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Description
Scattering amplitudes involving unstable particles often exhibit kinematical singularities when an intermediate particle in the t-channel becomes on shell. Such singularities appear in various phenomenologically important processes and require a careful treatment to avoid double counting with sequential production and decay processes. Conventional approaches typically introduce finite beam sizes or thermal effects to regularize the divergence, making the result dependent on external conditions rather than the underlying scattering amplitude.
In this work, we present an analytic prescription that isolates and subtracts the on-shell contribution directly at the amplitude level. Our method provides a finite scattering cross section without relying on beam-profile or thermal regularization and can be applied in a model-independent manner. The prescription is closely related to the factorization of the on-shell intermediate state and offers a transparent interpretation of the subtraction procedure.
As a phenomenological application, we study the process $\mu^+\mu^- \rightarrow W^\pm e^\mp \nu$, which is of particular interest for future muon colliders. We demonstrate that the proposed prescription yields finite and well-defined cross sections while preserving the physical interpretation of the scattering process. Owing to its generality, the method is expected to be applicable to a broad class of scattering processes involving t-channel on-shell singularities.