High energy neutrinos from choked GRBs and their flavor ratio measurement by the IceCube

The high energy neutrinos produced in a choked GRB can undergo matter oscillation before emerging out of the stellar envelope. Before reaching the detector on Earth, these neutrinos can undergo further vacuum oscillation and then Earth matter oscillation. In the context of IceCube we study the Earth matter effect on neutrino flux in the detector. For the calculation of track-to-shower ratio R in the IceCube, we have included the shadowing effect and the additional contribution from the muon track produced by the high energy tau lepton decay in the vicinity of the detector. We observed that R is different for different CP phases in vacuum but the matter effect suppresses these differences. We have also studied the behavior of R when the spectral index $\alpha$ varies.


I. INTRODUCTION
Gamma-Ray Bursts (GRBs) are cosmological events with the emission of very intense electromagnetic radiation in the energy range ∼ 100 keV -1 MeV. Phenomenologically GRBs come in two variants: the short-hard bursts and long-soft bursts. The long gamma-ray bursts (LGRBs, typically with duration longer than 2 seconds), which constitute about 3/4 of the total observed GRBs, are generally believed to be associated with deaths of massive stars [1,2]. In this scenario the gamma rays emitted by the collapsing star during a long GRB event should be the result of relativistic jets of radiation and matter breaking through the stellar envelope. Fermi-accelerated electrons would produce gamma rays by synchrotron and inverse Compton scattering in optically thin magnetized relativistic shocks. In this same shock protons should also be accelerated to relativistic velocities and interact with the photons producing neutrinos with an energy range from MeV-EeV [3,4]. Observationally, only a small fraction (≤ 10 −3 ) of core collapse SNe are associated with GRBs [5][6][7]. These correspond to the cases when the energetic jet successfully penetrates through the stellar envelope and reaches a highly relativistic speed (Lorentz factor Γ ≥ 100). It is possible that the larger fraction of the core collapse may not be able to punch through the massive envelope to launch a successful GRB. Irrespective of its failure to emerge out from the thick envelop, like the successful jet, these choked jet can also accelerate protons to very high energy and produce multi-TeV neutrinos through interaction with the keV photon background present in the jet environment [8]. The high energy neutrinos are produced from the decay of charged pions which lead to the neutrino flux ratio at the source Φ 0 νe : Φ 0 νµ : Φ 0 ντ = 1 : 2 : 0 (Φ 0 να corresponds to the sum of neutrino and antineutrino flux at the source). As is well known, matter effect can substantially modify the flux ratio due to neutrino oscillation, in a presupernova star scenario, high energy neutrinos propagating through a heavy envelope can oscillate to other flavors due to matter effects, resulting in flavor ratios at the surface of the star that can be significantly different from 1:2:0. In a previous paper [9] (Paper-I) we presented a detail calculation of the effects of matter inside the presupernova star on the neutrino fluxes, using a formalism that takes into account the three neutrino flavors and different density profiles for the presupernova star. Our results show that for neutrinos with E ν ≤ 10 TeV the fluxes on the surface of the star are different from the original one 1:2:0.
We have also calculated the fluxes of the these neutrinos on the surface of the Earth after they travel through the long baseline between the source and the Earth. We found that for neutrino energy E ν ≤ 10 TeV, the flux ratio is different from 1:1:1 and above this energy the ratio converges to 1:1:1 implying that matter effect does not play a significant role for high energy neutrinos.
The IceCube neutrino detector in South pole is fully operational since December 2010.
The IceCube collaboration has reported the observation of 37 neutrino events in the energy range 30 TeV-2 PeV and the sources of these events are unknown [10][11][12]. These neutrino events have flavors, directions and energies not compatible with the atmospheric neutrinos and it is believed that this is the first indication of extraterrestrial origin of high energy neutrinos. Recently, IceCube collaboration has presented results of 641 days data taken during 2010-2012 in the energy range 1 TeV-1 PeV from the southern sky which gives a new constraint on the diffuse astrophysical neutrino spectrum [13]. These high energy neutrino events have generated much interest and several models are proposed for their origin. The choked GRBs are potential candidates to produce the high energy neutrinos which can propagate hundreds of Mpc baseline to reach the Earth. So it is important to study these neutrinos and the matter effect on their propagation. The present work is an extension of Paper-I. Here we take into account the matter effect of both presupernova star medium and the Earth on the calculation of the flux ratio by a detector like IceCube which could be relavent to get information regarding the type of progenitor responsible for the choked GRBs. We also take into account the shadowing effect of Earth on these neutrinos.
