Analytic proof of nonnegative continuum entropy

Prove analytically that, for the log-polynomial Pauli–Villars regularization at infinite anisotropy factor, the normalized entanglement entropy is nonnegative and approaches zero smoothly in the continuum limit.

Background

The log-polynomial Pauli–Villars scheme is constructed to cancel the ultraviolet divergences of the energy-momentum tensor, and its leading area contribution to the entanglement entropy vanishes in the continuum limit. Numerical results at infinite anisotropy factor suggest that the normalized entropy remains nonnegative and tends smoothly to zero, whereas finite anisotropy factors exhibit regions of negative entropy that are interpreted as discretization artifacts.

The authors explicitly state that they cannot establish the nonnegativity and smooth limiting behavior analytically. A proof would clarify whether the observed behavior is a structural property of the regulator or merely a numerical feature of the explored parameter range.

References

Although we are unable to prove it analytically, there is numerical evidence that for an infinite anisotropy factor $c$, i.e.~when taking all $l$-modes into account, the normalized entropy is nonnegative and approaches zero very smoothly in the continuum limit.

— Entropy, area, and the choice of regulator during gravitational collapse  (2609.20663 - Guenther et al., 17 Sep 2026) in Section 4, subsection “Flat spacetime with log-polynomial Pauli-Villars regularization”