1 Introduction

Testing non-standard neutrino matter interactions in atmospheric neutrino propagation

[0.2cm]

Animesh Chatterjee 111Email: animesh.chatterjee@uta.edu, Poonam Mehta 222Email: pm@jnu.ac.in, Debajyoti Choudhury 333Email: debchou@physics.du.ac.in and Raj Gandhi 444Email: raj@hri.res.in

Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA

School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India

Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India

July 14, 2019

Abstract

We study the effects of non-standard interactions on the oscillation pattern of atmospheric neutrinos. We use neutrino oscillograms as our main tool to infer the role of non-standard interactions (NSI) parameters at the probability level in the energy range, GeV and zenith angle range, . We compute the event rates for atmospheric neutrino events in presence of NSI parameters in the energy range GeV for two different detector configurations - a magnetized iron calorimeter and an unmagnetized liquid Argon time projection chamber which have different sensitivities to NSI parameters due to their complementary characteristics.

## 1 Introduction

With the immense progress over the past few decades in establishing neutrino masses and mixings, it is fair to say that neutrino physics has entered an era of precision measurements. The first confirmation came in 1998 courtesy the pioneering experiment, Super Kamiokande (SK) [1]. With more data as well as with the aid of numerous other experiments, we have steadily garnered more and more precise information about the neutrino mixing parameters. As a result, the long list of unanswered questions in the standard scenario has become shorter (for recent global analyses of all neutrino oscillation data, see [2, 3, 4]). The focus of the ongoing and future neutrino experiments is on resolving the issue of neutrino mass hierarchy i.e., sign 111. , measuring the CP phase () and determining the correct octant of the mixing angle .

The minimal theoretical scenario needed to describe oscillations requires the existence of neutrino masses. The simplest way is to add right handed neutrino fields to the Standard Model (SM) particle content (something that the originators of the SM would, no doubt, have trivially done were nonzero neutino masses known then) and generate a Dirac mass term for neutrinos. However it is hard to explain the smallness of the neutrino mass terms via this mechanism. To overcome this, an attractive way is to add dimension-five non-renormalizable terms consistent with the symmetries and particle content of the SM, which naturally leads to desired tiny Majorana masses for the left-handed neutrinos222Terms such as are gauge invariant too and phenomenologically unconstrained. While they break lepton number, the latter is only an accidental symmetry in the SM. Thus, such terms, in conjunction with the usual Dirac mass terms, would generate tiny observable neutrino masses through the see-saw mechanism. It can be readily seen that, the aforementioned dimension-five term () essentially mimics this mechanism in the low energy effective theory.. However in the minimal scenario of this extension, the dominant neutrino interactions involving the light fields are still assumed to be described by weak interactions within the SM in which flavour changes are strongly suppressed.

Once we invoke new physics in order to explain the non-zero neutrino masses, it seems rather unnatural to exclude the possibility of non standard interactions (NSI) which can, in principle, allow for flavour changing interactions. Simultaneously, these are new sources of CP violation which can affect production, detection and propagation of neutrinos [5]. Some of the early attempts discussing new sources of lepton flavour violation (for instance, -parity violating supersymmetry) were geared towards providing an alternate explanation for the observed deficit of neutrinos in the limiting case of zero neutrino masses and the absence of vacuum mixing [6, 7]. In recent years, the emphasis has shifted towards understanding the interplay between the standard electroweak interactions (SI) and NSI and whether future oscillation experiments can test such NSI apart from determining the standard oscillation parameters precisely. This has led to an upsurge in research activity in this area (see the references in [5]). This is also interesting from the point of view of complementarity with the collider searches for new physics. There are other motivations for NSI as well such as (electroweak) leptogenesis [8], neutrino magnetic moments [9, 10, 11, 12], neutrino condensate as dark energy [13, 14].

