Toward extracting scattering phase shift from integrated correlation functions V: complex $\phi^4$ field model in $3+1$ dimensions

In Ref.~\cite{Guo:2024zal} and associated studies, a relativistic finite-volume formalism in $1+1$ dimensions is proposed to extract infinite-volume scattering phaseshift. It is based on the difference of integrated correlation functions (ICF) rather than energy spectrum in the finite volume, and can be regarded as complementary to the well-known L\"{uscher} formalism. In the present work, the formalism is further extended into $3+1$ dimensional spacetime. The aim is to explore and demonstrate the challenges in applying the formalism to more practical settings. Specifically, Monte Carlo simulations of a complex $\phi^4$ relativistic field model are carried out in both 2+1 and 3+1 dimensions on lattices of varying sizes, and phaseshifts for the contact interaction are extracted from the formalism using modest computing resources.