Weak deflection angle of charged signal in magnetic fields

We use the perturbative method to study the influence of the magnetic field on the weak deflection angle of charged signals in magnetized stationary and axisymmetric spacetimes within general electromagnetic potentials. The deflection angle is expressed as a series expansion of the inverse of the impact parameter $b$, with coefficients determined by the asymptotic expansions of the metric functions and the electromagnetic four-potential. It is found that in general, the deflection angle can always be separated into two parts, the usual gravitational part as for neutral particles, and the electromagnetic part due to the interaction between the (electro)magnetic field and the signal. The leading order of the gravitational, electrostatic (from nonzero spacetime charge) and magnetic (from nonzero magnetic dipole moment) contributions are $b^{-1},\,b^{-1}$ and $b^{-2}$ respectively. The entire electromagnetic part is enhanced by the large specific charge of elementary particles but suppressed by the reciprocal Lorentz factor. The deflection angle result is then applied to three spacetimes with intrinsic or externally enforced magnetic fields. Effects of the magnetic field on the deflection angle from various parameters, including the spacetime spin, magnetic dipole moment and magnetic parameters, are analyzed. In all these cases, it is found that in the weak deflection limit, these effects agree with the expectation for a Lorentz force; that is, an attractive (or repulsive) one will enlarge (or decrease) the deflection angle.