https://doi.org/10.1140/epjs/s11734-026-02137-w
Regular Article
Sunset integrals with up to three mass scales in chiral perturbation theory: a comparative study of the Mellin–Barnes representation technique
1
Centre for High Energy Physics, Indian Institute of Science, Bangalore, 560012, Bangalore, Karnataka, India
2
Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland
3
PSI Center for Neutron and Muon Sciences, 5232, Villigen PSI, Switzerland
4
CNRS/IN2P3, IJCLab, Université Paris-Saclay, 91405, Orsay Cedex, France
5
Univ Claude Bernard Lyon 1, CNRS/IN2P3 IP2I UMR5822, Univ Lyon, 69622, Villeurbanne Cedex, France
6
Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, 53115, Bonn, Germany
7
Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich, 52425, Jülich, Germany
8
Peng Huanwu Collaborative Center for Research and Education, International Institute for Interdisciplinary and Frontiers, 100191, Beijing, China
a
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Received:
1
November
2025
Accepted:
12
January
2026
Published online:
20
May
2026
Abstract
Sunset integrals are among the simplest of two-loop integrals that appear in perturbative quantum field theories and possess up to four distinct mass scales. By means of integration by parts identities, they can be written in terms of four distinct master integrals. In this article, we discuss the independent configurations of on-shell and off-shell sunset master integrals with one, two and three mass scales that arise in chiral perturbation theory. We derive Mellin–Barnes integral representations of these integrals and analytically solve them using various methods to obtain exact results in the form of single and double convergent series of the hypergeometric type, for the values of the mass parameters that allow us to do so. We then discuss how to analytically continue the results to other regions of the parameters and conclude by discussing a few applications in chiral perturbation theory.
The original online version of this article was revised:
Following the renumbering of appendices C → B, D → C, and E → D, three cross-references in the main text were not updated correctly and pointed to non-existent appendices. This has been amended: Section 7.1.1: “given in Table 2 and App. C” now reads “App. B”; Section 7.2.1: “given in Table 4 and App. E” snow reads “App. C”; Table 4 caption: “Expressions for the residues can be found in App. E” now reads “App. C”.
In the sentence ending with“…has been automated in a Mathematica package called MBConicHulls” has been corrected to read “…called MBConicHulls [23, 28]”.
Reference [59] was incomplete and should have included the thesis information and institutional affiliation. It has been corrected to “S. Ghosh, ‘Analytical Mellin-Barnes techniques with applications to two-loop SU(3) chiral perturbation theory and QED at higher loops,’ PhD thesis, Indian Institute of Science, Bangalore. Available at: https://etd.iisc.ac.in/handle/2005/5432”.
A correction to this article is available online at https://doi.org/10.1140/epjs/s11734-026-02421-9.
© The Author(s) 2026
modified publication 2026
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