. Subrahmanyam

Banaras Hindu University, Varanasi 221005, India


Shiba Prasad Behera, Vinay B. Chandratre, Nitali Dash, Vivek M. Datar,

V. K. S. Kashyap, Ajit K. Mohanty, Lalit M. Pant

Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India



Animesh Chatterjee, Sandhya Choubey, Raj Gandhi, Anushree Ghosh,

Deepak Tiwari

Harish Chandra Research Institute, Jhunsi, Allahabad 211019, India


Ali Ajmi, S. Uma Sankar

Indian Institute of Technology Bombay, Powai, Mumbai 400076, India


Prafulla Behera, Aleena Chacko, Sadiq Jafer, James Libby, K. Raveendrababu,

K. R. Rebin

Indian Institute of Technology Madras, Chennai 600036, India



D. Indumathi, K. Meghna, S. M. Lakshmi, M. V. N. Murthy, Sumanta Pal,

G. Rajasekaran, Nita Sinha

Institute of Mathematical Sciences, Taramani, Chennai 600113, India


Sanjib Kumar Agarwalla, Amina Khatun

Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India


Poonam Mehta

Jawaharlal Nehru University, New Delhi 110067, India


Vipin Bhatnagar, R. Kanishka, A. Kumar, J. S. Shahi, J. B. Singh

Panjab University, Chandigarh 160014, India


Monojit Ghosh, Pomita Ghoshal, Srubabati Goswami, Chandan Gupta,

Sushant Raut

Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India



Sudeb Bhattacharya, Suvendu Bose, Ambar Ghosal, Abhik Jash, Kamalesh Kar,

Debasish Majumdar, Nayana Majumdar, Supratik Mukhopadhyay, Satyajit Saha

Saha Institute of Nuclear Physics, Bidhannagar, Kolkata 700064, India


B. S. Acharya, Sudeshna Banerjee, Kolahal Bhattacharya, Sudeshna Dasgupta,

Moon Moon Devi, Amol Dighe, Gobinda Majumder, Naba K. Mondal,

Asmita Redij, Deepak Samuel, B. Satyanarayana, Tarak Thakore

Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India





C. D. Ravikumar, A. M. Vinodkumar

University of Calicut, Kozhikode, Kerala 673635, India



Gautam Gangopadhyay, Amitava Raychaudhuri

University of Calcutta, Kolkata 700009, India


Brajesh C. Choudhary, Ankit Gaur, Daljeet Kaur, Ashok Kumar, Sanjeev Kumar,

Md. Naimuddin

University of Delhi, New Delhi 110021, India


Waseem Bari, Manzoor A. Malik

University of Kashmir, Hazratbal, Srinagar 190006, India



Jyotsna Singh

University of Lucknow, Lucknow 226007, India


S. Krishnaveni, H. B. Ravikumar, C. Ranganathaiah

University of Mysore, Mysuru, Karnataka 570005. India


Swapna Mahapatra

Utkal University, Vani Vihar, Bhubaneswar, 751004, India


Saikat Biswas, Subhasis Chattopadhyay, Rajesh Ganai, Tapasi Ghosh,

Y. P. Viyogi

Variable Energy Cyclotron Centre, Bidhannagar, Kolkata 700064, India



Spokesperson, nkm@tifr.res.in

Currently: University of Texas at Arlington, USA.

Currently: Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil.

Currently: University of Bern, Switzerland.

Currently: Central University of Karnataka, Gulbarga, India.

Chennai Mathematical Institute, Chennai, India.

Homi Bhabha National Institute, Mumbai, India

Currently: Weizmann Institute of Science, Rehovot, Israel.

Currently: Tokyo Metropolitan University, Hachioji, Tokyo, Japan

Currently: LNM Institute of Information Technology, Jaipur, India.

Currently: Institute for Basic Science, Daejeon, Korea

Currently: National Institute of Science, Education and Research, Bhubaneswar, India.

Currently: University of Bristol, UK.

Currently: Virginia Tech, Blacksburg, VA, USA

Currently: Universidade Federal de Goias, Goiania, Brazil.

Currently: Louisiana State University, USA.

Currently: Tata Institute of Fundamental Research, Mumbai, India.

Raja Ramanna Fellow.

Abstract

The upcoming 50 kt magnetized iron calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO) is designed to study the atmospheric neutrinos and antineutrinos separately over a wide range of energies and path lengths. The primary focus of this experiment is to explore the Earth matter effects by observing the energy and zenith angle dependence of the atmospheric neutrinos in the multi-GeV range. This study will be crucial to address some of the outstanding issues in neutrino oscillation physics, including the fundamental issue of neutrino mass hierarchy. In this document, we present the physics potential of the detector as obtained from realistic detector simulations. We describe the simulation framework, the neutrino interactions in the detector, and the expected response of the detector to particles traversing it. The ICAL detector can determine the energy and direction of the muons to a high precision, and in addition, its sensitivity to multi-GeV hadrons increases its physics reach substantially. Its charge identification capability, and hence its ability to distinguish neutrinos from antineutrinos, makes it an efficient detector for determining the neutrino mass hierarchy. In this report, we outline the analyses carried out for the determination of neutrino mass hierarchy and precision measurements of atmospheric neutrino mixing parameters at ICAL, and give the expected physics reach of the detector with 10 years of runtime. We also explore the potential of ICAL for probing new physics scenarios like CPT violation and the presence of magnetic monopoles.

Preface

The past two decades in neutrino physics have been very eventful, and have established this field as one of the flourishing areas of high energy physics. Starting from the confirmation of neutrino oscillations that resolved the decades-old problems of the solar and atmospheric neutrinos, we have now been able to show that neutrinos have nonzero masses, and different flavors of neutrinos mix among themselves. Our understanding of neutrino properties has increased by leaps and bounds. Many experiments have been constructed and envisaged to explore different facets of neutrinos, in particular their masses and mixing.

The Iron Calorimeter (ICAL) experiment at the India-based Neutrino Observatory (INO) [1] is one of the major detectors that is expected to see the light of the day soon. It will have unique features like the ability to distinguish muon neutrinos from antineutrinos at GeV energies, and measure the energies of hadrons in the same energy range. It is therefore well suited for the identification of neutrino mass hierarchy, the measurement of neutrino mixing parameters, and many probes of new physics. The site for the INO has been identified, and the construction is expected to start soon. In the meanwhile, the R&D for the ICAL detector, including the design of its modules, the magnet coils, the active detector elements and the associated electronics, has been underway over the past decade. The efforts to understand the capabilities and physics potentials of the experiment through simulations are in progress at the same time.

We present here the Status Report of our current understanding of the physics reach of the ICAL, prepared by the Simulations and Physics Analysis groups of the INO Collaboration. It describes the framework being used for the simulations, the expected response of the detector to particles traversing it, and the results we expect to obtain after the 50 kt ICAL has been running for about a decade. The focus of the physics analysis is on the identification of the mass hierarchy and precision measurements of the atmospheric neutrino mixing parameters. The feasibilities of searches for some new physics, in neutrino interactions as well as elsewhere, that can be detected at the ICAL, are also under investigation.

The first such report [2] had been published when the INO was being proposed, and the ICAL Collaboration was at its inception. Our understanding of the detector has now matured quite a bit, and more realistic results can now be obtained, which have been included in this report. The work on improving several aspects of the detector, the simulations, the reconstruction algorithms and the analysis techniques is in progress and will remain so for the next few years. This Report is thus not the final word, but a work in progress that will be updated at regular intervals.

In addition to the ICAL detector, the INO facility is designed to accommodate experiments in other areas like neutrinoless double beta decay, dark matter search, low-energy neutrino spectroscopy, etc. Preliminary investigations and R&D in this direction are in progress. The special environment provided by the underground laboratory may also be useful to conduct experiments in rock mechanics, geology, biology etc. This Report focusses mainly on the ongoing physics and simulation related to the ICAL detector. The details of other experiments will be brought out separately.

The Government of India has recently (December 2014) given its approval for the establishment of INO. This is a good opportunity to present the physics capabilities of the ICAL experiment in a consolidated form.

Contents

Executive Summary

The INO and the ICAL detector

The India-based Neutrino Observatory (INO) is proposed to be built in Bodi West Hills, in Theni district of Tamil Nadu in South India. The main detector proposed to be built at the INO is the magnetised Iron CALorimeter (ICAL) with a mass of 50 kt. The major physics goal of ICAL is to study neutrino properties, through the observation of atmospheric neutrinos that cover a wide range of energies and path lengths. A special emphasis will be on the determination of the neutrino mass hierarchy, by observing the matter effects when they travel through the Earth. This would be facilitated through the ability of ICAL to distinguish neutrinos from antineutrinos.

Table 1 gives the salient features of the ICAL detector. The active detector elements in ICAL will be the Resistive Plate Chambers (RPCs). The detector is optimised to be sensitive primarily to the atmospheric muon neutrinos in the 1–15 GeV energy range. The structure of the detector, with its horizontal layers of iron interspersed with RPCs, allows it to have an almost complete coverage to the direction of incoming neutrinos, except for those that produce almost horizontally traveling muons. This makes it sensitive to a large range of path lengths for the neutrinos travelling through the Earth, while the atmospheric neutrino flux provides a wide spectrum in the neutrino energy .

ICAL
No. of modules 3
Module dimension 16 m 16 m 14.5 m
Detector dimension 48 m 16 m 14.5 m
No. of layers 151
Iron plate thickness 5.6 cm
Gap for RPC trays 4.0 cm
Magnetic field 1.5 Tesla
RPC
RPC unit dimension 2 m 2 m
Readout strip width 3 cm
No. of RPC units/Layer/Module 64
Total no. of RPC units  30,000
No. of electronic readout channels 3.9
Table 1: Specifications of the ICAL detector.

ICAL will be sensitive to both the energy and direction of the muons that will be produced in charged-current (CC) interactions of the atmospheric muon neutrinos (and antineutrinos) with the iron target in the detector. In addition, the fast response time of the RPCs (of the order of nanoseconds) will allow for a discrimination of the upward-going muon events and downward-going ones. (Once the starting point of the track is identified, the initial hits in the track determine the initial muon direction accurately.) This direction discrimination separates the neutrinos with short path lengths from those with longer ones. Such a separation is crucial since the neutrino oscillation probability is strongly dependent on the path length .

Moreover, since ICAL is expected to be magnetised to about 1.5 T in the plane of the iron plates, it will be able to discriminate between muons of different charges, and hence will be capable of differentiating events induced by muon neutrinos and muon antineutrinos. Through this sensitivity, one can probe the difference in matter effects in the propagation of neutrinos and antineutrinos that traverse the Earth before they reach the detector. This in turn will allow for a sensitivity to the neutrino mass hierarchy, which is the primary goal of the ICAL experiment.

The magnetic field is also crucial for reconstructing the momentum of the muon tracks in the case of partially contained events. When the muon track is completely contained inside the detector, the length of the track can determine the energy of the muon reliably, and the magnetic field plays a supplementary role of improving the momentum resolution. However for the partially contained track events, the bending of the track in the local magnetic field is crucial to reconstruct the muon momentum in the energies of interest. The good tracking ability and energy resolution of ICAL for muons makes it very well suited for the study of neutrino oscillation physics through the observation of atmospheric neutrinos.

In addition, ICAL is also sensitive to the energy deposited by hadrons in the detector in the multi-GeV range, a unique property that enables a significant improvement in the physics reach of ICAL, as will be clear in this report. In the present configuration the sensitivity of ICAL to electrons is very limited; however, this is still under investigation.

Though the ICAL is yet to be built, its putative properties have been simulated using the CERN GEANT4 [3] package. The details of these simulations have been presented in Chapter 3. This report presents results on the response of ICAL to particles traversing through it in Chapter 4. The resultant physics potential of the detector, obtained from these simulations, is given in later chapters, where we focus on the identification of neutrino mass hierarchy, and the precise determinations of the atmospheric neutrino parameters: and , as well as the octant of . In addition, we also discuss some novel and exotic physics possibilities that may be explored at ICAL.

The simulation framework

For the results presented in this report, the atmospheric neutrino events have been generated with the NUANCE [4] neutrino generator using the Honda 3d fluxes [5] for the Kamioka site in Japan. The details of the fluxes have been presented in Chapter 2. The Honda atmospheric neutrino fluxes at Theni, the INO site, are expected to be finalised soon and will be used when available. A preliminary comparison of the fluxes at the two sites is also presented in Appendix A of this report. The number of muon track events are expected to be similar, within statistical errors, for both fluxes, for energies more than 3 GeV. We therefore do not expect the reach of ICAL, especially for the mass hierarchy, to change significantly with the use of the Theni fluxes.

