Second order formalism for spin 1/2 fermions and Compton scattering

Abstract

We develop a second order formalism for spin 1/2 fermions based on the projection over Poincar\'{e} invariant subspaces in the $(1/2,0)\oplus(0,1/2)$ representation of the homogeneous Lorentz group. Using $U(1)_{em}$ gauge principle we obtain second order description for the electromagnetic interactions of a spin 1/2 fermion with two free parameters, the gyromagnetic factor $g$ and a parameter $\xi$ related to odd-parity Lorentz structures. We calculate Compton scattering in this formalism. In the particular case $g=2, \xi=0$ and for states with well defined parity we recover Dirac results. In general, we find the correct classical limit and a finite value $r_{c}^{2}$ for the forward differential cross section, independent of the photon energy and of the value of the parameters $g$ and $\xi$. The differential cross section vanishes at high energies for all $g, \xi$ except in the forward direction. The total cross section at high energies vanishes only for $g=2, \xi=0$. We argue that this formalism is more convenient than Dirac theory in the description of low energy electromagnetic properties of baryons and illustrate the point with the proton case.