Cosmogenic neutrino fluxes under the effect of active-sterile secret interactions

Ultra High Energy cosmogenic neutrinos may represent a unique opportunity to unveil possible new physics interactions once restricted to the neutrino sector only. In the present paper we study the observable effects of a secret active-sterile interactions, mediated by a pseudoscalar, on the expected flux of cosmogenic neutrinos. The results show that for masses of sterile neutrinos and pseudoscalars of hundreds MeV, necessary to evade cosmological, astrophysical and elementary particle constraints, the presence of such new interactions can significantly change the energy spectrum of cosmogenic neutrinos at Earth in the energy range from PeV to ZeV. Interestingly, the distortion of the spectrum results to be detectable at GRAND apparatus if the scalar mediator mass is around 250 MeV and the UHECRs are dominated by the proton component. Larger mediator masses or a chemical composition of UHECRs dominated by heavier nuclei would require much larger cosmic rays apparatus which might be available in future.


I. INTRODUCTION
The neutrino sector still represents a partially unknown territory. Fundamental questions like the nature of neutrinos (Dirac or Majorana) or the possible connection of their small masses with physics Beyond the Standard Model (BSM) can represent possible windows on new physics. In the last decade increasing attention has been devoted to high energy astrophysical neutrinos, after the observation of the first events at the IceCube detector [1]. These astrophysical fluxes are of extreme relevance for the neutrino sector since they provide a powerful tool of investigation for Beyond Standard Model physics, such as sterile neutrinos, Lorentz violations and non standard model interactions. In this context, cosmogenic neutrinos, that are mainly expected to have energies up to 10 12 GeV, could represent a unique opportunity.
Cosmogenic neutrinos are produced by the photohadronic interactions of Ultra High Energy Cosmic rays (UHECrs), whose precise chemical composition and origin is still unknown, with the Cosmic Microwave Background (CMB) [2]. The same process is also responsible for a depletion in the flux of UHECRs, which is known in literature as the GZK cutoff [3,4]. For this reason cosmogenic neutrinos are also known as GZK-neutrinos and they have been extensely studied in a number of works, see for instance .
The desirable observation of high-energy cosmogenic neutrinos would be particularly relevant in order to determine the origin of UHECRs. Several cosmic rays apparatus, like HiRes [30] and Auger [31,32] for example, have already tried to perform such a measurement, and others are planned to do it in future with much better chances, like for instance GRAND [33] and JEM-EUSO [34]. Unfortunately, a possible additional difficulty lies in the fact that cosmogenic neutrino fluxes have a strong dependence on the chemical composition of cosmic rays, as shown for instance in [35]. This is particularly important since recent results suggest that the chemical composition is actually mixed, containing significant amounts of heavier nuclei rather than simply protons [36], and unfortunately for heavier nuclei one expects a suppression in the neutrino production. For this reason, in this work we have analyzed two benchmark scenarios for the chemical composition of the UHECRs, namely the cases in which the dominant component is either protons or Helium nuclei.
In view of the above considerations a future detection of cosmogenic neutrinos would allow us to infer crucial astrophysical informations concerning UHECRs chemical composition, interactions and origin. However, at the same time the shape of the cosmogenic neutrino spectrum, that could be strongly distorted by the presence of new physics, could unveil the presence of these new interactions at least in the neutrino sector. In particular, any kind of BSM interactions involving active neutrinos ν could modify the expected spectrum and rate of cosmogenic neutrinos at Earth. Examples of such new couplings discussed in literature are: i) Non Standard Interactions (NSI) for active neutrinos νν → f f [37][38][39][40][41][42][43][44][45][46][47][48], where f denotes quarks or charged leptons; ii)Secret Interactions (SI) mediated by some new boson (scalar or vector), and just restricted to the active neutrino sector νν → νν [24,[49][50][51][52][53][54][55]; iii)Secret Interactions just restricted to the sterile neutrino sector ν s ν s → ν s ν s for different sterile neutrino mass