A typical CC interaction of in the detector gives rise to a charged muon that leaves a track, and single or multiple hadrons that give rise to shower-like features. The simulations of the propagation of muons and hadrons in the detector have been used to determine the response of the detector to these particles. This leads to the determination of detection efficiencies, charge identification efficiencies, calibrations and resolutions of energies and directions of the particles. The results of these simulations have been presented in Chapter 4.

In order to perform the physics analysis, we generate a large number (typically, an exposure of 1000 years) of unoscillated events using NUANCE, which are later scaled to a suitable exposure, and oscillations are included using a reweighting algorithm. The typical values of oscillation parameters used are close to their best-fit values, and are given in Table 2.

(eV) (eV)
0.3 0.5 0.1 0
Table 2: True values of the input oscillation parameters used in the analyses, unless otherwise specified. For more details, including 3 limits on these parameters, see Table 1.1.

Here is the effective value of relevant for the two-neutrino analysis of atmospheric neutrino oscillations [6, 7]. The energies and directions of the relevant particles are then smeared according to the resolutions determined earlier. This approach thus simulates the average behaviour of the measured quantities. At the current stage of simulations, we also assume that the muon track and the hadron shower can be separated with full efficiency, and that the noise due to random hits near the signal events in the short time interval of the event is negligible. While these approximations are reasonable, they still need to be justified with actual detector, possibly by collecting data with a prototype. In the meanwhile, a complete simulation, which involves passing each of the generated events through a GEANT4 simulation of the ICAL detector, is in progress.

The different analyses then determine the relevant physics results through a standard minimization procedure, with the systematic errors included through the method of pulls, marginalizations over the allowed ranges of parameters, and including information available from other experiments using priors. The details of the analysis procedures for obtaining the neutrino mixing parameters have been given in Chapter 5, which presents the results using only the information on muon energy and angle, as well as the improvement due to the inclusion of information on hadron energies. Chapter 6 further includes combined analyses of the reach of ICAL with other current and near-future detectors such as T2K and NOA, for the mass hierarchy and neutrino oscillation parameters. Chapter 7 discusses the reach of ICAL with respect to exotic physics possibilities such as the violation of CPT or Lorentz symmetries, the detection of magnetic monopoles, etc..

We now list the highlights of the results compiled in this report. Many of these results have appeared elsewhere [8, 9, 10, 11, 12, 13, 14, 15, 16], however some have been updated with more recent information.

Detector response to propagating particles

Response to muons

The ICAL detector is optimised for the detection of muons propagating in the detector, identification of their charges, and accurate determination of their energies and directions. The energies and directions of muons are determined through a Kalman filter based algorithm. The reconstructed energy (direction) for a given true muon energy (direction) is found to give a good fit to the Gaussian distribution, and hence the resolution is described in terms of the mean and standard deviation of a Gaussian distribution. The reconstruction efficiency for muons with energies above 2 GeV is expected to be more than 80%, while the charge of these reconstructed muons is identified correctly on more than 95% occasions. The direction of these muons at the point of their production can be determined to within about a degree. The muon energy resolution depends on the part of the detector the muon is produced in, but is typically 25% (12%) at 1 GeV (20 GeV), as can be seen in the left panel of Fig. 1 [8].

Figure 1: Left panel shows the momentum resolution of muons produced in the region (see Sec. 4.1), as functions of the muon momentum in different zenith angle bins [8]. Right panel shows the energy resolution of hadrons (see Sec. 4.2) as functions of , where events have been generated using NUANCE in different bins. The bin widths are indicated by horizontal error bars [9].

Response to hadrons

The detector response to hadrons is quantified in terms of the quantity for the CC processes that produce a muon, which is calibrated against the number of hits in the detector. The hit distribution for a given hadron energy is found to give a good fit to the Vavilov distribution. Hence is calibrated against the mean of the corresponding Vavilov mean for the number of hits, and the energy resolution is taken to be the corresponding value of . The energy resolution is shown in Fig. 1. The complete description of Vavilov distributions needs a total of four parameters, the details of which may be found in Chapter 4. The presence of different kinds of hadrons, which are hard to distinguish through the hit information, is taken care of by the generation of events through NUANCE, which is expected to produce the hadrons in the right proportions. The energy resolution of hadrons is found to be about 85% (36%) at 1 GeV (15 GeV) [9]. The information on the shape of the hadron shower is not used for extracting hadron energy yet; the work on this front is still in progress.

Physics reach of ICAL

Sensitivity to the Mass Hierarchy

In order to quantify the reach of ICAL with respect to the neutrino mass hierarchy, a specific hierarchy, normal or inverted, is chosen as the true (input) hierarchy. The CC muon neutrino events are binned in the quantities chosen for the analysis, and a analysis is performed taking the systematic errors into account and marginalising over the ranges of the parameters , and . The significance of the result is then determined as the with which the wrong hierarchy can be rejected (see Chapter 5).

The analysis for mass hierarchy identification using only the muon momentum information [11] yields with 10 years of exposure of the 50 kt ICAL, as can be seen from Fig. 2 (black dashed curve), which also shows (red solid curve) that a considerable improvement in the physics reach is obtained if the correlated hadron energy information in each event is included along with the muon energy and direction information; i.e. the binning is performed in the 3-dimensional parameter space . The same exposure now allows the identification of mass hierarchy with a significance of [14] for maximal mixing angle () and . The significance depends on the actual value of and , and increases with the values of these mixing angles. When and are varied in their allowed ranges, the corresponding signifiance varies in the range 7–12.

Figure 2: The hierarchy sensitivity of ICAL with input normal (left) and inverted (right) hierarchy including correlated hadron energy information, with , and marginalised over their ranges [14]. Improvement with the inclusion of hadron energy is significant.

Precision Measurements of oscillation parameters

Figure 3: The precision reach of ICAL in the plane, in comparison with other current and planned experiments [14]. Information on hadron energy has been included.

The precision on the measurements of the neutrino oscillation parameters and is quantified in terms of , where is the parameter under consideration. The precision on the measurement of is essentially a function of the total number of events, and is expected to be about 12-14%, whether one includes the hadron energy information or not. The precision on , however, improves significantly (from 5.4% to 2.9%) if the information on hadron energy is included.

Figure 3 shows the comparison of the 10-year reach of 50 kt ICAL in the plane, with the current limits from other experiments. It is expected that the precision of ICAL would be much better than the atmospheric neutrino experiments that use water Cherenkov detectors, due to its better energy measurement capabilities. However the beam experiments will keep on accumulating more data, hence the global role of ICAL for precision measurements of these parameters will be not competitive, but complementary.

Sensitivity to the octant of

While the best-fit value for is close to maximal, it is not fully established whether it deviates from maximality, and if so, whether is less than or greater than 0.5, that is, whether it lies in the first or second octant. ICAL is sensitive to the octant of through two kinds of effects: one is through the depletion in atmospheric muon neutrinos (and antineutrinos) via the survival probability and the other is the contribution of the atmospheric electron neutrinos to the observed CC muon events through the oscillation probability . Both effects are proportional to , however act in opposite directions, thereby reducing the effective sensitivity of ICAL to the octant. The reach of ICAL alone for determining the octant is therefore limited; it can identify the octant to a significance with 500 kt-yr only if [14]. The information from other experiments clearly needs to be included in order to identify the octant.

Synergies with other experiments

Neutrino Mass Hierarchy determination

The ability of currently running long baseline experiments like T2K and NOA to distinguish between the mass hierarchies depends crucially on the actual value of the CP-violating phase . For example, if is vanishing, this ability is severely limited. However if one adds the data available from the proposed run of these experiments, a preliminary estimation suggests that even for vanishing , the mass hierarchy identification of may be achieved with a run-time as low as 6 years of the 50 kt ICAL, for maximal mixing [17]. This may be observed in Fig. 4. Note that this improvement in the ICAL sensitivity is not just due to the information provided by these experiments on the mass hierarchy, but also due to the improved constraints on and .

Figure 4: Preliminary results on the hierarchy sensitivity with input normal (left)and inverted (right) hierarchy when ICAL data is combined with the data from T2K (total luminosity of protons on target in neutrino mode) and NOA (3 years running in neutrino mode and 3 years in antineutrino mode) [17].

Identifying mass hierarchy at all values

The large range of path length of the atmospheric neutrinos makes ICAL insensitive to the CP phase , as a result its reach in distinguishing the hierarchy is also independent of the actual value of [18]. On the other hand the sensitivity of fixed-baseline experiments such as T2K and NOA is extremely limited if and the true hierarchy is normal. However adding of the ICAL information ensures that the hierarchy can be identified even in these unfavoured regions [11]. Of course, in the regions favourable to the long baseline experiments, the ICAL data can only enhance the power of discriminating between the two hierarchies.

Determination of the CP phase

Though ICAL itself is rather insensitive to , data from ICAL can still improve the determination of itself, by providing input on mass hierarchy. This is especially crucial in the range , precisely where the ICAL data would also improve the hierarchy discrimination of NOA and other experiments [19].

Other physics possibilities with ICAL

ICAL is a versatile detector, and hence could be employed to test for a multitude of new physics scenarios. For example, the violation of CPT or Lorentz symmetry in the neutrino sector [15] can be probed to a great precision, owing to its excellent energy measurement capability. The passage of magnetic monopoles through the detector may be looked for by simply looking for slowly moving, undeflecting tracks [16]. Dark matter annihilation inside the Sun may be constrained by comparing the flux from the sun with the flux from other directions. Many such scenarios are under investigation currently.

Concluding remarks

A strong and viable physics program is ready for ICAL at INO. The simulations based on the incorporation of the ICAL geometry in GEANT4 suggests that the detector will have excellent abilities for detection, charge identification, energy measurement and direction determination for charged muons of GeV energies. The magnetic field enables separation of from , equivalently that of from , thus increasing the sensitivity to the difference in matter effects on neutrino and antineutrino oscillations. It will also be sensitive to hadrons, an ability that will increase its physics reach significantly and will offer advantages over other atmospheric neutrino detectors. Apart from its main aim of identifying the neutrino mass hierarchy, ICAL can also help in precision measurements of other neutrino mixing parameters, and can probe exotic physics issues even beyond neutrinos.

Chapter 1 Introduction

The earth is just a silly ball

To them, through which they simply pass

Like dustmaids through a drafty hall

-John Updike

Many important developments have taken place in neutrino physics and neutrino astronomy in recent years. The discovery of neutrino oscillations and consequent inference about the non-vanishing mass of the neutrinos, from the study of neutrinos from the Sun and cosmic rays, have had far-reaching consequences for particle physics, astroparticle physics and nuclear physics. The observation of neutrinos from natural sources as well as those produced at reactors and accelerators have given us the first confirmed signals of physics beyond the Standard Model of particle physics. They have also enabled us access to the energy production mechanisms inside stars and other astrophysical phenomena.

Experimental observations of neutrino interactions began in the mid 1950s at Savannah river reactor by Reines and Cowan [20] followed by experiments deep in the mines of Kolar Gold Fields (KGF) in India [21] and in South Africa [22]. The pioneering solar neutrino experiments of Davis and collaborators in the USA [23, 24], the water Cherenkov detector Kamiokande [25] and its successor the gigantic Super-Kamiokande (SK) [26, 27], the gallium detectors SAGE [28] in Russia and Gallex [29], GNO [30] at the Laboratorio National di Gran Sasso (LNGS) in Italy, the heavy-water detector at the Sudbury Neutrino Observatory (SNO) in Canada [31, 32], the KamLAND [33] and K2K [34] experiments in Japan, etc. have together contributed in a very fundamental way to our knowledge of neutrino properties and interactions. The observation of solar neutrinos has given a direct experimental proof that the Sun and the stars are powered by thermonuclear fusion reactions that emit neutrinos. The recent results from reactor neutrino experiments, beginning with Double CHOOZ [35] in France and culminating in the results from Reno [36] in Korea and Daya Bay [37] in China, and from accelerator experiments like MINOS [38], T2K [39], and NOA  [40, 41] have further revealed properties of neutrinos that not only serve as windows to physics beyond the Standard Model of particle physics, but also provide possibilities of understanding the matter-antimatter asymmetry in the universe through the violation of the charge-parity (CP) symmetry in the lepton sector.

