Effects of Majorana Physics on the UHE $\nu_{\tau}$ Flux Traversing the Earth

We study the effects produced by sterile Majorana neutrinos on the $\nu_{\tau}$ flux traversing the Earth, considering the interaction between the Majorana neutrinos and the standard matter as modeled by an effective theory. The surviving tau-neutrino flux is calculated using transport equations including Majorana neutrino production and decay. We compare our results with the pure Standard Model interactions, computing the surviving flux for different values of the effective lagrangian couplings, considering the detected flux by IceCube for an operation time of ten years, and Majorana neutrinos with mass $m_N \thicksim m_{\tau}$.


I. INTRODUCTION
The discovery of neutrino flavor oscillations still remains as one of the most compelling evidence for physics beyond the Standard Model (SM ). While many proposals have been posed to explain the tiny ordinary neutrino masses, the seesaw mechanism stays as one of the most straightforward ideas for solving the neutrino mass problem [1][2][3][4][5][6]. This mechanism introduces right handed sterile neutrinos that, as they do not have distinct particle and antiparticle degrees of freedom, can have a Majorana mass term leading to the tiny known masses for the standard neutrinos, as long as the Yukawa couplings between the right handed  [7][8][9]. Thus, as suggested in [9], the detection of Majorana neutrinos (N ) would be a signal of physics beyond the minimal seesaw mechanism, and its interactions could be better described in a model independent approach based on an effective theory, considering a scenario with only one Majorana neutrino N and negligible mixing with the ν L .
On the other hand, in the recent years the observation of ultra high energy (UHE) astrophysical neutrinos in the IceCube telescope [10], with a yet unknown specific origin, spectral shape and flavor composition, as well as the non-finding of tau neutrinos in primary searches performed within these data [11], raise the question on tau-neutrino detection in neutrino telescopes, and make plausible to surmise that this might be connected to the neutrino mass problem.
In addition, anomalies found in short baseline (SBL) neutrino oscillation experiments [12][13][14][15] have driven the introduction of light, almost sterile neutrinos, that mix poorly with the known light mass states and could help to accommodate the data introducing a third mass splitting. The IceCube collaboration has searched for these sterile neutrinos [16] probing light sterile neutrino 3 + 1 models [17,18], and recently led to new bounds for the sterile-active muon-neutrino mixing.
As the mixing parameters for the second fermion family with a sterile Majorana neutrino ν µ −ν s are strongly constrained within the framework of 3+1 scenarios [7,19], and motivated by the intriguing lack of tau neutrinos in the UHE cosmic flux in IceCube, in this work we study the possibilities that UHE tau-neutrinos from astrophysical sources may provide a signature for Majorana neutrino production by giving a surviving flux after traversing the Earth which may differ from the standard one. In particular the non-observation of ν τ going up signal could be a manifestation of a sterile neutrino modifying the ν τ flux.
Many recent works have studied the IceCube UHE astrophysical events with explanations involving dark matter models with right-handed neutrinos [20][21][22] and non standard effective interactions [23], and also the chances to probe sterile-tau neutrino mixings have been considered [24]. The effects of dark matter and new physics on ν τ propagation has been studied by our group in refs. [25][26][27], and the Majorana neutrino effective phenomenology regarding the relevant N decay modes and interactions is treated in [28,29].
In Sec.II, we briefly describe the effective operator approach and the bounds on the effective couplings we take into account. In Sec.III A we discuss the relevant processes and the results obtained for the cross-sections and decay rates. In Sec.III B, we review the passage of high energy tau neutrinos through the Earth using transport equations including the effects of Majorana neutrinos. We solve these equations taking into account the neutral current regeneration and the regeneration by the decay of the Majorana neutrino, for different values of ζ which, as we will define in the next section, is a combination of the effective couplings and the energy scale associated with the new interactions. This enables us to compare the surviving flux with the one obtained using SM physics only. In particular we go to a Majorana mass region where the couplings are less constrained maximizing the effect on the survival flux. Finally, in Sec.IV we present the results and in Sec.V a short discussion with our conclusions.