scales, [56][57][58][59][60][61][62][63][64][65][66][67][68]. iii) Secret interactions involving active and sterile neutrinos simultaneously νν → ν s ν s [69] In this paper we consider a different scheme of SI, where in the new interaction, mediated by a new pseudoscalar boson, intervene both active and sterile neutrinos. In particular we study the distortion implied by such new coupling on the expected cosmogenic neutrino flux arXiv:2002.10125v1 [hep-ph] 24 Feb 2020 estimating the possibility to measure this effect in apparatus like GRAND [33]. For seek of simplicity we will assume just one sterile neutrino (hereafter denoted by ν s ) coupling with the active sector via this new interaction. Moreover, with the aim to catch the main implications of such scenario on cosmogenic neutrino flux we consider an active neutrino only (hereafter denoted by ν), hence focussing on a more simple 1 + 1 framework. However it would be straightforward to extend our analysis to three active neutrinos even though we do not expect that the main results, here obtained, would be drastically changed in a more realistic 3 + 1 framework. The interaction term then becomes where λ is a dimensionless free coupling.
II. COSMOGENIC NEUTRINO FLUX AT EARTH WITHOUT SECRET INTERACTIONS As described above, cosmogenic neutrinos are produced by the scattering of high energy protons from the cosmic rays with the CMB photons. The cosmogenic neutrino flux φ ν , expected to be isotropic, can be parameterized in the form where F [z , E(1 + z )] is the number of neutrinos produced per unit time per unit energy interval per unit solid angle per unit volume at redshift z and with comoving energy E(1 + z ). The quantity F [z , E(1 + z )] depends of course on the proton spectrum, which is itself the solution of a Boltzmann equation (see [35]), which takes into account the proton energy losses due to Bethe-Heitler processes and their depletion due to pγ processes. This calculation has been performed for example in [35], which provides the neutrino spectrum expected at Earth. It is important to notice that they assume a cosmic ray spectrum purely made of protons and emitted by sources whose density follows the star formation evolution. In other words, the proton luminosity can be written as is the proton injection spectrum from the single sources and H(z) is the Star Forming rate [70] where N 1 = 2 3.7 and N 4 = 5 3.2 . The proton luminosity L p (z, E) works as an input to the Boltzmann equation which provides the function F [z , E(1 + z )] in Eq. (2). Since Ref. [35] only furnishes the final neutrino spectrum at Earth, and does not provide the proton spectrum Q p (E) at each redshift, in principle one would not be able to reproduce the function F [z , E(1 + z )] in Eq. (2). However, at sufficiently high energies the mean free path for pγ interaction becomes so small that neutrinos can be assumed to be produced exactly at the same place in which the emission of the protons occurs. If we make the further hypothesis that at each redshift the injection spectrum of the protons has exactly the same form, with a redshifted energy, the function F [z , E(1 + z )] takes on the form where ρ(z ) is proportional to the Star Forming Rate given in Eq. (3). After this simplification, there is a unique function f [E(1 + z )] which reproduces the spectrum obtained by Ref. [35]. While the effects of these simplifications may be relevant at lower energies, they should be almost irrelevant in the high energy part of the cosmogenic spectrum, where the hypothesis of a mean free path is natural. In particular, the mean free path for pγ interactions is typically of the order of 50 pc, which is very small compared to the cosmological distances.
The inversion of Eq. (2) under the ansatz of Eq. (4) is in principle possible in an exact way through the use of a Mellin transform. In fact, the equation takes the form However, due to the differential properties of the functions involved, this method is very hard to apply, because of the very fast oscillations of the Mellin transform. It is however possible to obtain a very good approximation by observing that the integral kernel connecting f (E) to the observed spectrum, which is ρ(z)/H(z), is a peaked function around z 1. Thus, under the assumption that f [E(1 + z)] depends on the redshift more slowly than ρ(z)/H(z), we may take it out of the integral evaluating it at redshift z = 1, finding We have numerically checked that this approximation gives good results by comparing the expected spectrum at Earth computed with Eq. (6) with the input spectrum.