The organization of the papers is as follows: In Sec.2 we discuss about the neutrino propagation in the Earth by considering the realistic density profile of it. Here we also take into account the shadowing effect which is important for high energy neutrinos. In Sec. 3, the signature of shower and track events are discussed. The detailed calculation of track-toshower ration is discussed in Sec. 4. Finally we present our results in Sec. 5 followed by a summary in Sec. 6.

II. MATTER EFFECT ON NEUTRINOS GOING THROUGH THE EARTH
The energy spectra of the gamma rays produced by long GRBs have been measured and they follow power laws, or broken power laws [14]. In the GRB jet (both successful and choked), neutrinos are produced with varying energy depending on the distance from the central engine. The one which are closer to the central engine are in the MeV range and it increases as the distance increases. This happens because the protons are Fermi accelerated within the jet and gain energy as the distance increases up to a maximum, where neutrinos of ∼ EeV energy can be produced. In this environment the high energy γ-rays and neutrinos are produced through pp and/or pγ interaction within the jet environment and the fluxes of these GeV-TeV neutrinos and the γ-rays are related. Both the γ-rays and the neutrinos have power-law spectrum. Here we assume a simple power-law spectrum for the high energy neutrinos as: where α ≥ 2 is the spectral index and N ν l is the normalization constant in units of High energy neutrinos reaching the detector on Earth from the opposite side can experience absorption due to neutrino-nucleon CC and NC interactions. For very high energy neutrinos the interaction cross sections are large enough so that the absorption effects become very important and have to be taken into account. The shadowing factor due to this absorption is given by [15]: where σ T OT is the total neutrino-nucleon cross section, N A = 6.0221 × 10 23 mol −1 is the Avogadro's number, and X is the column depth traveled by the neutrino inside the Earth before interaction. The column depth is the product of the distance traveled and the density of matter inside the Earth ρ e . Since the Earth's density depends on position, ρ e = ρ e (r) and X is given by: where the integral is a path integral along the trajectory of the neutrino, from the entrance point to the Earth up to the detector, and can be parametrized in terms of the zenith angle θ of the neutrino track at the detector. The cross section σ T OT is a function of the neutrino energy E ν . Then the shadowing factor P shad depends on both E ν and θ and can be expressed as P shad = P shad (E ν , θ). We consider the most realistic density profile of the Earth, which is given by [15]: where x = r/R Earth and ρ e is given in units of g/cm 3 . The Earth density profile is shown in The values of the total cross sections, for neutrino and antineutrino interaction with matter (nuclei) at high energies, have to be extrapolated from low energy data, since no measurements have been performed yet. In this work we use the cross sections reported in Ref. [16] and present in Figs. 2 and 3 respectively for ν − N andν − N. Comparison of the total cross sections ν − N andν − N shows that in the low energy limit E ν ≤ 10 TeV there is a very small difference between these two which can be seen in Fig. 4.  In Fig. 5, P shad is plotted as a function of E ν , for a zenith angle θ = 180 • (neutrinos arriving to the detector from underneath). From the graph it can be noticed that the shadowing factor decreases as the neutrino energy increases beyond ∼ 1 TeV and the Earth becomes opaque for neutrinos with energies above ∼ 1000 TeV. There is a small difference between neutrino and antineutrino shadowing factor above 1 TeV. Since we are interested in TeV neutrinos, the shadowing effect has to be taken into account properly in the calculation of neutrino fluxes arriving at the detector. Depending on the energy of the neutrinos, the interaction of the neutrinos with the medium inside the Earth will also result on flavor oscillations. Since in this work we will account for those neutrinos that go through the Earth before undergoing deep inelastic collision with the surround medium to the detector, we must take into account the flavor oscillation. In Paper-I we have already used the analytic formalism developed by T. Ohlsson and H. Snellman (OS) to calculate three-flavor neutrino oscillations [18,19] in the presupernova star [9] and then calculate the flavor ratio of neutrinos arriving on Earth. Here we are extending the calculation by taking into account the matter effect of the Earth to calculate the flavor ratio at the IceCube detector. For this calculation we use the Earth density profile are described in detail in [9].
