The two and three-loop matter bispectrum in perturbation theories

We evaluate for the first time the dark matter bispectrum of large-scale structure at two loops in the Standard Perturbation Theory and at three loops in the Renormalised Perturbation Theory (MPTbreeze formalism), removing in each case the leading divergences in the integrals in order to make them infrared-safe. We show that the Standard Perturbation Theory at two loops can be employed to model the matter bispectrum further into the quasi-nonlinear regime compared to one loop, up to $k_{\text{max}} \sim 0.1 \, h/{\text{Mpc}}$ at $z = 0$, but without reaching a high level of accuracy. In the case of the MPTbreeze method, we show that its bispectra decay at smaller and smaller scales with increasing loop order, but with smaller improvements. At three loops, this model predicts the bispectrum accurately up to scales $k_{\text{max}} \sim 0.17 \, h/{\text{Mpc}}$ at $z = 0$ and $k_{\text{max}} \sim 0.24 \, h/{\text{Mpc}}$ at $z = 1$.