A. Effective operators and lagrangian
In this paper we study the effects of the possible existence of a heavy sterile Majorana neutrino N in the ν τ propagation through the Earth. Being the N a SM singlet, its only possible renormalizable interactions with SM fields involve the Yukawa couplings. But as we discussed in the introduction, these couplings must be very small in order to accommo-date the observed tiny ordinary ν masses. In this work we take an alternative approach, considering that the sterile N interacts with the light neutrinos by higher dimension effective operators, and take this interaction to be dominant in comparison with the the mixing through the Yukawa couplings. In this sense we depart from the usual viewpoint in which the sterile neutrinos mixing with the standard neutrinos is assumed to govern the N production and decay mechanisms [30,31].
We parameterize the effects of new physics by a set of effective operators O constructed with the standard model and the Majorana neutrino fields and satisfying the SU (2) L ⊗U (1) Y gauge symmetry [9]. The effect of these operators is suppressed by inverse powers of the new physics scale Λ, which is not necessarily related to the Majorana neutrino mass m N .
The total lagrangian is organized as follows: For the considered operators we follow [9] starting with a rather general effective lagrangian density for the interaction of right handed Majorana neutrinos N with bosons, leptons and quarks. We list the dimension 6 operators that can be generated at tree level or one-loop level in the unknown fundamental ultraviolet theory, and are baryon-number conserving.
The first subset includes operators with scalar and vector bosons (SVB), and a second subset includes the baryon-number conserving 4-fermion contact terms: where l, u, d and L, Q denote, the right handed SU (2) singlet and the left-handed SU (2) doublets, respectively. The following one-loop level generated operators coefficients are naturally suppressed by a factor 1/16π 2 [9,32]: In order to study the effects on the ν τ propagation through the Earth due to the existence of Majorana neutrinos N , we consider the dominant processes responsible for the change in the ν, τ and N fluxes. Besides the SM processes involving ordinary neutrinos, we have new contributions related with the production and scattering of the Majorana neutrinos N interacting with matter nucleons (N ): Also, we will take into account the Majorana neutrino decay contribution to the different fluxes. For the low Majorana neutrino mass region, the dominant decay was found to be N → γ ν [28]. For completeness we include in Fig.1 a plot with the N Branching Ratios in the low mass region.
The shown reactions (5) contribute to different terms in the transport equations to be presented in Sec.III B, where the relative relevance of the different terms for the considered mass region will be discussed.
In order to obtain the above interactions we derive the effective lagrangian terms involved in the calculations, taking the scalar doublet after spontaneous symmetry breaking as φ = 0 v+h √ 2 . We have contributions to the effective lagrangian coming from (2), related to the spontaneous symmetry breaking process: (6) and the four-fermion interactions involving quarks and leptons from (3) In Eqs. (6) and (7) a sum over the family index i, j is understood, and the constants α are associated to specific operators: In this work we allow for family mixing in the interaction involving two or more different SM leptons.
The one-loop generated operators are suppressed by the 1/(16π 2 ) factor but, as we show in [28], these play a major role in the N -decay. In particular for the low m N range studied here, the dominant channel N → νγ is produced by terms coming from the operators in (4) where P (a) is the 4-moment of the incoming a-particle and a sum over the family index i is understood again. The constants α (i) L j with j = 1, 3 are associated to the specific operators: The complete lagrangian for the effective model is presented in an appendix in our recent work [29].
In order to maintain the discussion as simple as possible we will consider the contributions where v = 250 GeV represents the Higgs field vacuum expectation value. Previous analysis [34,35] refer in general to similar heavy neutrino-standard boson interaction structures that modify the weak currents and lead to variations in the weak bosons decay rates, and W and Z mediated processes involved in the existing experimental tests:  (2) lead to a term in the effective lagrangian (6) that can be compared to the interaction in (12). The relation between the coupling α (i) W and the mixing U lN was derived in [9]: U lN . In our current notation this would be: The couplings ζ SV B can be bounded taking into account LEP and τ lepton universality tests results. We consider the LEP bounds on single Z → νN and pair Z → N N sterile neutrino production searches [36]. Conservative limits for any m N mass [36] are This result is model-independent and holds for the production of a single and a pair of heavy neutral objects decaying into a photon and a light invisible particle.
