Constraining the energy spectrum of neutral pions in ultra-high-energy proton-air interactions

Fluctuations in the muon content of extensive air showers are anticorrelated to the fluctuations of the energy taken by the neutral pions which emerge from the first interaction of the cosmic ray in the atmosphere. We show that the high-energy tail of the neutral pion spectrum produced in the first proton-air interaction can be constrained, within the uncertainties of present cosmic ray experiments, through the analysis of the shower-to-shower distribution of the muon content of the air showers, $P(N_{\mu })$.

Figure 1

Shower-to-shower distribution of ${\alpha }_1$ (large figure) and $\ln N_{\mu }$ (inset figure) for proton-induced showers of $E_0={10}^{19}\mathrm{eV}$ and $\theta =67°$. The different lines are the predictions for distinct hadronic interaction models: epos-lhc (green); qgsjet-II.04 (red), and sibyll 2.3c (yellow).

Figure 2

Conversion between ${\mathrm{\Lambda }}_{\mu }$ and ${\mathrm{\Lambda }}_{\alpha }$. The filled circles indicate the predictions by the different interaction models. The lines show how ${\mathrm{\Lambda }}_{\mu }$ changes if ${\mathrm{\Lambda }}_{\alpha }$ is changed (see text for details).

Figure 3

Distribution of the fraction of energy in the laboratory frame carried by the highest energy ${\pi }^0$ in the first interaction of a proton with $E_0={10}^{19}\mathrm{eV}$. The gray distribution is the standard distribution in sibyll 2.3c while the yellow is for a modified neutral pion energy spectrum. The corresponding muon-number distributions for the air showers are shown in the inset plot.

Figure 4

Conversion between ${\mathrm{\Lambda }}_{\mu }$ and ${\mathrm{\Lambda }}_{\pi }$. The filled circles indicate the predictions by the different interaction models. The lines show how ${\mathrm{\Lambda }}_{\mu }$ changes if ${\mathrm{\Lambda }}_{\pi }$ is changed (see text for details).

Figure 5

Muon number distribution (see text for details on primary composition). The showers have been generated using sibyll 2.3c, the primary energy and zenith angle are ${10}^{18.7}\mathrm{eV}$ and 67°, respectively. A resolution of 20% has been applied to the simulated muon number. The fit region chosen to extract ${\mathrm{\Lambda }}_{\mu }$ appears as a shaded band (see text for details).