Effective description of open quantum dynamics in relativistic scattering

We investigate the open dynamics of quantum particles in relativistic scattering. Specifically, we consider the scattering process of quantum particles coupled to an environment initially in a vacuum state. Tracing out the environment and using the unitarity of an S operator, we find the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator that describes the evolution of the particles. The GKSL generator is exemplified by focusing on the concrete processes: one is the decay of scalar particle ($\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$), and the others are the pair annihilation and the $2\ensuremath{\rightarrow}2$ scattering of scalar particles ($\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$ and $\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\phi}\ensuremath{\phi}$). The GKSL generator for $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$ has a parameter with the coupling between $\ensuremath{\phi}$ and $\ensuremath{\chi}$ and the masses of the fields. The GKSL generator associated with $\ensuremath{\phi}\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{\chi}\ensuremath{\chi}$ is characterized by a Lorentz-invariant function of initial momenta. Notably, in the pair annihilation process, we show that the probability of pair annihilation varies depending on the superposition state of incident scalar $\ensuremath{\phi}$ particles. Furthermore, we observe that the GKSL generators derived in this paper have Poincar\'e symmetry. This means that GKSL generators with Poincar\'e symmetry provide an effective description of the asymptotic behavior of open quantum dynamics in the long-term processes considered here.