Impelled by these discoveries and their implications for the future of particle physics and astrophysics, plans are underway worldwide for new neutrino detectors to study such open issues as the hierarchy of neutrino masses, the masses themselves, the extent of CP violation in the lepton sector, the Majorana or Dirac nature of neutrinos, etc.. This involves R&D efforts for producing intense beams of neutrinos at GeV energies, suitable detectors to detect them at long baseline distances, and sensitive neutrinoless double beta decay experiments. A complementary approach to these is the use of atmospheric neutrinos, whose fluxes are more uncertain than beam neutrinos, but which provide a wider range of energies, and more importantly, a wider range of baselines.

The India-based Neutrino Observatory (INO) is one such proposal aiming to address some of the challenges in understanding the nature of neutrinos, using atmospheric neutrinos as the source. The unique feature of ICAL, the main detector in INO, will be the ability to distinguish neutrinos from antineutrinos, which enables a clearer distinction between the matter effects on neutrinos and antineutrinos travelling through the Earth, leading to the identification of neutrino mass hierarchy. In this Chapter, we shall introduce the INO laboratory, the ICAL detector, and describe the role of such a detector in the global context of neutrino physics experiments.

1.1 The ICAL detector at the INO facility

1.1.1 Neutrino experiments in India: past and present

Underground science in India has a long history. The deep underground laboratory at Kolar Gold Fields (KGF), where Indian scientists conducted many front ranking experiments in the field of cosmic rays and neutrinos, was a pioneering effort. The KGF mines are situated at about 870m above sea level near the city of Bangalore in South India. It has a flat topography around the area surrounding the mines. The mines have extended network of tunnels underground which permitted experiments up to a depth of 3000 m below the surface. Initially attempts were made to find the depth variation of muon fluxes starting from the surface up to the deepest reaches. The absence of any count around a depth of 8400 hg/cm lead to the conclusion that the atmospheric muon intensity is attenuated to such a level where one could search for very weak processes like the interactions of high energy neutrinos. This was in the beginning of the sixties when very little was known about the interaction of neutrinos at high energies ( a few GeV) from accelerators, and that too with only muon neutrino beams. Nothing was known about the electron neutrino or antineutrino interactions.

Thus began the neutrino experiments in KGF in the early sixties, conducted by a collaboration consisting of groups from Durham University (UK), Osaka City University (Japan) and TIFR in India. The techniques used were perfected during the years of muon experiments and involved a basic trigger with scintillation counters and Neon Flash Tubes (NFT) for tracking detectors initially. Seven such detectors were placed in a long tunnel at a depth of 2.3 km, in the Heathcote shaft of Champion reef mines, in three batches over a period of two year starting from the end of 1964 [42].

The first ever atmospheric neutrino event was recorded underground was in early 1965 [21]. Two well defined tacks emerging from the rock in an upward direction indicated unambigiously a clear inelastic neutrino event. Later this collaboration put together the first experiment that searched for proton decay. Nature did not oblige and experiments are still looking for proton decay. The KGF laboratory operated for nearly four decades, almost till the end of 1980’s, collecting data on atmospheric muon and neutrino interactions at various depths, starting from about 300 metres all the way down to 2700 metres. In the process they also detected some anomalous events which could not be attributed to neutrinos at depths around 2000 metres [43, 44, 45]. Such events have neither been proved wrong nor have they been confirmed by other experiments.

The INO project, the discussions about which were formally held first in the Workshop on High Energy Physics Phenomenology (WHEPP-VI) in Chennai [46] is an ambitious proposal to recapture this pioneering spirit and do experiments in neutrino physics at the cutting edge. The immediate goal of INO project is the creation of an underground laboratory which will house a large magnetised iron calorimeter (ICAL) detector to study the properties of naturally produced neutrinos in the earth’s atmosphere. Apart from experiments involving neutrinos, in the long term the laboratory is envisaged to develop into a full-fledged underground laboratory for studies in Physics, Biology and Geology as well. The INO is the first basic science laboratory planned on such a large scale in India.

1.1.2 Location and layout of the INO

The INO will be located at the Bodi West Hills (BWH), near Pottipuram village, in the Theni district of Tamil Nadu, India. The site has been chosen both for geotechnical reasons as well as from environmental considerations. It is near the historic city of Madurai, as shown in Fig. 1.1. Madurai is about 120 km from the INO site, and will also be the location for the Inter-Institutional Centre for High Energy Physics (IICHEP), where activities related to the INO will be carried out. The figure shows the location and also the features of the local terrain. The construction of the laboratory below the Bodi Hills involves building a horizontal tunnel, approximately 1900 m long, to reach the laboratory that is located under a mountain peak. One large and three small laboratory caverns are to be built with a rock burden of 1000 m or more all around (with a vertical overburden of 1300 m) to house the experiments. The reduction of cosmic ray background at this site is almost the same as at the Gran Sasso laboratory, as can be seen from the bottom right panel, which shows the cosmic ray muon flux as a function of depth, with the locations of other major laboratories.

In addition to the main Iron CALorimeter (ICAL) detector whose prime goal is the determination of the neutrino mass hierarchy, the laboratory is designed to accommodate experiments in other areas like neutrinoless double beta decay, dark matter search, low-energy neutrino spectroscopy, etc. Preliminary investigations and R&D in this direction are in progress. The special environment provided by the underground laboratory may also be useful to conduct experiments in rock mechanics, geology, biology etc.

Figure 1.1: The location of the INO site and the nearby major landmarks. The IICHEP is located about 120 km east of the INO site, in the city of Madurai, as shown in the top right panel. The photo in the bottom right panel shows the view of the hill under which the cavern will be located. The terrain is totally flat with minimal undergrowth as seen in the picture (Photo: M V N Murthy). The photo is taken before the start of any construction. The bottom left panel shows the suppression in intensity of atmospheric muon flux at various underground sites, compared to the INO cavern [2].

The present configuration of the laboratory caverns is shown in Fig. 1.2. The largest cavern that will house the main iron calorimeter detector (ICAL) is 132 m (L) 26 m (W) 32.5 m (H). This cavern, called “UG-Lab 1”, is designed to accommodate a 50 kt ICAL (planned) and a second possible ICAL-II neutrino detector of equal size. Each ICAL consists of three modules with dimensions of 16 m (L) 16 m (W) 14.5 m (H), so that the total footprint of both the detectors would be 96 m (L) 16 m (W). Fig. 1.2 shows three modules of the 50 kt ICAL.

Figure 1.2: Underground caverns layout showing the footprints of proposed experiments and other components

1.1.3 The ICAL detector

Figure 1.3: Schematic view of the 50 kt ICAL detctor

The ICAL detector is similar in concept to the earlier proposed Monolith [47, 48] detector at Gran Sasso. The layout of the proposed ICAL detector is shown in Fig. 1.3. The detector will have a modular structure of total lateral size , subdivided into three modules of area . It will consist of a stack of 151 horizontal layers of 5.6 cm thick magnetized iron plates interleaved with 4 cm gaps to house the active detector layers, making it 14.5 m high. Iron spacers acting as supports will be located every 2 m along both X and Y directions; the 2 m wide roads along the transverse ( direction will enable the insertion and periodic removal of RPCs, when required.

The active detector elements, the resistive plate chambers (RPCs) made up of a pair of 3 mm thick glass plates of area separated by 2 mm spacers, will be inserted in the gaps between the iron layers. These will be operated at a high voltage of about 10 kV in avalanche mode. A high energy charged particle, passing through the RPCs, will leave signals that will be read by orthogonal X and Y pickup strips, about 3 cm wide, one on each side of an RPC. Detailed R & D has shown that the RPCs have an efficiency of around 90–95% with a time resolution of about a nanosecond. This will allow the determination of the X and Y coordinates of the track of the charged particles passing through the RPC. The layer number of the RPC will provide the Z coordinate. The observed RPC time resolution of 1 ns will enable the distinction between upward-going particles and downward-going particles. From the hit pattern observed in the RPCs, the energies as well as directions of the charged particles produced in the neutrino interactions can be reconstructed.

Each module will have two vertical slots cut into it to enable current-carrying copper coils to be wound around as shown in Fig. 1.3. Simulation studies [49] have shown that the iron plates can be magnetized to a field strength of about 1.5 T, with fields greater than 1 T over at least 85% of the volume of the detector. The bending of charged particles in this magnetic field will enable the identification of their charge. In particular, the sign of the charge of the muon produced by neutrino interactions inside the detector will help in identifying and studying the and induced events separately. The magnetic field will also help the measurement of the momentum of the final state particles, especially the muons.

With about 14000 iron plates of 2 m 4 m area and 5.6 cm thickness, 30000 RPCs of 2 m 2 m area, 4,000,000 electronic readout channels, and a magnetic field of 1.5 T, the ICAL is going to be the largest electromagnet in the world, and is expected to play a pivoting role in our understanding of neutrino properties.

1.2 Role of ICAL in neutrino mixing and beyond

In this section, we briefly discuss our present understanding of neutrino oscillation parameters and identify the fundamental issues in the neutrino sector that can be addressed by the ICAL detector.

1.2.1 Current status of neutrino mixing parameters

The neutrino flavour states (where ) are linear superpositions of the neutrino mass eigenstates (with ), with masses :

(1.1)

Here is the unitary mixing matrix. A physically motivated form of the mixing matrix that is conventionally used is [50, 51, 52]

where , , and denotes the CP violating (Dirac) phase, also called . Note that the Majorana phases are not included in the above parameterization, since they do not play a role in neutrino oscillation experiments.

The probability of an initial neutrino of flavour and energy being detected as a neutrino of the same energy but with flavour after travelling a distance in vacuum is

(1.2)

where (eV)(km)(GeV), with the mass squared differences between the th and th neutrino mass eigenstates. Oscillation measurements are not sensitive to the individual neutrino masses, but only to their mass-squared differences. Note that the above expression is valid only for propagation through vacuum. In matter, the probabilities are drastically modified. The relevant expressions may be found in Appendix B.

The neutrino flavour conversion probabilities can be expressed in terms of the two mass squared differences, the three mixing angles, and the single CP-violating phase. Also of crucial importance is the mass ordering, i.e., the sign of (the same as the sign of ). While we know that is positive so as to accommodate the observed energy dependence of the electron neutrino survival probability in solar neutrino experiments, at present is allowed to be either positive or negative. Hence, it is possible to have two patterns of neutrino masses: , called normal ordering, where is positive, and , called inverted ordering, where is negative. Determining the sign of is one of the prime goals of the ICAL experiment. Note that, though the “mass ordering” is perhaps the more appropriate term to use in this context, the more commonly used term in literature is “mass hierarchy”. In this report, therefore, we will use the notation “normal hierarchy” (NH) to denote normal ordering, and “inverted hierarchy” (IH) to denote inverted ordering. The word “hierarchy” used in this context has no connection with the absolute values of neutrino masses.

Table 1.1 summarises the current status of neutrino oscillation parameters [53, 54] based on the world neutrino data that was available after the NOW 2014 conference. The numbers given in Table 1.1 are obtained by keeping the reactor fluxes free in the fit and also including the short-baseline reactor data with [53, 54].

Parameter Best-fit values ranges Relative
precision
(eV)
(eV) (NH) (NH)
(eV) (IH) (IH)
0.3 [0.27,   0.34]
0.45 (NH), 0.58 (IH) [0.38,   0.64]
0.022 [0.018,  0.025]
[0,   ]
Table 1.1: The values of neutrino oscillation parameters used for the analyses in this paper [53]. The second column shows the central values of the oscillation parameters. The third column depicts the ranges of the parameters with the relative errors being listed in the last column. Note that the parameter () is used while performing the fit with normal (inverted) hierarchy. The current best-fit values and allowed ranges of these parameters may be found in [55, 56, 54].

Table 1.1 also provides the relative precision111Here the precision is defined as 1/6th of the variations around the best-fit value. on the measurements of these quantities at this stage. The global fit suggests the best-fit value of with a relative precision of 2.4%. In a three-flavor framework, we have the best-fit values of for NH and for IH. A relative precision of 2.5% has been achieved for the atmospheric mass-squared splitting.

As far as the mixing angles are concerned, is now pretty well measured with a best-fit value of and a relative precision of 4% has been achieved for the solar mixing angle. Our understanding of the atmospheric mixing angle has also improved a lot in recent years. Combined analysis of all the neutrino oscillation data available so far disfavours the maximal mixing solution for at confidence level [57, 53, 54, 58, 59]. This result is mostly governed by the MINOS accelerator data in and disappearance modes [60]. The dominant term in survival channel mainly depends on the value of . Now, if turns out to be different from 1 as suggested by the recent oscillation data, then it gives two solutions for : one whose value is less than half, known as the lower octant (LO) solution, and the other whose value is greater than half, known as the higher octant (HO) solution. This creates the problem of octant degeneracy of  [61]. At present, the best-fit value of in LO (HO) is 0.45 (0.58) assuming NH (IH). The relative precision on is around 8.7% assuming maximal mixing as the central value. Further improvement in the knowledge of and settling the issue of its octant (if it turns out to be non-maximal) are also important issues that can be addressed by observing atmospheric neutrinos.