III. CROSS SECTIONS
With the inclusion of Secret Interactions given in Eq. (1), the cosmogenic neutrino spectra at Earth could change, depending on the free parameters of the new interaction, namely the coupling λ, the masses M ϕ of the scalar ϕ and of the sterile neutrino m s . In the following, we provide the cross sections for the processes considered in this work. In the computation of the cross sections we should in principle consider initial and final states in the form of mass eigenstates. However, in the hypothesis that the active-sterile mixing angle θ as is sufficiently small θ as 1, the effects coming from taking this into account will be small corrections only, proportional at least to the square of the mixing angle θ 2 as . The cross sections are therefore computed without any correction coming from mixing angles. Therefore the initial and final states may be taken directly as mass eigenstates.
At tree level the new processes introduced by our new interaction are the four particle collisions ν +ν → ν s +ν s , ν + ν s → ν + ν s and ν s + ν s → ν + ν as shown in the diagram of Fig In fact, since the Cosmic Neutrino Background (CNB) only involves active neutrinos, all collisions must involve at least one active neutrino in their initial state 1 . The process ν + ν s → ν + ν s is still relevant, even though the cosmogenic neutrinos are active in flavor, because a sterile neutrino originated from mixing or from a previous collision of an active neutrino with the background might still in principle produce a relevant active flux.
Considering the process ν + ν → ν s + ν s , the squared amplitude written in terms of the Mandelstam invariants s = (p + l) 2 , t = (p − k) 2 and u = (p − q) 2 (see Figure 1) is where m is the mass of the active neutrino ν of CNB, Γ is the decay rate of the scalar mediator given below, and M ϕ is its mass. We remind that the squared amplitude depends on two Mandelstam invariants only, since the third is connected to the others by the relation s+t+u = 2(m 2 +m 2 s ). The total cross section for the ν+ν → ν s +ν s interaction is then given by where and The differential cross section for the production of a sterile neutrino with energy E s in the process ν +ν → ν s +ν s is dσ aa→ss Here E is the energy of the incident cosmogenic active neutrino.
Let us discuss now the second process ν + ν s → ν + ν s . Again, the initial momentum of the background neutrino is l, the momentum of the incident sterile neutrino is p, the momentum of the final sterile and active neutrinos are respectively k 1 and k 2 as shown in Fig. (2). Notice that in this case, since the two final particles are distinguishable, the choice of how to define the Mandelstam parameters is not equivalent: we choose the convention that t = (p − k 2 ) 2 = (l − k 1 ) 2 . With this choice, the squared amplitude |M as→as | 2 is identical to Eq. (7) with the s and the u parameters exchanged in the corresponding equation. The total cross section for the process is with and again the energy of the incident sterile neutrino is E and J is defined as The differential cross section for the production of an active neutrino of energy E 2 is then The differential cross section for the production of a sterile neutrino of energy E 1 is: Concerning the scalar mediator, its decay rate is given by where ξ = A point to emphasize is the fact that, for m s ≥ M ϕ , the decay rate of the scalar mediator vanishes, since there is no decay channel kinematically allowed. This means that the resonances in the cross sections become unregulated. While this is not a problem for the s-resonance, which can never be reached in the physical space of parameters of the collision, the t-and u-resonance exhibit instead a singular behavior. This behavior needs to be regulated taking into account the finite transverse amplitude of the scattering beams, in a way analogous to [71]. In order to avoid this difficulty, we have restricted to the case M ϕ > m s .

