Energy and Direction Estimation of Neutrinos in muonless events at ICAL
Abstract
In this paper, we study events without identifiable muon tracks in the Iron Calorimeter detector at the Indiabased Neutrino Observatory. Such events are dominated by high energy (E1 GeV) charged current interactions, which have been studied only in a few experiments so far. The charged particles, produced in these neutrino interactions, give rise to a set of hits in the detector. We attempt to reconstruct the energy and the direction of the neutrino in such events. We study the energy distribution for a given pattern of hits of these events and find that the Landau distribution provides a good fit. We define two kinematic variables based on the hit distribution and use them to determine the cosine of the polar angle of the neutrino direction (). There is a moderate correlation between these variables and the . These provide us enough information to prepare calibration charts for looking up the energy and direction of the incident neutrino.
1 Introduction
Indiabased Neutrino Observatory (INO) [1] is an upcoming experimental facility which aims to study atmospheric neutrinos. It houses a gigantic magnetised Iron Calorimeter (ICAL) neutrino detector. ICAL consists of 3 modules, each of which contains 151 horizontal iron layers. These iron layers, each of thickness 5.6 cm, are interspersed with resistive plate chambers (RPCs) [2, 3, 4]. The total number of RPCs in the detector is 30,000, and the total mass of the detector is approximately 50 kilotons. When a neutrino interacts with an iron nucleus, it produces a set of charged particles. Depending on the particle type and its energy, these charged particles pass through one or more RPCs. Whenever a charged particle passes through an RPC, it produces a hit. These hits are our primary signals. The layer number of RPC gives the zcoordinate of the hit. The x and ycoordinates are given by the copperstrips of the pickup panels which are orthogonally oriented at the top and the bottom of the RPCs [5].
The ICAL detector at INO will observe different types of neutrino interactions. The charged current (CC) interactions of the muon neutrinos ( and ) produce events with muons (CC events). If the energy of the muon is 0.5 GeV, and its direction not close to the horizontal, it passes through a number of layers and produces a track in the detector. Such events are easy to identify and reconstruct [6]. A significant part of ICAL physics program is based on these CC events. In addition, there will be a number of events without observable muon tracks [7]. They will consist of (i) charged current events of electron neutrinos ( and ) called CC events, (ii) charged current events of tau neutrinos ( and ) called CC events, (iii) neutral current events of all three types of neutrinos and antineutrinos called NC events and (iv) those CC events for which muon track cannot be reconstructed. The last type of events occur due to low neutrino energy (E 0.5 GeV) and/or due to near horizontal direction of neutrino ( 0.5 where is the zenith angle of the neutrino).
The magnetised ICAL detector can easily indentify the charge of the muon. The energy and direction of the muon can be estimated from muon track information [6]. In a typical CC event, there are likely to be some hadrons also. The energy of these hadrons can be estimated using the techniques described in [8]. By combining the information from the muon and the hadrons, it is possible to obtain an estimate of the neutrino energy [9] and direction.
The CC events produce electrons (positrons) which create a shower in the detector. They lose energy very fast and are not able to travel through many layers. The CC events [10] are rather small in number because of the large mass of the lepton. The lepton decays mostly into hadrons and hence these events also look like a shower of hadrons [11]. The only visible part of an NC event consists of hadrons, because the final state neutrino escapes detection. Hence, the NC events also look like a shower of hadrons. In the sample of showerlike events, we must include those CC events for which the muon track cannot be reconstructed.
We classify all these events, without a clear muon track, to be muonless events. In this paper, we attempt to develop methods to estimate the energy and direction of the neutrinos which produce such muonless events.
Neutrino interactions are typically described in terms of neutrino energy and direction. The direction enables us to calculate the distance the neutrino has travelled [12]. Thus an estimate of the incident atmospheric neutrino energy and direction will allow us to perform a more quantitative analysis of any physics topic.
Using the Nuance neutrino event generator [13], we generated 500 years data for ICAL. In generating this data, we assumed normal hierarchy and used the following values for neutrino parameters: eV, eV, eV, =0.31, =0.09, =0.5 and =0. The generated events are then simulated in the ICAL detector using GEANT4 [14].
