Phenomenology of upward tau showers in atmosphere

Nuclear Physics B (Proc. Suppl.) 136 (2004) 129–136 www.elsevierphysics.com

Phenomenology of upward tau showers in atmosphere S. Bottaia a

INFN Firenze, via G. Sansone 1, I-50019 Sesto F.no (FI), Italy

Tau neutrinos interacting inside the Earth produce τ leptons which thereafter can decay inside the atmosphere. The propagation of extremely energetic ντ ’s and τ ’s through the Earth is studied by means of a detailed Monte Carlo simulation and results are compared with analytical approximation. The rates of τ ’s emerging from the Earth are determined as a function of τ ’s energy for several cosmic neutrino models.

1. Introduction Cosmic neutrinos give a unique opportunity to open a new observational window in the field of Astrophysics and Cosmology. The detection of such particles, not deflected or absorbed during their intergalactic path, could be revealing for the identification of sources of Ultra and Extreme High Energy Cosmic Rays (UHECR and EHECR). The detection of such ultra and extreme high energy neutrinos is a challenge as it demands a very large sensitive area. The experiments of the upcoming generation plan to use underwater-underice Cherenkov detectors [1] and large field-of-view atmospheric fluorescence detectors [2] [3]. One of the best evidence of cosmic neutrinos would be the detection of upstream showers/particles emerging from the Earth. For this kind of events the atmospheric muon and primary charged cosmic ray background would be completely suppressed. On the other hand this signature can hardly be observed at extreme energy because the rise of weak cross sections entails the opacity of the Earth with respect to neutrino propagation [4] [5] [6]. The behaviour of τ -neutrinos, whose existence should be guaranteed in a neutrino-oscillation scenario, is significantly different from νµ and νe [7] [8] [6] [9] [6]. Whilst muon and electron neutrinos are practically absorbed after one charged current (CC) interaction, the τ lepton created by the ντ CC interaction may decay in flight before losing too much energy, thereby generating a new ντ with comparable energy. Hence ultra high en0920-5632/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2004.10.026

ergy τ -neutrinos and τ ’s should emerge from the Earth instead of being absorbed. As pointed out in ref. [10] [11] [12] [13] [14] [15] the tau leptons, created by CC ντ interactions inside the Earth, because of their relative long decay length at high energy, could emerge from the Earth surface and eventually decay inside the atmosphere (fig. 1). Such kind of events could be detected

νe

ντ

e

τ

τ

ντ EARTH

AT M

τ

OS PH ER E

ντ

Figure 1. Schematic drawing of atmospheric showers induced by neutrinos. Upward going ντ ’s could be detected as upward going showers induced by τ decay.

by atmospheric shower detectors as upward going showers. In reference [10] an overview of the physics processes involved in case of τ ’s emerging

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from Earth and mountains is presented, suggesting the possibility of the detection of such events. In reference [11] the rate of expected events in the Auger detector is given on the basis of a Monte Carlo simulation while in reference [12] the results of a detailed Monte Carlo simulation are presented in a general framework, giving the rate of emerging τ ’s for unit Earth surface above a given minimum energy, from 1014 eV to 1022 eV and for a large variety of neutrino fluxes. In this work is performed a comparison within Monte-Carlo results and the approximated analytical approach proposed in Reference [13]. 2. Lepton interactions Deep inelastic neutrino-nucleon scattering is the dominant interaction of energetic neutrinos into conventional matter. Charged and neutral current differential cross sections [5] used for this simulation [6] have been calculated in the framework of QCD improved parton model. Since a large contribution to cross sections comes from the very low x region, laying beyond present accelerator domain, an extrapolation of parton distribution at very low x (x ≈ 10−8 ) is necessary. We have used the parton distribution set CTEQ3-DIS [16], available in the program library PDFLIB [17], with NLO-DGLAP formalism used for the evolution with the invariant momentum transfer between the incident neutrino and outgoing lepton (−Q2 ) and assuming for very low x the same functional form measured at x = O(10−5 ). The electromagnetic interactions of muons with matter, at the considered energies, are dominated by radiative processes rather than ionisation [18]. Cross sections for electromagnetic radiative processes of τ are lower than muon’s, yet radiative interactions remain the dominant process for τ energy loss. The cross sections used for radiative interactions of τ leptons are based on Petrukhin-Shestakov QED calculation for bremsstrahlung [19] and on KokoulinPetrukhin calculation for direct pair production [20]. Photonuclear (PH) interaction cross sections are taken from I. Dutta at al. [22]. For all aforementioned processes we have implemented stochastic interactions for ν = (Ei − Ef )/Ei ≥