IV. PROPAGATION: TRANSPORT EQUATION
The effect of the secret interaction on the neutrino flux produced through the pγ interactions is described by a Boltzmann equation. The relevant physical processes are the collisions of a cosmogenic neutrino with a neutrino from the CNB, and two different effects are in principle possible: on the one hand, the collision of an astrophysical neutrino produces a depletion in the flux, described by an absorption term; on the other hand, after the collision two new daughter sterile neutrinos are produced. If the incident neutrinos are highly relativistic, as we are assuming, the collisions will be strongly forward, with collinear emission of the daughter neutrinos, and we can assume that in principle they will replenish the original flux. Nonetheless, we will see that, even though they are produced with the correct angle, their energies will be too low to be relevant to our work. The interplay between these two processes is described by the transport equation which we show below.
A subtle point which has to be taken into account is the effect of oscillations. In principle, we should write a differential evolution equation for each of the components of the density matrix in flavor space. However, at the extremely high energies of interest to us, the De Broglie wavelength of the neutrinos, which characterizes the distances over which neutrino oscillates, is much smaller than the characteristic distance of propagation, which is the mean free path for the interaction. With this consideration, we are assured that between two successive collision the oscillations have averaged out. This means that, even though neutrinos are produced as eigenstates of flavor, during their propagation their density matrix averages to a form which is diagonal in the space of the mass eigenstates. For this reason, we have studied the propagation equation for the fluxes of neutrinos in the mass eigenstates.
Let Φ a (z, E) be the flux of active neutrinos per unit energy interval per unit solid angle at a redshift z, Φ s the analogous flux of sterile neutrino: we will collectively denote them by Φ l where Φ l ≡ dφ l /dE dΩ. The transport set of equations is: Here σ l is the total cross section for collision of a neutrino of type l = a, s with a neutrino from the CNB, where σ a ≡ σ aa→ss of Eq. (8) and σ s ≡ σ as→as of Eq. (12) (we are assuming a single active flavor for the latter, since the CNB is composed only of active neutrinos). Moreover in Eq. (19) the quantity dσ ml dE (E → E) denotes the partial cross section of Eq.s (15) and (16) for the production of an l-th neutrino with energy E after the collision of an m-th neutrino of energy E with the CNB one. The quantity n denotes the number density of CNB neutrinos, which we have taken to be n(z) = n 0 (1 + z) 3 with n 0 = 116cm −3 . The function f (E) is the number of neutrinos emitted per unit energy interval per unit time per unit solid angle, which has been described above. ρ(z) is the density of sources which has been taken to evolve with the Star Formation Rate.
Equation (19) is a system of two partial differential coupled equations, which should in principle be solved numerically. However, some physical considerations allow us to obtain the most interesting results with a simplified approach. If there is no mixing from oscillations between the active and the sterile neutrinos, then we have no interest in the sterile flux at Earth, which could not be detected in any case. In the equation for the active flux, the term describing the replenishment of the flux by the process ν s + ν → ν s + ν is weighted by the differential cross section for production of an active neutrino. As we show in Figure 3, however, this differential cross section is substantially peaked at energies much lower than the cosmogenic ones, between 10 5 GeV and 10 12 GeV, interesting for us. For this reason we do not consider this process. Therefore, we can neglect this term in the equation for the active neutrinos, which becomes: This equation contains only an absorption term, and admits now an analytical solution for the flux at Earth: The validity of this approximation has been verified by explicitly finding the numerical solution to the full system (19) for some benchmark values of the sterile mass and the mediator mass between 250 MeV and 1 GeV, and comparing it to (21). We found a perfect agreement between the two.

V. CONSTRAINTS
The model we are assuming is in principle subject to a number of constraints from cosmology, astrophysics and laboratory experiments. We did not perform a detailed study of the unconstrained region spanned by the parameters λ, m s , M , since this would be beyond the scope of this work. We focused on a particular region which turns out not to be affected by these constraints.

Laboratory bounds
Mesons decays, in particular Kaon decays, can be quite restrictive for models with Secret Interactions, since the latter can introduce new decay channels. For secret interactions among active neutrinos severe constraints come from meson decays in the (λ, M ) plane, as shown for instance in [72], where it is roughly found that the region up to M 250 MeV is insensitive to the exclusions. For our model one should in principle make a similar analysis, since through the secret interaction the kaon can decay into K → µν s ϕ or K → µν s ν s ν. If, however, we restrict to the range of masses m s ≥ 250 MeV and M ≥ 250 MeV, these reactions are kinematically forbidden, justifying the absence of constraints from these decays.

