the trilinear Higgs coupling at leading order (LO) [@Glover:1987nx; @Plehn:1996wb; @Dawson:1998py; @Djouadi:1999rca; @Baglio:2012np]. These processes are complementary to indirect effects induced by the Higgs self-interactions in radiative corrections to electroweak observables and single-Higgs processes [@Degrassi:2016wml; @Degrassi:2017ucl] that are plagued by unknown interference effects with other kinds of New Physics.

The Higgs self-interactions are uniquely described by the SM Higgs potential $$V = \frac{\lambda}{2} \left( \phi^\dagger \phi -
  \frac{v^2}{2}\right)^2\, ,$$ where $\lambda$ defines the self-interaction strength of the SM Higgs field. In unitary gauge, the Higgs doublet $\phi$ is given by $$\phi = \left( \begin{array}{c} \displaystyle 0 \\ \displaystyle
\frac{v+H}{\sqrt{2}} \end{array} \right)$$ with $v\approx 246$ GeV denoting the vacuum expectation value (vev) and $H$ is the physical Higgs field. In the SM, the self-interaction strength is given in terms of the Higgs mass $M_H$ by $\lambda =
M_H^2/v^2$. Expanding the Higgs field around its vev, the Higgs self-interactions, including the corresponding permutations, are uniquely determined as $$\lambda_{H^3} = 3 \frac{M_H^2}{v}, \qquad \lambda_{H^4} = 3
\frac{M_H^2}{v^2}\, ,$$ where $\lambda_{H^3}$ ($\lambda_{H^4}$) denotes the trilinear (quartic) Higgs self-coupling.

While the quartic Higgs coupling $\lambda_{H^4}$ cannot be probed directly at the LHC, due to the tiny size of the triple-Higgs production cross section [@Plehn:2005nk; @Binoth:2006ym; 