The input neutrino fluxes at the surface of the Earth, as functions of neutrino energy E ν , are those calculated in Paper-I, for three different models of the presupernova star, which we will refer to as model A, B and C and are discussed throughly in Paper-I. For reference we present the density profile of these three models in Fig. 6 and a detail description is given in paper-I [9]. In Figs Table I. We also consider two sets of parameters Set-I and Set-II corresponding to two different presuprenova star radii R * as shown in Table I and analyze our results.   the interacting neutrino is an electron type, the resulting electron will quickly interact with the medium, producing an electromagnetic shower, which will overlap with the hadronic shower. If the neutrino is muon type, the resulting muon will produce a long track that emerges from the shower. Finally, if the neutrino is tau type, the resulting tau lepton may or may not produce a track depending on its energy. But when the tau decays into muon, τ → ν µ µ ν τ the later will produce a long track, just like in the case of a muon-neutrino CC interaction, this modifies the number of track events, which has to be accounted for.
Since in this work we are considering neutrinos coming from underneath the detector, those with energies above 1 PeV will be drastically suppressed, and therefore the lollipop and double-bang events that are associated with very energetic ν τ will also be suppressed [21]. In  (14), and the Monte Carlo results presented in reference [23].
this work we will not consider these kinds of events, however, we will include the µ-track events induced by tau neutrinos, as explained above.
In conclusion, the ratio of track events to shower events is related in a convoluted way to the neutrino flavor ratios. However, given a set of flavor ratios, like 1:1:1 in the "standard picture", or any other set, like in the case we are presenting in this work, the ratio of tracksto-showers R can be calculated. In the next section we discuss in detail the track-to-shower ratio calculation.

IV. THE TRACK-TO-SHOWER RATIO
The calculation of the track-to-shower ratio R presented in this section is based on the calculations from references [21,22]. Here we have included the shadowing effect due to the neutrino absorption by the Earth, P shad (E ν , θ). Since we are considering neutrinos coming from underneath, θ = 180 o , then P shad (E ν ) = P shad (E ν , θ = 180 o ). The ratio R is defined as: R = Number of µ-track events Number of shower-like events .
The µ-track events have two components: N µµ from µ-tracks induced by muon neutrinos, and N µτ from µ-tracks induced by tau neutrinos. The number of shower-like events have three components: N sh had from hadronic showers associated with NC interaction, N shem from electromagnetic showers produced by CC interaction of ν e and N shτ from showers produced by CC interaction of ν τ decaying hadronically. So we can express R as The µ-tracks induced by ν µ (ν µ ) result from the CC interaction of the neutrinos with the rock or the ice underground. The muons can travel a long distance before decaying; the effective muon range R µ depends on the initial energy E µ and the detection energy threshold E th µ ; in the case of IceCube this threshold is ∼ 100 GeV. The µ-track induced by ν τ (ν τ ) result from the decay of a τ produced in a CC interaction into a µ; this decay has a probability density f (E τ , E µ ) and a branching ratio B = 17.8%. The expressions for N µµ and N µτ are given by where the muon range is defined as and its probability density is given by The expression for f (E τ , E µ ) is an approximation valid for β → 1 (γ ≫ 1), where β = The number of shower-like events for the different kinds of processes are given by: where ρ is the density of the detector medium, A is the effective area of the detector, L is the length of the detector, N A is the Avogadro's number and dF ν l /dE ν l is defined in Eq. (1). The normalization for this equation, N ν l , is proportional to the neutrino flux, for the different flavors. Since dF ν l /dE ν l is evaluated in the quotient of equation (5), the proportionality constant cancels out. The total cross sections for CC (σ CC ) and NC (σ N C ) shown in Figs. (2) and (3) are used to evaluate the N sh had and N shem .
In order to evaluate dσ CC /dE l we performed an empirical fit to the differential cross section presented in Fig. 4 of reference [23], which is given as: where N 0 is the normalization, and The parameters in Eq. (14) are as follows: and x = log 10 (E ν l /GeV).
The normalization is set such that We compare our fit with the data presented in reference [23] which are shown in Fig. 9.