Neutrino oscillation experiments can probe NSI by exploiting the interference with the Standard Model amplitude. In view of the excellent agreement of data with standard flavour conversion via oscillations, we would like to explore the extent to which NSI (incorporated into the Lagrangian phenomenologically via small parameters) is empirically viable, with specific focus on atmospheric neutrino signals in future detectors. NSI in the context of atmospheric neutrinos has been studied by various authors [15, 16, 17, 18, 19]. Also there are studies pertaining to other new physics scenarios using atmospheric neutrinos such as CPT violation [20, 21], violation of the equivalence principle [22], large extra dimension models [23] and sterile neutrinos [24, 25, 26].

The plan of the article is as follows. We first briefly outline the NSI framework in Sec. 2 and subsequently discuss the neutrino oscillation probabilities in presence of NSI using the perturbation theory approach (in Sec. 3). We describe the features of the neutrino oscillograms in Sec. 4. We give the details of our analysis in Sec. 5 and the discussion on events generated for the two detector types in Sec. 6. Finally, we conclude in Sec. 7.

## 2 Neutrino NSI Framework: relevant parameters and present constraints

As in the case of standard weak interactions, a wide class of “new physics scenarios” can be conveniently parameterised in a model independent way at low energies (, where is the electroweak scale) by using effective four-fermion interactions. In general, NSI can impact the neutrino oscillation signals via two kinds of interactions : (a) charged current (CC) interactions (b) neutral current (NC) interactions. However, CC interactions affect processes only at the source or the detector and these are are clearly discernible at near detectors (see for example, [27, 28]). On the other hand, the NC interactions affect the propagation of neutrinos which can be studied only at far detectors. Due to this decoupling, the two can be treated in isolation. Usually, it is assumed that the CC NSI terms (e.g., of the type with being the components of a weak doublet) are more tightly constrained than the NC terms and, hence, are not considered. It turns out, though, that, in specific models, the two can be of comparable strengths [29]. However, since we are interested in NSI that alter the propagation of neutrinos, we shall consider the NC type of interactions alone.

The effective Lagrangian describing the NC type neutrino NSI of the type is given by333One could think that other Dirac structures generated by intermediate scalar (), pseudoscalar () or tensor () fields may also be there. However, these would only give rise to subdominant effects.

 LNSI=−2√2GFεfCαβ [¯ναγμPLνβ] [¯fγμPCf] , (1)

where is the Fermi constant, are neutrinos of different flavours, and is a first generation SM fermion (444Coherence requires that the flavour of the background fermion () is preserved in the interaction. Second or third generation fermions do not affect oscillation experiments since matter does not contain them.. The chiral projection operators are given by and . If the NSI arises at scale from some higher dimensional operators (of order six or higher), it would imply a suppression of at least (for , we have ). However, such a naive dimensional analysis argument breaks down if the new physics sector is strongly interacting as can happen in a variety of models. We shall, hence, admit even larger as long as these are consistent with all current observations. In general, NSI terms can be complex. Naively, invariance would dictate that operators involving must be accompanied by ones containing the corresponding charged lepton field, thereby leading to additional CC interactions. This, however, can be avoided by applying to breaking and/or invoking multiple fields and interactions in the heavy (or hidden) sector. Rather than speculate about the origin of any such mechanism, we assume here (as in much of the literature) that no such CC terms exist.

The new NC interaction terms can affect the neutrino oscillation physics either by causing the flavour of neutrino to change () i.e., flavour changing (FC) interaction or, by having a non-universal scattering amplitude of NC for different neutrino flavours i.e., flavour preserving (FP) interaction. At the level of the underlying Lagrangian, NSI coupling of the neutrino can be to . However, from a phenomenological point of view, only the sum (incoherent) of all these individual contributions (from different scatterers) contributes to the coherent forward scattering of neutrinos on matter. If we normalize555If we normalize to either up or down quark abundance (assume isoscalar composition of matter) instead, there is a relative factor of 3 which will need to be incorporated accordingly. to , the effective NSI parameter for neutral Earth matter666For neutral Earth matter, there are 2 nucleons (one proton and one neutron) per electron. For neutral solar matter, there is one proton for one electron, and is

 εαβ = ∑f=e,u,dnfneεfαβ=εeαβ+2εuαβ+εdαβ+nnne(2εdαβ+εuαβ)=εeαβ+3εuαβ+3εdαβ , (2)

where is the density of fermion in medium crossed by the neutrino and refers to neutrons. Also, which encodes the fact that NC type NSI matter effects are sensitive to the vector sum of NSI couplings.

