High Energy Particle Production from Proton Synchrotron Radiation in Strong Magnetic Fields in Relativistic Quantum Field Theory

TL;DR

This paper tackles proton synchrotron radiation in ultra-strong magnetic fields by solving the Dirac equation in a uniform $B$-field, constructing the proton propagator, and computing photon, pion, and vector-meson emission within a fully relativistic quantum framework that accounts for Landau quantization and recoil. It introduces a general scaling rule via the overlap integral ${\cal M}$ and a curvature parameter $\chi_p = e B E_i / M_p^3$, enabling accurate predictions of decay widths and luminosities for extremely large Landau numbers and high proton energies. The results show that decay widths scale with $\sqrt{eB}$ while luminosities depend only on $\chi_p$, saturating at high $E_i$ and displaying substantial deviations from semiclassical theories, especially for meson channels and recoil effects. The findings have significant implications for magnetar and ultra-high-energy cosmic-ray sources, providing quantitative predictions for momentum distributions and radiative outputs in regimes where Landau quantization is essential and classical approaches fail.

Abstract

We investigate photon, pion, and rho-meson production from proton synchrotron radiation in the presence of strong magnetic fields. The proton decay widths and the luminosities of the emitted particles are calculated within a relativistic quantum framework that incorporates Landau quantization. A scaling rule is derived for the transition probability between different Landau levels. This allows an evaluation of transitions for extremely high Landau numbers exceeding $10^{15}$. Furthermore, we calculate the momentum distribution of the emitted particles by properly including the proton recoil effect associated with particle emission. The results differ significantly from conventional semiclassical approaches.

Figures (10)

• Figure 1: The scaling relation of ${\cal W}_{if}$ in Eq. (\ref{['Wif']}) when $\chi_p=0.01$ (a), and $\chi_p=0.1$ (b) for $\pi^0$, $\chi_p=0.1$ (c), and $\chi_p=1$ (d) for $\rho^0$. Black filled circles indicate the energy of the adiabatic limit. The numbers shown in the key correspond to the Landau levels represented by each line.
• Figure 2: Total decay width of a proton at $p_{iz} = 0$ for the direct $\gamma$ (a), $\pi^0$- (b) and $\rho^0$- meson emission versus $\chi_p$. The solid, dashed and dotted lines represent the results at $B=10^{15}$, $10^{14}$ and $10^{13}$ G, respectively.
• Figure 3: Total synchrotron luminosity of the direct $\gamma$, $\pi^0$- and $\rho^0$-mesons versus $\chi_p$ (a) amd $E_i$ (b) The red, blue and green lines represent the results for $\gamma$, $\pi$ and $\rho$-mesons, and the solid, dashed and dotted lines indicate those at $B=10^{15}$, $10^{14}$ and $10^{13}$ G, respectively.
• Figure 4: Differential decay width $d \Gamma / d e_q$ (multiplied by $[(10^{15}~{\rm G})/B]^2$) for proton to pion decay at $p_{iz} = 0$ versus the emitted pion energy $e_q$ normalized by the incident proton energies $E_i$ for $\chi_p =0.717$.
• Figure 5: Differential decay width $d \Gamma / d e_q$ of protons for direct $\gamma$ (a), $\pi^0$ (b), and $\rho^0$ emission (c) at $B=10^{15}$ G.. The blue dotted, purple dashed, red solid, brown dot-dashed and green long-dashed lines represent the results for $E_i = 1$ TeV, 10 TeV, 100 TeV, 1 PeV, and 10 PeV.