Electroweak Scattering at the Muon Shot

It has long been recognized that the scattering of electroweak particles at very high energies is dominated by vector boson fusion, which probes the origin of electroweak symmetry breaking and offers a unique window into the ultraviolet regime of the SM. Previous studies assume SM-like couplings and rely on the effective $W$ approximation (or electroweak parton distribution), whose validity is well-established within the SM but not yet studied in the presence of anomalous Higgs couplings. In this work, we critically examine the electroweak production of two Higgs bosons in the presence of anomalous $VVh$ and $VVhh$ couplings. We compute the corresponding helicity amplitudes and compare the cross section results in the effective $W$ approximation with the full fixed-order calculation. In particular, we identify two distinct classes of anomalous Higgs couplings, whose effects are not captured by vector boson fusion and effective $W$ approximation. Such very high energy electroweak scatterings can be probed at the Muon Shot, a multi-TeV muon collider upon which we base our study, although similar considerations apply to other high energy colliders.