Very-forward energy measurements with the CASTOR and HF calorimeters of CMS at s=13\sqrt{s}=13 TeV show that no current hadronic interaction model or tune reproduces all forward energy observables simultaneously: the spread of model predictions exceeds tuning uncertainties, Sibyll 2.3 overestimates the fraction of low-energy-deposition events, and the hadronic energy is slightly overestimated at the edge of the uncertainties, while the electromagnetic spectrum is well described. These data also exclude, within the tested model space, increasing the forward hadronic energy in proton-proton collisions as a remedy for the air-shower muon deficit. The open problem is to determine which phenomenological model ingredients (multiparton-interaction regularisation, beam-remnant fragmentation, diffraction, elasticity, forward secondary-particle composition) must be revised so that a single model simultaneously describes the forward energy data, and to establish whether the resulting air-shower simulations resolve the muon deficit or require physics beyond current model classes. Later literature (2022-2026), including the Muon Puzzle white paper, Auger's report of a more-than-5-sigma inconsistency of unmodified models with combined shower data, constrained strangeness-enhancement scenarios, and the EPOS.LHC-R and QGSJET-III model developments, confirms that both threads remain open.

Background

Ultra-high-energy cosmic rays (UHECRs) are observed indirectly: each primary particle initiates an extensive air shower, a cascade of hadronic and electromagnetic interactions in the atmosphere, whose interpretation relies on Monte Carlo simulations of hadron production. In each hadronic generation, neutral pions decay promptly to photons and feed the electromagnetic cascade, while charged pions and kaons either interact again or decay into muons; the number of muons reaching the ground is therefore controlled by the composition and energy of hadrons produced close to the collision axis of the equivalent lab-frame proton-air interaction.

Pseudorapidity, η=−ln⁡tan⁡(θ/2)\eta=-\ln\tan(\theta/2), is the standard collider measure of angle relative to the beam direction; ∣η∣≈6|\eta|\approx 6 corresponds to about half a degree. The CASTOR calorimeter of the CMS experiment at the CERN LHC is a tungsten-quartz sampling calorimeter covering −6.6<η<−5.2-6.6<\eta<-5.2 on one side of the interaction point, in the phase space where the peak of the energy is deposited in proton-proton (pp) collisions, and it is the only LHC detector with dedicated acceptance there; the Hadron Forward (HF) calorimeter extends coverage to ∣η∣≈5|\eta|\approx 5. Particle production at such forward angles is dominated by soft, non-perturbative QCD: multiparton interactions (MPI, additional low-momentum-transfer parton-parton scatterings in the same pp collision) and the fragmentation of beam remnants (the spectator partons), both modeled phenomenologically in event generators whose parameters are tuned to data. Collider generators (PYTHIA 8 and its tunes) and the generators used for air-shower physics (EPOS-LHC, QGSJET II, Sibyll) implement these ingredients differently.

CMS measured, at s=13\sqrt{s}=13 TeV: the energy density dE/dηdE/d\eta over 3.15<∣η∣<6.63.15<|\eta|<6.6 (Eur. Phys. J. C 79, 391 (2019)); the differential cross sections as functions of the total energy deposited in CASTOR and, for the first time using direct electromagnetic/hadronic separation, of its electromagnetic component (energy of electrons and photons, dominated by neutral-pion decay photons) and hadronic component (charged and neutral hadrons) (JHEP 08 (2017) 046); and the average very-forward energy versus central track multiplicity (Eur. Phys. J. C 79, 893 (2019)). None of the tested models describes everything: the model spread exceeds the tuning-uncertainty band, the evolution of energy density with η\eta is poorly described especially in the HF region 3.15<∣η∣<5.203.15<|\eta|<5.20, Sibyll 2.3 overestimates the fraction of events with little deposited energy (possibly indicating too large a collision elasticity), and the hadronic energy is slightly overestimated at the edge of the uncertainties while the electromagnetic spectrum is well described.

In parallel, the Pierre Auger Observatory found that simulated showers based on the same post-LHC models produce too few muons at ground level relative to data (Phys. Rev. D 91, 032003 (2015)); this is the muon deficit, or muon puzzle, which begins to appear at primary energies corresponding to TeV-scale centre-of-mass interactions. Because the electromagnetic/hadronic energy separation in CASTOR constrains the balance between neutral pions and charged hadrons that governs muon production, the forward measurements link directly to the muon puzzle: the CASTOR data leave, within the tested model space, no room to increase the air-shower muon number by increasing the forward hadronic energy in pp collisions. Whether alternative model ingredients, or physics beyond the current model classes, can reconcile simulations with cosmic-ray data is unresolved.

Problem Statement

Determine the revisions of hadronic interaction models required by the very-forward LHC energy measurements and their consequences for the air-shower muon deficit. Concretely: (1) identify which phenomenological ingredients of hadronic event generators, including multiparton-interaction regularisation, beam-remnant fragmentation, diffraction fractions, overall collision elasticity, and forward secondary-particle composition, must be changed, and by what physics, so that a single model or tune simultaneously reproduces, within the published experimental uncertainties, the measured very-forward observables at s=13\sqrt{s}=13 TeV: the energy density dE/dηdE/d\eta over 3.15<∣η∣<6.63.15<|\eta|<6.6, and the differential cross sections as functions of the total, electromagnetic, and hadronic energy deposited in −6.6<η<−5.2-6.6<\eta<-5.2, without degrading the established description of central-rapidity minimum-bias and underlying-event data; and (2) establish, using air-shower simulations based on the revised model, whether the muon deficit observed in ultra-high-energy cosmic-ray showers is thereby removed, or whether ingredients or effects beyond the current model classes are required, given that the CASTOR data already exclude, within the tested model space, increasing the forward hadronic energy in pp collisions as the remedy. The two parts share one objective: a hadronic interaction model consistent with the very-forward collider data whose air-shower predictions, including ground-level muon content and the depth of shower maximum Xmax⁡X_{\max}, agree with cosmic-ray measurements. Both parts are required. A resolution is a physics-level identification and assessment, argued against the measured values and quoted uncertainties of the very-forward measurements cited above, not a demonstrated generator retune or a full air-shower simulation campaign.

References

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