For many years, we only had an upper bound on the value of the 1-3 mixing angle [62, 63, 64, 65]. A nonzero value for this angle has been discovered rather recently [35, 36, 37, 66, 67], with a moderately large best-fit value of , which is mostly driven by the high-statistics data provided by the ongoing Daya Bay reactor experiment [37, 67]. It is quite remarkable that already we have achieved a relative precision of 5.3% on . On the other hand, the whole range of is still allowed at the level.

1.2.2 Unravelling three-neutrino mixing with ICAL

As has been discussed earlier, the main advantage of a magnetised iron calorimeter is its ability to distinguish from , and hence to study and separately. This allows a cleaner measurement of the difference in the matter effects experienced by neutrinos and antineutrinos. However, this difference depends on the value of . The recent measurement of a moderately large value of therefore boosts the capability of ICAL for observing these matter effects, and hence its reach in addressing the key issues related to the neutrino masses and mixing. In this section, we shall highlight the role that an iron calorimeter like ICAL will have in the context of global efforts to measure neutrino mixing parameters.

The moderately large value has opened the door to the fundamental measurements of (i) the neutrino mass ordering, (ii) the deviation of 2-3 mixing angle from its maximal value and hence the correct octant of , and (iii) the CP phase and to look for CP violation in the lepton sector, for several experiments which would have had limited capability to address these questions had this parameter been significantly smaller. Central to all these measurements are effects which differ between neutrinos and antineutrinos. These could either be matter related effects which enhance or suppress the oscillation probabilities (relevant for i and ii above), or those induced by a non-zero value of (relevant for iii). The ICAL will be sensitive to the matter effects, but will have almost no sensitivity to the actual value of .

Prior to summarizing the role of ICAL, it is useful to mention several other experiments which are underway or will come online in the next couple of decades with the aim of making the three important measurements mentioned above. The T2K experiment has observed electron neutrino appearance in a muon neutrino beam [66], thus showing a clear evidence of neurino oscillations. The accelerator based long-baseline beam experiments NOvA [68, 69, 70] has already started taking data that will be sensitive to mass hierarchy, and the first results have been presented [41]. The IceCube DeepCore experiment has also recently [71] published their results on atmosphetic neutrino oscillations. Future large atmospheric neutrino detectors on the cards are Hyper-Kamiokande (HK) [72], Precision IceCube Next-Generation Upgrade (PINGU) [73] and Oscillation Research with Cosmics in the Abyss (ORCA) [74]. The Deep Underground Neutrino Experiment (DUNE) [75], a combined initiative of the earlier Long Baseline Neutrino Experiment (LBNE) [76, 77] and Long Baseline Neutrino Oscillation (LBNO) [78, 79, 80] collaborations, is also slated to aim at the mass hierarchy identification. Additionally, the medium-baseline reactor oscillation experiments [81], JUNO [82] and RENO-50 [83] aim to determine the hierarchy by performing a very precise, high statistics measurement of the neutrino energy spectrum. The CP phase can be measured (in principle) by accelerator experiments like T2K [84, 85], NOvA [86, 85, 87, 88] T2HK [72], and DUNE [75]. These experiments, if they run in both the neutrino and the antineutrino mode, would additionally be sensitive to the octant of [89, 90], and so would the large-mass atmospheric experiments like ICAL [14] and Hyper-K [91].

The ICAL detector at the INO cavern will provide an excellent opportunity to study the atmospheric neutrinos and antineutrinos separately with high detection efficiency and good enough energy and angular resolutions in the multi-GeV range in the presence of the Earth’s matter effect. There is no doubt that the rich data set which would be available from the proposed ICAL atmospheric neutrino experiment will be extremely useful to validate the three flavor picture of the neutrino oscillation taking into account the Earth’s large matter effect in the multi-GeV range. The first aim of the ICAL detector would be to observe the oscillation pattern over at least one full period, in order to make a precise measurement of the atmospheric oscillation parameters. The ICAL detector performs quite well in a wide range of and can confirm the evidence of the sinusoidal flavor transition probability of neutrino oscillation already observed by the Super-Kamiokande detector by observing the dips and peaks in the event rate versus [92], as well as by the IceCube DeepCore [71]. In the case of Super-Kamiokande, the sub-GeV events have played an important role to perform this analysis, while for IceCube the very high energy events ( GeV) have contributed significantly. The ICAL detector is sensitive mainly to the energy range 1–10 GeV, which fills the gap between the other two large Cherenkov detetors. In its initial phase, the ICAL experiment will also provide an independent measurement of by exploring the Earth’s matter effect using the atmospheric neutrinos. This will certainly complement the ongoing efforts of the reactor and the accelerator experiments to learn about the smallest lepton mixing angle .

The relevant neutrino oscillation probabilities that the ICAL will be sensitive to are , , , and , especially the former two. These probabilities have a rich structure for neutrinos and antineutrinos at GeV energies, traveling through the Earth for a distance of several thousands of km (for a detailed description, see Appendix B). The matter effects on these neutrinos and antineutrinos lead to significant differences between these oscillation probabilities, which may be probed by a detector like ICAL that can distinguish neutrinos from antineutrinos. This feature of ICAL would be instrumental in its ability to distinguish between the two possible mass hierarchies.

Detailed simulations of the ICAL detector performance, as discussed in the following chapters, show that ICAL would be an excellent tracker for muons. The energy and direction of a muon would also be reconstructed rather accurately, with the muon direction resolution of better than a degree at high energies. Furthermore, the capability of ICAL to study the properties of the final state hadrons in multi-GeV neutrino interactions would be one of its unique features. This would allow the reconstruction of the neutrino energy in every event, albeit with large error. (Note that the extraction of hadronic information at multi-GeV energies in currently running or upcoming water or ice based atmospheric neutrino detectors is quite challenging; the efficiency of reconstruction of multi-ring events is rather small in such detectors.) As a result, the ICAL would have a significant stand-alone sensitivity to the mass hierarchy, which, when combined with data from experiments like NOvA and T2K, would significantly enhance the overall sensitivity to this important quantity.

Although the ICAL would not be sensitive to the value of , this very feature would make it an important supporting experiment for others that are sensitive to , in a unique manner. Note that in experiments where event rates are sensitive simultaneously to both matter and CP phase effects, disentangling one from the other restricts the sensitivities to individual and unambiguous measurements of each of the three quantities (i), (ii) and (iii) mentioned above. The virtue of ICAL here would lie in its ability to offer a data-set that is free of entanglements between matter enhancements and dynamical CP violating effects due to a non-zero . Thus, when used in combination with other experiments, the ICAL measurements will facilitate the lifting of degeneracies which may be present otherwise. In particular, the ICAL data, when combined with that from NOA  and T2K, would make a significant difference to their discovery potential of CP violation [19].

1.2.3 Addressing new physics issues with ICAL

The full role of an iron calorimeter in the global scenario of neutrino physics is rich and complex. In addition to what is described here, it can add to our knowledge on very high energy muons [93, 94] on hitherto undiscovered long range forces [95], on CPT violation [15, 96] and on non-standard interactions [97], among other issues. The future of neutrino physics is, in our opinion, crucially dependent on the synergistic combination of experiments with differing capabilities and strengths. A large iron calorimeter brings in unique muon charge identification capabilities and an event sample independent of the CP phase. Both these aspects will play an important role in our concerted global effort to understand the mysteries of neutrino physics and consequently understand physics beyond the Standard Model.

Though the ICAL has been designed mainly with neutrino physics in view, it is expected that many non-neutrino issues may find relevance with this detector. For example, a few decades ago, both in the cosmic ray neutrino experiments [43, 44] and later in the proton decay experiment [45] at Kolar Gold Fields (KGF) in south India, some unusual events were seen. These so-called Kolar events were multi-track events with some unusual features which could not be explained away by any known processes of muons or neutrinos. Recently it has been speculated that such events may be caused by the decay of unstable cold dark matter particles with mass in the range of 10 GeV with a life time approximately equal to the age of the universe [98]. Such an interpretation may be easily tested with ICAL at INO, even without further modifications [99]. Signals of dark matter annihilation inside the Sun can also be detected at ICAL. The possible observation of GUT monopoles is another such issue that can be addressed at ICAL with its current setup.

Chapter 2 Atmospheric Neutrino Fluxes

Everything is in a state of flux, Even the status quo.

- Robert Byrne

2.1 Introduction to Atmospheric Neutrinos

Atmospheric neutrinos are produced in the cosmic ray interactions with the nuclei of air molecules in the atmosphere. The first report of cosmic ray induced atmospheric neutrinos was from the deep underground laboratories at Kolar Gold Field (KGF) in India by TIFR, Osaka University and Durham University [21], and immediately afterwards by Reines et al. [22] in an experiment conducted in South African mines in 1965. Atmospheric neutrinos have been studied since then in several other underground laboratories, and important discoveries such as the evidence for neutrino oscillations [27], have been made. We will briefly review the atmospheric neutrinos in this section.

Primary cosmic rays are high energy particles impinging on the Earth from galactic and extragalactic sources. Their origins are still clouded in mystery. In the GeV energy range, the cosmic ray particles are made up of mainly protons and about 9% helium nuclei, with a small fraction of heavy nuclei. Although the energy spectrum of cosmic rays extends to very high energies, beyond even GeV, it falls rapidly as energy increases. When cosmic rays enter the atmosphere, interactions with the nuclei in air molecules produce secondary particles. These secondary particles are mainly pions with a small admixture of kaons. These mesons decay mainly to muons and their associated neutrinos following the decay chain

(2.1)

Kaons also decay in a similar manner producing the two neutrino flavours, but their contribution to the atmospheric neutrino flux is small compared to the pions for neutrinos of a few GeV. We call the neutrinos produced in this manner as atmospheric neutrinos. It may be noted that only the and neutrinos, along with their antiparticles, are produced in the atmosphere. The flux of requires the production of mesons with heavy quarks, as a result their flux is extremely small and we do not consider these neutrinos here. A schematic illustration of this cascading neutrino production is shown in Fig. 2.1.

Figure 2.1: A schematic illustration of the production of neutrinos due to cosmic rays.

From Eq. 2.1 it is clear that the ratio

(2.2)

where denotes the flux of neutrinos. The ratio is only approximate, since at high energies muon may not decay before reaching the surface of the Earth. It, however, remains greater than 2, which may be observed from Fig. 2.2. The figure displays the direction-integrated neutrino fluxes in various models, as well as the ratios of fluxes of different kinds of neutrinos.

Figure 2.2: The direction integrated neutrino fluxes in various models are shown on the left panel. The ratios of fluxes of different neutrino species as functions of energy are shown on the right panel. Figures are reproduced from Honda et al. [5], based on the analysis of cosmic ray neutrino fluxes from [100], [101] and [102].

An important property of the atmospheric neutrino flux is that it is symmetric about a given direction on the surface of the Earth, that is

(2.3)

where is the zenith angle. This result is robust above 3 GeV, though at lower energies the geomagnetic effects result in deviations from this equality. Therefore, at higher energies, any asymmetry in the fluxes of the upward-going and downward-going neutrinos can be attributed to the flavour changes during propagation. Even at lower energies, large deviations from the above equality are not expected, except from neutrino oscillations. This up-down asymmetry is thus the basis of atmospheric neutrino analysis, and it was effectively used by the SuperKamiokande collaboration to establish the first confirmed signal of neutrino oscillations [27]. Of course detailed analyses need the calculations of atmospheric neutrino fluxes as functions of energies, zenith angles as well as azimuthal angles.

The atmospheric neutrinos not only provide neutrinos in two distinct flavours, but also over a whole range of energies from hundreds of MeV to TeV and beyond. Yet another advantage over the conventional accelerator neutrino beams is the fact that the atmospheric neutrinos traverse widely different distances in different directions: from 10 km on the way downwards to more than 12000 km on the way upwards through the centre of the earth. They also traverse matter densities varying from very small (essentially air) to almost 13 g/cc when passing through the earth’s core.

These facts make the analysis of atmospheric neutrinos not only interesting but also unique. The ICAL is expected to exploit the advantages of the freely available atmospheric neutrino flux, not only to explore the neutrino oscillation parameters but also determine the hierarchy of neutrino masses, and perhaps probe new physics.

