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Vanishing cycles and analysis of singularities of Feynman diagrams

Stanislav Srednyak, Vladimir Khachatryan

TL;DR

This work develops a topological framework to study singularities of perturbative Feynman loop integrals by deploying vanishing_cycles analyzed via the Mayer--Vietoris spectral sequence. It provides a complete classification of vanishing-geometries and derives an asymptotic expansion near Landau-type singularities, including explicit coefficient formulas. The authors illustrate the method through one- and two-loop vertex diagrams with arbitrary complex masses and specialized QED cases, extracting leading asymptotics controlled by Landau polynomials. They propose a Γ-series–based outlook for future cross-section calculations in lepton–proton scattering, aiming to enable precise radiative-correction computations and potential exact results for certain diagram classes, with implications for upcoming experiments.

Abstract

In this work, we analyze vanishing cycles of Feynman loop integrals by means of the Mayer-Vietoris spectral sequence. A complete classification of possible vanishing geometries are obtained. We employ this result for establishing an asymptotic expansion for the loop integrals near their singularity locus, then give explicit formulas for the coefficients of such an expansion. The 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.

Vanishing cycles and analysis of singularities of Feynman diagrams

TL;DR

This work develops a topological framework to study singularities of perturbative Feynman loop integrals by deploying vanishing_cycles analyzed via the Mayer--Vietoris spectral sequence. It provides a complete classification of vanishing-geometries and derives an asymptotic expansion near Landau-type singularities, including explicit coefficient formulas. The authors illustrate the method through one- and two-loop vertex diagrams with arbitrary complex masses and specialized QED cases, extracting leading asymptotics controlled by Landau polynomials. They propose a Γ-series–based outlook for future cross-section calculations in lepton–proton scattering, aiming to enable precise radiative-correction computations and potential exact results for certain diagram classes, with implications for upcoming experiments.

Abstract

In this work, we analyze vanishing cycles of Feynman loop integrals by means of the Mayer-Vietoris spectral sequence. A complete classification of possible vanishing geometries are obtained. We employ this result for establishing an asymptotic expansion for the loop integrals near their singularity locus, then give explicit formulas for the coefficients of such an expansion. The 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.
Paper Structure (18 sections, 6 theorems, 100 equations, 5 figures)

This paper contains 18 sections, 6 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homology Vanishing Cycles
  4. Vanishing Cycles and Generic Polynomials
  5. Non-Local Vanishing Cycles
  6. Introduction of Pinch Map
  7. Pinch Map I: General Polynomials
  8. Pinch Map II: Feynman Loop Integrals
  9. Examples with Feynman Vertex Loop Diagrams
  10. Vertex Diagram at One Loop: General Case
  11. Propagator at Two Loops: General Case
  12. Vertex Diagram at Two Loops
  13. General Case
  14. Five-Pinch Point for Arbitrary Mass Vertex
  15. QED Case
  16. ...and 3 more sections

Key Result

Theorem 4.1

For the generic stratum of the GZK discriminant for a polynomial, $P(q)$, the vanishing cycles become contractibe to the point where $Q_{i}$ are rational functions defined everywhere on the main stratum of the GZK discriminant ($Q_{i}(p_{\omega})$ must not be confused with $Q(q)$, where the latter is just a local symbol for a generic polynomial, while the former is actually the solution for a pin

Figures (5)

  • Figure 1: The upper vertex of the diagram depicted in the right panel of Figure \ref{['fig:vertex1']} but with arbitrary complex masses, $m_{i}$, considered. The arrows indicate the flow of momenta.
  • Figure 2: Feynman diagrams describing the QED one-loop corrections at the lepton line in $l-p$ scattering. The arrows indicate the flow of momenta. This figure is from Ref. Bucoveanu:2018soy and is used with the kind permission of The European Physical Journal A.
  • Figure 3: A diagram describing the two-loop propagator but with arbitrary complex masses, $m_{i}$, considered.
  • Figure 4: The vertex diagram shown in Figure \ref{['fig:vertex2']}b but with arbitrary complex masses considered.
  • Figure 5: Feynman diagrams describing QED two-loop corrections at the lepton line in $l-p$ scattering. The diagrams in the subfigures (a)-(g) constitute complete list of two-loop corrections to the QED vertex, and are important in the scattering theory because those constitute the sef of model independent corrections. We refer to Ref. Bucoveanu:2018soy for more details. This figure is from Ref. Bucoveanu:2018soy and is used with the kind permission of The European Physical Journal A.

Theorems & Definitions (11)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Lemma 4.4
  • Lemma 4.5
  • Remark 4.6
  • Theorem 4.7
  • ...and 1 more