For the decay Z → N N , we have a direct contribution from the tree level operator For the low m N values considered in this work, we can take Br(N → ν(ν)γ) 1 and then the corresponding bound is The process Z → νN has no contributions from the operators O SV B .
Another observable that can put restrictive bounds on the O SV B is the universality test from the τ -decay, in the mass range m µ ≤ m N ≤ m τ . Following [7] we define the quotient R τ in the effective model as The function g( is the SM result for the τ → ν τ + eν and µ → ν µ + eν decays, with y τ = (m e /m τ ) 2 and y µ = (m e /m µ ) 2 respectively. The τ → N + eν with y N = (m N /m τ ) 2 . The observed value for the quotient is R obs τ = (1.349 ± 0.004) × 10 6 [7,37]. This imposes stringent bounds on both the SV B ζ W and the 4-fermion ζ S0 couplings, but if we take the mass of the sterile N to be right below m τ , these bounds can be relaxed because the partial decay width is kinematically canceled.
We study now the case of the effective 4-fermion interactions. Here we have again contributions to the LEP process e − e + → νN and e − e + → N N but, in our case without the Z resonance, and then we expect weaker bounds than those imposed on the ζ SV B couplings.
We consider first the reaction e + e − → νN calculated at the Z-pole where Br(Z → e + e − ) = 3.4 × 10 −2 and the upper bound for the Branching ratio for the channel Z → νN obtained from (15) is Br(Z → νN ) ≤ 2.7 × 10 −5 [36]. Thus, we have and, as the 4-fermion contribution to the cross section is the bound for the corresponding coupling is ζ S0 2.85 × 10 −1 .
On the other hand, we have the reaction e − e + → N N with the bound obtained by LEP and shown in (14). Using the general expression for the cross section at the Z-pole and in the low mass limit where Br(N → ν(ν)γ) = 1 we have  In the case of 1-loop operators we have contributions to the Z-decay Z → νN and with the experimental bound for the Branching ratio [36] Br we obtain Γ(Z → νN ) ≤ 6.7 × 10 −5 , and thus For the Lepton-Flavor-Violating processes e.g. µ → eγ, µ → eee and τ → eee, induced by the quantum effect of the heavy neutrinos, we have very weak bounds for m N < m W [33,38].
In the case of the heavy Majorana neutrino with effective interactions we are considering, the clear dominance of the neutrino plus photon decay channel, as we show in Fig.1   region. The expressions for the partial widths are presented in [28]. The labels represent

A. Relevant processes
In this section we study the different reactions taking place in the transport of tauneutrinos in their journey through the Earth. We classify the produced effects as absorption and regeneration processes.
Absorption effects are all the processes that take out of the flux tau-neutrinos of energy E, and regeneration effects are those adding tau-neutrinos with energy E to the flux. We must consider that beside ordinary neutrinos we have Majorana neutrinos and tau-leptons produced by the former when they pass through the Earth.
The standard interactions of ordinary neutrinos with the nucleons N forming the Earth are ν τ N → l ± X and ν τ N → ν τ X. Here the charged-current and neutral-current reactions contribute to the absorption effects of ordinary tau-neutrinos and neutral-current reactions contribute to the regeneration effects as we will discuss in the next section.
The production of Majorana neutrinos is driven by the collision of ordinary neutrinos with nucleons in the Earth, ν τ N → N X. This reaction absorbs ordinary tau-neutrinos producing Majorana neutrinos.
We also have to take into account the interaction of Majorana neutrinos with the nucleons forming the Earth, N N → N X, N N → l ± X and N N → ν τ X. In the same way that for the ordinary neutrinos, these reactions produce absorption effects of Majorana neutrinos, as well as regeneration effects for ordinary neutrinos and charged leptons.
Finally we have the Majorana neutrino dominant decay: N → ν γ causing Majorana neutrino absorption and ordinary neutrino regeneration, and the standard τ -decay leading to τ absorption and ν τ regeneration.