Let us, now, discuss the constraints on the NC type NSI parameters. As mentioned above, the combination that enters oscillation physics is given by Eq. (2). The individual NSI terms such as or are constrained in any experiment (keeping only one of them non-zero at a time) and moreover the coupling is either to individually [30]. In view of this, it is not so straightforward to interpret those bounds in terms of an effective . There are two ways : (a) One could take a conservative approach and use the most stringent constraint in the individual NSI terms (say, use ) to constrain the effective term (say, ) in Eq. (2) and that leads to

 |εαβ|<⎛⎜⎝0.060.050.270.050.0030.050.270.050.16⎞⎟⎠ . (3)

The constraints involving muon neutrinos are at least an order of magnitude stronger (courtesy the NuTeV and CHARM scattering experiments) than those involving electron and tau neutrino [31]. (b) With the assumption that the errors on individual NSI terms are uncorrelated, the authors in Ref. [29] deduce model-independent bounds on effective NC NSI terms

 εαβto0.0pt ∼<⎧⎨⎩∑C=L,R[(εeCαβ)2+(3εuCαβ)2+(3εdCαβ)2]⎫⎬⎭1/2 , (4)

which, for neutral Earth matter, leads to

 |εαβ|<⎛⎜⎝4.20.333.00.330.0680.333.00.3321⎞⎟⎠ . (5)

Note that the values mentioned in Eq. (5) are larger by one or two orders of magnitude than the overly restrictive bounds of Eq. (3), which, of course, need not be applicable.

Apart from the model independent theoretical bounds, two experiments have used the neutrino data to constrain NSI parameters which are more restrictive. The SK NSI search in atmospheric neutrinos crossing the Earth found no evidence in favour of NSI and the study led to upper bounds on NSI parameters [32] given by (at 90% CL) in a two flavour hybrid model [5]777The SK collaboration uses a different normalization () while writing the effective NSI parameter (see Eq. (2)) and hence we need to multiply the bounds mentioned in Ref. [32] by a factor of 3.. The off-diagonal NSI parameter is constrained (at 90% CL) from MINOS data in the framework of two flavour neutrino oscillations [33, 34]. It should be noted, though, that the derivation of these bounds (the SK one in particular [32]) hinge upon certain assumptions. The primary theoretical assumption relates to the simplification of the system onto a (hybrid) two-flavour scenario. Within the SM paradigm, this approximation is expected to be a very good one. The situation changes considerably, though, once NSI are introduced. As we shall see shortly, the major effect of NSI accrues through matter effects (even in the limit of the decoupling entirely). However, there exists a nontrivial interplay between such effects and the corresponding matter effects induced by canonical three-flavour oscillations. Consequently, approximations pertaining to the neutrino mixing matrix can significantly alter conclusions reached about NSI. Similarly, the very presence of NSI can leave its imprint in the determination of neutrino parameters. A second set of imponderables relate to statistical and systematic uncertainties, including but not limited to earth density and atmospheric neutrino profiles. Thus, it is quite conceivable that the constraints quoted by the SK collaboration could be relaxed to a fair degree, though perhaps not to the extent of those in Eq. (5). In view of this, and following several other studies [35], we will use a value of (for the parameters , and ) in our oscillogram diagrams. This value is eminently in agreement with Eq. (5). Note, though, that this choice is essentially to aid visual appreciation of the differences in the oscillogram structures wrought by NSI. Indeed, the experimental sensitivities that we shall be deriving are comparable to (and often significantly better than) those achieved by the SK collaboration. Furthermore, we shall not be taking recourse to two-flavour simplifications to reach such sensitivities. Additionally, the allowed ranges of NSI parameters have been recently extracted using global analysis of neutrino data in Ref. [36].