After performing the necessary change of variable from E l to y, one can evaluated the integrals numerically. The neutrino-flavor ratios, R, obtained after propagating the neutrinos from the source, all the way up to the detector, for different combinations of the parameters involved, and for different energies are used as input for the calculation.

V. RESULTS
As can be seen from Figs In Figs. 12 to 15, we have shown R as a function of the spectral index α for models A and C. In these figures we also include no matter effect which implies: at the source we consider the flux ratio 1:2:0 and these neutrinos propagate up to the detector in vacuum. For our convenience we define the track-to-shower ratio for no matter effect as R 0 . For δ CP = 0 we found that R 0 ≤ R for any given value of α. Also the gap between R and R 0 is small. On the other hand, for δ CP = π, always we found R 0 > R and the gap is bigger. The value of R is minimum around α = 2.6 which is independent of whether we consider matter effect or not. We have also shown for three different ∆m 2 32 values, which shows that there is very little variation in R. This minimum value of R is also independent of ∆m 2 32 . The order in which R is arranged for different ∆m 2 32 values reverses by going from δ CP = 0 to π, which can be seen by comparing Fig. 12 with Fig. 13 in model A and similarly Fig. 14  In Figs. 16 to 19 we have shown the variation of R as a function of sin 2 θ 13 in models A and C for three different values of the spectral index α. In these plots we observe that the ratio R is almost constant for a given α and for both δ CP = 0 and π, as we vary sin 2 θ 13 for all the models. Also the value of R is higher for smaller α.
We have also shown the R as a function of sin 2 θ 13 for no matter effect in Fig.20. This shows a clear difference between δ CP = 0 (lower curve) and δ CP = π (upper curve) for each α. These two curves diverge from the point θ 13 = 0 as can be seen from the plots in

VI. SUMMARY
A very small fraction (≤ 10 −3 ) of the core collapse supernovae can produce GRBs by launching a successful jet. Although the majority of these core collapse can not produce GRBs, very high energy neutrinos can easily be produced in their choked jets. These neutrinos propagating through the over burden matter can undergo oscillation and the flux ratio on the surface of the star can be different from the point where these neutrinos were produced. The Mpc long baseline, from the surface of the star to the surface of the Earth, these neutrinos will have vacuum oscillation. Before reaching to the detector from the opposite side of the Earth, these neutrinos will cross the dimeter of the Earth and can undergo again matter oscillation. By considering the realistic density profile of the Earth we have extended our previous work to study numerically the three neutrino oscillation and evaluate the change in the flux ratio in the detector. Depending on the energy of these neutrinos, there can also be shadowing effect and neutrinos above few PeV can be completely absorbed.
In this work we have done a through analysis of the high energy neutrino propagation in the Earth before reaching to the detector by taking into account the shadowing effect. The track-to-shower ratio R is calculated for these high energy neutrinos. In the calculation of R  we have included the shadowing effect and the contribution of muon track produced by the high energy τ lepton decay around the IceCube detector. These τ leptons are produced due to the CC interaction of ν τ with the surround rock and ice of the detector. We have studied the variation of R when the spectral index α and the mixing angle sin 2 θ 13 vary. We found   16. The track-to-shower ratio R as a function of sin 2 θ 13 in Model-A for the parameter Set-I with ∆m 2 32 = 3.2 × 10 −3 eV 2 . The black curve is for δ CP = 0 and red one is for δ CP = π.      19. The track-to-shower ratio R as a function of sin 2 θ 13 in Model-C for the parameter Set-II with ∆m 2 32 = 3.2 × 10 −3 eV 2 . The black curve is for δ CP = 0 and red one is for δ CP = π.   20. The track-to-shower ratio R as a function of sin 2 θ 13 with no matter effect. Hre also the black curve is for δ CP = 0 and red one is for δ CP = π. We take ∆m 2 32 = 3.2 × 10 −3 eV 2 .
that R has a minimum around α = 2.6 and is independent of whether we consider matter effect or not. This minimum value of R is also independent of ∆m 2 32 value. We observed that the ratio R is different for δ CP = 0 and π when no matter effect is considered. But when Earth matter contribution is taken into account, the R value is almost blind to these different CP phases.
S.S. is thankful to Departamento de Fisica de Universidad de los Andes, Bogota, Colombia, for their kind hospitality during his several visits. This work is partially supported by