Big Bang Nucleosyntesis bounds
It is interesting to note that with the choice M, m s 250 MeV, the whole sector of the active and sterile neutrino and the scalar mediator remains in equilibrium at temperatures even lower than the BBN. For example, if λ = 1, the interactions are relevant down to temperatures of 100 keV. Of course, after around 1 MeV, the active neutrino decouples from the Standard Model plasma, so below this temperature we cannot speak of thermal equilibrium, but the interactions between active and sterile neutrinos and scalar mediator remain effective. Because of this, the sterile neutrino and the scalar mediator are Boltzmann suppressed at temperatures below essentially T ∼ 100 MeV, since at these temperatures the reactions of production of sterile neutrinos are kinematically suppressed. Therefore at the BBN temperatures, around 1 MeV, the newly introduced particles have disappeared and they do not count as radiative degrees of freedom, without in fact influencing the nucleosynthesis.

Cosmic Microwave Background bounds
At the time of formation of the Cosmic Microwave Background, the sterile and scalar particles have long disappeared. The active neutrinos can interact through the four point reactions νν → νν. It can be seen that this reaction has run out of equilibrium at temperatures much higher than the ones of interest for the CMB (at least 100 keV, as mentioned in the previous section), so no essential change is introduced in the features of the Cosmic Microwave Background.

Astrophysical bounds
Another constraint might in principle come from the analysis of neutrino fluxes from supernovae. In fact, since neutrinos in the core of supernovae have energies of order of tenth or hundredth of MeV, they are sufficiently energetic to produce sterile neutrinos which could escape the supernova, giving rise to an energy loss with observable consequences. However, due to the interactions we are introducing, even though sterile neutrinos can in principle be produced, they are not able to escape the supernova, trapped by the secret interaction with the active neutrinos inside the core. In order to verify this statement, we have computed the order of magnitude of the mean free path of a sterile neutrino inside the core. We have obtained, for example, for a benchmark mass of 250 MeV for the sterile neutrino, 300 MeV for the scalar mediator and 1 for the coupling, a mean free path of 10 −11 m, clearly much smaller than the characteristic distances of a supernova.
In view of the above considerations we restrict in the following our analysis to the case of massive sterile neutrinos, with m s of few hundreds MeV and a mass of the pseudoscalar mediator M ϕ ≥ m s .

