Contribution of various spin configurations to the radiationless decay of PsH

Positronium hydride (PsH) is a weakly bound (molecular) state of two electrons (e^-), a positron (e⁺), and a proton (p⁺). Due to the e^--e⁺ annihilation, it decays to a number of processes, where the radiationless decay is one of the most important channels. In this decay process, the photons released from the e^--e⁺ annihilation are absorbed by the remaining e^- and p⁺ in PsH, giving us PsH→e^-p⁺. For a particular spin (singlet) configuration of the constituents of PsH, the radiationless decay rate of PsH was determined by Aslam et al. [Phys. Rev. A 104, 052803 (2021)]. In this work, we calculate the decay rate of PsH→e^-p⁺ by exercising all possible spin orientations of PsH and the final states e^- and p⁺. By summing all the contributions, we verify that the result is the same if we calculate it using the standard fermion spin-sum technique of quantum field theory.