## 3 Neutrino oscillation probability in matter with NSI

The purpose of the analytic expressions presented here is to understand the features in the probability in the presence of NSI. All the plots presented in this paper are obtained numerically by solving the full three flavour neutrino propagation equations using the PREM density profile of the Earth, and the latest values of the neutrino parameters as obtained from global fits (see Table 1).

The analytic computation of probability expressions in presence of SI [37, 38, 39, 40, 41, 42, 43] as well as NSI [44, 45, 46, 47, 48, 49, 50, 35] has been carried out for different experimental settings by various authors. Note that, for atmospheric neutrinos, one can safely neglect the smaller mass squared difference in comparison to since for a large range of values of and (especially above a GeV). This “one mass scale dominant” (OMSD) approximation allows for a relatively simple exact analytic formula for the probability (as a function of only three parameters and ) for the case of constant density matter [42] with no approximation on , and it works quite well888This approximation breaks down if the value of is small since the terms containing can be dropped only if they are small compared to the leading order term which contain . After the precise measurement of the value of by reactor experiments, this approximation is well justified. For multi-GeV neutrinos, this condition ( kmGeV) is violated for only a small fraction of events with GeV and km.. In order to systematically take into account the effect of small parameters, the perturbation theory approach is used. We review the necessary formulation for calculation of probabilities that affect the atmospheric neutrino propagation using the perturbation theory approach [35].

In the ultra-relativistic limit, the neutrino propagation is governed by a Schrdinger-type equation (see [51]) with an effective Hamiltonian

 H = Hvac+HSI+HNSI , (6)

where is the vacuum Hamiltonian and are the effective Hamiltonians in presence of SI alone and NSI respectively. Thus,

 H = (7)

where is the standard CC potential due to the coherent forward scattering of neutrinos and is the electron number density. The three flavour neutrino mixing matrix [ with and ] is characterized by three angles and a single (Dirac) phase and, in the standard PMNS parameterisation, we have

 U = ⎛⎜⎝1000c23s230−s23c23⎞⎟⎠⎛⎜⎝c130s13e−iδ010−s13eiδ0c13⎞⎟⎠⎛⎜⎝c12s120−s12c120001⎞⎟⎠ , (8)

where . While, in addition, two Majorana phases are also possible, these are ignored as they play no role in neutrino oscillations. This particular parameterisation along with the fact of commuting with , allows for a simplification. Going over to the basis, , we have and[47]

 ~H = λ[⎛⎜ ⎜⎝rA+s2130c13s13e−iδ000c13s13eiδ0c213⎞⎟ ⎟⎠+rλ⎛⎜ ⎜⎝s212c213c12s12c13−s212c13s13e−iδc12s12c13c212−c12s12s13e−iδs212c13s13eiδ−c12s12s13eiδs212s213⎞⎟ ⎟⎠ (9) +rA⎛⎜ ⎜⎝~εee~εeμ~εeτ~ε⋆eμ~εμμ~εμτ~ε⋆eτ~ε⋆μτ~εττ⎞⎟ ⎟⎠] ,

where we have defined the ratios

 λ≡δm2312E;rλ≡δm221δm231;rA≡A(x)δm231 . (10)

Once again, and the last term in Eq. (9) is

 λrA⎛⎜ ⎜⎝εeec23εeμ−s23εeτs23εeμ−c23εeτc23ε⋆eμ−s23ε⋆eτεμμc223+εττs223−(εμτ+ε⋆μτ)c23s23εμτc223−ε⋆μτs223+(εμμ−εττ)c23s23s23ε⋆eμ−c23ε⋆eτε⋆μτc223−εμτs223+(εμμ−εττ)c23s23εμμs223+εττc223+(εμτ+ε⋆μτ)c23s23⎞⎟ ⎟⎠

where are complex. For atmospheric and long baseline neutrinos, holds and for a large range of the and values considered here. The small quantities are and . We decompose into two parts : such that the zeroth order term provides the effective two flavour limit with and but , i.e.,

 ~H0 = λ⎛⎜ ⎜⎝rA(x)+s2130c13s13e−iδ000c13s13eiδ0c213⎞⎟ ⎟⎠ , (11)

while contains the other two terms (on the RHS of Eq. (9)) which represent corrections due to non-zero and the non-zero NSI parameters respectively. Upon neglecting terms like , we get an approximate form for , viz.,

