Constraints on hadronic models in extensive air showers with the Pierre Auger Observatory

Extensive air showers initiated by ultra-high energy cosmic rays are sensitive to the details of hadronic interactions models, so we present the main results obtained using the data of the Pierre Auger Observatory. The depth at which the maximum of the electromagnetic development takes place is the most sensitive parameter to infer the nature of the cosmic rays. However, the hadronic models cannot describe consistently the maximum and the muon measurements at energies higher than those reached at the LHC.


The EM cascade
The longitudinal development of the energy deposit in air by the showers can be measured in the FD, as a function of the atmospheric slant depth. The profile is dominated by the electrons belonging to the EM cascade and the profile maximum is denoted by X max . The depth of shower maximum is mostly sensitive to the first interaction position and the primary particle type, i.e. showers induced by heavy primaries will develop higher (at a shallower depth), faster and with less shower-to-shower fluctuations than those induced by lighter nuclei (due to a larger interaction cross-section and higher number of nucleons). The primary composition can be inferred statistically from the distribution of shower maxima, due to the fluctuations in the first few hadronic interactions in the cascade. The X max distributions can be a superposition of different nuclei, so four component (proton, iron, helium and nitrogen) were considered to fit the data distributions. For both models considered, QGSJetII-04 and EPOS-LHC, the fraction of protons is decreasing and the fraction of helium is increasing for energies above a few EeV, with no significant fraction of iron for all energies (see [5]).
In figure 1 left, the energy evolution of the average X max is shown for the data and predictions of the hadronic interaction models. Different selection cuts are applied to ensure good data-taking conditions [6]. The elongation rate (D 10 = d X max /d log 10 E) would be constant if the composition were pure. The best fit to our data comprises two linear fits (in units of log 10 (E/eV)) with a breaking point at log 10 (E/eV) = 18.27±04(stat) +0.06 −0.07 (sys). Considering the model predictions, above the breaking point, the observed rate of change of X max becomes significantly smaller (∼ 26 g.cm −2 /decade) indicating that the composition is becoming heavier. The observed width of the X max distribution is corrected by subtracting the detector resolution in quadrature to obtain σ(X max ) (figure 1 right). It also shows that above the breaking point the composition changes from light elements to heavier ones. Syst.

The hadronic cascade
In highly-inclined events, the electromagnetic component of a shower is largely absorbed in the atmosphere, while the muons still survive to the ground, which is suitable for direct studies of the EAS muon content. Hybrid events have independent energy and muon measurements, and with zenith angles in the range 62 • < θ < 80 • , the SD signals can be considered as muons. The whole procedure can be seen in [3], schematically: for each event, the muon density at the ground ρ rec µ is recorded with the SD and adjusted to the shape ρ map µ and relative muon content R µ , as ρ rec µ = R µ ρ map µ . The footprint ρ map µ is derived from simulations of protons with QGSJetII-03 at an energy of E = 10 19 eV. A power-law dependence of the mean muon content with energy is expected as R µ = a(E/10 19 eV) b . The data shown in figure 2 left, gives a = 1.84 ± 0.03(stat) ± 0.32(sys) and b = 1.03 ± 0.02(stat) ± 0.03(sys). Even with the large systematic uncertainty (square brackets) data are not compatible with a pure proton composition, and, at higher energy, are marginally compatible with iron. The energy evolution slope also suggests a transition from a lighter to a heavier composition. The problems with the hadronic models becomes evident when the muon results are compared to the depth of shower maximum (electromagnetic). In figure 2 right, the average X max and R µ are shown for the data and simulations at E = 10 19 eV, where the Auger data fall completely out of the model's phase-space. The average values of R µ and the logarithmic slope d ln R µ d ln E are shown in figure  3 and compared to the models predictions using the EM X max composition. In all cases the models fail to match both measurements, requiring a substantial increase in muon production. Additionally, the large value of the measured slope favors a changing composition.
The EM component is highly absorbed in the atmosphere, so under certain conditions (large zenith angles, large distances to the shower core), the SD signal is dominated by muons. In those cases, the temporal structure and distance to the shower axis of muons arriving at the ground can be measured.  [3,7]. d ln E (right) between 4 × 10 18 eV and 5 × 10 19 eV with predictions for air-shower simulation models for a pure proton, a pure iron and a mixed composition compatible with the FD measurements (labeled as ln A ) [3,7].
Using a set of simple assumptions [8], they can be used to obtain the production point of the muons along the shower axis. Summing all muons, the longitudinal development of muon production depth (MPD) can be measured on an event-by-event basis. The evolution with energy of the corresponding maximum X µ max is shown in figure 4 left. Both muonic X µ max and electromagnetic X max parameters can be translated into ln A as shown in figure 4 right, for two interaction models, allowing a direct comparison of both observables. The QGSJetII-04 model and the EPOS-LHC model give incompatible mass values, from the electromagnetic and muonic components.  Figure 4. Evolution of X µ max with energy [8](left), the number of events is indicated in each energy bin and brackets representing the systematic uncertainty. ln A from X µ max and X µ max as a function of energy [7,8](right).

Conclusions
Using data from the Auger Observatory we have found that the average primary composition from the X max observable is not consistent with a pure composition, assuming the overall correctness of the hadronic interaction models considered. Under the assumption that there is no new physics affecting the air-shower development trans-LHC energies, we can see a transition from light to heavy UHECR composition. Nevertheless, muon measurements are inconsistent with hadronic interaction model predictions. The models yield a deficit in the produced number of muons and the interpretation of the longitudinal maxima X µ max and X max leads to an inconsistent average primary mass ln A . Neither electromagnetic or muonic measurements can be described consistently by the hadronic interaction models, so they can be used to constrain them at energies beyond existing terrestrial accelerators.