We study pion production by proton synchrotron radiation in the presence of a strong magnetic field. In this study we find that the decay width satisfies a robust scaling relation. This scaling implies that one can infer the decay width in more realistic magnetic fields of 10 15 G, where n i, f ∼ 10 12 − 10 13 , from the results for n i, f ∼ 10 4 − 10 5 . Then, we present the resultant pion intensity and angular distributions for realistic magnetic field strengths.
KEYWORDS: Pion Production, Strong Magnetic Field, Relativistic Quantum ApproachIt is widely accepted that soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs) correspond to magnetars [1]. The magnetars have also been proposed [2] as an acceleration site for ultra high-energy (UHE) cosmic rays (UHECRs) and a possible association between magnetar flares [3] and UHECRs has also been observed. Synchrotron radiation can be produced by highenergy protons accelerated in a strong magnetic field. Particularly, the pion process is expected to exceed photon synchrotron emission in the GeV -TeV energy range [4].However, theoretical calculations were performed approximately in a semi-classical way [5][6][7] and each model gave a different result. In addition, these model could not give a momentumdistribution of a final pion.In this work we study pion production from proton synchrotron radiation in the presence of strong magnetic fields in the relativistic quantum approach. In addition, we show the energy and angular distributions of emitted pions.We assume a uniform magnetic field along the z-direction, B = (0, 0, B), and take the electromagnetic vector potential A µ to be A = (0, 0, xB, 0) at the position r ≡ (x, y, z) . The relativistic proton wave function ψ is obtained from the following Dirac equation:where M p is the proton mass, κ p is the proton anomalous magnetic moment (AMM), e is the elementary charge, and E is the single particle energy written asBy using the pseudo-vector coupling for the πN-interaction, we obtain the decay width of the proton with the above wave functions.