IR-Safe and IR-Resummed Bispectra Before and After Reconstruction in Unified Lagrangian Perturbation Theory

Abstract: We develop a unified analytic framework for modeling the real-space dark matter bispectrum, including both auto and cross bispectra constructed from pre- and post-reconstruction density fields, based on Unified Lagrangian Perturbation Theory (ULPT). ULPT reorganizes the standard Lagrangian approach by separating the Jacobian deviation, which generates the nonlinear source bispectrum, from the displacement-mapping effect that determines how long-wavelength bulk flows distort observed configurations. Within this structure, we derive general one-loop ULPT expressions for the bispectrum and analyze their infrared (IR) behavior, demonstrating exact, nonperturbative IR cancellation and the natural emergence of an IR-resummed description of baryon acoustic oscillation (BAO) damping. In particular, ULPT enables a detailed and fully analytic treatment of the wiggle-wiggle contribution to the IR-resummed bispectrum, whose structure has remained comparatively unexplored in previous approaches. We further construct IR-resummed models for the cross bispectra of all pre- and post-reconstruction combinations. For configurations in which the displacement fields differ, ULPT captures not only the overall exponential damping but also the more intricate BAO-scale modulation characteristic of mixed pre/post bispectra. Our results clarify the physical origin of nonlinear BAO suppression and provide a compact theoretical framework in which the full ULPT bispectrum, once implemented numerically through displacement-mapping convolution integrals, will automatically encode all nonlinear BAO and IR effects. The framework developed here therefore offers a unified and IR-safe foundation for interpreting next-generation bispectrum measurements.