Complementary Probes of the Dimension‐6 Operator via Gravitational Waves and Multi‐Higgs Production

We investigate the phenomenological consequences of a dimension‐6 operator , Λ is the effective scale at which new physics takes over, which modifies the scalar potential of the standard model and can induce a strong first‐order electroweak phase transition (SFOEWPT). We estimate the lower limit on the effective scale Λ to be 515 GeV from the di‐Higgs production cross section measured by the LHC. We analyze the associated collider and cosmological signatures in a unified framework, emphasizing the complementarity between gravitational wave (GW) observations and multi‐Higgs production at hadron colliders. Using one‐loop and finite‐temperature effective potential techniques, we identify the parameter space where SFOEWPT occurs and compute the resulting stochastic GW spectra. We show that for cutoff scales Λ ≲750 GeV, the predicted GW signal lies within the sensitivity reach of future interferometers such as LISA, DECIGO, and BBO. At the same time, such low‐scale scenarios also enhance di‐Higgs and tri‐Higgs production cross sections, making them testable at the HL‐LHC, HE‐LHC, and FCC‐hh. We derive current bounds on Λ from Run‐II di‐Higgs searches and demonstrate how future multi‐Higgs measurements can probe the same region favored by GW observability. Our results highlight the power of combining collider and GW experiments to probe extended scalar interactions and the nature of electroweak symmetry breaking.