10−3 and a continuous energy loss approximation for ν = (Ei − Ef )/Ei ≤ 10−3 , where Ei and Ef are τ energies before and after the interaction respectively. It is worth mentioning that bremsstrahlung cross section scales as the inverse square of lepton mass ml whereas direct pair production and photonuclear interaction cross sections approximately scale according to 1/ml [22] [24]. As a consequence the dominant processes for τ lepton energy loss are direct pair production and even more photonuclear interaction rather than bremsstrahlung photon radiation. Photonuclear interactions are the most important interactions for tau energy loss at UHE. The cross sections used here are taken from ref. [22] and are consistent to cross section calculation by Bugaev and Y.V. Shlepin [23], while they differ by an order of magnitude from previous calculation by Bezrukov and Bugaev [21]. The τ decay has been simulated by using the TAUOLA package [25]. The Monte Carlo simulation has been performed following neutrinos and charged leptons along their path inside the Earth. The Earth model considered is the preliminary Earth model of ref. [26], approximated with 8 shells of constant density. A perfectly spherical smooth Earth surface is also considered. For what concerns the Earth composition we have used “standard rock” (Z = 11, A = 22) for the crust and the mantle and iron (Z = 26, A = 55.8) for the core. 3. Neutrino induced τ ’s inside the Earth 3.1. Effective aperture In this section the calculation of the fluxes of τ ’s emerging from the Earth has been performed by means of a full simulation of ντ − τ system propagation through the Earth. For zenith angles corresponding to a path length in the Earth larger than ντ interaction length in the Crust (LνCC ≈ 70 km for Eν = 1020 eV), neutrinos interact far away from the Earth surface, and the produced τ ’s lose energy and decay before getting out. Owing to this mechanism, the energy of the ντ -τ system decreases as zenith angle increases and the probability for a ντ to emerge as a τ drops at large

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131






dEν (2)

2

β (cm /g)

·

Pντ →τ (Θ, Eν , Eτmin ) |cos(Θ)| dΩ

where S is a portion of Earth surface. One can now define an effective aperture Aef f (sr) containing all the physics of leptons propagation : 10

-6


Aef f (Eν , Eτmin )

=

Pντ →τ (Θ, Eν , Eτmin ) · ·|cos(Θ)| dΩ

10

so that Eq. 2 reads :

-7


Nτ (Eτ ≥ Eτmin ) = S

12

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21

·Aef f (Eν , Eτmin ) dEν (3)

22

log(E(eV))

1 Figure 2. β = − dE/dX E ρ in case of standard rock (A=22, Z=11) for τ PH interactions in I. Dutta et al. [22] compared to previous calculation of Bezrukov and Bugaev [21].

zenith angles. The number of emerging τ ’s with energy larger than Eτmin (minimum energy) from an Earth infinitesimal surface dS can be calculated in a unidimensional approach as :
d2 N min dNτ (Eτ ≥ Eτ ) = (Eν ) · Eν ≥Eτmin dEν dΩ · Pντ →τ (Θ, Eν , Eτmin ) dS |cos(Θ)| dΩ dEν (1) 2

where dEd νNdΩ (Eν ) is the cosmic τ neutrino flux, Θ is the zenith angle of the emerging particles, Pν→τ (Θ, Eν , Eτmin ) is the probability that a ντ impinging on the Earth surface will emerge as a τ with an energy Eτ ≥ Eτmin . Assuming a spherical shaped Earth and an isotropic neutrino flux, Eq. 1 turns into :
d2 N (Eν ) · Nτ (Eτ ≥ Eτmin ) = S Eν ≥Eτmin dEν dΩ

Eν ≥Eτmin

d2 N (Eν ) · dEν dΩ

The effective aperture Aef f does not depend on neutrino fluxes or detector performance and allows a simple and straightforward calculation of the number of emerging τ ’s for any cosmic neutrino flux and detector sensitive area. It must be pointed out that in the definition of Pντ →τ (Θ, Eν , Eτmin ) we also constrain the emerging τ to decay within an atmospheric slant depth larger than 1000 g/cm2 . This condition guarantees the full size formation of an EAS after τ decay. The results for the effective aperture are shown in figs. 3. They seem to be in fairly good agreement with those in ref. [11] for the Auger detector, though the contribution of the effective Auger sensitivity can not be completely unfolded from quoted results using available informations. An effective aperture Aνefe f for downward going νe CC interactions in the atmosphere is also shown in figs. 3. For this kind of events, the energy of produced EAS is the same as νe . For an isotropic neutrino flux, the number of EAS induced by νe CC interactions within the atmospheric mass overhanging a portion of Earth surface S is given by :
·

