Multi-Higgs Amplitudes Bootstrapped: Dissecting SMEFT and HEFT

The precise measurement of the Higgs boson properties requires a robust framework to parametrize possible deviations from Standard Model (SM) predictions in the most model-independent way possible. The Effective Field Theory (EFT) framework has become the most widely used since it offers a broad scope and a consistent path to increase the precision of the computations. Two prominent EFTs are the Standard Model Effective Field Theory (SMEFT) and the Higgs Effective Field Theory (HEFT). While similar in many aspects, their phenomenological differences are nowhere more pronounced than in multi-Higgs production. To precisely chart the separation between both EFTs, we study gluon-fusion double and triple Higgs production using bootstrapped on-shell amplitudes. This allows us to get the kinematic dependence of the gauge-invariant amplitude without field-redefinition ambiguities. As part of our study, we develop a technique that allows to build tree-level five-point on-shell amplitudes from lower-point on-shell amplitudes and bootstrapped contact terms. We then match the bootstrapped on-shell scattering amplitudes to the amplitudes computed in SMEFT (up to order $1/\Lambda^4$) and HEFT (at NNLO) and analyze the EFT order at which each kinematic structure appears. We also show how certain structures in $gg\to hhh$ appear only at dimension-12 in SMEFT or N$^3$LO in HEFT.