2.2 Calculations of atmospheric neutrino fluxes

The neutrino oscillation studies with atmospheric neutrinos can be put on a firm foundation provided the atmospheric neutrino flux estimates and their interaction cross sections are known as precisely as possible. The main steps in the determination of atmospheric neutrino fluxes are:

  • The energy spectrum of primary cosmic rays: The flux of primary cosmic rays decreases approximately as in the 10 GeV to TeV region. Consequently the flux of neutrinos decreases rapidly in the high energy region. The flux of cosmic rays outside the atmosphere is isotropic and constant in time. These are well measured experimentally up to tens of GeV. The primary spectrum of cosmic ray protons can be fitted to a form

    (2.4)

    where  [103].

  • The energy spectrum of secondary muons: The interactions of primary cosmic rays with the air nuclei produce in pions and kaons, which in turn yield muons. An important input needed for this calculation is the hadronic cross sections. These are well measured in accelerator experiments from low energy up to hundreds of GeV. Beyond the range of accelerator energies these cross sections are model-dependent. Hence the composition of the secondary cosmic rays and their energy spectrum is well known up to TeV energies. The muons are produced by the decay of these mesons.

  • The energy spectrum of neutrinos: This needs modelling the altitude dependence of interactions in the atmosphere, the geomagnetic effect on the flux of cosmic rays and secondaries, and the longitudinal dependence of extensive air showers.

Uncertainties in each one of the above steps limit the precision in the determination of neutrino fluxes on the surface. Typically these introduce an uncertainty of the order of 15-20% in the overall normalization.

Figure 2.3: Zenith angle dependence of the neutrino flux averaged over azimuthal angles are shown for three different energies. This has been calculated for the Kamioka location. Figures are taken from Honda et al. [104]. Note that the flux scales are different for different energies.

Some typical features of the zenith angle distributions of atmospheric neutrino fluxes may be seen in Fig. 2.3, which show the fluxes (averaged over azimuthal angles) for three neutrino energies, calculated for Kamioka, the location of the Superkamiokande detector. The figure shows that the flux is typically maximum near , i.e. for horizontal neutrinos, where the muons have had the maximum proper time to decay. Also, the ratio of muon to electron neutrino flux is observed to increase at higher energies and at more vertical (down-going or up-going) neutrinos, where muons have less proper time to decay, so the second reaction in Eq. 2.1 is less efficient. It can also be seen that at GeV, the fluxes are essentially symmetric in zenith angle. However at lower energies, there is some asymmetry, arising mainly from the bending of muons in the geomagnetic field.

Recently Honda et al. [105] have calculated the atmospheric neutrino spectrum at the INO location (Theni). It is observed that the total flux at INO is slightly smaller than that at Kamioka at low energies ( GeV), but the difference becomes small with the increase in neutrino energy. Also, at low energies ( GeV), the up-down asymmetries are larger at the INO site. These asymmetries decrease with the increase in neutrino energy. The detailed characteristics of these fluxes have been given in Appendix A. The analyses presented in this report use fluxes at the Kamioka location. We plan to use the recently compiled fluxes [104, 105] for the Theni site in our future analysis.

Chapter 3 The ICAL Simulation Framework

A good simulation gives us

a sense of mastery over experience.

Heinz R. Pagels

The broad simulation framework for the ICAL, starting with event generation, is indicated schematically in Table 3.1. The events in the detector are generated using the NUANCE Monte Carlo generator [4]. This uses the atmospheric neutrino fluxes as described in Chapter 2 along with various possible neutrino-nucleus interaction cross-sections to generate the vertex and the energy-momentum of all final states in each event; these are then propagated through the virtual ICAL detector using the GEANT4 simulation tool. The GEANT4 simulates the propagation of particles through the detector, including the effects of the iron, the RPCs, and the magnetic field. The information in the events is then digitised in the form of coordinates of the hits in the RPCs and the timing corresponding to each of these “hits”. This is the information available for the event reconstruction algorithms, which attempt, from the hit pattern, to separate the muons tracks from the showers generated by the hadrons, and reconstruct the energies and directions of these particles. The process is described in detail below.

NUANCE Neutrino Event Generation  . Generates particles that result from a random interaction of a neutrino with matter using theoretical models for both neutrino fluxes and cross-sections. Output: (i) Reaction Channel (ii) Vertex and time information (iii) Energy and momentum of all final state particles
GEANT Event Simulation through simulated ICAL Simulates propagation of particles through the ICAL detector with RPCs and magnetic field. Output: (i) of the particles as they propagate through detector (ii) Energy deposited (iii) Momentum information
DIGITISATION Event Digitisation of final states on including noise and detector efficieny Add detector efficiency and noise to the hits. Output: (i) Digitised output of the previous stage
ANALYSIS Event Reconstruction of , (total hadrons) Fit the muon tracks using Kalman filter techniques to reconstruct muon energy and momentum; use hits in hadron shower to reconstruct hadron information. Output: (i) Energy and momentum of muons and hadrons, for use in physics analyses.
Table 3.1: The simulation frame-work as implemented in the ICAL simulation package.

3.1 Neutrino interactions and event generation

Neutrino and antineutrino interactions in the ICAL detector are modelled using NUANCE neutrino generator version 3.5 [4]. Some preliminary studies and comparisons have also been initiated using the GENIE neutrino generator [106], but are not a part of this Report. The interactions modelled in NUANCE include (i) quasi-elastic scattering (QE) for both charged and neutral current neutrino interactions with nucleons, which dominate below neutrino energies of 1 GeV, (ii) resonant processes (RES) with baryon resonance production mainly from neutrinos with energy between 1 and 2 GeV, (iii) deep inelastic scattering (DIS) processes with considerable momentum (squared) transfer from the neutrino to the target, with the nuclei breaking up into hadrons, which is the dominant contribution in the multi-GeV region, (iv) coherent nuclear processes on nuclei, and (v) neutrino-electron elastic scattering and inverse muon decay. These are the main neutrino interaction processes of relevance for our simulation frame work, with the contribution of the last two being the least in the few GeV energy region of interest. A simple ICAL geometry has been described within NUANCE, including mainly the iron and glass components of the detector, as most of the interactions will occur in these two media. NUANCE identifies these bound nucleons (with known Fermi energies) differently from free nucleons and also applies final state nuclear corrections.

The NUANCE generator calculates event rates by integrating different cross sections weighted by the fluxes for all charged current (CC) and neutral current (NC) channels at each neutrino energy and angle. Some typical total cross sections for different CC processes are illustrated in Fig. 3.1. Based on the interaction channel, there can be 10-40 uncertainty in cross sections in the intermediate energy ranges [107].

Figure 3.1: Muon neutrino (left) and muon antineutrino (right) total charged current cross sections in (cm/GeV), obtained from NUANCE, are shown (smooth lines) as a function of incident neutrino energy, , in comparison with the existing measurements of these cross sections along with their errors [108]. Note that the -axis scale of the two panels is different.

As mentioned in Chapter 2, for the present we have used the Honda fluxes [5] generated at the location of Kamioka. This will be changed soon to that at the actual location of INO. We choose to generate only unoscillated neutrino events using NUANCE for the simulations, even though there is a provision for generating oscillated events in it. The oscillations are applied externally, separately in each analysis.

3.2 Simulation of the ICAL Detector

We now describe the ICAL detector geometry within the GEANT4 [3] simulation framework. This includes the geometry itself, and the magnetic field map and the RPC characteristics that are inputs to the simulation.

3.2.1 The Detector Geometry

The simulations have been performed for the 50 kt ICAL detector, which has a modular structure with the full detector consisting of three modules, each of size 16 m (length) 16 m (width) 14.5 m (height), with a gap of 20 cm between the modules. The ICAL coordinates are defined as follows. The direction along which the modules are placed is labelled as the -direction with the remaining horizontal transverse direction being labelled . At present, is also considered to coincide with South, since the final orientation of the INO cavern is not yet decided. The -axis points vertically upwards so that the polar angle equals the zenith angle while the zero of the azimuthal angle points South. The origin is taken to be the centre of the second module. Each module comprises 151 horizontal layers of 5.6 cm thick iron plates. The area of each module is 16 m 16 m, while the area of each iron plate is 2 m 4 m. There is a vertical gap of 4 cm between two layers. The iron sheets are supported every 2 m in both the and directions, by steel support structures. The basic RPC units have dimensions of 1.84 m 1.84 m 2.5 cm, and are placed in a grid format within the air gaps, with a 16 cm horizontal gap between them in both and directions to accommodate the support structures.

Vertical slots at m (where is the central value of each module) extending up to m and cutting through all layers are provided to accommodate the four copper coils that wind around the iron plates, providing a magnetic field in the - plane, as shown in Fig. 3.2. The detector excluding the coils weighs about 52 kt, with 98% of this weight coming from iron where the neutrino interactions are dominantly expected to occur, and less than 2% from the glass of the RPCs. In the central region of each module, typical values of the field strength are about 1.5 T in the -direction, as obtained from simulations using MAGNET6.26 software [109].

3.2.2 The Magnetic Field

Fig. 3.2 depicts the magnetic field lines in the central iron plate near the centre of the central module. The arrows denote the direction of magnetic field lines while the length of the arrows (and the shading) indicates the magnitude of the field. The maximum magnitude of the magneic field is about 1.5 T. Notice that the field direction reverses on the two sides of the coil slots (beyond 4 m) in the -direction. In between the coil slots (an 8 m 8 m square area in the - plane) the field is maximum and nearly uniform in both magnitude (to about 10%) and direction; we refer to this as the central region. Near the edges in the direction (outside the coil slots) the field is also fairly uniform, but in the opposite direction; this is called the side region. Near the edges in the direction, i.e., in the regions  m  m, both the direction and magnitude of the magnetic field vary considerably; this region is labelled as the peripheral region.

In our simulations, the field has been assumed to be uniform over the entire thickness of an iron plate at every position, and has been generated in the centre of the iron plate, viz., at . In the 4 cm air gap between the iron plates, the field is taken to be zero since it falls off to several hundred gauss in these regions, compared to more than 1 T inside the iron plates. The magnetic field is also taken to be zero in the (non-magnetic) steel support structures. These support structures, along with the coil slot, form the bulk of the dead spaces of the detector.

Figure 3.2: Magnetic field map in the central plate of the central module (), as generated by the MAGNET6 software. The length and direction of the arrows indicate the magnitude and direction of the field; the magnitude (in T) is also shown according to the colour-coding indicated on the right.

The side and peripheral regions are beset by edge effects as well as by non-uniform and lower magnetic field. We confine ourselves, in the present study, to tracks generated only in the central region ( m m and m m), although the particle may subsequently travel outside this region or even exit the detector.

3.2.3 The Resistive Plate Chambers

In order to appreciate the hit pattern in the simulated detector it is necessary to describe the active detector elements, the RPCs. These are glass chambers made by sealing two 3 mm thick glass sheets with a high DC voltage across them, with a uniform gap of 2 mm using plastic edges and spacers through which a mixture of R134A ( 95%), isobutane, and trace amounts of SF gas continually flows. In brief, the working principle of an RPC is the ionisation of the gaseous medium when a charged particle passes through it. The combination of gases keeps the signal localised and the location is used to determine the trajectory of the charged particle in the detector. For more details, see Ref. [110].

A 150 micron thick copper sheet is the component most relevant to the simulation and track reconstruction as it inductively picks up the signal when a charged particle traverses the chamber. This copper sheet is pasted on the inside of a 5 mm thick foam (used for structural strength and electrical insulation) placed both above and below the glass chamber. It is pasted on the side of the foam facing the glass and is insulated from the glass by a few sheets of mylar. This layer is scored through with grooves to form strips of width 1.96 cm in such a way that the strips above and below are transverse to each other, that is, in the and directions111Note that the strip width in the current ICAL design is 3 cm; however this is subject to change.. These pick-up strips thus provide the and location of the charged particle as it traverses the RPC while the RPC layer number provides the information. A timing resolution of about 1.0 ns is assumed as also an efficiency of 95%, consistent with the observations of RPCs that have been built as a part of the R&D for ICAL [110].

3.3 Event Simulation and Digitisation

Muons and hadrons, generated in neutrino interactions with the detector material, pass through dense detector material and an inhomogeneous magnetic field. Simulation of such particles through the detector geometry is performed by a package based on the GEANT4 [3] toolkit. Here the ICAL geometry is written to a machine readable GDML file — which includes the RPC detector components, the support structure, and the gas composition as described above — that can be read off by other associated packages, like the event reconstruction package. The pickup strips are considered as a continuous material for GEANT4 simulations, however for signal digitization the strips are considered independently.