Eν ≥Eτmin

νe NEAS (EEAS ≥ Eτmin ) = S ·

d2 N (Eν )σCC (Eν ) NA p 2π dEν (4) dEν dΩ

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Aeff (sr)

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10 10 10 10 10 10

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21 22 Eν (eV)

10 10 10 10 10 10 10 10 10

Figure 3. Effective aperture for different minimum energy Eτmin (or Ethr ) and standard rock considered for the outer Earth layer. Dotted line shows the effective aperture for events induced by downward going νe CC interactions in the atmosphere.

where p is the atmospheric pressure (column density) at ground level, NA is the Avogadro num2 ber, dEd νNdΩ (Eν ) is the cosmic electron neutrino flux and σCC (Eν ) is the neutrino CC cross section on isoscalar target. Again, we can define an effective aperture : Aνefe f (Eν ) = 2π NA p σCC (Eν )

(5)

such that the number of νe CC interactions can be written as: νe NEAS (EEAS ≥ Eτmin ) =


=S

Eν ≥Eτmin

d2 N (Eν ) Aνefe f (Eν ) dEν dEν dΩ

(6)

Aνefe f (Eν ) increases with energy following the rise of neutrino cross section. At low energies (Eν  1018 eV) the rise with energy of Aef f for

emerging τ ’s is steeper than the rise of Aνefe f (Eν ) because both the increase of ν cross section and the increase of τ decay length contribute to enlarge Pν→τ (Θ, Eν , Eτmin ). Aef f (Eν ) keeps increasing as long as the increase of ν cross section enhances the production of τ ’s in the Earth. Around Eν  1018 eV the cross section of neutrinos is so high that the probability for neutrinos to have at least one interaction inside the Earth is close to one for most zenith angles and the point of first ντ interaction in the Earth is likely to be far from the emerging point (this latter effect is responsible for the slow decrease of Aef f between 1019 eV and 1020 eV in fig 3). Moreover, the increase of τ decay length with energy prevents some of the extreme high energy events from producing a detectable shower into the atmosphere (only τ ’s decaying within 1000 g/cm2 slant depth are retained). Due to these mechanisms, the value of Aef f at extreme high energies shows an almost flat dependence on energy and strongly depends on the required Eτmin . The angular distributions of emerging τ ’s are shown in reference [12]. As the initial neutrino energy increases, the maximum of the angular distribution moves toward 900 zenith angle. 3.2. Comparison between Monte Carlo and analytical approach Following references [13] [14] the differential ν ,Eτ ) for a neutrino of energy probability dK(Θ,E dEτ Eν to produce a tau lepton which emerge from Earth with energy Eτ , can be written with a simple and approximated analytical formula. Neglecting neutral current, regeneration due to tau decay, using approximation that the neutrino interaction length is always much longer than tau range, continuous energy loss approximation dEτ /dx = −β(Eτ )Eτ , Eτ = 0.8Eν in any CC neutrino interaction one get : 1 dK(Θ, Eν , Eτ ) · ≈ ν dEτ LCC (Eν )  2Rcos(Θ) −N σν ρ(r(Θ,z))dz ·e A CC 0 · mτ 1 1 1 ·e cττ βρ { 0.8Eν − Eτ } · · Pdecay (Θ, Eτ ) (7) Eτ βρs where ττ is the lifetime of tau lepton , c the light velocity, β = β(Eτ ) is the energy loss due to pair

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Pντ →τ (Θ, Eν , Eτmin ) = dK(Θ, Eν , Eτ ) = dEτ dEτ min Eτ ≤Eτ ≤0.8Eν


unrealistically absorbed. Again the difference at higher value of Eτmin and high Eν is due partially to regeneration and partially to the approximation of LνCC much greater than tau range. Such latter approximation is used in the analytical approach to use all the available path inside the Earth for neutrino flux suppression, hence it results in an over estimation of neutrino suppression and under estimation of probability to generate emerging tau’s. The cross check between Monte Carlo and analytical approach shows that the results are well under control.