2 Incident Energy of the neutrino
A more energetic neutrino is expected to give more number of hits in the detector, distributed among more number of layers compared to a less energetic neutrino. For a given neutrino energy, an event with neutrino travelling in vertical direction will have hits in more layers compared to one travelling close to horizontal direction [15]. Also, for a given neutrino energy, the NC events have less number of hits compared to CC events. Therefore, the two quantities,

number of hits,

number of layers having one or more hits,
are the basic kinematical variables to be used in the determination of the neutrino energy. From these we can define additional variables such as average hits per layer (hpl), hit distribution, etc. We use combinations of these variables to obtain an estimate of the neutrino energy and direction.
We found that there is negligible correlation between the hits and the neutrino energy, or the layers and the neutrino energy.
The variable ‘hpl’ does have some dependence on the neutrino energy but the correlation is too small for it to be an effective parameter for neutrino energy determination. This is depicted in figure 1, where we have considered all events which have number of hits greater than 10. We expect muonless events to have hits only in a few layers. Therefore, we further classify events according to the number of layers in which the hits are observed. For example, events with hits only in one layer, events with hits only in two layers etc. upto events with hits only in five layers. A majority of muonless events have hits in five layers or less. If two neutrino interactions of different energies give hits in same number of layers, then the more energetic neutrino should give more hits than the less energetic neutrino. That is, there should be a correlation between neutrino energy and hpl, if the number of layers is held fixed. This is illustrated in figure 2, for the case of hits in four layers.
The trends observed in figure 2 for CC events are also evident in the case of the NC events and those CC events which are confined to a few layers.
2.1 Calibration of neutrino Energy
As mentioned before, we classify events according to the number of layers containing hits. We have considered values of L (number of layers with one or more hits) ={2,…,5}. For a given L, we further subdivide the events in bins of hpl (hits per layer) = {(1), (12), (23), …, (910), …}. In each of these bins, we have plotted the neutrino spectrum, with the neutrino energy taken from Nuance. A sample of these plots is shown in figure 3, for L=4. These spectra show a gradual shift in the peak towards the right along the energy axis, with increasing value of hits per layer. Hence, it appears to provide a reasonable calibration of the neutrino energy. One requires a distribution function to represent each of these spectra. A number of fitting functions have been attempted. Landau distribution is certainly the more obvious one, because it is related to energy loss of charged particles in any medium [16, 17].
We describe the Landau distribution by its most probable value (MPV) and its sigma ()
Other distribution functions like the Vavilov and a few nonstandard functional forms have also been attempted. The Vavilov distribution, which is the more general form of the Landau function, fits the spectra well in the bins with moderate values of hits per layer. In fact, the fit is slightly better than the Landau distribution. But the fit to Vavilov distribution is very sensitive to the limits on the fit parameters. Moreover, at very low values of hits per layer and at very high values, Landau distribution gives a much better fit. Here, we consider the calibration of energy over a very wide range of hits per layer. So, we prefer to use one common distribution function over the entire range.
The nature of the energy deposition is different in each of the three types of neutrino events considered here. They are grouped under the generic name “muonless” events but each type has its characteristic interaction properties. The NC events contain outgoing neutrinos which give no hits. The electrons present in the CC events give more hits, in addition to those by the hadrons. The CC events contain those muons which do not give identifiable tracks. Since the energy loss in iron is the smallest for muons, CC events are likely to have more number of hits in comparison to CC events of the same energy.
The correlation between the neutrino energy and the hpl is shown in figure 4, for NC events with hits in exactly four layers. The central points are the MPVs of the corresponding Landau distributions and the error bars are s of those distributions. Figure 5 shows this correlation for CC and CC events also in addition to NC events. The NC events as expected give less hits than the other two event types. A chart of energy correlation is prepared in figure 6, with “number of layers” counting from 2 to 5. The curve in green is more relevant if we are dealing with an events sample rich in NC events. The curve in red is to be referred to if we have an events sample rich in CC events [7]. We have verified that a sample of pure CC events and a sample of events rich in CC events (with purity) both obey the same correlation plot. The curve in blue focuses on the CC events. It is worth noting that the energy scale increases with increasing L.
