Parametrization of the nucleus-nucleus $\gamma$-ray production cross sections below 100 GeV/nucleon: Subthreshold pions and hard photons

‘‘Subthreshold pions” and so-called “hard photons” are two important channels for producing less than 1 GeV $\gamma$ rays and $e^{\pm }$ pairs from nuclear collisions with energy per nucleon below the $\pi$-meson production threshold. I use publicly available experimental data to parametrize these two channels’ $\gamma$-ray and $e^{\pm }$ production cross sections and extend the pion contribution to these particles spectra at higher energies using their corresponding spectra from $pp$ interactions. These parametrizations are valid for collision energy $T_p\le 100A$ GeV and agree reasonably well with the available experimental data. The new parametrizations allow, for the first time, accurate studies of astrophysical $\gamma$ rays below 1 GeV.

Figure 1

Hard photon inverse slope parameter $E_0^{\gamma }$ for $E_{\gamma }>30\mathrm{MeV}$ as a function of ${\epsilon }_p=\left(T_p-V_c/A_p\right)/m_p$, where $T_p$ is the projectile kinetic energy per nucleon, $V_c$ is the Coulomb energy, $A_p$ the projectile mass number, and $m_p$ is the nucleon mass in units of energy. The experimental data that are used here are listed in Table 1. Data in grey color correspond to light projectiles $p, \alpha$, and Li, whereas, data in black color belongs to heavier projectiles. The grey thick-dashed line shows the Eq. (7) parametrization for projectiles $p, \alpha$, and Li which is $E_0^{\gamma }=(60\pm 10){\epsilon }_p^{0.54\pm 0.06}\mathrm{MeV}$; whereas the black thick-line shows the parametrization for projectiles heavier than Li $E_0^{\gamma }=(182\pm 1){\epsilon }_p^{0.805\pm 0.002}\mathrm{MeV}$. The thin dash-dotted line is the parametrization given in Cassing et al. [4] and the dots curve is the parametrization given in Schutz et al. [5].

Figure 2

Hard photon emission probability $P_{\gamma }$ for $E_{\gamma }>E_{\gamma }^{\min }=30\mathrm{MeV}$ as a function of ${\epsilon }_p$. The experimental data points are compiled by Cassing et al. [4]. The black line is the parametrization shown in Eqs. (5) and (7). The thin dash-dot line and the dotted line are the parametrizations given in Cassing et al. [4] and Schutz et al. [5], respectively.

Figure 3

$\gamma$-ray production differential cross section for ${}^{16}\mathrm{O}+{}^{184}\mathrm{W}$ interactions at $T_p=15A$ MeV [21]. The thick gray line is the hard photon fit using the parametrization in Eq. (8).

Figure 4

The universal pion production probability as a function of ${\epsilon }_p$. All experimental data shown here are listed in Tables 2, 3, 4. The fitting curve is the parametrization given in Eq. (11), $P_{\pi }/({\mathcal{I}}_{\pi }\times \zeta )={\epsilon }_p^{-1/4}\times exp\left[{\epsilon }_p^{-1/4}(0.0057x^4+0.019x^3-0.19x^2+1.07x-3.7)\right]$, where $x=log\left({\epsilon }_p\right)$.

Figure 5

The pion temperature as a function of ${\epsilon }_p$. The experimental data points are listed in Table 5. The line represents the parametrization given in Eq. (16) and the shaded area is the $1\sigma$ fitting bounds.

Figure 6

Neutral pion production cross section as a function of the projectile kinetic energy per nucleon for $p+{}^4\mathrm{He}, p+{}^{12}\mathrm{C}$, and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ interactions. The experimental data points are described in the text, whereas the curves are the predictions of Eq. (9). The $p+p\to {\pi }^0$ production cross section is added for comparison.

Figure 7

Negative pion production cross section as a function of projectile kinetic energy per nucleon for $p+{}^4\mathrm{He}$ and $p+{}^{12}\mathrm{C}$ interactions. The experimental data points are described in the text, the full lines are described by Eq. (9), whereas the dash lines represent the superposition model [83] (see the text).

Figure 8

Comparison of the hard photon $\gamma$-ray production differential cross section Eq. (8) with the available experimental data for $p+{}^{12}\mathrm{C}\to \gamma$ at 124 MeV [34] and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}\to \gamma$ at $84A$ MeV [29].

Figure 9

Comparison of the subthreshold ${\pi }^0$ production differential cross section for ${}^{14}\mathrm{N}{+}^{27}\mathrm{Al}\to {\pi }^0$ at $35A$ MeV [45] and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}\to {\pi }^0$ for 60, 74, and 84 $A$ MeV [48]. The histogram line represent the experimental data, whereas the full line is the parametrization presented in Eq. (15).

