Charged pion chain decay and the cosmic ray positron flux

TL;DR

This paper addresses a conceptual and numerical issue in modeling the cosmic-ray secondary positron flux: an error in the commonly used positron spectrum from charged-p pion decay reported in Moskalenko and Strong (1997). It presents exact analytic expressions for the chain decay spectra $\pi^\pm \to \mu^\pm \to e^\pm$, including muon polarization, in both the pion rest frame and boosted frames, and clarifies that the $e^+$ and $e^-$ spectra are actually identical when polarization is treated correctly. The authors quantify the impact of the MS97 error, finding a modest overestimation of about 10% in the positron spectrum, which is small compared to other uncertainties in CR propagation. Extending the analysis to a power-law pion spectrum, they derive Z-factors that connect parent pion spectra to final-state $e^\pm$ and neutrino spectra, showing that the slope is preserved and providing explicit expressions for use in CR background modeling and neutrino predictions.

Abstract

The study of the cosmic ray positron flux has attracted intense attention in recent years, especially because the observations suggest that it could receive contributions from sources such as pulsars or the self--annihilation or decay of dark matter particles. The main known source of relativistic positrons, that form the background to possible additional contributions, is the chain decay of $π^+$ produced in the inelastic interactions of cosmic rays with interstellar and circumstellar gas. The shape of the energy spectrum of positrons produced in these pion decays can be calculated exactly and is well known. However, surprisingly, some estimates of the contribution of the standard mechanism to the positron flux have adopted an incorrect spectral shape of the positron produced in these decays, following an error present in a 1997 paper of Moskalenko and Strong. In this work we report this error, discuss its origin, and estimate its impact on the numerical results.

Figures (6)

• Figure 1: Energy spectra (in the rest frame of the parent) of the particles created in the chain decay of charged pions: $\pi^\pm \to \mu^\pm ~\nu_\mu (\overline{\nu}_\mu)$ followed by $\mu^+ \to e^+ \, \nu_e \, \overline{\nu}_\mu$ (or charge conjugate mode). The spectra are calculated neglecting the electron mass, and shown as a function of the adimensional variable $y = 2 \, E/m_\pi$. The different lines are for $e^ \pm$, and $\nu_e (\overline{\nu}_e)$. The spectra of the $\overline{\nu}_\mu (\nu_\mu)$ created in the $\mu^\pm$ decay are identical to the $e^\pm$ spectra. The vertical line shows the (monochromatic) spectrum of the $\nu_\mu (\overline{\nu}_\mu)$ created in the first pion decay.
• Figure 2: Spectra of the final state particles in the chain decay of charged pions. The spectra are calculated in a frame where the pion is ultrarelativistic ($\beta_\pi \to 1$) and plotted as a function of the ratio $E/E_\pi$.
• Figure 3: Energy spectra of the electrons and positrons created in the chain decay $\pi^\pm \to \mu^\pm \to e^\pm$. The spectra are calculated in a "laboratory frame" where the parent pion has velocity $\beta_\pi = 0$, 0.2, 0.4, 0.6, 0.8 and 1, and plotted as a function of the ratio $z = E_e/E_\pi$.
• Figure 4: The top panel shows as a solid line the spectrum of the $e^\pm$ (and also the $\nu_\mu$ or $\overline{\nu}_\mu$) emitted in the chain decay of charged pions ($\pi^\pm \to \mu^\pm \to x$) plotted as a function of the adimensional variable $y = 2 \, E/m_\pi$. The dashed line is the positron spectrum used in reference Moskalenko:1997gh. The bottom panel shows the same spectra for a frame where the parent pion is ultrarelativistic (plotted as a function of the quantity $z = E/E_\pi$.
• Figure 5: The thick solid line shows the $Z$--factor that corresponds to the spectra of the $e^\pm$ and $\nu_\mu$ ($\overline{\nu}_\mu$) produced in the chain decay $\pi^\pm \to \mu^\pm \to x$. The dashed line is the $Z$-factor obtained using the incorrect spectrum for the positrons emitted the chain pion decay in reference Moskalenko:1997gh. The thin line shows the $Z$--factor that corresponds to describing the spectrum as a sharp line at energy $E = E_\pi/4$.