For Landau distribution, of the events fall within the range (MPV  1.02 , MPV + 4.65 ) [19]. This leads to the definitions and . The plots of and are shown in figure 7 for . The lower values are about whereas the higher values are about for all the cases. For given values of L and hpl, we can assign the event the most probable value of the neutrino energy. We can also estimate the probability of the event having a given energy based on the MPV and of the corresponding Landau distribution.
3 Direction of the neutrino
The direction of a neutrino is given in terms of the polar angle and the azimuthal angle . The neutrino flux is expected to be symmetric in , except for a small eastwest asymmetry arising due to the earth’s magnetic field. The polar angle of neutrino is certainly of greater importance because it determines the distance travelled by the neutrino, given by the expression where is the radius of the earth and is the atmospheric height at which the neutrino is produced [12]. We see from the above equation that =1 (1) for the vertically up (down) going neutrinos. To do oscillation physics, we need to distinguish between upgoing neutrinos which travel thousands of km and down going neutrinos which travel only tens of km. In addition, it will be useful to distinguish between neutrino events in horizontal and vertical directions.
In muonless events, there are no tracks to serve as an easy handle in determining the neutrino direction. Since these events have hits in five layers or less, the timing information is not useful in distinguishing between upward going and downward going events. Here we attempt to develop a method to find a correlation between the hit pattern and the neutrino direction. The following methods are tested with the CC events and are found to work. Both NC events and CC events without muon tracks (those for which hits are limited to five layers or less) also behave in a similar fashion. Minimal selection criteria may be used to select such event sample [7].
We have checked that the neutrino direction is not particularly correlated to the number of hits or the number of layers hit. Hence, devising criteria based on hits and layers or even hits per layer is not effective in finding a way to recognize the angular information of an incident neutrino.
3.1 Algorithms devised for neutrino angle depiction
We have considered the problem in two parts: Firstly, we try to distinguish between the vertical to nearvertical events from the horizontal to near horizontal neutrino events. Secondly, we attempt to tell apart the upgoing neutrinos from the downgoing neutrinos. We have devised a number of algorithms and tested their efficacy in determining the direction.
Horizontal or Vertical Direction
The vertical or near vertical events should have shorter average horizontal spread than the horizontal or near horizontal events. Having taken this cue, the maximum total spread of a neutrino event in horizontal plane is studied. This parameter is defined as follows: A hit point in ICAL detector refers to the pair of (x,y) coordinates of the hit in an RPC. The layer number of the RPC gives the z coordinate. The signals from the detector are read as the strip numbers along the X or Y direction. Hence, the hit points are devised by making all possible combinations of the signal giving XY strip numbers. Now, the distance between any two points in a given layer is given by ), for a pair of hits in a layer.
We can also define the horizontal distance between any two hit points by a similar formula, even if the hits are in different layers. This horizontal distance is the projection, of the total distance between the two points, on the horizontal plane. The maximum horizontal distance, the variable we use here, is defined to be the maximum value of the horizontal distance of an event. It is the maximum of projected lengths, on the horizontal plane, of the distances between any two hits in the event.
From figure 8, we see that there is a small correlation between direction of the neutrino (vertical or horizontal) and the maximum horizontal distance. The neutrinos incident in the vertical cone have less horizontal spread than the ones in the horizontal cone. For example, an upper cut of 30 cm on the maximum horizontal distance, gives above 75% vertical events ( or those within a cone of angle about the vertical direction) in the selected sample.
Upgoing or Downgoing neutrinos
The muonless events do not travel through many layers, unlike an event with a muon track. So, one needs to find a method/way to tell apart the upgoing neutrinos from the downgoing ones, in order to extract better physics information.
Finally, an algorithm using the hit distribution pattern can select upgoing neutrinos to an efficiency of 70%, and is described as follows. Assuming the number of hits across the layers as a type of distribution, their mean and the standard deviation from the mean (rms) are calculated. The details of this calculation are explained in [7]. These values of the mean and the rms of the layerhits distribution, hardly show any dependence on the direction of the neutrino. But the ratio of layerhits mean to layerhits rms (“MRratio”) shows a dependence on whether a neutrino is upgoing or downgoing, as seen in figure 9. The figure clearly shows that the lower values of this variable called MRratio selects mostly upgoing events and vice versa.
