Measuring the trilinear Higgs self-coupling in Higgs boson pair production at multi-TeV muon colliders

The trilinear Higgs self-coupling determines the shape of the Higgs potential, and its measurement is a central goal of future colliders. We assess the sensitivity of multi-TeV muon colliders to the coupling modifier $\kappa_3$ in Higgs boson pair production via vector boson fusion, using the $b\bar{b}b\bar{b}$ final state at $\sqrt{s}=3$ TeV with $1$ ab$^{-1}$ and at $10$ TeV with $10$ ab$^{-1}$. Events are analyzed in two complementary regions, a resolved region with four jets and a boosted region with two large-radius jets. To extract the signal from backgrounds a few orders of magnitude larger, we combine a supervised jet-to-Higgs pairing network based on the SPANet approach, topological data analysis of the event energy flow, and two dedicated classifiers, $D_{\rm HH}$ for the signal-to-background separation and $D_{\kappa_3}$ for the $\kappa_3$ shape information. The coupling is extracted from a two-dimensional likelihood fit to the distribution of the two classifier outputs. Combining the two regions, we obtain $0.80<\kappa_3<1.29$ at $3$ TeV and $0.96<\kappa_3<1.05$ at $10$ TeV at $68\%$ confidence level. The $10$ TeV determination reaches the few-percent level in this statistics-limited projection, which surpasses by a large amount the precision projected for the HL-LHC.