As we will discus in the next section, only some processes are relevant for the ν τpropagation in the Earth. We do not show explicitly expressions for the different crosssections because it is a standard calculation. We prefer to show the results as plots for the interaction and decay lengths for the SM scatterings ν τ N → lX, ν τ N → ν τ X, the τ -decay, the Majorana production process ν τ N → N X, and the N -decay, as they give the relevant contributions to the transport equations. In Fig.2  These lengths are defined in terms of the associated process cross-sections as: Here ρ n is the average number density along the column depth on the path with inclination θ with respect to the nadir direction as it will be shown in (43). The number density is defined as ρ n = N a ρ where N a is the Avogadro number and ρ is the Earth mass density. The decay functions Σ N (τ ) are defined in (38).
In order to take into account the contribution of the N decay to the regeneration through the N → ν τ γ channel, we follow the approach of Gaisser [39] which is developed in the appendix A. We also take into account the regeneration effects coming from the τ -decay according to expressions obtained by Gaisser [39] and also shown in [40]. In Fig.2b we show for comparison the interaction length for the other invloved interactions.
In the following section we will discuss the relative importance between the different contributions to the ν τ propagation. The neutrinos traveling through the Earth may suffer charged current (CC) and neutral current (NC) interactions with the nucleons in their path (see [41,42] and references therein).

L/REarth
Neutrino oscillation within the Earth can be neglected for energies higher than 1 TeV [43].
As we mentioned above, the change in neutrino flux Φ ντ (E, χ) as it traverses the Earth can be divided into two effects: absorption and regeneration. Absorption is a decrease in the neutrino flux of a given energy. In the SM we have the total cross-section σ SM tot ντ (E) = σ CC (E) + σ NC (E), which represents a probability of CC or NC standard τneutrino interactions. When neutrinos pass through an amount of matter dχ = ρ n (z)dz in a distance dz along the neutrino beam path, where ρ n (z) is the Earth's number density, the change in the flux Φ ντ (E, χ) due only to absorption is proportional to Φ ντ (E, χ) and to the cross section: Here χ(z) is the amount of material found up to a depth z, that is, where the number density is the Avogadro's constant times the density, ρ n (z ) = N A ρ(z ).
In order to consider the complete transport effect for UHE neutrinos, we have to add to (29) the effect of regeneration, which accounts for the possibility that neutrinos of energies E > E may end up with energy E due to NC interactions with the nucleons, adding neutrinos to the flux of energy E. Then, the SM transport equation for neutrinos reads Here the usual change of variables y = (E − E)/E has been made.
On the other hand, taking into account the sterile Majorana neutrinos production and decay processes, we have new contributions to the absorption and regeneration effects on the ν τ -flux. Moreover, as it is well known, if we consider the ν τ transport, it is important to take into account the transport of the τ -lepton which regenerates the ν τ by τ -decay. In the same way, the Majorana neutrino decay can regenerate the ν τ flux. So in principle one needs to simultaneously solve a system of three coupled integro-differential equations: In (32) the right hand side terms correspond to the absorption, neutral-current regeneration, charged current regeneration by τ interaction, regeneration by Majorana neutrino interaction, and regeneration by τ and N decays.
The Eq.(33) corresponds to the τ transport with absorption by interaction and decay, and regeneration by τ neutral current, ν τ charged current and by the interaction of Majorana neutrino through the N N → τ X reaction. The third term represents the energy loss due to electromagnetic interactions.
In the Eq.(34) we have absorption terms by interaction and by N decay. The other terms represent the N -flux regeneration by the ν τ , τ and N interactions with nucleons.
In the above equations the cross-sections in the absorption by interaction terms are: and σ t N (E) and σ t τ (E) include all the Majorana neutrino interactions.
The different cross-section regeneration kernels are: The decay kernels for the Majorana neutrino N or τ -lepton are calculated in the appendix A and in [40] respectively: and the decay-length functions are where T N (τ ) = (Γ tot rest N (τ ) ) −1 is the N (τ ) lifetime in its rest frame, with as for the N low mass range the dominant decay is N → νγ.
Some of the terms in the equations above can be neglected in the considered energy range.
