Effect of fermionic components on trion-electron scattering

To test the validity of replacing a composite fermion by an elementary fermion, we here calculate the transition rate from a state made of one free electron and one trion to a similar electron-trion pair, through the time evolution of such a pair induced by Coulomb interaction between elementary fermions. To do it in a convenient way, we describe the trion as one electron interacting with one exciton, and we use the tools we have developed in the new composite-exciton many-body theory. The trion-electron scattering contains a direct channel in which in'' and out’’ trions are made with the same fermions, and an exchange channel in which the in'' free electron becomes one of the out’’ trion components. As expected, momenta are conserved in these two channels. The direct scattering is found to read as the bare Coulomb potential between elementary particles multiplied by a form factor which depends on the in'' and out’’ trion relative motion indices $\eta$ and ${\eta }^{\mathrm{\prime }}$, this factor reducing to ${\delta }_{\eta {\eta }^{\mathrm{\prime }}}$ in the zero momentum transfer limit: In this direct channel, the trion at large distance reacts as an elementary particle, its composite nature showing up for large momentum transfer. On the contrary, the fact that the trion is not elementary does affect the exchange channel for all momentum transfers. We thus conclude that a 3-component fermion behaves as an elementary fermion for direct processes in the small momentum transfer limit only.