When a charged particle, for example, a muon, passes through an RPC, it gives a signal which is assigned or values from the respective pick-up strip information, a -value from the layer information, and a time stamp . The minimum energy deposited in the RPC gap which will produce an electron-ion pair, and hence give a hit, is taken to be 30 eV, with an average efficiency of 95%. The global coordinates of the signals are then translated through digitisation into information of the strip and the strip at the plane. These digitized signals along with the time stamp form what are called “hits” in the detector as this is precisely the information that would be available when the actual ICAL detector begins to take data.

The spatial resolution in the horizontal plane is of the order of cm (due to the strip width) while that in the direction is of the order of mm (due to the gas gap between the glass plates in the RPCs). The effect of cross-talk, i.e., the probability of either or both adjacent strips giving signals in the detector, is also incorporated, using the results of the on-going studies of RPCs [110]. Finally, since the and strip information are independent, all possible pairs of nearby and hits in a plane are combined to form a cluster.

A typical neutrino CC interaction giving rise to an event with a muon track and associated hadron shower is shown in Fig. 3.3. It can be seen that the muon track is clean with typically one or two hits per layer, whereas the hadron hits form a diffused shower.

Figure 3.3: Sample track of a neutrino event with a muon track and hadron shower in the ICAL detector, where indicates the central layer of the detector.

3.4 Event Reconstruction

The reconstruction of individual hadrons in the hadronic showers is not possible since the response of the detector to different hadrons is rather similar. Only an averaged information on the energy and direction of the hadrons is in principle possible; furthermore, hadrons, due to the different nature of their interactions, propagate relatively short distances in the detector. The response of ICAL to hadrons as determined by simulations studies is described in Sec. 4.2.

Muons, on the other hand, being minimum ionizing particles, leave long clean tracks, and hence the ICAL detector is most sensitive to them. The muon momentum can be determined from the curvature of its trajectory as it propagates in the magnetized detector, and also by measuring its path length. The nanosecond time resolution of the RPCs also allows the distinction between up-going and down-going muons. The muon momentum reconstruction is achieved by using a track finder package, followed by a track fitting algorithm that reconstructs both the momentum and charge of the muon, using the information on the local magnetic field.

The track finder uses clusters, i.e., the combinations of all possible pairs of nearby and hits in a -plane, as its basic elements. A set of clusters generated in three successive layers is called a tracklet. The track finder algorithm uses a simple curve fitting algorithm to find possible tracklets by finding clusters in three adjacent planes. It includes the possibility of no hit (due to inefficiency) in a given plane, in which case the next adjoining planes are considered. Typically, charged current muon neutrino interactions in ICAL have a single long track due to the muons and a shower from the hadrons near the vertex. Since typically muons leave only about one or two hits per layer they traverse ( on average) as opposed to hadrons that leave several hits per layer, the hadron showers are separated by using criteria on the average number of hits per layer in a given event.

Ends of overlapping tracklets are matched to form longer tracks, and the longest possible track is found [111] by iterations of this process. The track finder package thus forms muon tracks as an array of three dimensional clusters. In the rare cases when there are two or more tracks, the longest track is identified as the muon track. The direction (up/down) of the track is calculated from the timing information which is averaged over the and timing values in a plane. For muon tracks which have at least 5 hits in the event, the clusters in a layer are averaged to yield a single hit per layer with , and timing information; the coordinates of the hits in the track are sent to the track fitter for further analysis. (This translates to a minimum momentum of about 0.4 GeV/c for a nearly vertical muon, below which no track is fitted.)

The track fitter, a Kalman-filter based algorithm, is used to fit the tracks based on the bending of the tracks in the magnetic field. Every track is identified by a starting vector which contains the position of the earliest hit as recorded by the finder, with the charge-weighted inverse momentum taken to be zero. Since the tracks are virtually straight in the starting section, the initial track direction (the slopes ) is calculated from the first two layers. This initial state vector is then extrapolated to the next layer using a standard Kalman-filter based algorithm, which calculates the Kalman gain matrix using the information on the local magnetic field and the geometry, the composition of the matter through which the particle propagates, and the observed cluster position in that later.

In the existing code, the state prediction is based upon the Kalman filter algorithm and the corresponding error propagation is performed by a propagator matrix [111]. The state extrapolation takes into account process noise due to multiple scattering as described in [112] and energy loss in matter, mostly iron, according to the Bethe formula [113]. A new improved set of formulae for the propagation of the state and errors [114], optimised for atmospheric neutrinos with large energy and range, have also been developed, and are being used in the Kalman filter. The extrapolated point is compared with the actual location of a hit in that layer, if any, and the process is iterated.

The process of iteration also obtains the best fit to the track. The track is then extrapolated backwards to another half-layer of iron (since the interaction is most likely to have taken place in the iron) to determine the vertex of the interaction and the best fit value of the momentum at the vertex is returned as the reconstructed momentum (both in magnitude and direction). Only fits for which the quality of fit is better than are used in the analysis.

While determines the magnitude of the momentum at the vertex, the direction is reconstructed using and , which yield the zenith and the azimuthal angles, i.e., and . The results on the quality of reconstruction are presented in the next chapter.

Chapter 4 ICAL Response to Muons and Hadrons

“Muons are clean because they leave a trail,

Hadrons are dirty because they shower.”

– M. V. N. Murthy

In this chapter, we discuss the simulations response of ICAL to the final state particles produced in neutrino-nucleus interactions as discussed in the previous chapter. Being minimum ionising particles, muons typically register clean long tracks with just about one hit per RPC layer in the detector while hadrons produce a shower with multiple hits per layer, due to the very different nature of their interactions. Multiple scattering further affects the quality of the track.

First we discuss the detector response with respect to single particles (muons or hadrons) with fixed energies. In order to simulate the neutrino events fully, we then use the particles generated in atmospheric neutrino events using the NUANCE [4] event generator, for calibration. For the case of single muons, we study the response of the detector to the energy/momentum, direction and charge of muons propagated with fixed energy/momentum and direction () from the central region of the detector (described in Chapter 3). Next we propagate the hadrons, mainly single pions, also with fixed energy and direction, through the central region of the detector and determine the energy response of the detector with respect to hadrons. Amongst the particles generated via NUANCE, the muons tracks can be separated while the hits from all the hadrons in the event are treated as just one shower.

4.1 Response of ICAL to Muons

In this Section, we present the results of the reconstruction of the charge, energy and direction of muons [8]. For this study, we confine ourselves to tracks generated only in the central region of the ICAL detector, i.e. , and , with the origin taken to be the center of the central detector module. The particle may subsequently travel outside this region or even exit the detector: both fully contained and partially contained events are analyzed together. At low energies, the tracks are fully contained while particles start to leave the detector region for GeV/c, depending on the location of the vertex and the direction of the paticles.

The and are analysed separately for fixed values of the muon momentum and direction: while is kept fixed for a set of typically 10000 muons, the azimuthal angle is smeared over all possible values from . The distribution of reconstructed muon momentum for the particular choice is shown in the left panel of Fig. 4.1. Since the results are almost identical, as can be seen from the figure, only the results for are presented in the further analysis.

The mean and the rms width are determined by fitting the reconstructed momentum distribution; the momentum resolution is defined as . Apart from the intrinsic uncertainties due to particle interactions and multiple scattering effects, the distribution—especially its width—is also sensitive to the presence of detector support structures, gaps for magnetic field coils, etc., that have been described in the previous chapter.

Figure 4.1: The left panel shows the reconstructed momentum distributions for smeared over the central volume of the detector for and particles [8]. The right panel shows the same distribution, but for , for , fitted with the Landau-convoluted-with-Gaussian distribution.

In addition, the reconstructed momentum distribution of low energy muons has a clear asymmetric tail, as can be seen in the right panel of Fig. 4.1 for muons with . It can be seen that the distribution at low energies is significantly broader, and there is also a shift in the mean. It is therefore fitted with a convolution of Landau and Gaussian distributions. For muons with GeV/c, the distributions are fitted with purely Gaussian distributions. In the case of Landau-Gaussian fits, the width is defined as , in order to make a consistent and meaningful comparison with the Gaussian fits at higher energies, where the square root of the variance, or the rms width, equals FWHM/2.35. Before we present results on the muon resolution, we first discuss the impact on the resolutions of the muon angle and location within the detector.

4.1.1 Momentum resolution in different azimuthal regions

The number of hits in the detector by a muon with a fixed energy will clearly depend on the zenith angle, since muons traversing at different angles travel different distances through each iron plate. As a result, the momentum resolution would depend on the zenith angle. However, it also has a significant dependence on the azimuthal angle for two different reasons. One is that the magnetic field explicitly breaks the local azimuthal symmetry of the detector geometry. There is an additional effect due to the coil gaps that are located at m where is the centre of each module. The second reason is that the support structures are also not azimuthally symmetric; moreover, the length of track “lost” within these dead spaces is also a function of the location from where the muon was propagated and the zenith angle at which it traverses these spaces. The cumulative dependence on the azimuthal angle is a complex consequence of all these dependences and impacts low momentum and large zenith angle muons more than higher energy, small angle ones.

For instance, a muon initially directed along the -axis experiences less bending since the momentum component in the plane of the iron plates (henceforth referred to as in-plane momentum) is parallel to the magnetic field. Furthermore, upward-going muons that are in the negative (positive) direction experience a force in the positive (negative) direction (the opposite is be true for ) and so muons injected with traverse more layers than those with the same energy and zenith angle but with and hence are better reconstructed. This is illustrated in the schematic in Fig. 4.2 which shows two muons () injected at the origin with the same momentum magnitude and zenith angle, one with positive momentum component in the direction, and the other with negative momentum component. The muon with (initially directed in the positive direction) bends differently than the one with (along negative direction) and hence they traverse different number of layers, while having roughly the same path length. Hence, muons with different elicit different detector response. Because of these effects, the momentum resolution is best studied in different azimuthal angle bins. We separate our muon sample into four regions/bins: bin I with , bin II with , bin III with , and bin IV with . The resolutions of in the above -regions, for six values of the zenith angle, are shown in Fig. 4.3.

Figure 4.2: Schematic showing muon tracks (for ) in the - plane for the same values of but with and (momentum component in the direction positive and negative, respectively). The different bending causes the muon to traverse different number of layers in the two cases.
Figure 4.3: Muon resolution as a function of input momentum and , in bins of as described in the text [8].

It may be noticed that, at lower energies the resolution improves as increases, as expected, but at higher energies the behaviour is rather complicated. At higher energies, the muons injected at larger zenith angles in regions I and IV, have better resolutions than their counterparts at more vertical angles (for instance, versus ) because larger portions of the tracks, being more slanted, are still contained within the detector. In contrast in regions II and III, muons with larger zenith angles have worse resolution than those at smaller zenith angles because the former exits the detector from the direction and are partially contained. In general, at angles larger than about (), the resolution is relatively poor since there are several times fewer hits than at more vertical angles.

Finally, note that the simulations studies of the physics reach of ICAL discussed in the later chapters use the azimuth-averaged resolutions for muons. This is because the main focus of ICAL is the study of neutrino oscillations using atmospheric neutrino fluxes that are azimuthally symmetric for GeV. While studying the neutrinos from fixed astrophysical sources, for example, the azimuthal dependence of the detector needs to be taken into account.

In the rest of this section, we present simulations studies of the azimuth-averaged response of ICAL to the muon direction and its charge, since these are not very sensitive to the azimuth. We also present the overall reconstruction efficiency for muons; this determines the overall reconstruction efficiency of the entire neutrino event as there may or may not be an associated hadron shower in the event.

4.1.2 Zenith Angle Resolution

The events that are successfully reconstructed for their momenta are analysed for their zenith angle resolution. The reconstructed zenith angle distribution for muons with GeV/c, at zenith angles and , are shown in Fig. 4.4, where the time resolution of the RPCs is taken to be ns. (Muons with are up-going). It can be seen that a few events are reconstructed in the opposite (downward) direction i.e., with zenith angle (). For muons with GeV/c at large (small) angles with , this fraction is about 4.3 (1.5) %. As energy increases, the fraction of events reconstructed in the wrong direction drastically comes down. For example, this fraction reduces to 0.3% for muons with GeV/c at . Comparison of the goodness of fits to a track, assuming it to be in upward and downward direction, reduces this uncertainty further. The analysis incorporating this technique is in progress.