 ~HI ≈ λ[rλ⎛⎜⎝s212c12s120c12s12c2120000⎞⎟⎠+rA⎛⎜ ⎜⎝~εee~εeμ~εeτ~ε⋆eμ~εμμ~εμτ~ε⋆eτ~ε⋆μτ~εττ⎞⎟ ⎟⎠] . (12)

In what follows, we use the perturbation method described in [43] to compute the oscillation probabilities. The exact oscillation probability is given by

 Pαβ = |Sβα(x,x0)|2 , (13)

where is the evolution matrix defined through with and satisfying the same Schrdinger equation as . It can, trivially, be seen to be given by where is independent of . We first evaluate using

 ~S(x,x0) = ~S0(x,x0) ~S1(x,x0) . (14)

Here, and satisfy

 iddx~S0(x,x0) = ~H0(x) ~S0(x,x0) ;~S0(x0,x0)=I , iddx~S1(x,x0) = [~S0(x,x0)−1 ~HI(x) ~S0(x,x0)]~S1(x,x0);~S1(x0,x0)=I . (15)

where is given by Eq. 12. To the first order in the expansion parameter, we have

 ~S(x,x0) ≃ ~S0(x,x0)−i~S0(x,x0)∫xx0[~S0(x′,x0)−1 ~HI(x′) ~S0(x′,x0)]dx′ . (16)

Finally, the full evolution matrix can be obtained by going back to the original basis from the tilde basis using . The oscillation probability for can be obtained as

 PNSIeμ ≃ 4s213s223[sin2(1−rA)λL/2(1−rA)2] (17) +8s13s23c23(|εeμ|c23cχ−|εeτ|s23cω)rA[sinrAλL/2rA sin(1−rA)λL/2(1−rA) cosλL2] +8s13s23c23(|εeμ|c23sχ−|εeτ|s23sω)rA[sinrAλL/2rA sin(1−rA)λL/2(1−rA) sinλL2] +8s13s223(|εeμ|s23cχ+|εeτ|c23cω)rA[sin2(1−rA)λL/2(1−rA)2] ,

where we have used to the leading order in , and , . Only the parameters and enter in the leading order expression [45], as terms such as have been neglected. Let us discuss the two limiting cases, and . When , we recover the vacuum limit (given by the first term on the RHS of Eq. (17)). When , we are close to the resonance condition ( since is small) and the probability remains finite due to the and terms in the denominator of Eq. (17).

The survival probability for is given by

 PNSIμμ ≃ 1−s22×23[sin2λL2] (18) + (|εμμ|−|εττ|)s22×23c2×23[rAλL2sinλL−2rAsin2λL2] ,

where and . Note that the NSI parameters involving the electron sector do not enter this channel and the survival probability depends only on the three parameters  [47, 50, 45]. Once again, the vacuum limit is recovered for . Of these three NSI parameters, is subject to the most stringent constraint (Eq. 5). If we look at , the phase factor results in minima of probability for (vacuum dip) and maxima for (vacuum peak) where is any integer. The oscillation length for the NSI terms, though, is different, and this changes the positions of the peaks and dips.

In order to quantify the impact of NSI, it is useful to define a difference999The difference used in [35] has an overall sign compared to our definition.

 ΔPαβ=PSIαβ−PNSIαβ , (19)

where is probability of transition assuming standard interactions (i.e., with being set to zero in Eqs. (17) and (18)) and is the transition probability in presence of NSI parameters. For the different channels that are relevant to our study, the quantities are given by

 ΔPeμ ≃ −8s13s23c23(|εeμ|c23cχ−|εeτ|s23cω)rA[sinrAλL/2rA sin(1−rA)λL/2(1−rA) cosλL2] (20) −8s13s23c23(|εeμ|c23sχ−|εeτ|s23sω)rA[sinrAλL/2rA sin(1−rA)λL/2(1−rA) sinλL2] −8s13s223(|εeμ|s23cχ+|εeτ|c23cω)rA[sin2(1−rA)λL/2(1−rA)2] .
 ΔPμμ ≃ |εμτ|cosϕμτs2×23(s22×23(rAλL)sinλL+4c22×23rAsin2λL2) (21) − (|εμμ|−|εττ|)s22×23c2×23[rAλL2sinλL−2rAsin2λL2] .