VI. RESULTS AND DETECTION CHANCES
Among the most relevant experiments that have performed a detection campaign for cosmogenic neutrinos, and the future apparatus that will undertake such measurements the Pierre Auger Observatory (PAO) and the planned GRAND set up, among the others, deserve a particular mention. In the following, we will compare our results with the sensitivity of such experiments.
The main goal of PAO, located in Argentina, was the measurement of extensive air showers, produced in the interaction of UHECrs with the atmosphere. The Collaboration started to take data in 2004. Particles produced from UHECrs interaction in atmosphere are detected at ground level with an array of water-Cherenkov detectors that are located on a large area of about 3000 km 2 arranged in a hexagonal pattern.
Although the primary goal of Auger detector was to study UHECrs, it has been shown in [73] that it can also study cosmogenic neutrinos. Indeed neutrinos arriving at large zenith angle (horizontal with respect to the detector) produce at the sea level extensive air showers with small radius of curvature [73], in contrast with other showers from large zenith angles. The electromagnetic component of ordinary air showers at large zenith angles from hadronic cosmic rays is attenuated by the atmosphere before arriving at the sea level. Deeply penetrating particles like neutrinos come instead unattenuated [74]. Recent upgrades of the Auger sensitivity can be found in [75] and an integrated sensitivity to cosmogenic neutrino fluxes is found of about 4 · 10 −9 GeV cm −2 s −1 sr −1 . Such a sensitivity will be improved by the Giant Radio Array for Neutrino Detection (GRAND) [33] that is planned to take data in 2025 and should reach, after 10 years of data, an integrated sensitivity to cosmogenic neutrino fluxes of 1 · 10 −10 GeV cm −2 s −1 sr −1 , thus improving of a factor of ten the current Auger sensitivity. GRAND will detect the radio emission coming from large particle showers, namely extensive air showers like Auger. However, GRAND will use an array of 10.000 radio antennas deployed over an area of 10.000 km 2 .
In Figure 4 we provide the expected cosmogenic spectra both in the absence of secret interactions and in their presence, for some benchmark values of the scalar mediator masses. These spectra have been obtained under the assumption of a purely protonic UHECrs flux (top panel) and purely helium cosmic rays (bottom panel). The coupling has been fixed to λ = 1, while the sterile mass is taken to be 250 MeV. The dashed, dotdashed and dotted curve respectively correspond to a mass of the scalar mediator of 300 MeV, 500 MeV and 1 GeV. In agreement with the expectations, larger masses for the mediator correspond to weaker absorption.
The sensitivities of the Grand and the Auger experiments are shown as well. It appears that already the Grand experiment after 3 years of data taking would be able to distinguish the presence of the secret interaction, even for masses as large as 500 MeV, at least in the case of purely protonic UHECRs. In fact, while in the absence of the secret interaction the cosmogenic flux should be detected up to energies of ∼ 10 10 GeV, being above the sensitivity of the experiment, the interaction causes its absorption, making the flux undetectable already at energies of ∼ 10 9 GeV.
This conclusion depends very strongly, however, on our choice for the coupling. In fact, since the cross section, appearing in the exponent of the absorption coefficient, grows as the fourth power of the coupling, already a choice of λ ∼ 0.1 renders the absorption effect completely negligible. A milder dependence is the one expected on the sterile neutrino masses. Raising this mass causes an increase in the threshold energy for the production. In fact, the latter is E th m 2 s /m, which is the energy at which we expect the absorption effect to begin. All of our conclusions are of course based on the assumption of a purely protonic UHECrs flux. Recent data [36] suggest a mixed chemical composition, tending towards Helium and Nitrogen as the dominant components. It is therefore of interest to determine how our results change for a different chemical composition. In the case of Helium dominated cosmic rays, the spectrum without interaction is already lower than the Grand sensitivity at 3 years, while it is slightly higher than the integrated sensitivity after 10 years of data taking.
For Nitrogen dominated cosmic rays, the spectrum, with or without secret interaction, is not amenable to detection at Grand, being even below the integrated sensitivity at 10 years. Since our partial informations about the chemical composition of UHECrs suggest a mixed composition, it is reasonable to expect a cosmogenic spectrum somewhere in between the extreme cases analyzed.

VII. CONCLUSIONS
In this paper we have investigated the experimentally observable effects coming from a simple form of activesterile secret interactions in neutrino sector, mediated by a pseudoscalar, at air shower experiments like Grand and Auger. Due to the presence of such interactions, the flux of cosmogenic neutrinos results to be strongly distorted. In this scenario, in order to evade the constraints coming from cosmology, astrophysics and particle physics, we need to take quite a large mass for the sterile neutrinos of the order of few hundreds MeV. Observable effects are then seen in the cosmogenic neutrino fluxes of ultrahigh energy, produced through the GZK interaction. In fact, these extremely energetic neutrinos can, during their path to Earth, collide with neutrinos from the Cosmic Neutrino Background, producing a flux of sterile neutrinos and depleting the flux of active ones. We showed that this results in an absorption of the cosmogenic neutrino fluxes and then in a peculiar distortion of the expected flux. We also showed that, in the case of GZK neutrinos coming from cosmic rays dominated by protons, the effects of these absorptions are sufficiently important to be revealed by a forthcoming experiment like Grand. In the case of different chemical compositions of the cosmic rays, the neutrino flux is generally lower even without absorption and the possibility of cosmogenic neutrino detection with the Grand experiment is only marginal. In this case, the cosmogenic neutrino detection would demand for future a much larger generation of cosmic rays apparatus. by the Italian Ministero dell'Università e della Ricerca (MUR). The authors acknowledge partial support by the research project TAsP (Theoretical Astroparticle Physics) funded by the Instituto Nazionale di Fisica Nucleare (INFN).