High-energy neutrino interaction physics with IceCube

Although they are best known for studying astrophysical neutrinos, neutrino telescopes like IceCube can study neutrino interactions, at energies far above those that are accessible at accelerators. In this writeup, I present two IceCube analyses of neutrino interactions at energies far above 1 TeV. The first measures neutrino absorption in the Earth, and, from that determines the neutrino-nucleon cross-section at energies between 6.3 and 980 TeV. We find that the cross-sections is 1.30 $^{+0.21}_{-0.19}$ (stat.) $^{+0.39}_{-0.43}$ (syst.) times the Standard Model cross-section. We also present a measurement of neutrino inelasticity, using $\nu_\mu$ charged-current interactions that occur within IceCube. We have measured the average inelasticity at energies from 1 TeV to above 100 TeV, and found that it is in agreement with the Standard Model expectations. We have also performed a series of fits to this track sample and a matching cascade sample, to probe aspects of the astrophysical neutrino flux, particularly the flavor ratio.


Introduction
The IceCube observatory has observed neutrinos with energies well above 2 PeV, far beyond the 500 GeV reached by the most energetic terrestrial neutrino beams.These neutrinos have given us significant insight into the highenergy universe, particularly regarding astrophysical accelerators.However, these same neutrinos also offer us the opportunity to extend neutrino interaction studies to much higher energies.
IceCube consists of 86 vertical strings of optical sensors (digital optical modules, or DOMs) which were lowered into holes drilled into the Antaractic ice cap at the South Pole [1,2].The array covers a surface area of 1 km 2 .Each string is instrumented with 60 DOMs.On 78 strings, the DOMs are positioned every 17 m, between a depth of 1450 and 2450 m.On the remaining strings, the DOMs are deployed with a 7 m spacing near the bottom of the array.
Each DOM [3] consists of a 25 cm photomultiplier tube and data acquisition (DAQ) electronics, in a clear glass pressure vessel.Two waveform digitizer systems record the arrival times of most photoelectrons.A calibration system maintains the timing calibration for all of the DOMs to within 3 nsec [4].All of the data is sent to the surface, where software triggers find groups of timecorrelated hits, and send the data to a processor farm for on-line reconstruction.
One of the major challenges in studying neutrino interactions with IceCube is understanding the beam, and accounting for its uncertainties.The beam has three components, conventional and prompt atmospheric neu-trinos, and astrophysical neutrinos, each with different, often energy-dependent flavor composition and neutrino:antineutrino ratios.Here, we focus on ν µ , so are not limited by the uncertainties in flavor composition.All three components have somewhat different ν : ν ratios.Since ν and ν are indistinguishable in neutrino telescopes, but interact with different cross-sections and produce different inelasticity distributions, the ν : ν ratio is a significant systematic uncertainty.