Figure 10

The energy distribution of the ${\pi }^{\pm }$ mesons and their decay products produced in $p+C$ interactions with collision momentum $P_p=3$, 5, 8, and 12 GeV/$c$. The left column shows the ${\pi }^{\pm }$ energy distributions. The data points plotted therein are the angle integrated HARP experiment forward data [86, 87]. The middle column shows the $e^{\pm }$ energy distributions from the corresponding ${\pi }^{\pm }$ decay and the right column shows the $\gamma$-ray energy distribution from the ${\pi }^0$ decay. The histogram lines are the geant4 predictions, the dash line are the predictions from the angle integrated Sanford-Wang (SW) parametrization [86] and the full line is the respective parametrization provided here (see the text). The black color in the left and middle columns show the ${\pi }^{+}$ and $e^{+}$ results, whereas the gray represents ${\pi }^{-}$ and $e^{-}$ results. For visual effects, the ${\pi }^{-}$ results are divided by ten and the $e^{-}$ results are divided by one hundred. The gray areas represent the uncertainties of the $\gamma$-ray and $e^{\pm }$ production spectra when computed directly from the experimental pion data.

Figure 11

The energy distribution of the ${\pi }^{\pm }$ mesons and their decay products produced in $p+C$ interactions with collision momentum $P_p=31$ GeV/$c$. The left panel shows the ${\pi }^{\pm }$ energy distributions obtained from angle integration of the NA61/SHINE forward data [88, 89]. The middle panel shows the $e^{\pm }$ energy distributions from the corresponding ${\pi }^{\pm }$ decay and the right panel shows the $\gamma$-ray energy distribution from the ${\pi }^0$ decay. The experimental ${\pi }^0$ differential cross section is obtained from the ${\pi }^{\pm }$ ones through the isospin relations. The histogram lines are the geant4 predictions, the dash line are the predictions from interpolating the ${\pi }^{\pm }$ experimental data and the full line is the respective parametrization provided here (see the text). The black color in the left and middle columns show the ${\pi }^{+}$ and $e^{+}$ results, whereas the gray represents ${\pi }^{-}$ and $e^{-}$ results. For visual effects, the ${\pi }^{-}$ results are divided by ten and the $e^{-}$ results are divided by 100. The gray areas represent the uncertainties of the $\gamma$-ray and $e^{\pm }$ production spectra when computed directly from the experimental pion data.

Figure 12

Secondary particles energy distribution for $p+{}^{12}\mathrm{C}$ interaction at $T_p=100$ GeV. The histogram line represent the geant4 results, whereas the full line is the parametrization presented here (see the text). The black lines in the left panel represent the $e^{+}$ results, whereas, the gray color represents the $e^{-}$ results that for visual effects are divided by 100.

Figure 13

$\gamma$-ray production spectra for $p+{}^{12}\mathrm{C}$ (left) and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ (right) interactions. The projectile fluxes have a power-law index $\alpha =3$ and in one case $T_p^{\mathrm{cut}}\to \infty$ (top) and in the other case $T_p^{\mathrm{cut}}=1A$ GeV (bottom), see Eq. (23). The thick black line is the ${\pi }^0\to 2\gamma$ contribution, doted line is the hard photon contribution and the dashed line is the sum. The thin gray line correspond to $pp\to {\pi }^0$ scaled according to the wounded nucleon model. The small panel in each plot show the spectral ratio between the nucleus-nucleus and $pp$.

Figure 14

Electron and positron production spectra for $p+{}^{12}\mathrm{C}$ (left) and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ (right) interactions. The projectile fluxes have a power-law index $\alpha =3$ and in one case $T_p^{\mathrm{cut}}\to \infty$ (top) and in the other case $T_p^{\mathrm{cut}}=1A$ GeV (bottom), see Eq. (23). The thick black line corresponds to $e^{+}$ production and the thick dashed line correspond to $e^{-}$ production. The thin gray line shows the contribution from $pp\to e^{+}$ and the thin dash-doted line shows the contribution from $pp\to e^{-}$ which are scaled for each case according to the wounded nucleon model.

Figure 15

$\gamma$-ray production spectrum from the hard photons and ${\pi }^0$-meson decay for a solar-like composition of elements. The exponential cut-off energies are $T_p^{\mathrm{cut}}\to \infty$ (left) and $T_p^{\mathrm{cut}}=1A$ GeV (right), whereas the power-law index varies between $2\le \alpha \le 5$. The solid lines represent the contribution from all nuclei, whereas, the thin dash lines represent the $\gamma$-ray spectra from $pp$ collisions which are scaled according to the wounded nucleon model.