Elliptic master integrals and the one-loop cosmological wavefunction
Abstract
Loop corrections to the cosmological wavefunction involve similar integrals to those required for scattering amplitudes. For the one-loop triangle of a conformally coupled scalar, the master integrals are elliptic, and so the triangle wavefunction coefficient was expected to be elliptic as well. Using a bootstrap approach, we compute this triangle coefficient exactly and show that it is not: the elliptic periods cancel, and the flat-space wavefunction coefficient is a combination of dilogarithms. The cancellation occurs because the residue of the loop integrand on the elliptic sector is a flat-space amplitude, rational in the loop momentum. The one-loop box, by contrast, is not polylogarithmic, even with a single external leg at each site. Its threshold discontinuity involves elliptic integrals, which we also give in closed form. For regular-tetrahedron momenta the periods of a K3 surface enter its loop integrand.
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BibTeX
@misc{mccune2026elliptic,
title = {{Elliptic master integrals and the one-loop cosmological wavefunction}},
author = {Amara McCune and Matthew D. Schwartz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38725},
eprint = {2610.38725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38725},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Elliptic master integrals and the one-loop cosmological wavefunction
AU - Amara McCune
AU - Matthew D. Schwartz
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38725
UR - https://arxiv.org/abs/2610.38725
AB - Loop corrections to the cosmological wavefunction involve similar integrals to those required for scattering amplitudes. For the one-loop triangle of a conformally coupled scalar, the master integrals are elliptic, and so the triangle wavefunction coefficient was expected to be elliptic as well. Using a bootstrap approach, we compute this triangle coefficient exactly and show that it is not: the elliptic periods cancel, and the flat-space wavefunction coefficient is a combination of dilogarithms. The cancellation occurs because the residue of the loop integrand on the elliptic sector is a flat-space amplitude, rational in the loop momentum. The one-loop box, by contrast, is not polylogarithmic, even with a single external leg at each site. Its threshold discontinuity involves elliptic integrals, which we also give in closed form. For regular-tetrahedron momenta the periods of a K3 surface enter its loop integrand.
ER -