Aeff (sr)

production and photonuclear interactions (see fig 2) ; R is the radius of the Earth ; ρ(r(Θ, z)) is the density at distance r from the Earth center and r2 (Θ, z) = R2 + z 2 − 2Rzcos(Θ) where z is the distance the incoming neutrino travels through Earth before converting to a tau near the Earth’s surface. ρs and LνCC (Eν ) are the density and the neutrino interaction lenght evaluated in the Earth near the its surface. The factor Pdecay (Θ, Eτ ), not present in [13] [14], is the probability that the emerging tau will decay inside the atmosphere. In order to fully develops a shower we select only events where at least 1000 g/cm2 slant depth remain after the decay. Such condition reflects in allowing a decay length Lmax after the emerging point of at least Lmax ≈ 300km for Θ = 88o , Lmax ≈ 400km for Θ = 89o , Lmax ≈ 500km for Θ = 90o . Here, considering the fact the effect is more important for Eτ ≥ 1019 eV where the expected [12] emerging angles are between 88o and 90o , for simplicity we choose a constant value of Lmax corresponding to Θ = 89o . The neutrino cross sections and tau energy loss parameters used here are the same than those used for the Monte Carlo simulation. ν ,Eτ ) is evaluated then is possible Once dK(Θ,E dEτ to calculate the probability Pντ →τ (Θ, Eν , Eτmin ) defined in Eq. 1 by integration over dEτ :

10

10

10

10

10

10

(8)

than the evaluation of the effective aperture Aef f (Eν , Eτmin ) defined in Eq. 9 is straightforward :
min Aef f (Eν , Eτ ) = Pντ →τ (Θ, Eν , Eτmin ) · ·|cos(Θ)| dΩ Where the integration is limited to the lower hemisphere. The comparison between Monte Carlo results and the above analytical approach is described in fig. 4. The difference for very low value of Eτmin can be explained considering that no regeneration is present in the analytical approach and hence at the occurence of a neutrino CC interaction far from the surface the neutrino is

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log(E ν(eV))

Figure 4. Effective aperture for different minimum energy Eτmin and standard rock considered for the outer Earth layer. Red stars are the Monte Carlo results while blue lines are from approximated analytical approach

3.3. Fluxes of emerging τ ’s According to the hypothesis of νµ −ντ neutrino oscillations suggested by the SuperKamiokande experiment [27], half of the original cosmic νµ ’s appear as ντ ’s after their intergalactic path toward the Earth [28].

Within this scenario, we have used the same fluxes for parent ντ and νe : Φντ = Φνe = Φinitial /2 and we have calculated the rate of νµ emerging τ ’s by using equation (3).

Nτ /1000km2 /year

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1

Nτ /1000km2 /year Eτ > Emin

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log(E τ(eV))

-4 -5 -6

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log(E min(eV))

Figure 5. Rates of emerging τ ’s versus tau minimum energy using standard rock for the outer Earth layer (red line). Green line show the rate of νe CC interactions in the atmosphere above the selected surface. The flux of neutrinos considered here is the GZK neutrinos from ref. [29].

Fig. 5, 6, 7, 8 show the flux of emerging τ ’s compared with the rate of νe CC interactions in the atmosphere overhanging 1000km2 surface. The signal induced by the decay of emerging τ ’s dom17 inate for Eτmin < ∼ 5 × 10 eV while for higher minmin imum energy Eτ , Earth matter strongly suppresses the upgoing τ fluxes. It must be emphasized that, in this analysis, we have not taken into account the difference between induced EAS generated by different τ decay channels. It is worth pointing out that leptonic decays τ − → µ− ν¯µ ντ do not produce any EAS.

Figure 6. Rates of emerging τ ’s versus tau energy (red line). Green line show the rate of νe CC interactions in the atmosphere above the selected surface. The flux of neutrinos considered here is the GZK neutrinos from ref. [29].

4. Conclusions A detailed study of ντ -τ leptons propagation through the Earth indicates the likely existence of a quite intense flux of upgoing τ ’s emerging from its surface. The rate of EAS induced by τ ’s decay is larger than the rate of νe ’s induced EAS 17 for minimum energy Eτmin < ∼ 5 × 10 eV, while it drops for larger values of minimum energy. The detection of such τ induced events is challenging at a detector’s energy threshold Ethr ≈ 1019 eV even for new generation experiments, while for Ethr  1018 eV the detection seems to be realistic for detectors capable of recording signals induced by very inclined showers with sensitive areas > 103 km2 .

10 2

Nτ /1000km2 /year

Nτ /1000km2 /year Eτ > Emin

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10 1

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log(E min(eV))

Figure 7. Rates of emerging τ ’s versus tau minimum energy using standard rock for the outer Earth layer (red line). Green line show the rate of νe CC interactions in the atmosphere above the selected surface. The flux of neutrinos considered here is the Topological Defect neutrinos from ref. [30].

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log(E τ(eV))

Figure 8. Rates of emerging τ ’s versus tau energy (red line). Green line show the rate of νe CC interactions in the atmosphere above the selected surface. The flux of neutrinos considered here is the Topological Defect neutrinos from ref. [30].

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