Energy spectra of secondaries in proton-proton interactions

We compare the predictions of AAfrag for the spectra of secondary photons, neutrinos, electrons, and positrons produced in proton-proton collisions to those of the parameterisations of Kamae et al., Kelner et al., and Kafexhiu et al. We find that the differences in the normalisation of the photon energy spectra reach 20$-$50% at intermediate values of the transferred energy fraction $x$, growing up to a factor of two for $x\rightarrow\,$1, while the differences in the neutrino spectra are even larger. We argue that LHCf results on the forward production of photons favor the use of the QGSJET-II-04m model on which AAfrag is based. The differences in the normalisation have important implications in the context of multi-messenger astronomy, in particular, for the prediction of neutrino fluxes based on gamma-ray flux measurements, or regarding the inference of the cosmic ray spectrum, based on gamma-ray data. We note also that the positron-electron ratio from hadronic interactions increases with energy towards the cutoff, an effect which is missed using the average electron-positron spectrum from Kelner et al. Finally, we describe the publicly available python package aafragpy, which provides the secondary spectra of photons, neutrinos, electrons, and positrons. This package complements the AAfrag results for protons with energies above 4 GeV with previous analytical parameterizations of particle spectra for lower energy protons.