Depolarization of Spin-½ Particles by Electromagnetic Scatterings

A study is made of the depolarization of polarized, relativistic fermions (spin ½) passing through matter. The final polarization of the projectile shows two features, (i) a rotation of the polarization vector so that it does not have the same direction as the initial polarization with respect to the initial or final momenta: rotation; (ii) an unpolarized component so that the magnitude of the polarization has diminished: shrinkage. We consider the scattering of the incident polarized fermion off unpolarized target electrons and nuclei to lowest order in $\alpha$. Whereas to this order no polarization can be produced, i.e., the magnitude of the polarization vector cannot increase, the magnitude of the spin vector can decrease if the target has spin. General formulas are presented for the spin-½ particles scattered electromagnetically from an unpolarized target with arbitrary spin in terms of form factors. Numerical results are presented for processes (i) and (ii) in the cases of positrons and muons scattered by unpolarized electrons. Process (ii) is proportional to $t^2$ (for small momentum transfer $t$). If one expands the expressions for the polarization phenomena keeping only the linear term in $t$, then the shrinkage (ii) vanishes and the rotation effects (i) all reduce to those for the pure Coulomb scattering case. (As is well-known the depolarization due to Coulomb scattering is negligible for small-angle scattering.) However, if one is concerned with particles scattered into a sizeable solid angle, then (a) the rotation effects in, e.g., positron-electron scattering become enormously larger than that given by Coulomb scattering; (b) they become strongly dependent on the relative orientation of the incident polarization vector: much larger rotations occur for transversely polarized beams; (c) one cannot omit the contribution from the annihilation diagram compared to that from the direct one-photon exchange; (d) and most important the depolarization due to shrinkage is comparable to the rotational effects. In multiple scattering, the shrinkage is a cumulative effect whereas the rotational contribution to depolarization is a random walk process.