Quantum scale invariance on the lattice

TL;DR

The paper proposes a non-perturbative lattice framework to realize quantum scale invariance by promoting the lattice spacing to a dynamical dilaton field, thereby replacing the renormalization scale with a field-dependent combination $\omega^2=\xi_\chi \chi^2+\xi_h h^2$ and constructing a scale-invariant action with $\sigma^2=\chi^2+h^2$. It introduces a scale-invariant lattice formulation and an effective potential $V_{\mathrm{eff}}(\chi,h)$ to select parameter regions where SI is spontaneously broken, potentially yielding a massless dilaton and a massive Higgs-like state. The approach is extended to gravity via a scale-invariant Regge/Dynamical Triangulation framework, with a dilaton-modified gravitational action $S_G = G \sum_{hinges} \delta_i V_i \sigma_i^2$ that can lead to emergent or unimodular-like gravity in the continuum limit. If viable non-perturbative continuum limits exist, this framework could address the hierarchy problem and the cosmological constant, linking ultraviolet regularization to quantum gravity, though substantial numerical exploration is required.

Abstract

We propose a scheme leading to a non-perturbative definition of lattice field theories which are scale-invariant on the quantum level. A key idea of the construction is the replacement of the lattice spacing by a propagating dynamical field -- the dilaton. We describe how to select non-perturbatively the phenomenologically viable theories where the scale invariance is broken spontaneously. Relation to gravity is also discussed.