Entanglement entropy and monotones in scattering process

In this paper, we study the entanglement property of a 4-particle system. In this system, two initially entangled electrons A and C are scattered by two uncorrelated positrons B and D, respectively. We calculate the entanglements among the particles both before and after the double QED scattering ($AB\rightarrow AB, CD\rightarrow CD$). We find that the change of entanglement entropy between subsystems A and B during the scattering processs is proportional to the total cross section, $\sigma_{tot}=\sigma_{AB}\times\sigma_{CD}$. Even though there is no direct interaction between subsystems A and C (or B and D), the scattering process induces entanglement change among them which is also proportional to $\sigma_{tot}$. This result shows some kind of entanglement sharing property in multipartite system. In order to further investigate the entanglement sharing, we calculate the entanglement monotones which quantify the genuine multipartite entanglement in a multipartite system. For our chosen scattering process, $e^+e^-\rightarrow\mu^+\mu^-$, however, we find that the outgoing state is a W-type 4-partite entangled state which has no genuine 4-partite entanglement.