Suppression exponent for multiparticle production in $\lambda\phi^{4}$ theory

We compute the probability of producing $n$ particles from few colliding particles in the unbroken $(3+1)$-dimensional $\lambda\phi^4$ theory. To this end we numerically implement semiclassical method of singular solutions which works at ${n \gg 1}$ in the weakly coupled regime ${\lambda \ll 1}$. For the first time, we obtain reliable results in the region of exceptionally large final state multiplicities ${n\gg \lambda^{-1}}$ where the probability decreases exponentially with $n$, ${{\cal P}(\mbox{few} \to n) \sim \exp\{f_\infty(\varepsilon) \, n\}}$, and its slope $f_{\infty}< 0$ depends on the mean kinetic energy $\varepsilon$ of produced particles. In the opposite case ${n\ll \lambda^{-1}}$ our data match well-known tree-level result, and they interpolate between the two limits at $n \sim \lambda^{-1}$. Overall, this proves exponential suppression of the multiparticle production probability at ${n\gg 1}$ and arbitrary $\varepsilon$ in the unbroken theory. Using numerical solutions, we critically analyze the mechanism for multiple Higgs boson production suggested in the literature. Application of our technique to the scalar theory with spontaneously broken symmetry can eradicate (or confirm) it in the nearest future.