The τ and N interactions are neglected against their decays. The τ interactions begin to be dominant at an energy around E ντ = 10 8 GeV [40]. For the N interactions, the contributions of the different processes are compared in Fig.2 as a plot for the ratio between the interaction and decay lengths and the Earth radius. We neglect the regeneration terms coming from the τ and N interactions, which are proportional to the τ and N flux, in comparison with those due to NC interactions of ν τ and those originated in the τ and N decay. In these conditions, for the ν τ transport equation (32) we take into account the absorption and neutral-current regeneration terms and the regeneration by τ and N -decay. For the τ -transport equation, we consider absorption by τ -decay and regeneration by ν τ scattering by nucleons, which is the source of τ leptons appearance. In the case of the Majorana neutrino transport equation, absorption by N -decay and also regeneration by ν τ scattering by nucleons are included, the last process being the source for the N -flux.
Finally, the equations we need to solve are: The system of transport equations (Eqs.40-42) must be solved with the initial conditions Taking the column depth on the path with inclination θ respective to the nadir direction taken from the down-going normal to the neutrino telescope as T (θ): with R as the Earth radius, we define ρ n as the average number density along the column The Earth density is given by the Preliminary Reference Earth Model (PREM) [44]. In Fig.3 we present the Earth density profile. In accordance with [45], and following the treatment made in [26,40,46], we solve Eqs. (41) and (42) considering the terms dependent of the ν τ flux Φ ντ (E, χ) as non-homogeneities, and replace those solutions in (40), dividing by Φ ντ (E, χ). Then we make an approximation, taking the fluxes quotient (45) as the ones solving the corresponding homogeneous equations [45]. Finally, we write the solution for the surviving τ -neutrino flux traversing a path of length T (θ) through the Earth in terms of σ eff (E, T (θ)) as: with Here E yy = E/((1 − y)(1 − y )) and the flux quotients are with

IV. NUMERICAL RESULTS
In this section we present our results assuming a Majorana neutrino contribution with m N ∼ m τ . In order to obtain numerical results for the surviving ν τ flux including Majorana neutrino effects, we consider a particular choice for the effective coupling constants ζ, with the upper values presented in (27). Also, for the initial ν τ flux we have considered an astrophysical flux of the order of the IceCube sensitivity E 2 Φ ντ = 1 × 10 −8 GeV cm −2 s −1 sr −1 [10].
The idea is to see whether the effect of the Majorana neutrino modifies the ν τ surviving flux and to what extent it should be distinguishable from the standard surviving flux when a detection is performed in a neutrino telescope, which will clearly depend on the uncertainty involved.
First we compare the surviving ν τ flux with Majorana neutrino effects, taking into account both absorption and regeneration, with the SM prediction. In Fig.4, we show the comparison with the SM , showing the quotient R(θ, E ντ ) = Φ ντ /Φ SM ντ for different nadir angles θ. In order to calculate the capability of IceCube to detect the effects of Majorana neutrinos physics, we have considered an aproximated number of events as given by where n T is the number of target nucleons in the effective volume, and the cross section includes neutral and charged currents, in order to consider both cascade and track events.
The function Φ ντ (E, θ) is the τ -neutrino flux in the vicinity of the detector. We consider the neutrino telescopes are planned to be working in the Northern hemisphere. In particular the European project KM3NeT [47][48][49], originated in the projects ANTARES, NEMO and NESTOR will be installed in the Mediterranean sea with an instrumented volume of several cubic kilometers. This telescope along with the Baikal-GVD upgrade [50,51] will improve the statistics, increasing the significance of the observations to bound new physics effects as the ones we discussed in this work.
We first integrate over θ ν and next we integrate over x in the interval (x min , x max ) with x min = z/(1 + β N ) and x max = min(1, z/(1 − β N )), obtaining For the low mass range considered in this work the clearly dominant decay channel is the neutrino plus photon mode, and Γ tot LAB (E) = i=e,µ,τ Γ N →ν i γ LAB (E). Then we consider the ν τ decay channel, leading to the final ν τ neutrinos distribution in the laboratory frame: Thus, after the indicated integrations in the evolution equations, the useful expression that we obtain is where z = 1 − y, x 0 = 1/2 and P = +1 for the right-handed Majorana neutrinos.