Cross-section measurement
In the Standard Model neutrinos interact via chargedcurrent (CC) and neutral-current (NC) deep inelastic scattering.The cross-sections for these process increase with increasing energy.As Volkova and Zatsepin [5] first pointed out, at energies above about 40 TeV, the Earth becomes opaque to neutrinos, so one can use energetic neutrinos to probe the interior of the Earth.Or, one can turn this around, assume that the Earth density profile is known, and use absorption to measure the neutrino-nucleon crosssection.For long chords through the Earth, as are studied here, the standard "Preliminary Earth Reference Model" [6] contributes less than a 1% systematic uncertainty to the cross-section measurement.
Figure 1 shows the transmission probability, as a function of zenith angle and neutrino energy.Absorption manifests itself as a change in the zenith angle distribution with increasing neutrino energy.Beyond Standard Model (BSM) processes might also contribute to the crosssection, further increasing the rise with energy.Two phenomena that can increase the cross-section are models that involve leptoquarks [7] and those with additional, tightlyrolled-up spatial dimensions [8].We measured the cross-section using 1 year of data from IC-79, a sample of 10,784 upward-going ν µ events [9] with measured muon energy above 1 TeV [10].These events were then binned in two dimensions, cos(θ z ) and muon energy.The resulting histogram was then fitted to a model that included the three fluxes from conventional and prompt atmospheric neutrino and astrophysical neutrinos.
The cross-section was determined by a maximumlikelihood fit which had the cross-section as a free parameter [11,12].It was assumed to be a multiple of the Standard Model cross-section, σ ν , with R = σ ν /σ CSMS , where σ CSMS is the Standard Model cross-section, as computed in [13].This calculation is done in next-to-leading order perturbative QCD, with DGLAP evolution used to extrapolate the parton distribution functions to the low-x region.A similar calculation, by Connolly, Thorne and Waters, found similar results [14].
In the analysis, the charged-current (CC) and neutralcurrent (NC) cross-sections are assumed to vary in parallel [15].In NC interactions, the neutrinos lose energy, but are not absorbed.The fit accounted for these interactions, by treating propagation through the Earth as a twodimensional problem, with one dimension for the entering neutrino energy, and the other for its energy when it reaches IceCube.The propagation was calculated for different cross-sections, as a function of these energies and zenith angle, and the fitter interpolated between the nearest cross-sections.The absorption calculation neglected nuclear shadowing, which can reduce the per-nucleon crosssection of heavy nuclei [16].It also neglected electromagnetic interactions, whereby the neutrino fluctuates to a µ and a W ± , with the µ then interacting with the Coulomb field of the target nucleus.Overall, these should both be less than 10% effects.
The fit also included seven nuisance parameters to account for uncertainties in the neutrino fluxes, plus one for the DOM efficiency.These were the normalizations for the conventional and prompt atmospheric fluxes and the astrophysical flux, the cosmic-ray spectral index, and the K/π and ν/ν ratio for conventional neutrinos, and the astrophysical flux spectral index.An additional nuisance parameter accounts for uncertainties in the overall DOM optical sensitivity.
The fit found a cross-section multiplier of R = 1.30+0.30  −0.26 .This is the statistical uncertainty, plus some systematic uncertainty due to the nuisance parameters.We isolated the statistical uncertainty by fixing the nuisance parameters to their preferred values and rerunning the fit.We then determined the systematic uncertainties associated with the fit by subtracting, in quadrature, the statistical uncertainty from the total fit uncertainty.The total systematic uncertainty includes some contributions from factors which were not included in the fit: uncertainties about the optical properties of the ice ( +0.30 −0.38 ), uncertainties in the density distribution of the Earth (±0.01), latitude-dependent variations in production rate due to temperature ( +0.00 −0.04 ), uncertainties in the angular acceptance of the IceCube DOMs ( +0.04 −0.00 ) and finally uncertainties in the spectral indices of the prompt and astrophysical spectral indices.The latter were already included in the fit; this additional uncertainty was included to account for the tension between the spectral indices observed by contained event studies [17] and those from through-going muons [18].We then found the total systematic uncertainty by adding these factors, in quadrature to the systematic error associated with the fit.This led to the final result, that the cross-sections are 1.30 +0.21  −0.19 (stat.)+0.39 −0.43 (syst.)times the Standard Model cross-section.
We determined the energy range for which this measurement applied by studying the change in likelihood as we turned off Earth absorption, first starting from very low energies, working upward, and then starting from very high energies, working downward.The points where the likelihood worsened by 2∆LLH = 1 gave us the minimum and maximum energies respectively, 6.3 TeV and 980 TeV. Figure 2 shows this result, along with previous lower-energy results from accelerator experiments.The data does not show a large rise, as would be expected in some BSM theories, particularly those involving leptoquarks or additional rolled-up spatial dimensions.
Similar analyses have been done using contained cascade events [19,20].These analyses suffer from much more limited statistics, so the statistical errors are much larger.On the other hand, the cascade energies are much better known, so it is easier to measure the cross-section in multiple energy bins.

Inelasticity measurement
If BSM processes contribute to the cross-section, there is no reason to expect the form of the interactions to be similar to those from Standard Model processes, so they are

Figure 1 .
Figure 1.The ν µ transmission probability as a function of neutrino energy and zenith angle.The horizontal dashed white line shows the core mantle boundary.The high core density produces a noticeable inflection at that point.From Ref. [11].