The Higgs field sits in a Mexican-hat potential, and the shape of that hat is set by how strongly the Higgs interacts with itself. The only direct way to measure it is to catch the LHC making two Higgs bosons at once, more than a thousand times rarer than making one. Drag κλ (1 = the Standard Model) and watch what changes.
Solid: your κλ. Dashed: Standard Model.
Horizontal axis: distance of the Higgs field from today's vacuum, in GeV. Cartoon: only the h³ term is scaled, h⁴ kept at its SM value. A real model that shifts κλ also brings new physics, so read the shape, not the numbers.
Two diagrams make a Higgs pair: a box (top-quark loop, no self-coupling) and a triangle (one Higgs splits into two, ∝ κλ). They interfere destructively, so the rate bottoms out near κλ ≈ 2.3 instead of at 0. Shaded: excluded by the 2026 ATLAS+CMS combination.
Double-Higgs production has not been observed yet: 1.1σ is far from the 5σ discovery standard. The High-Luminosity LHC, with about 20 times more collisions, is where that race gets decided.
Each Higgs decays almost instantly, so experiments hunt its products. The big channels are common but messy; the rare ones are clean.
| final state | share of HH | why it's hard or useful |
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