For the case of anti-neutrinos, (which implies that ) while . Similarly for IH, .

In the present work, for the sake of simplicity, the NSI parameters are taken to be real () and also .

## 4 Neutrino oscillograms in presence of NSI :

Within the SM, for a given hierarchy (NH or IH) and best-fit values of the oscillation parameters (as given in Table 1), the oscillation probability depends on only two quantities : the neutrino energy and the zenith angle of the direction of the neutrino, namely , with the vertically downward direction corresponding to . The oscillation pattern can, then, be fully described by contours of equal oscillation probabilities in the plane. We use these neutrino oscillograms of Earth to discuss the effect of neutrino–matter interactions on the atmospheric neutrinos passing through the Earth (see Refs. [52, 53] for a more detailed discussion of the general features of the SI oscillograms).

disappearance channel :

In Fig. 1, we reproduce the neutrino oscillograms in the channel for the case of NH (left panel) and IH (right panel) in the - plane. As expected, the muon neutrino disappearance probability experiences matter effects (MSW effects as well as parametric resonances) for the case of NH but not for the case of IH where it is essentially given by the vacuum oscillation probability (which depends on ). For SI in the channel, and the plots for NH and IH get interchanged [42]. In vacuum, the positions of the peaks () and dips () can be calculated from the first line on RHS in Eq. (18) as

 (L/E)dip≃(2p−1)π1.27×2×δm231 km/GeV;(L/E)peak≃kπ1.27×δm231 km/GeV (22)

where . The first dip and peak, then, are at

 (L/E)dip≃499 km/GeV;(L/E)peak≃998 km/GeV (23)

which means that for a given (say km or ), we can predict the values of peak energy GeV and dip energy GeV. This can be seen clearly from the right panel of Fig. 1 which corresponds to the IH as the probability in this case is dominated by vacuum oscillations.

The MSW matter effect can occur both in the mantle region as well as the core [54, 55]. The energy at which the MSW resonance takes place in the 13 sector is

 ρER≃δm2310.76×10−4×cos2θ13 GeV g/cc . (24)

Using the values of and from Table 1, we get for which is the average density for a neutrino traversing km through the earth101010Note, though, that neutrinos of such energies but travelling a smaller path through the earth would also hit regions with and, thus, suffer resonant conversion. to reach the detector. As the neutrino path nears the core, the energy at which the MSW resonance effects occur decreases (see Table 2). As discussed in Ref. [42], when coincides with or , one expects a large change in the probability. We see this feature in the left plot of Fig. 1 around GeV where the probability is reduced from the peak value by almost . (Note that km () implies that the neutrino has passed only through the crust and the mantle regions, without penetrating the core.). Also the pattern in the left oscillogram changes abruptly at a value of demarcating two regions : for , the neutrinos pass through both mantle and core which allows for parametric effects while for , the neutrinos cross only the mantle region where only the usual MSW effects operate. On the other hand, the parametric resonance occurs when neutrinos traversing the Earth pass through layers of alternating density (mantle-core-mantle) [52, 53].

Having described the case of SI, let us now address the impact of NSI on neutrinos and antineutrinos traversing the Earth. To best illustrate the features, we consider only one NSI parameter to be nonzero. In the leading order expression only two combinations of the three NSI parameters () appear. Let us discuss these in turn.

(a) : In Fig. 2, we show the corresponding for the case of NH (top row) and IH (bottom row) and two specific values of the NSI parameter consistent with the current bounds. Note that the case of NH and is grossly similar to the case of IH and (and, similarly, for NH and vs. IH and ). From Eq. (18), we see that there are two terms proportional to , one where the oscillating function is with the other being . Thus, the first term can be positive or negative depending upon the value of the phase, while the second term is always positive. It is the interplay of these two terms that leads to the features in these plots. The mass hierarchy dependence comes from the first term since we have which changes sign when we go from NH to IH. As noted earlier, near the vacuum dip , this term will be dominant. Consequently, for NH and , the oscillatory pattern is a modification of the standard one. For IH and , the term proportional to will have a negative overall sign and this leads to washout to a certain extent of the oscillation pattern.

