IR/UV mixing from higher-order interactions in a Scalar Field

Abstract

The observed vacuum energy lies far below quantum-field-theoretic estimates. Weinberg's theorem shows that no field can dynamically relax the cosmological constant to zero in a local theory with a translationally invariant vacuum. Approaching this question from a different point of view, the Cohen-Kaplan-Nelson (CKN) bound ties an effective theory's ultraviolet cutoff to its infrared size - however, it has lacked a concrete field-theoretic realization. Our central idea is that anharmonic field oscillations with a supra-linear per-mode ground-state energy reach the Planck scale at a much smaller wavenumber than linearly dispersing modes. Under the standard single-pole assumptions stated below, a supra-linear one-particle pole law cannot arise from a Lorentz-invariant self-energy. We start with a Lorentz-invariant, nonlocal action that breaks Weinberg's locality assumption. We then assume a vacuum that spontaneously breaks boost invariance and preserves spatial isotropy in a preferred frame. A smooth-kernel nonlocal quartic interaction, inserted as our ansatz, yields an instantaneous (in the preferred frame) diagonal reduced Hamiltonian whose high-wavenumber modes are quartic oscillators with ground-state energy $\mathcal{E}_0(k)\sim|\vec k|^{8/3}$. The interaction is diagonal at leading order in the operative regime, with its single coupling's magnitude fixed by a CKN-inspired closure. We establish stability of the reduced theory and the regime of controlled unitary evolution. Imposing the per-mode Planck ceiling together with CKN saturation gives a closure scale $k_{ cutoff}\sim k_{ Pl}^{1/3}\,k_{ box}^{2/3}$, independent of the mode-energy power up to an order-one prefactor. We carry this forward to deduce an equation of state parameter for this vacuum energy.