Production cross-section of γ-rays in p-p collisions in the energy range 1 GeV-1 PeV for the study of γ-ray astronomy

We present a semi-empirical formula for the production cross-section of γ-rays in p-p collisions covering a very wide energy range, 1 GeV-1 PeV, based on both accelerator and cosmic-ray data. The formula is applicable to studies of the energy spectrum of the diffuse γ-ray background from the Galaxy, observed up to ∼50 GeV by EGRET, and constrained at TeV energies by ground-based γ-ray telescopes such as Whipple and HEGRA. As high energy γ-ray astronomy will develop more and more with the increasing scale of both on-board and ground-based telescopes, it is vitally important to determine a reliable production cross-section of γ-rays in p-p collisions. We assume that both the energy and the transverse momentum distributions of the γ-rays have exponential forms in the center of mass system, exp[-(Eγ∗/T0+pT/p0)], with additional correlation terms related to (Eγ∗,pT) introduced so that the so-called “cigar”-type pseudo-rapidity distribution is reproduced. There are three critical parameters in this empirical formula; the average γ-ray transverse momentum p¯T, the average γ-ray multiplicity N¯γ, and the ratio of T0 to p0, τ ≡ T0/p0. Another important parameter, σpp(E0), the inelastic cross-section of p-p collision, has been provided by our previous work and reliably reproduces experimental data from the threshold energy of pion production, E0 ∼ 2mπ, to the FNAL energy, ∼1.7 PeV, in the laboratory system. Despite the relatively simple nature of our empirical formula, with only three parameters, (p¯T,N¯γ,τ), it reproduces surprisingly well the experimental data on the energy and angular distribution of γ-rays in the reaction process p + p → γ + anything over the six decades of energy from 1 GeV to 1 PeV with an appropriate choice of the above three parameters.