Multi-Higgs production in the Higgs Effective Theory

Abstract
Present experimental data are consistent with the Standard Model (SM) Higgs mechanism for electroweak symmetry breaking (EWSB); yet its underlying dynamics of EWSB \textemdash therefore, the exact nature of the Higgs itself \textemdash remains open. Within experimental uncertainties, deviations from the SM predictions are possible, and can be captured in the model-independent framework provided by Effective Field Theories (EFTs). At energies $m_W \ll \sqrt{s} \ll \Lambda$, the Equivalence Theorem (ET) allows to probe the scalar sector --- namely, the Higgs and the Nambu-Goldstone modes --- through longitudinal vector-boson scattering (VBS), $V_L V_L$.  In particular, VBS provides access to the strenghts of $h V_L V_L$ and $hh V_L V_L$ interactions. In the Higgs Effective Field Theory (HEFT), these are controlled, respectively, by parameters $a_1$ and $a_2$ ($a_1 = 2$, $a_2 = 1$ in the SM). The parameter $a_1$ is relatively well constrained, whereas bounds on $a_2$ are looser, inferred mostly from double-Higgs VBS ($V_L V_L \rightarrow hh$). In this work, we study how the so-far unexploited process{ $V_L V_L \rightarrow V_L V_L h$} can sharpen constraints on $(a_1, a_2)$. Focusing on the charged channel $W^+_L W^-_L \rightarrow Z_L Z_L h$, we compute the leading-order (LO) tree-level amplitude using the ET, treating all external states as massless and neglecting $\mathcal{O}(m_W/\sqrt{s})$ effects; transverse gauge interactions and Yukawas are switched off so that the scalar sector can be treated in isolation. The resulting amplitude is controlled by a definite combination of $a_1$ and $a_2$. We obtain the result in two equivalent ways: (i) from the canonical HEFT Lagrangian and (ii) after applying nonlinear field redefinitions that remove vertices involving three fields from the original Lagrangian and reduce to one operator only the relevant contribution to the process in consideration. In the field-redefined basis, the set of tree-level Feynman diagrams collapses to a single contact diagram. Subsequently, we derive the cross-section and compare the result with the Standard Model Effective Theory (SMEFT) prediction. Within SMEFT, correlations (which are absent in HEFT) among couplings make the channel more strongly suppressed than in HEFT. Finally, we perform a toy joint analysis that combines $V_L V_L\to hh$, external knowledge of $a_1$, and two different scenarios for the uncertainty on $V_L V_L \rightarrow V_L V_L h$. The obtained 68\% confidence regions suggest that even conservative improvements on this process could lead to a reduction of the allowed $(a_1, a_2)$ region.