The difference between SI and NSI contributions to the probability is shown in Fig. 3. can be as large as 1 for regions in the core and in mantle for some choice of and hierarchy. We also note large changes in probability (the regions where the difference is large ) along the diagonal line.

(b) : This case will correspond to the case of diagonal FP NSI parameter, . As mentioned above, is tightly constrained (see Eq. (5)) while the bound on is loose. If we choose (and ), we see that effects due to in are insignificant for most baselines except for a tiny region in the core (see Fig. 4). From Eq. 21, only the terms in second line contribute in this case and the minus sign between the two terms lowers the value of .

(c) Subdominant effects due to : For the case of NH, we compare the cases of non-zero in Fig. 4. From Eq. (5), we see that the bounds for and are similar () while that on is rather loose (). It is seen that the other parameters involving the electron sector play only a sub-dominant role in this channel. This can also be understood from the fact that, in the expression for (see Eq. (18)), these terms appear only at the second order [47].

appearance channel :

In Fig. 5, we have shown the standard neutrino oscillograms in the channel for the case of NH (left panel) and IH (right panel) in the (-) plane. In this case, the probability is negligible in most parts of the parameter space (especially for the case of IH). The appearance probability in matter differs from that in vacuum in the leading order itself and also the position of peaks and dips of the vacuum curves do not, in general, coincide with those in presence of matter (unlike in the case of muon survival probability). In order to analyse the plots, let us look at the OMSD expression [42] (since our analytic expression is valid to first order in )

 POMSDeμ = sin22~θ13sin2θ23sin2δ~m231L4E (25)

where

 sin2~θ13 = sin2θ13δm231~δm231 δ~m231 ≡ √(δm231cos2θ13−A)2−(δm231sin2θ13)2 (26)

The peak energy in matter will be given by [42],

 (L/E)peak≃(2p−1)π1.27×2×~δm231 km/GeV (27)

where . One would expect to be large when the matter peak coincides with the resonance energy, which gives GeV. However, the resonance condition which implies that also leads to taking its minimum value at resonance energy . Hence, the probability becomes large when is satisfied. This gives a value of km for  [42]. Note that the maximum value of is given by the value of . The range of and where is close to its maximal value due to MSW effect is given by GeV and in the mantle region. In the core region, the MSW peak will occur at smaller energies and the parametric resonance leads to large changes.

Having described the case of SI, let us now address the impact of NSI on neutrinos and antineutrinos traversing the Earth. In the leading order expression for (see Eq. (17)) there are only two NSI parameters () that appear whereas does not appear at all. We discuss them in turn.

(a) Subdominant effects due to : In Fig. 6, we show the effect of on the oscillograms of . Since the parameter does not appear at all in the first order expression (Eq. (17)), naturally its impact is expected to be small. Consequently, only in very tiny regions and can at best be as large as .

(b) Comparison of effects due to , , and : In Fig. 7, we compare effects due to four NSI parameters for the case of NH, allowing only one of them to be non-zero at a time. Since the parameters appear in the first order expression (Eq. (17)), they naturally have a larger impact as compared to the other two and, in the favourable situation, can be as large as . This is to be contrasted with which could take values as large as under favourable conditions. Also, if we look at Eq. (17), we note that and appear on equal footing as far as is concerned.

## 5 Simulating an experiment

### 5.1 Atmospheric events

The neutrino and anti-neutrino CC events are obtained by folding the incident neutrino fluxes with the appropriate probabilities, relevant CC cross sections, the detector efficiency, resolution, mass and the exposure time.

The event rate in a specific energy bin of width and the angle bin of width can be written as

 d2NμdΩdE=12π [(d2ΦμdcosθdE)Pμμ+(d2ΦedcosθdE)Peμ