Neutrino-Nucleon Interactions

In the Standard Model (SM) description of particles and their interactions, neutrinos interact through the weak force which is mediated by the Z⁰ -boson andW^±-bosons. Two kinds of interaction processes are distinguished, charged current (CC) interactions mediated byW^±, where a charged lepton is produced and neutral current (NC) interactions, mediated by Z⁰, where the neutrino from the initial state can also be found in the final state. The deep-inelastic-scattering process between the nucleon N and the neutrinoν_l with flavor l, produces in both cases a hadronic cascade X as the result of

the hadronization of the debris of the nucleon:

ν_l+N −→^W± l+X (CC) (3.1)

ν_l+N −→^Z0 ν_l+X (N C). (3.2) The charged secondaries of these interactions are detectable with Cheren-kov light detectors, as described below. The cross sections for these processes have been calculated in [62]. It involves the evaluation of quark distribu-tion funcdistribu-tions of the target nucleon, which are measured at collider exper-iments [22, 42]. Tabulated versions of these functions are provided by the cteq collaboration [102]. At energies above ∼1 PeV no experimental data is available and extrapolations have to be used. Uncertainties from different extrapolation models can reach a factor of two at energies around 100 EeV [63]. Also, new physics could step in and considerably change the cross sec-tions at the highest energies [43]. Relevant for the discussion in this work is the behavior of the integrated cross section:

σ_CC/NC∝

where−Q²is the invariant momentum transfer between the incident neutrino and the outgoing lepton; E_ν the energy of the neutrino; and M_W/Z is the mass of mediating boson M_W = 80 GeV and M_Z = 91 GeV, respectively.

Figure 3.1 shows the integrated neutral and charged current cross sections forνN and ¯νN. The NC cross sections are lower than the CC cross sections, both rise linearly with energy up to ∼10⁴GeV. As the propagator term of the interaction 1/(Q²+M_W/Z² ) is then dominated by Q² which exceeds the mass of the boson, the rise is damped andσ ∝E^0.36 [95]. At low energies the cross section for ¯νN is smaller, but it becomes equal to the νN cross section above ∼1 PeV where the contribution from sea quarks dominates the quark distribution functions of the nucleon.

The νe⁻ interactions can generally be neglected, because owing to the small electron mass the cross section is much smaller. However, for ¯νee⁻ in-teractions resonant W⁻ production occurs, leading to a cross section about 300 times higher than that for CC neutrino-nucleon interactions (Glashow resonance) at energies around E¯νe =M_W² /2m_e≈6.3 PeV [66]. The inter-actions ¯ν_ee→W⁻→ν¯_µµ and ¯ν_ee→W⁻→X_hadr dominate in the energy range from 3 PeV to 10 PeV (see Figure 3.1). This offers a good chance to detect a signal from electron-neutrinos.

In contrast to muon searches, where a muon is required in the final state, in cascade analyses as presented in this work, the neutral-current channel

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹²

Figure 3.1: Neutrino-nucleon cross sections from 10 GeV to 100 EeV (data from [62]). The solid lines are the total cross sections, including CC (dashed) and NC (dotted) interactions. Anti-neutrino (red) and neutrino (blue) cross sections differ at energies below 1 PeV but are equal above. The resonant W⁻production in ¯νee⁻interactions (black) with a peak at 6.3 PeV (Glashow resonance) is also shown.

Figure 3.2: Mean of the inelasticity parameter of the charged current (solid) and neutral current (dashed) νN (blue) and ¯νN (red) cross sections as a function of the neutrino energy (data from [62]).

offers also detection potential. For the assessment of a measured energy spectrum from cascade events, it is important to know the energy transfer to the nucleon and the corresponding energy deposit of the hadronic cascade.

This is given by the inelasticity y= 1−E_l/E_ν where E_l is the energy of the outgoing lepton. The mean value hyi as a function of the neutrino energy E_ν is shown in Figure 3.2. For high energieshyi approaches a value of∼0.2.

However, as neutral-current and charged-current interactions have the same event signatures, it can only be taken into account for an error estimate of the energy reconstruction of the primary neutrino.

The mean scattering angle between the incident neutrino and the outgoing lepton is less than a degree for neutrino energies above 1 TeV [104], which is an important measure for point-source searches, as it is the upper limit of the angular resolution of the detector.

Absorption in the Earth

In the standard muon neutrino search one exploits the small neutrino nucleon cross section by looking for particles coming from below the horizon. Other particles like muons from air showers, which form the major background, are absorbed after propagating through a few kilometers of dense material. How-ever, at high energies the steadily rising neutrino interaction cross sections interfere. The interaction length is given by:

L_int= 1

σ_tot(E_ν)^N_A^Aρ, (3.4) where N_A = 6.022·10⁻²³mol⁻¹ is Avogadro’s number, A the atomic weight which is 1 g/mol for nucleons, and ρ is the density of the medium, here taken to be water (ρ = 1 g/cm³). The interaction length is proportional to 1/σ and hence decreases with energy. Figure 3.3 shows the interaction length for neutrino-nucleon interactions and ¯ν_ee-scattering as a function of the energy of the incident neutrino. Above ∼50 TeV the earth diameter, shown as the dashed line, exceeds L_int. The interaction probability after traversing a column depthx is given by:

P(x) = 1−exp(−x/Lint). (3.5) It is obvious that high-energetic neutrinos are absorbed by the Earth.

Hence, searches for ultra-high energy neutrinos are restricted to the upper hemisphere. Figure 3.4 shows the rate for neutrino interactions in the vicinity of the IceCube detector from a hypothetical E⁻² ν_e flux as a function of energy and zenith angle. Above 100 TeV the contribution from neutrinos

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹²

Figure 3.3: Interaction length for neutrino-nucleon interactions as a function of energy for charged-current (dashed), neutral-current (dotted), and total (solid) νN (blue) and ¯νN (red) interactions.

Also shown is the interaction length for ¯νe⁻-scattering (black).

10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹²

Figure 3.4: The interaction rate in a 1 km³target volume from a νe+ ¯νe flux (1·10⁻⁷E⁻²GeV s⁻¹sr⁻¹cm⁻²) per energy and zenith angle bin. The angle bin width is equally spaced in cosine(zenith) to compensate the angle dependence of the integration area.

from the lower hemisphere gradually decreases, because the column depth x = ^R ρdl is larger than the interaction length at these energies. At the highest energies only neutrinos with zenith angles smaller than 95^◦ reach the detector. The increased interaction probability for ¯ν_e interactions due to the Glashow resonance is clearly visible for neutrinos coming from above the horizon. The increased absorption probability is barely visible, though.

Tau-neutrinos are affected differently by absorption. In case of CC inter-actions the tau-lepton decays rapidly due to its short lifetime ofτ = 2.9·10⁻¹³s.

The final state of this decay will again have a tau-neutrino with less energy.

This process is commonly known as “tau-regeneration”. In addition, due to the energy decrease, the interaction length increases. Details can be found in [74].