Prospective study for the development of an analysis to measure the tri-linear Higgs coupling using the Matrix Element Method with the ATLAS experiment at the HL-LHC.

The measurement of the Higgs boson self-coupling $\lambda$, also referred to as the Higgs boson tri-linear coupling $\lambda_{HHH}$, is of great importance to experimentally reconstruct the Higgs potential and assess whether the Higgs boson discovered in 2012 at CERN is the one predicted by the Brout-Englert-Higgs mechanism. The Higgs boson potential was postulated ad hoc in the Standard Model (SM) and various extensions of the SM predict changes of the value of $\lambda$. The Higgs boson pair production, the most direct and stringent way to constrain $\lambda$, is a very rare process which has not yet been observed in the most recent data of the LHC (run 2). In this document, we present the first developments of an analysis to measure $\lambda$ based on the Matrix Element Method (MEM) and assuming data from the HL-LHC. This method uses the likelihood to observe a sample of events given a theoretical model and a set of parameters. The MEM therefore makes it possible to determine which value of $\lambda$ is the most likely to have produced a sample of di-Higgs event candidates. The MEM establishes a direct link between theoretical and experimental observables and is a statistically optimal method. The computation of the likelihoods was performed using the publicly available MadWeight and MoMEMta tools. Promising prospects were obtained using MoMEMta and a proof-of-principle of the analysis was established.