Emerging Nonlocal K\"{a}ll\`{e}n-Lehmann Higgs Spectra at the LHC

Electroweak symmetry breaking may arise from emergent nonlocal K\"{a}ll\`{e}n-Lehmann spectral densities in Hamiltonians with multiscalar interactions. The nonlocality scale $\Lambda_{NL}$ emerges naturally from the exponentially increasing degeneracy of mass eigenstates in the Higgs two-point function at scales $p^2 \geq \Lambda^2_{NL}$. Following the renormalization of the nonlocal Higgs propagator, we provide a framework for deriving analytic expressions for non-perturbative scattering amplitudes. We demonstrate that for energies exceeding the nonlocality scale, scattering amplitudes are exponentially suppressed. Furthermore, the real part of the Higgs self-energy is suppressed at deep spacelike momenta ($p^2 \sim -\Lambda^2_{NL}$), offering a solution to the Hierarchy problem. Such nonlocal scalar sectors are accessible to current and future LHC runs. We argue that the nonlocal K\"{a}ll\`{e}n-Lehmann spectral density can be constrained through a simultaneous global fit of LHC measurements in exclusive channels, including di-Higgs, electroweak di-boson, and di-photon production. This approach represents a paradigm shift in the search for new physics at high-energy colliders.