Energy spectra of gamma rays, electrons, and neutrinos produced at proton-proton interactions in the very high energy regime

We present new parameterizations of energy spectra of secondary particles, $\pi$ mesons, gamma rays, electrons, and neutrinos produced in inelastic proton-proton collisions. The simple analytical approximations based on simulations of proton-proton interactions using the public available SIBYLL code provide very good accuracy for energy distributions of secondary products in the energy range above 100 GeV. Generally, the recommended analytical formulas deviate from the simulated distributions within a few percent over a large range of $x=E_i/E_p$—the fraction of energy of the incident proton transferred to the secondaries. Finally, we describe an approximate procedure of continuation of calculations towards low energies, down to the threshold of $\pi$-meson production.

Figure 1

Energy spectra of neutral pions from the numerical simulations of the QGSJET code (histograms) and from the presentation given by Eq. (6) for two energies of primary protons: (a) 0.1 TeV and (b) ${10}^3\mathrm{TeV}$.

Figure 2

Energy spectra of $\pi$ and $\eta$ mesons from the numerical simulations of the SIBYLL code (histograms) and from the analytical presentations given by Eqs. (12, 13) for four energies of primary protons: (a) 0.1 TeV, (b) 10 TeV, (c) 100 TeV, (d) 1000 TeV.

Figure 3

Comparison of the energy distributions of secondary $\pi$ mesons given by Eq. (4) (dashed curves) and Eq. (12) (solid curves) calculated for two energies of parent protons $E_p=1\mathrm{TeV}$ and 100 TeV.

Figure 4

Energy distributions of the secondary products (photons, electrons, muonic and electronic neutrinos) of decays of monoenergetic ultrarelativistic neutral and charged pions. All distributions are normalized, ${\int }_0^{1}dw=1$.

Figure 5

The ratio of the number of leptons to photons of same energy for power-law distributions of pions with spectral index $\alpha$. It is assumed that ${\pi }^0$, ${\pi }^{+}$, and ${\pi }^{-}$ have identical distributions.

Figure 6

Energy spectra of gamma rays from p-p interactions corresponding to four different energies of incident protons: (a) 1 TeV, (b) 30 TeV, (c) 300 TeV, (d) 3000 TeV. Solid lines: numerical calculations based on simulations of production of $\pi$ and $\eta$ mesons using the SIBYLL code; dotted-dashed line: analytical presentation given by Eq. (58). The partial contributions from ${\pi }^0$ and $\eta$ meson decays are also shown.

Figure 7

Energy spectra of gamma rays described by Eq. (58) for three energies of incident protons: 0.1 TeV (curve 1), 100 TeV (curve 2), and 1000 TeV (curve 3). The dashed curve corresponds to the Hillas parameterization of the spectra obtained for proton energies of several tens of TeV.

Figure 8

Energy spectra of electrons described by Eq. (62) for three energies of incident protons: 0.1 TeV (curve 1), 100 TeV (curve 2), and 1000 TeV (curve 3).

Figure 9

Energy spectra of all muonic neutrinos described by Eqs. (62, 66) for three energies of incident protons: 0.1 TeV (curve 1), 100 TeV (curve 2), and 1000 TeV (curve 3).

Figure 10

Energy spectra of all decay products produced at p-p interactions for two energies of incident protons: (a) 0.1 TeV and (b) 1000 TeV. The spectrum of electronic neutrinos is not shown because it practically coincides with the spectrum of electrons.

Figure 11

Inelastic cross section of p-p interactions approximated by Eq. (79). The experimental data are from http:wwwppds.ihep.su:8001/c5-5A.HTML, the open points correspond to the cross sections which are used in the SIBYLL code.

Figure 12

Energy spectra of gamma rays and leptons from p-p interactions calculated for the distribution of protons given by Eq. (74) with parameters $E_0=1000\mathrm{TeV}$, $\beta =1$ and (a) $\alpha =2$, (b) $\alpha =1.5$. The dashed curves are calculated in the $\delta$-functional approximation.

Figure 13

Comparison of energy spectra of gamma rays calculated for the nominal inelastic cross section given by Eq. (73) and a formally assumed energy-independent cross section ${\sigma }_{\mathrm{inel}}=34\mathrm{mb}$. The proton distribution is given by Eq. (74); the relevant parameters are shown in the figure.

Figure 14

Energy spectra of gamma rays and leptons from p-p interactions in the very low energy band. The energy distribution of parent protons is given by Eq. (74); the relevant parameters $\alpha$, $\beta$, $E_0$ are shown in figures.

Figure 15

Energy spectra of electrons (dashed line) and positrons (dotted-dashed line) from decays of charged $\pi$ mesons based on Eqs. (C2)–(C6) of Ref. 25 compared with accurate distribution of electrons ($\equiv \mathrm{\text{positrons}}$) given by Eq. (36) (solid line).