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

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.

💡 Research Summary

The paper presents a comprehensive quantum‑field‑theoretic study of high‑energy particle emission from proton synchrotron radiation in ultra‑strong magnetic fields typical of magnetars (B ≈ 10¹³–10¹⁵ G). The authors start by emphasizing that magnetars are promising sources of ultra‑high‑energy cosmic rays, gamma‑ray bursts, and fast radio bursts, and that conventional semiclassical treatments of synchrotron emission neglect essential quantum effects such as Landau quantization and proton recoil.

A uniform magnetic field along the z‑axis is assumed, with the vector potential A = (0, 0, xB, 0). Solving the Dirac equation in this background yields proton wavefunctions quantized into Landau levels nₗ, with the relativistic energy spectrum
E² = p_z² + 2 nₗ e B + M_p².
The proton Green’s function is constructed from these eigenstates, providing the basis for transition amplitudes between initial (n_i) and final (n_f) Landau levels when a particle (photon, neutral pion π⁰, or ρ⁰ meson) is emitted.

Interaction Lagrangians are taken as
L_π = g_π ψ̄ γ⁵ ψ φ_π for the pseudoscalar pion and
L_V = g_V ψ̄ γ^μ ψ φ_μ^V for vector particles (γ, ρ⁰).
The decay width is derived from Fermi’s golden rule, involving the transition matrix T_if, which in turn depends on the proton current J^α, the Green’s function ρ_M, and overlap integrals M_α(n_i,n_f) of the Landau wavefunctions.

In the ultra‑high‑energy (UHE) regime (E_i ≳ 10⁸ GeV) the Landau numbers become enormous (n_i ≈ 10⁸–10¹⁵). The authors exploit this by approximating all overlap integrals M_α as a single common factor M, which greatly simplifies the algebra. They introduce a dimensionless curvature parameter

χ_p = (e B E_i)/M_p³,

which encapsulates the combined dependence on magnetic field strength and proton energy. Numerical evaluation shows that the scaled transition probability W_if(χ_p) is essentially a universal function of χ_p, independent of the absolute values of B and E_i, provided χ_p lies in the range 0.01–1. This universality is demonstrated for both pion and ρ⁰ emission; the photon case follows a similar pattern but with a different overall coefficient.

The final expression for the decay width of particle type A (A = γ, π⁰, ρ⁰) reads

Γ_A = d_A (4 g_A²/√eB) ∑_{n_f} A_if W_if(χ_p) √n_i,

where d_A = 1 for pions and 2 for vector particles, and A_if contains the remaining kinematic factors. The proton recoil is treated exactly, leading to a δ(q_z) factor that forces the emitted particle’s longitudinal momentum to vanish in the proton rest frame; consequently the emitted particles are collinear with the original proton momentum in the laboratory frame.

Numerical results are presented for three magnetic field strengths (10¹³, 10¹⁴, 10¹⁵ G). The total decay widths for γ, π⁰, and ρ⁰ are plotted versus χ_p, revealing that when divided by eB/M_p the curves collapse onto a single universal line. The peak of the curves occurs at χ_p ≈ 1–10, corresponding to proton energies of 0.1–1 PeV for B = 10¹⁵ G. The pion and ρ⁰ widths are roughly two orders of magnitude larger than the photon width, reflecting the much stronger strong‑interaction couplings (g_π ≈ g_ρ ≈ 13 GeV⁻¹) compared with the electromagnetic coupling.

The authors also compute differential decay widths dΓ/dE_q and synchrotron luminosities I = ∫dq E_q dΓ/dq. The luminosities scale as B² e_q/E_i when expressed in terms of the dimensionless ratio e_q/E_i, confirming that the overall power output is proportional to B² and independent of the absolute magnetic field once χ_p is fixed. This scaling contrasts sharply with semiclassical treatments, which predict a weaker B‑dependence.

A notable discussion concerns the choice of pion‑nucleon coupling. Using a pseudoscalar (PS) coupling destroys the χ_p scaling, whereas a pseudovector (PV) coupling preserves it. This sensitivity suggests that the microscopic interaction structure can dramatically affect observable spectra in magnetar environments.

In summary, the paper establishes a robust quantum‑mechanical framework for proton synchrotron radiation in extreme magnetic fields, derives a universal scaling law based on the curvature parameter χ_p, and demonstrates that meson emission (π⁰, ρ⁰) can dominate over photon emission at ultra‑high energies. The results provide a new theoretical basis for interpreting high‑energy astrophysical phenomena associated with magnetars and set the stage for future observational tests and more detailed magnetospheric modeling.