Proton-air interaction properties at √s 100 TeV from shower-depth measurements with the Pierre Auger Observatory and their connection to the Muon Puzzle

Hybrid measurements at the Pierre Auger Observatory indicate that most high-energy hadronic interaction models underestimate the average depth of the shower maximum, Xₘₐₓ, at a center-of-mass energy of √s=97.7 ± 0.4+6.6-6.2\,TeV. In this Letter, the hadronic interaction models are shown to follow a universal relation between the predicted Xₘₐₓ and the mean values of variables characterizing the energy spectra of secondary particles produced in the first interaction of proton-induced air showers. Assuming the validity of these relations in Nature, we map the values of Xₘₐₓ favored by Auger data into mean values of these variables. All models favor an increase in the mean elasticity and in the fraction of hadronic energy in proton--air interactions. The latter must be amplified by a factor of $2.8$ to $4.6$ to account for the muon puzzle.