  We study two-to-two scattering amplitudes of a scalar particle of mass $m$.
For simplicity, we assume the presence of $\mathbb{Z}_2$ symmetry and that the
particle is $\mathbb{Z}_2$ odd. We consider two classes of amplitudes: the
fully nonperturbative ones and effective field theory (EFT) ones with a cut-off
scale $M$. Using the primal numerical method which allows us to impose full
non-linear unitarity, we construct novel bounds on various observables in $2
\leq d \leq 4$ space-time dimensions for both classes of amplitudes. We show
that our bounds are much stronger than the ones obtained by using linearized
unitarity or positivity only. We discuss applications of our bounds to
constraining EFTs. Finally, we compare our bounds to the amplitude in $\phi^4$
theory computed perturbatively at weak coupling, and find that they saturate
the bounds.
