$SU(2)$ symmetry and conservation of helicity for a Dirac particle in a static magnetic field at first order

Abstract

We investigate the spin dynamics and the conservation of helicity in the first order $S-$matrix of a Dirac particle in any static magnetic field. We express the dynamical quantities using a coordinate system defined by the three mutually orthogonal vectors; the total momentum $\mathbf{k}=\mathbf{p_f}+\mathbf{p_i}$, the momentum transfer $\mathbf{q}=\mathbf{p_f-p_i}$, and $\mathbf{l}=\mathbf{k\times q}$. We show that this leads to an alternative symmetric description of the conservation of helicity in a static magnetic field at first order. In particular, we show that helicity conservation in the transition can be viewed as the invariance of the component of the spin along $\mathbf{k}$ and the flipping of its component along $\mathbf{q}$, just as what happens to the momentum vector of a ball bouncing off a wall. We also derive a ``plug and play" formula for the transition matrix element where the only reference to the specific field configuration, and the incident and outgoing momenta is through the kinematical factors multiplying a general matrix element that is independent of the specific vector potential present.