The muons that contribute to mass hierarchy identification have energies greater than about 4 GeV and the time interval between the first and the last hit of such muons is more than 5 ns, so that the up-going vs. down-going muons would be easily identified. The time resolution of the detector therefore is not expected to affect the mass hierarchy determination.

(a) ;
(b) ;
Figure 4.4: Reconstructed theta distribution for  GeV/c at two values of [8]. The time resolution of the RPCs has been taken to be 1 ns. Note that the fraction of muons reconstructed in the wrong hemispehere decreases sharply with energy.

The events distribution as a function of the reconstructed zenith angle is shown in the left panel of Fig. 4.5 for a sample . It is seen that the distribution is very narrow, indicating a good angular resolution for muons. The right panel of Fig. 4.5 shows the resolution as a function of input momentum for different zenith angles.

Figure 4.5: The left panel shows the reconstructed distribution for input . The right panel shows the resolution as a function of input momentum.

As noted earlier, due to multiple scattering and the smaller number of layers with hits, the momentum resolution is worse at lower energies. This is also true for the zenith angle, whose resolution improves with energy. For a given energy, the resolution is worse for larger zenith angles since again the number of layers with hits decreases. Even so, it is seen that the angular resolution for (i.e. ) is better than for muon momenta greater than 4 GeV/c.

4.1.3 Reconstruction efficiency

The reconstruction efficiency for muons is defined as the ratio of the number of reconstructed events (irrespective of charge) to the total number of events, (typically 10000). We have

(4.1)

The left panel of Fig. 4.6 shows the muon reconstruction efficiency as a function of input momentum for different bins.

Figure 4.6: The left (right) panel shows the reconstruction efficiency (the relative charge identification efficiency) as a function of the input momentum for different values [8].

When the input momentum increases, the reconstruction efficiency also increases for all angles, since the number of hits increases as the particle crosses more number of layers. At larger angles, the reconstruction efficiency for small energies is smaller compared to vertical angles since the number of hits for reconstructing tracks is less. But as the input energy increases, above almost 4 GeV since the particle crosses more number of layers, the efficiency of reconstructing momentum also increases and becomes comparable with vertical angles. At higher energies the reconstruction efficiency becomes almost constant. The drop in efficiency at high energies for vertical muons is due to the track being partially contained, their smaller bending in the magnetic field, as well as the impact of the detector support structure. it is expected that this may improve as the track recognition algorithms are refined and better tuned.

The fraction of muon-less charged current events / neutral current events that get misidentified as charged current muon events is , as long as the energy of the reconstructed muon is GeV. This fraction may be further reduced with a proper choice of cuts, and for high energy muons that are relevant for mass hierarchy determination, this is expected to be negligible. Work in this direction is in progress.

4.1.4 Relative charge identification efficiency

The charge of the particle is determined from the direction of curvature of the track in the magnetic field. Relative charge identification efficiency is defined as the ratio of number of events with correct charge identification, , to the total number of reconstructed events:

(4.2)

Figure 4.6 shows the relative charge identification efficiency as a function of input momentum for different bins. As seen earlier, there is a very small contribution to the set with the wrongly identified charge from the events where the track direction is wrongly identified (); such events will also reconstruct with the wrong charge as there is a one-to-one correspondence between the up-down identification and the muon charge.

When a low energy muon propagates in the detector it undergoes multiple scattering. So the number of layers with hits is small, and the reconstruction of charge goes wrong, which results in poor charge identification efficiency as can be see from Fig. 4.6. As the energy increases, the length of the track also increases, due to which the charge identification efficiency also improves. Beyond a few GeV/c, the charge identification efficiency becomes roughly constant, about 98%.

4.2 Response of ICAL to hadrons

An important feature of ICAL is its sensitivity to hadrons over a wide energy range. This allows the reconstruction of the energy of the incoming muon neutrino in a charged-current event by combining the energies of the muon and the hadrons. It also enables the detection of neutral-current events, charged-current DIS events generated by interactions, and charged-current events where the decays hadronically. The information contained in all these events adds crucially to our knowledge of neutrino oscillations. The charged-current events is a direct measure of the oscillation probabilities among the three active neutrino species. On the other hand, the neutral-current events are not affected by active neutrino oscillations, and hence help in flux normalization, as well as in the search for oscillations to sterile neutrinos. It is therefore important to characterize the response of the ICAL to hadrons.

The hadrons generated from the interactions of atmospheric neutrinos consist mainly of neutral and charged pions, which together account for about 85% of the events. The rest of the events consist of kaons and nucleons, including the recoil nucleons that cannot be distinguished from the remaining hadronic final state. The neutral pions decay immediately giving rise to two photons, while the charged pions propagate and develop into a cascade due to strong interactions. For the neutrino-nucleon interaction , the incident neutrino energy is given by

(4.3)

where is the energy of the initial nucleon which is taken to be at rest, neglecting its small Fermi momentum. The visible hadron energy depends on factors like the shower energy fluctuation, leakage of energy, and invisible energy loss mechanisms, which in turn affect the energy resolution of hadrons . We choose to quantify the hadron response of the detector in terms of the quantity [9]

(4.4)

As the first step in understanding the ICAL response to hadrons, single charged hadrons of fixed energies are generated via Monte Carlo and propagated through the detector to compare its response to them. The response to charged pions is then studied in more detail, and the pion energy is calibrated against the number of hits. Next, the multiple hadrons produced through atmospheric neutrino interactions are generated using NUANCE [4], and the quantity is used to calibrate the detector response. This should take care of the right combination of the contributions of different hadrons to the hits, on an average. It would of course be dominated by neutral and charged pions, and hence we expect it to be similar to the response to fixed-energy pions.

4.2.1 Energy response to fixed-energy hadrons

In an RPC, the X- and Y-strip information on a hit is independently obtained from the top and bottom pick-up strips, as described in Sec. 1.1.3 and then combined to give the coordinates of the hit. For a muon, the positions of the hits in a given layer can easily be identified since a muon usually leaves only one or two hits per layer. However a hadron shower consists of multiple hits per layer, and combining all possible X and Y strip hits leads to overcounting, resulting in what are termed as “ghost hits”. To avoid the ghost hit counts, the variables “x-hits” and “y-hits”—the number of hits in the X and Y strips of the RPC, respectively—can be used. We choose to perform the energy calibration with the variable “orig-hits”, which is the maximum of x-hits or y-hits.

Figure 4.7 shows the comparison of these three types of hit variables for of energy 3 GeV. Clearly, there is no significant difference among these variables, however orig-hits has been used as the unbiased parameter. It is also observed that the detector response to the positively and negatively charged pions is identical, so we shall not differentiate between them henceforth.

Figure 4.7: The comparison of the distributions of x-hits, y-hits and orig-hits for (left) and (right) of energy 3 GeV [9].

Fixed-energy single pion events in the energy range of 1 to 15 GeV were generated using the particle gun in GEANT4. The total number of events generated for each input energy value is 10000 in this section, unless specified otherwise. Each event is randomly generated to have vertices over a volume 2 m 2 m 2 m in the central region of the ICAL detector. As in the earlier section, the reference frame chosen has the origin at the centre of the detector, the -axis pointing vertically up, and the - plane along the horizontal plates, with the three modules lined up along the axis. The hadron direction is uniformly smeared over the zenith angle and azimuth of . This serves to smear out any angle-dependent bias in the energy resolution of the detector by virtue of its geometry which makes it the least (most) sensitive to particles propagating in the horizontal (vertical) direction.

Figure 4.8: The direction-averaged hit distributions at various energies for , , and protons propagated from vertices smeared over the chosen detector volume [9].

Figure 4.8 shows the hit distributions in the detector for pions, kaons, and protons at various energies in the range of 1 to 15 GeV. It is observed that the hit patterns are similar for all these hadrons, though the peak positions and spreads are somewhat dependent on the particle ID. Hence the detector cannot distinguish the specific hadron that has generated the shower. The large variation in the number of hits for the same incident particle energy is mainly a result of different strong interaction processes for different hadrons (for the interactions are electromagnetic since it decays immediately to a pair), and partly an effect of angular smearing.

4.2.1.1 The ratio

The NUANCE [4] simulation suggests that the fraction of the different types of hadrons produced in the detector is , with the remaining 3% contribution coming mainly from kaons [10]. While the response of the detector to and is almost identical as seen earlier, its response to the electromagnetic part of the hadron shower that originates from is different. This may be quantified in terms of the ratio, i.e. the ratio of the electron response to the charged-pion response. This ratio would help us characterise the effect of neutral hadrons on the energy resolution.

In order to study this ratio, we generated 100,000 electron events at fixed energies in the energy range 2–15 GeV, propagating in arbitrary directions (with smeared from and from ) from vertices within a volume of 2 m 2 m 2 m in the central region of the ICAL detector. The hit distributions averaged over all directions for and 14 GeV electrons is shown in Fig. 4.9. This may be compared with the hit distributions shown in Fig. 4.8. The response is almost the same as that for , with narrower high-energy tails than those for charged pions.

Figure 4.9: The left panel shows the hit distributions of fixed energy single electrons at 2, 5, 10 and 14 GeV, averaged over all directions. The right panels shows the variation of the ratio with the particle energies [10].

The ratio is obtained as

(4.5)

where is the arithmetic mean of the electron hit distribution and is the arithmetic mean of the hit distribution for , for a given fixed energy of the two particles. If = 1, then the detector is said to be compensating. The variation of the ratio with incident energy is shown in the right panel of Fig. 4.9.

It can be seen that the value of decreases with energy. However, it should be noted that there is no direct measurement of the energy deposited in ICAL. Here the energy of a shower is simply calibrated to the number of hits, and electrons which travel smaller distances in a high Z material like iron have lower number of hits compared to charged pions. At lower energies the electron as well as pion shower hits are concentrated around a small region. The mean of the electron hit distribution is roughly the same or slightly larger than that of the hit distribution. With the increase in energy, the charged pions travel more distance and hence give more hits (as they traverse more layers) since the hadronic interaction length is much more than the electromagnetic interaction length at higher energies and hence the ratio of hits in the two cases drops with energy.

In a neutrino interaction where all types of hadrons can be produced (although the dominant hadrons in the jet are pions), the response of ICAL to hadrons produced in the interaction depends on the relative fractions of charged and neutral pions. Using the relative fractions as mentioned above, the average response of hadrons obtained from the charged current muon neutrino interaction can be expressed as:

(4.6)

where is the electron response, the charged hadron response and is the neutral pion fraction in the sample.

The atmospheric neutrino events of interest in ICAL are dominated by low energy events with hadrons typically having energies GeV for which the average value of is . Using in Eq. (4.6), we get the average hadron response for NUANCE-generated events to be which is not very different from . For this reason, the analysis of response with multiple hadrons in NUANCE-generated events sample is not expected to be very different from that of the single pions sample. However we shall confirm this by first focusing on the detector response to fixed-energy charged pions in Sec. 4.2.2, and then moving on to a more general admixture of different hadrons in Sec. 4.2.3.

4.2.2 Analysis of the charged pion hit pattern

The hit distributions for charged pions, at sample values of GeV, are shown in Fig. 4.10. The distributions are asymmetric with long tails, with a mean of about two hits per GeV. In addition, at low energies several events yield zero hits in the detector.

Figure 4.10: The hit distributions at 3 GeV (left) and 8 GeV (right), for pions propagating in the detector, starting from randomized vertices over a volume of 2 m2 m2 m in the detector. The red curve denotes a fit to the Vavilov distribution [9].

A search for a good fitting function for the distribution was made, and it was found that the Vavilov distribution function gives a good fit for all energies, as is illustrated in Fig. 4.10. This distribution (see Appendix C) is described by the four parameters , , and , which are energy-dependent [9]. The Vavilov distribution reduces to a Gaussian distribution for , which happens for GeV. However at lower energies, it is necessary to use the full Vavilov distribution.

Figure 4.11: The mean hit distribution (left) and the energy resolution (right) for fixed-energy charged pion events, as a function of pion energy. The right panel also shows a fit to Eq. (4.10) [9].

The mean of the number of hits from the Vavilov fit at different energies is shown in the left panel of Fig. 4.11. It increases with increasing pion energy, and saturates at higher energies. It may be approximated by

(4.7)

where and are constants. This fit has to be interpreted with some care, since and are sensitive to the energy ranges of the fit. The value of is found to be 30 GeV when a fit to the energy range 1–15 GeV is performed. Since the energies of interest for atmospheric neutrinos are much less than , Eq. (4.7) may be used in its approximate linear form . A fit to this linear form is also shown in Fig. 4.11.