3.2 Estimation of the neutrino cos(theta):
The ratio of layerhits mean to rms (“MRratio”) gives a way to distinguish the upgoing neutrino events from the downgoing ones. The maximum horizontal distance (“mxdist”) provides us with the ability to separate vertical events from the horizontal events. We have attempted to find some correlation between the two of them. 2D histograms of these variables show gradual shifts in the peak positions of such distributions, in varying bins of neutrino direction. Figure 10 shows some of them. The 2D projection of the surface plot, of each of these distributions, show them to be symmetric along the MRratioaxis and asymmetric along the mxdistaxis, with a tail towards higher values of mxdist.
The left hand side panel in figure 10 shows distributions of the upgoing neutrinos in a few selected bins of cos. The right hand side panel shows the same for the down going neutrinos. The cometlike distribution (tail more notably) has shifted from left (in figures of the left hand side panel) to the right (in figures on the right hand side panel), along Xaxis, i.e. in the direction of increasing MRratio. The head of the “comet” moves downwards (towards smaller values of mxdist), as one goes from the top to the bottom of these set of panels (as cos varies from vertical to horizontal) in figure 10.
The directional information conveyed by these two variables independently, may be combined to obtain the resultant neutrino direction. Hence, one can present a calibration of cos vs. MRratio and mxdist in 3dimensions.
The 2Dimensional histograms can be fitted with appropriate surface distribution functions. The peak of such a fitted function gives us the coordinates for the 3Dcalibration plot. The errors can be quoted from the sigma of those peaks. Looking at the cometlike distribution from figure 10, one may propose a Gaussian distribution fit along Xaxis (MRratio), and a Landau distribution fit along Yaxis (mxdist). Figure 11 shows such an example.
The angular estimation can thus be done in terms of a 3Dcalibration plot of MRratio along Xaxis, mxdist along Yaxis and costheta along Zaxis, as in figure 12. The Xaxis contains the Gaussian mean of the MRratio with Gaussian sigma as the standard deviation, in that costheta bin. The Yaxis contains the Landau peak position of the maximum spread with Landau sigma as the standard deviation. The costheta is along Zaxis with binwidth of 0.05.
The XZ and YZ projections have been shown in figure 13 and 14 to help in visualising figure 12. But the estimation of the angle is preferably made from the 3DCalibration Plot in figure 12. As we see from figure 14, mxdist gives a reasonable estimate of the modulus of cos. Then we can use MRratio to break the degeneracy between the up and down going events.
4 Summary and Conclusions
An effective method has been devised to estimate the energies of the neutrino events which give no clear muon track. The energy of the incident neutrino can be well estimated by following the approximate calibration curves in figure 6. If the event sample contains equal proportions of CC, NC and CC events, then the uncertainty in the estimated energy is rather large. However, it is possible to choose event samples which are rich in CC or in NC events [7]. For those muonless samples, it is possible to get a good energy estimate.
One of the important types of such muonless events is the NC interaction. It must be remembered that NC events have an outgoing neutrino, which does not leave any signature. So, this method serves as an eventual solution to estimating the energy of such incident neutrinos.
The energies of the neutrinos in case of the muontrack containing events can be determined from the muon track informations added to that from the hadron energy calibration. Now, with the present method, one can estimate the energies of neutrinos that do not give muon tracks, i.e., the Neutral Current interactions, CC interactions as well as CC events which lack identifiable muons. Hence, to sum up, we can estimate the energies of all kinds of neutrinos (active) that are detected by the INOICAL Detector.
We can even make a crude estimation of the angle or direction of the incident neutrino for muonless events. The two variables, mxdist and MRratio have been defined to get this estimation. Mxdist can distinguish the vertical/nearvertical events from the horizontal/nearhorizontal events, which leads to a degeneracy between upgoing and downgoing events. MRratio removes this degeneracy and makes it possible to get an approximate estimate of the neutrino direction.
5 Acknowledgement
We express our deep gratitude to all our comembers of the INO Collaboration. We are thankful for the suggestions from Prof. Gobinda Majumder and discussions with our fellow members during the meetings have gone a great way in guiding us rightly through the project. We are also grateful to the Department of Atomic Energy (DAE) for financial support.
Footnotes
 The correction 0.22 to the peak position which is recommended for rootfitting at lower energies is neglected here since the error bar covers it [18].
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