Parameterization of the energy and rapidity distributions of secondary   pions and kaons produced in energetic proton-proton collisions
  The possibility of proton acceleration to very high energies in astrophysical sources may have unique observational consequences. In particular, the decay of secondary mesons created in the interaction of energetic protons with photons or nucleons gives rise to high-energy gamma rays and neutrinos with potentially observable fluxes. Recently, there has been considerable interest in the parameterization of the energy spectra of these secondaries. Less attention has been paid to the angular distributions, which may have an important effect on observational quantities and are required to address collisions between protons with different energies and an arbitrary scattering angle. In this work, we study the complete particle distributions of secondary mesons created in proton-proton collisions. We present parameterizations of the energy and rapidity distributions of secondary pions and kaons that reproduce results generated with the event generator PYTHIA to within ~10% in the bulk of the parameter space. The parameterizations are based on incident proton energies from 1 TeV to 1 PeV and are suited for extrapolation to higher energies. We present several applications of these parameterizations. Energy spectra and angular distributions of stable decay products (electrons, neutrinos and gamma rays) follow readily. We give an example of the gamma-ray spectrum that results from the decay of pi0 mesons created in a proton-proton collision. We show that there is a strong correlation between the energy of secondary mesons and the degree of collimation around the direction of the colliding protons. This effect may have important implications for the detection possibility of neutrinos created in the interaction of a developing GRB with its surroundings. 
Hylke B. J. Koers, Asaf Pe'er, and Ralph A. M. J. Wijers