What is the Higgs trilinear self-coupling?
In plain words
The Higgs interacts with itself, and the SM predicts the exact strength from the Higgs mass. Measuring it requires seeing Higgs bosons produced in pairs, an extremely rare process.
Precise statement
Measure $\kappa_{\lambda} = \lambda_3/\lambda_3^{\mathrm{SM}}$, where the SM gives $\lambda_3 = m_h^2/(2 v^2) \sim 0.13$ for the cubic term $\lambda_3 v h^3$. An answer is $\kappa_{\lambda}$ with total uncertainty below about 0.3 (30 percent), sufficient to distinguish the SM from models with a strongly first-order phase transition.
What would settle it
Combined HH production at the HL-LHC (projected about 30 to 50 percent precision) and at a future collider (FCC-hh projected $\sim 5$ percent).
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
The 2026 ATLAS plus CMS Run 2 HH combination (arXiv:2602.23991) gives $-0.71 < \kappa_{\lambda} < 6.1$ at 95 percent CL (expected -1.3 to 6.7) and HH signal strength $0.8 +0.9/-0.7$, an observed significance of $1.1\sigma$.
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