Using publicly available Monte Carlo codes as well as compilation of published data on interactions for proton kinetic energy below 2 GeV, we parametrize the energy spectra and production rates of -rays by simple but quite accurate () analytical expressions in a broad range from the kinematic threshold to PeV energies.
DOI:https://doi.org/10.1103/PhysRevD.90.123014
© 2014 American Physical Society
The total inelastic cross section as a function of the proton kinetic energy in the laboratory system. The data points are taken from the PDG [47] cross section data. The red line is our -fit formula shown in Eq. (1). The dashed green line is the formula given in Kelner et al. [44] and the dot-dashed blue line is the formula given in Kamae et al. [43].
Reuse & PermissionsCross section data (see Table 1) for the channel compared with Eq. (2) (top panel). In the bottom panel we compare data with Eq. (2) and the parametrization given in Dermer [67]. Notice that near the kinematic threshold the parametrization of Dermer [67] can be up to 70% larger than the data.
Reuse & PermissionsThe two panels on the top show our fit to the experimental data given in Tables 2 and 3. Left panel on the top corresponds to the channel and right panel on the top to the sum of the and channels. Experimental data reveal that the cross section has a dip around . Our parametrization of these exclusive channels cross sections is . For the channel, , , and , . For the channel, , , , and . The panel on the bottom shows the sum of these two-pion production channels multiplied with their respective multiplicities [i.e. ]. They are denoted in the figure with open circles. The full red line is the parametrization given in Eq. (5). The two-pion channel contributions to the total inclusive production become non-negligible for , thus the accuracy below this energy is not necessarily important.
Reuse & PermissionsThe average production multiplicity as a function of proton kinetic energy. GEANT 4.10.0 multiplicity for is calculated using the fit formula in Eq. (6), whereas, for is calculated using fit formula of Eq. (7) with the appropriate coefficients in Table 4. For PYTHIA 8.18, SIBYLL 2.1 and QGSJET-I the fit formula is Eq. (7) with the corresponding coefficients presented in Table 4. Note that SIBYLL 2.1 and QGSJET-I multiplicities were calculated using the parametrizations of the and spectra that are provided in Kelner et al. [44]. The accuracy of the fit formulas that are presented here is better than 3%. An exception to this is the GEANT 4.10.0 description, which for has an accuracy of less than 10%. We, however, do not use the GEANT 4.10.0 multiplicity in this energy interval, instead adopting our own fit to the experimental cross sections in this energy range.
Reuse & PermissionsInclusive production cross section as a function of proton kinetic energy. The left panel compares the parametrization given in Dermer [67], our parametrization and the experimental data at 1.25, 2.2 and 3.5 GeV from the HADES Collaboration [60, 68, 69]. The panel on the right shows the differences between high energy models using the parametrization we have introduced here. Notice that GEANT 4.10.0 works for ; in the plot we have extrapolated its multiplicity up to .
Reuse & PermissionsComparison of the center-of-mass system -spectrum between data (open circles), our -fit of the data (full red line) and the isobar calculations from Dermer [67] (dash blue line). Our fit of the pion differential cross section is based on the partial wave expansion and the FSI. This function is a sum of three terms , and which are a function of , where is the kinetic energy in the center-of-mass system, while, is the maximum kinetic energy in the center-of-mass frame allowed by the kinematics, see Eq. (10). The function , where . Function and . , and are fitted using the method. The pion differential cross section is given by , with the inclusive production cross section.
Reuse & PermissionsThe peak value of the -ray production differential cross section as a function of proton kinetic energy. Open circles represent the calculations from -spectrum data; whereas, the line represents our fit function . \The accuracy of the fit is better than 10% as it is seen from the ratio.
Reuse & PermissionsGamma-ray spectrum for different proton collision energies. The open circles (“Data”) are calculations using experimental spectra. Full red line is the fit formula in Eq. (14). Dash blue lines are calculations from Dermer [67]. Dash-dot green lines are calculations from Kamae et al. [43] fits. Kamae et al. [43] calculations are valid for and they do not respect -ray kinematic limit. Moreover, they do not satisfy the expected symmetry with respect to that decay create. The ratios are taken with respect to calculations from data and the accuracy of the fit is better than 20%.
Reuse & PermissionsThe value of the peak of the -ray production differential cross section over the total inelastic cross section () as a function of proton kinetic energy. Open circles give GEANT 4.10.0, PYTHIA 8.18, SIBYLL 2.1 and QGSJET-I calculations; whereas, the red line is the fit formula of Eq. (12) with coefficients listed in Table 7. As the ratio shows, the accuracy of the fit is better than 15% for and better than 3% for . It is remarkable that , for most of the models, does not have a strong energy dependence especially for . This means that will have the same energy dependence, at high energies, as the total inelastic cross section.
Reuse & PermissionsGEANT 4.10.0 -ray production differential cross section for some specific proton kinetic energies. The open circles are GEANT 4.10.0 calculations; whereas, the full red lines are the fit formula shown in Eq. (11) of Sec. 3, with , and summarized in Table 5. The dash-blue line are calculations from Dermer [67] which are based on the isobar model. The dash-dot green line is the parametrization given by Kamae et al. [43]. The ratio between the fit and the calculations show that the accuracy of the fit is better than 20%.
Reuse & PermissionsPYTHIA 8.18 -ray production differential cross section for some specific proton kinetic energies. The open circles are PYTHIA 8.18 calculations, whereas, full red line is the fit formula shown in Eq. (11) with the corresponding , and listed in Table 5. The dash-green line is the fit given in Kamae et al. [43] which works for . Although not clearly shown in these plots, the fit provided by Kamae et al. [43] violates the symmetry that decay spectra has with respect to . It also violates the that is allowed by the kinematics. The ratio between the fit and the calculations shows that the accuracy of the fit is better than 20%.
Reuse & PermissionsSIBYLL 2.1 -ray production differential cross section for some specific proton kinetic energies. The open circles are SIBYLL 2.1 calculations, the full red line is the fit formula shown in Eq. (11) with the corresponding , and listed in Table 5 and the dash green line is the fit given in Kelner et al. [44]. The ratio between the fit and the calculations shows that the accuracy of the fit is of the order 20%.
Reuse & PermissionsQGSJET-I -ray production differential cross section for some specific proton kinetic energies. The open circles are QGSJET-I calculations, the full red line is the fit formula shown in Eq. (11) with the corresponding , and listed in Table 5 and the dash green line is the QGSJET-II fit calculated using the ppfrag code given in Kachelrieß and Ostapchenko [75]. The ratio between the fit and the calculations shows that the accuracy of the fit is of the order 20%.
Reuse & PermissionsThe nuclear enhancement factor as a function of cosmic ray kinetic energy. For comparison we include Sihver et al. [86] cross sections and two more cross section models Letaw et al. [82] and Tripathi et al. [89], which take into account the energy dependence of -nucleus inelastic cross sections at low energies. Broken lines represent in the case when , whereas, the full lines are calculations with given in Eq. (19).
Reuse & PermissionsAn archetypal spectral plot (black line) demonstrating the quality of fits to both the “threshold” (red dotted-line region on left of the plot) and “cutoff” (green dotted-line region on right of the plot) regions. The model parameters adopted are , and . The blue dash-dot-line is the same result including the nuclear enhancement factor calculated in Sec. 4. The magenta dashed-line shows the equivalent result adopting the -approximation (no enhancement factor).
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