Vanishing Cycles and Analysis of Singularities of Feynman Diagrams

Srednyak, Stanislav; Khachatryan, Vladimir

doi:10.3390/math13060969

Open AccessFeature PaperArticle

Vanishing Cycles and Analysis of Singularities of Feynman Diagrams

by

Stanislav Srednyak

¹

and

Vladimir Khachatryan

^1,2,*

¹

Department of Physics, Duke University, Durham, NC 27708, USA

²

Department of Physics, Indiana University, Bloomington, IN 47405, USA

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(6), 969; https://doi.org/10.3390/math13060969

Submission received: 1 February 2025 / Revised: 8 March 2025 / Accepted: 13 March 2025 / Published: 14 March 2025

(This article belongs to the Special Issue Advanced Approaches to Mathematical Physics Problems in High-Energy Physics)

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Abstract

In this work, we analyze the vanishing cycles of Feynman loop integrals by the means of the Mayer–Vietoris spectral sequence. A complete classification of possible vanishing geometries is obtained. We use this result for establishing an asymptotic expansion for the loop integrals near their singularity locus and then give explicit formulas for the coefficients of such an expansion. Further development of this framework may potentially lead to exact calculations of one- and two-loop Feynman diagrams, as well as other next-to-leading and higher-order diagrams, in studies of radiative corrections for upcoming lepton–hadron scattering experiments.

Keywords: algebraic geometry; homology classes; Mayer–Vietoris sequence; pinch map; quantum electrodynamics; Feynman loop integrals; Landau varieties; gamma series; radiative corrections

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MDPI and ACS Style

Srednyak, S.; Khachatryan, V. Vanishing Cycles and Analysis of Singularities of Feynman Diagrams. Mathematics 2025, 13, 969. https://doi.org/10.3390/math13060969

AMA Style

Srednyak S, Khachatryan V. Vanishing Cycles and Analysis of Singularities of Feynman Diagrams. Mathematics. 2025; 13(6):969. https://doi.org/10.3390/math13060969

Chicago/Turabian Style

Srednyak, Stanislav, and Vladimir Khachatryan. 2025. "Vanishing Cycles and Analysis of Singularities of Feynman Diagrams" Mathematics 13, no. 6: 969. https://doi.org/10.3390/math13060969

APA Style

Srednyak, S., & Khachatryan, V. (2025). Vanishing Cycles and Analysis of Singularities of Feynman Diagrams. Mathematics, 13(6), 969. https://doi.org/10.3390/math13060969

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Vanishing Cycles and Analysis of Singularities of Feynman Diagrams

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