Exact Two-Loop Effective Potentials in de Sitter Space via Finite-Difference Methods
Abstract
We compute the renormalised two-loop effective potential of the O(N)-symmetric scalar theory on four-dimensional de Sitter space. The main technical ingredient is the two-loop watermelon vacuum integral. Although a closed-form representation of this integral was previously available, extracting its finite part near four dimensions remained exceedingly difficult. We use exact finite-difference equations in the propagator spectral labels to obtain a representation in which the ultraviolet poles are isolated in Gamma-function factors and the remaining series are regular near four dimensions. We prove the boundary condition selecting the physical solution, extract the finite part, and combine the result with the tadpole reconstruction of the potential. The calculation includes the subtraction of subdivergences and the derivation of the matter-sector renormalisation-group functions. Holding the field-dependent masses and scalar curvature fixed makes the spectral labels regulator-dependent. We track this dependence explicitly and show that its derivative terms cancel after the forest subtraction. The resulting potential is exact in the mass-to-curvature ratios at two-loop order, away from propagator poles and subject to the usual limitations of perturbation theory. This is the first exact two-loop determination of the effective potential in four-dimensional de Sitter space.
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BibTeX
@misc{bassett2026exact,
title = {{Exact Two-Loop Effective Potentials in de Sitter Space via Finite-Difference Methods}},
author = {Bruce Bassett and Sergio Luigi Cacciatori and Ugo Moschella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31527},
eprint = {2610.31527},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31527},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Two-Loop Effective Potentials in de Sitter Space via Finite-Difference Methods
AU - Bruce Bassett
AU - Sergio Luigi Cacciatori
AU - Ugo Moschella
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31527
UR - https://arxiv.org/abs/2610.31527
AB - We compute the renormalised two-loop effective potential of the O(N)-symmetric scalar theory on four-dimensional de Sitter space. The main technical ingredient is the two-loop watermelon vacuum integral. Although a closed-form representation of this integral was previously available, extracting its finite part near four dimensions remained exceedingly difficult. We use exact finite-difference equations in the propagator spectral labels to obtain a representation in which the ultraviolet poles are isolated in Gamma-function factors and the remaining series are regular near four dimensions. We prove the boundary condition selecting the physical solution, extract the finite part, and combine the result with the tadpole reconstruction of the potential. The calculation includes the subtraction of subdivergences and the derivation of the matter-sector renormalisation-group functions. Holding the field-dependent masses and scalar curvature fixed makes the spectral labels regulator-dependent. We track this dependence explicitly and show that its derivative terms cancel after the forest subtraction. The resulting potential is exact in the mass-to-curvature ratios at two-loop order, away from propagator poles and subject to the usual limitations of perturbation theory. This is the first exact two-loop determination of the effective potential in four-dimensional de Sitter space.
ER -