Since in the linear regime () one has

(4.8)

The energy resolution may be written as

(4.9)

where is the variance of the distribution. The notation will be used for energy resolution throughout, and Eq. (4.9) will be taken to be valid for the rest of the analysis.

The energy resolution of pions may be parameterized by

(4.10)

where and are constants. The energy resolutions for charged pions as functions of the pion energy are shown in the right panel of Fig. 4.11. The parameters and extracted by a fit to Eq. (4.10) over the pion energy range 1–15 GeV are also shown. The values of and depend on the iron plate thickness; this dependence has been studied in detail in Appendix D.

4.2.2.1 Dependence of the energy resolution on hadron direction

Since the number of layers traveresed by a particle would depend on the direction of the particle, it is expected that the energy calibration and energy resolution for hadrons will depend on the direction of the hadron. To check this dependence, we simulate pions of fixed energies in the detector, which travel in different directions. The directions are binned into 5 zenith angle bins, and the distributions of number of hits is recorded. The ratio of the rms width of the distribution and its mean is used as a measure of the energy resolution [10]. Figure 4.12 shows the zenith angle dependence of the hadron energy resolution.

Figure 4.12: The dependence of the pion energy resolution on the zenith angle [10].

Since there is only a mild dependence on the hadron direction, and the direction of hadron itself cannot be determined yet with a good confidence, we continue to use the direction-averaged results in the future analyses in this Report.

4.2.3 Response to hadrons produced by atmospheric neutrinos

Atmospheric neutrino interactions in the detector may contain no hadrons (for quasi-elastic scattering events), one hadron or multiple ones (in resonance scattering and DIS events). While the former events dominate for GeV, at higher energies the DIS events dominate. In this section we focus on the charged-current interactions in the detector that produce hadrons in addition to the charged muons.

We assume here that the CC events can be clearly separated from the NC as well as CC events, and that the muon and hadron shower may be identified separately. (In our procedure, we determine the number of hadron hits by taking away the true muon hits, as in the Monte-Carlo simulation, from the total hits in the event.) Preliminary studies show that this is a reasonable assumption for GeV. At lower energies where the number of hits is small, the misidentification of a muon hit as a hadron one, or vice versa, can significantly affect the hadron energy calibration. The analysis of this effect is in progress.

The atmospheric neutrino () and antineutrino () events in ICAL have been simulated using the neutrino event generator NUANCE (v3.5) [4]. The hadrons produced in these interactions are primarily pions, but there are some events with kaons (about 3%) and a small fraction of other hadrons as well. As discussed earlier, it is not possible to identify the hadrons individually in ICAL. However since the hit distribution of various hadrons are similar to each other (see Fig. 4.8), and the NUANCE generator is expected to produce a correct mixture of different hadrons at all energies, it is sufficient to determine the hadron energy resolution at ICAL through an effective averaging of NUANCE events, without having to identify the hadrons separately.

Figure 4.13: The mean hit distribution (left) and the energy resolution (right) for hadron events generated by NUANCE, as a function of . The right panel also shows a fit to Eq. (4.10). The bin widths are indicated by horizontal error bars [9].

A total of 1000 kt-years of “data” events (equivalent to 20 years of exposure with the 50 kt ICAL module) were generated with NUANCE. The events were further binned into the various energy bins and the hit distributions (averaged over all angles) in these bins are fitted to the Vavilov distribution function. The mean values () of these distributions as a function of are shown in the left panel of Fig. 4.13. As expected, these are similar to the mean values obtained earlier with fixed energy pions. Since the mean hits grow approximately linearly with energy, the same linearized approximation used in section 4.2.2 can be used to obtain the energy resolution = . The energy resolution as a function of is shown in Fig. 4.13. The energy resolution ranges from 85% (at 1 GeV) to 36% (at 15 GeV).

The effective energy response obtained from the NUANCE-generated data is an average over the mixture of many hadrons that contribute to hadron shower at all energies. The fractional weights of different kinds of hadrons produced in neutrino interactions may, in principle, depend upon neutrino oscillations. In addition, the relative weights of events with different energy that contribute in a single energy bin changes because neutrino oscillations are energy dependent. In order to check this, events with oscillations using the best-fit values of standard oscillation parameters (mixing angles and mass-squared differences) [115] were also generated. The resolutions obtained without and with oscillations are very close to each other. Thus, the hadron energy resolution can be taken to be insensitive to oscillations.

4.2.4 Hadron energy calibration

To calibrate the hadron energy against the hit multiplicity, hadrons from the simulated NUANCE [4] “data” were divided into bins of different hit multiplicities . Even here, a good fit was obtained for the Vavilov distribution function at all values of . We show the mean, , and the standard deviation, , obtained from the fit in the calibration plot in Fig. 4.14.

Figure 4.14: Calibration plot for , where mean and from the Vavilov fits are represented by the black filled circles and error bars, respectively [9].

For charged-current events, the energy of the incident neutrino can be reconstructed through

(4.11)

where is reconstructed from the Kalman filter algorithm and is calibrated against the number of hadronic hits. The neutrino energy resolution will in principle depend on the energy and direction of the muon as well as that of the hadron shower. The poor energy resolution of hadrons also makes the energy resolution of neutrinos rather poor, and loses the advantage of an accurate muon energy measurement. Therefore, reconstructing neutrino energy is not expected to be the most efficient method for extracting information from the ICAL analysis. Indeed, we expect to use the muon and hadron information separately, as will be seen in Sec. 5.4.

4.2.5 Salient features of the detector response

The ICAL detector is mainly sensitive to muons produced in the charged-current interactions of atmospheric or . We have studied the response of the detector to muons [8] generated in the central volume of a module of the ICAL detector where the magnetic field is uniform. The momentum, charge and direction of the muons are determined from the curvature of the track in the magnetic field using Kalman filter algorithm. The response of the detector to muons in the energy range 1–20 GeV with is studied in the different azimuthal regions. The momentum resolution, reconstruction efficiency, charge identification efficiency and direction resolution are calculated. The momentum resolution is about 20% (10%) for energies of 2 GeV (10 GeV), while the reconstruction efficiency is about 80% for GeV. The relative charge identification efficiency is found to be 98% for almost all energies above the threshold. The direction resolution is found to be better than a degree for all angles for energies greater than about 4 GeV.

The hadron events of interest in the ICAL detector primarily contain charged pions. The hit pattern of pions and kaons in the detector is similar; hence it is not possible to separate different hadrons in the detector. Similarly, neutrino-nucleus interactions produce events with multiple hadrons in the final state (generated by the NUANCE neutrino generator), whose energies cannot be reconstructed individually. However, the total energy deposited in hadrons can be determined by a calibration against the hit multiplicity of hadrons in the detector [9].

The hit patterns in single and multiple hadron events are roughly similar, and may be described faithfully by a Vavilov distribution. Analyses, first with fixed-energy pions, and later with a mixture of hadrons from atmospheric interaction events, show that a hadron energy resolution in the range 85% (at 1 GeV) – 36% (at 15 GeV) is obtainable. The parameters of the Vavilov fit presented here as a function of hadron energy can be used for simulating the hadron energy response of the detector, in order to perform physics analyses that need the hadron energy resolution of ICAL. We have also presented the calibration for the energy of the hadron shower as a function of the hit multiplicity. This analysis will be improved upon by incorporating edge effects and noise in a later study, after data from the prototype detector is available.

The reconstruction of hadrons allows us to reconstruct the total visible energy in NC events. Combined with the information on the muon energy and direction in the CC events, it will also allow one to reconstruct the total neutrino energy in the CC events. As we shall see in Chapter 5, the correlated information in muon and hadron in a CC event will also help to enhance the capabilities of the ICAL. The ICAL will be one of the largest neutrino detectors sensitive to the final state muons as well as hadrons in neutrino interactions at multi-GeV energies, and this advantage needs to be fully exploited.

Note that the calibration of the hadron response presented in this section has been determined by Monte-Carlo simulations. To confirm its validity, we have compared in Appendix D the results of our simulations with the hadron response at MINOS and the baby MONOLITH detector at appropriate plate thicknesses. This appendix also studies the dependence of hadron response at ICAL as a function of the iron plate thickness.

Some muon events at ICAL will also arise from the ’s arising due to oscillations of . These ’s may produce ’s through charged-current interactions, which would further decay to muons within the detector. These events will then contaminate the direct muon signal [116]. The number of such events (indirect muon events through tau production) is however heavily suppressed, first due to the mass of the that implies a large threshold energy for the neutrino, then due to the small branching fraction of , and finally due to the three-body kinematics of decay that reduces the energy of the resultant muon even further. This results in only about 150 such indirect muon events in 5 years, as compared to a few thousands of direct muon events. Hence at this level of analysis, we have neglected these events. The charged current events which may be mistakenly reconstructed as charged-current muon events have also been neglected at this stage. These are in the process of being included in a more sophisticated analysis.

Chapter 5 Neutrino Oscillation Physics at ICAL

The pendulum of mind oscillates between sense and nonsense,

not between right and wrong.

– Carl Gustav Jung

In this chapter we will present the physics capabilities of ICAL for the mixing parameters within the three-generation flavor oscillation paradigm. We shall restrict ourselves to the charged-current events produced in the ICAL from and interactions, which produce and , respectively. We shall start by describing our analysis method in Sec. 5.1, and then proceed to present the results showing the physics reach of this experiment for various quantities of interest. We shall focus on the identification of the neutrino mass hierarchy, as well as on the precision measurements of and .

The results will be presented using three different analyses. First in Sec. 5.2, we use only the information on the measured muon energy and muon direction (), both of which should be rather precisely measured in this detector, as described in detail in Chapter 4. Note that the results for the muon reconstruction used in these physics analyses have been obtained with an averaging over the azimuthal angle, and with the vertex taken to be in the central (8m x 8m x 10m) region of each module of the detector. These muons may propagate out of this region into the peripheral regions, and even exit the detector. The latter “partially contained” events roughly form about 12% of our sample, and we have not analyzed them separately.

We next show the improvement expected in the precision measurement of the atmospheric mass squared difference and the mixing angle if we use the information on the hadron energy in addition, to reconstruct the neutrino energy in each event. In this analysis, first in Sec. 5.3 we analyse the data in terms of the reconstructed neutrino energy and the measured muon angle (). However the reconstruction of neutrino energy involves the addition of the rather coarsely known hadron energy information to the measured muon energy, which results in a dilution of the muon energy information, which is more accurately known due to the good tracking capabilities of the ICAL. In order to retain the benefits of the accurately measured muon energy, we separately use the information on the measured muon energy, muon direction and the hadron energy () corresponding to each atmospheric neutrino event at the ICAL detector. The results of this final analysis, which leads to the best physics reach for ICAL at this stage, are presented in Sec. 5.4.

Note that the detector characteristics used for the analyses presented in this Chapter have been determined in the central region of the central module of the ICAL detector, as mentioned in the previous Chapter. When the three modules are placed adjacent to each other along the -axis, similar detector response is seen in the extended central region that includes the central region of each module as well as the “side” regions that are sandwiched between two central regions. This comprises the region , , and the entire region, that is, about 42% of ICAL. As expected, the muon response is worse in the peripheral regions of the detector [117]. Studies show that the reconstruction efficiencies drop by about 10% while the charge identification efficiency drops from 98% in the central to about 96% in the peripheral region for few-GeV muons. Further, the momentum resolution worsens from 10% to about 12–15% while the direction resolution remains the same. Hence this would worsen the physics results that we would obtain, although not drastically. Note that the hadron resolutions are not altered on inclusion of the entire volume of ICAL, mainly since it is independent of the magnetic field. We do not comment further on this in this paper, and present all results using the central region resolutions described earlier.

5.1 Charged-current events in ICAL

We focus on the charged-current events from the atmospheric interactions, that produce muons in the ICAL. We shall start by dividing them into bins of energy and momenta, taking into account the efficiencies and resolutions obtained in Chapter 4. As its output, the generator provides the 4-momentum () of the initial, intermediate and the final state particles for each event. To reduce the Monte Carlo fluctuations in the events obtained, we generate an event sample corresponding to 1000 years of running of ICAL and and scale it down to the desired exposure for the analysis. The ICAL sensitivities presented here can then be interpreted as median sensitivities (in the frequentist sense), as described in [118]. Using 1000 years of data takes us closer to the ideal “Asimov” data set [119] that has no statistical fluctuations.

In the oscillated event sample, the total number of events come from the combination of the and the channels as

(5.1)

where is the number of targets and is the exposure time of the detector. Here and are the fluxes of and , respectively, and is the oscillation probability. The first term in Eq. (