Gauge-Invariant Entire-Function Regulators and UV Finiteness in NonLocal Quantum Field Theory

TL;DR

This work demonstrates how gauge- and diffeomorphism-invariant entire-function regulators $F(Box/M_*^2)$ can render nonlocal quantum field theories UV-finite while preserving gauge invariance and unitarity. Through spectral calculus and the background-field method, the regulator acts diagonally in the perturbative vacuum, reducing to a multiplicative form factor $F(-p^2/M_*^2)$ in Minkowski space and to an exponentially damped factor after Wick rotation to Euclidean space. Liouville's theorem does not obstruct physical amplitudes, since $F$ is chosen with no zeros in the finite complex plane, and Paley–Wiener bounds show that such regulators correspond to quasi-local kernels with noncompact exponential tails controlled by the scale $ ext{ell}_* oughly M_*^{-1}$. The framework naturally extends to gravity, yielding a damped graviton propagator and suggesting a path to renormalizability or finiteness at all orders within HUFT. Overall, the paper provides a rigorous covariant foundation for UV-finite gauge and gravitational theories with controlled nonlocality and preserved spectral structure.

Abstract

We clarify the status of gauge-invariant entire-function regulators in NonLocal Quantum Field Theory. The regulator is implemented as an entire function of the covariant Laplace--Beltrami operator. Working in the background-field formalism and expanding around flat, trivial backgrounds, we show that plane waves diagonalize the d'Alembertian, so that the entire function reduces to a multiplicative form factor in Minkowski momentum space. After Wick rotation, to the Euclidean axis, producing exponential ultraviolet damping in loop integrals without introducing additional poles or branch cuts. Our analysis provides a concise, gauge-covariant justification for the use of entire-function regulators in nonlocal quantum field theory.