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Higher-Loop Vacuum Integrals on SD

  • Varun Lochab
  • Frederik Denef
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Abstract

We develop a new framework for evaluating higher-loop integrals for scalar fields on SD. Starting from a Schwinger-parametric form for general higher-loop integrals on the sphere, it combines total-derivative identities with edge-contraction relations into a coupled linear system. To illustrate our method, we solve for all vacuum graphs through two loops. This includes, in particular, the two-loop three-melon, for which we construct the analytic solution for general D and obtain explicit dimensionally regularized results at all integer dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{lochab2026higherloop,
  title = {{Higher-Loop Vacuum Integrals on $S^D$}},
  author = {Varun Lochab and Frederik Denef},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30489},
  eprint = {2610.30489},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30489},
}
RIS
TY  - UNPB
TI  - Higher-Loop Vacuum Integrals on S^D
AU  - Varun Lochab
AU  - Frederik Denef
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30489
UR  - https://arxiv.org/abs/2610.30489
AB  - We develop a new framework for evaluating higher-loop integrals for scalar fields on S^D. Starting from a Schwinger-parametric form for general higher-loop integrals on the sphere, it combines total-derivative identities with edge-contraction relations into a coupled linear system. To illustrate our method, we solve for all vacuum graphs through two loops. This includes, in particular, the two-loop three-melon, for which we construct the analytic solution for general D and obtain explicit dimensionally regularized results at all integer dimensions.
ER  -