Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Landau Analysis in Momentum Space with Massless Particles: an Amuse Bouche

C. Vergu

TL;DR

This work extends Landau analysis to massless Feynman integrals in momentum space by incorporating blow-up resolutions and complex-structure deformations to handle square-root branch points. It develops a systematic asymptotic framework for massless configurations, refines the Landau exponent with Hessian corrections, and demonstrates the method across canonical examples (bubble, Sunrise, box, triangle) in various dimensions. The paper also integrates monodromy analysis for iterated integrals with square roots and introduces inversion techniques to access second-type singularities, providing a cohesive toolkit for predicting singularity structures and asymptotics in massless regimes. These advances offer a principled path toward a Landau bootstrap for massless amplitudes and clarify the interplay between IR behavior, kinematic thresholds, and analytic continuation in Feynman integrals.

Abstract

We illustrate how methods from Landau analysis that have been developed for studying the properties of massive Feynman integrals in momentum space can be generalized to massless integrals. We consider integrals with both massive and massless propagators in arbitrary dimensions, paying attention to square root branch points. By focusing on a number of well-chosen examples, we show how resolution of singularities (via blow-ups or complex structure deformation) can be used to predict how the behavior of these integrals is modified as different numbers of propagators are chosen to be massless.

Landau Analysis in Momentum Space with Massless Particles: an Amuse Bouche

TL;DR

This work extends Landau analysis to massless Feynman integrals in momentum space by incorporating blow-up resolutions and complex-structure deformations to handle square-root branch points. It develops a systematic asymptotic framework for massless configurations, refines the Landau exponent with Hessian corrections, and demonstrates the method across canonical examples (bubble, Sunrise, box, triangle) in various dimensions. The paper also integrates monodromy analysis for iterated integrals with square roots and introduces inversion techniques to access second-type singularities, providing a cohesive toolkit for predicting singularity structures and asymptotics in massless regimes. These advances offer a principled path toward a Landau bootstrap for massless amplitudes and clarify the interplay between IR behavior, kinematic thresholds, and analytic continuation in Feynman integrals.

Abstract

We illustrate how methods from Landau analysis that have been developed for studying the properties of massive Feynman integrals in momentum space can be generalized to massless integrals. We consider integrals with both massive and massless propagators in arbitrary dimensions, paying attention to square root branch points. By focusing on a number of well-chosen examples, we show how resolution of singularities (via blow-ups or complex structure deformation) can be used to predict how the behavior of these integrals is modified as different numbers of propagators are chosen to be massless.

Paper Structure

This paper contains 32 sections, 1 theorem, 260 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The Landau Equations with Blow-ups
  4. Pinches and permanent pinches
  5. Landau equations for massless particles in loop-momentum space
  6. Alternative Parametrization of the Blow-Up
  7. Asymptotic Analysis with Massless Particles
  8. Review: Expansions around generic-mass Landau varieties
  9. Massless bubble
  10. One-mass bubbles
  11. Bubble integral in two dimensions
  12. Bubble integral in three dimensions
  13. Sunrise
  14. Sunrise with three massive propagators
  15. Sunrise with a massless propagator
  16. ...and 17 more sections

Key Result

Theorem 1

Consider an iterated integral with forms $\omega_1, \dotsc, \omega_l$, such that the form $\omega_p$ has a pole along a codimension one variety $S$ and no other forms have a singularity there. Next, consider two paths $\gamma_\pm$ with the same end points and such that they go around $S$ in opposite where $\gamma'$ is the initial section of the path until $S$ and $\gamma"$ is the final section of

Figures (9)

  • Figure 1: The on-shell surface of a massive particle, together with a choice of complex detour which avoids all the complex points of the singular on-shell hypersurface.
  • Figure 2: The on-shell surface of a massless particle, together with a choice of complex detour which cannot avoid all the complex points of the singular on-shell hypersurface; there is a pinch at zero momentum.
  • Figure 3: Hierarchy of singularities for a bubble integral. The dot at the bottom represents the bubble integral itself, while the rest of the diagrams display the cut propagators.
  • Figure 4: Hierarchy of singularities for a sunrise integral. The dot at the bottom represents the sunrise integral while the rest of the diagrams display the cut propagators. Not all contractions are drawn, for example there are contractions from the sunrise Landau diagram to the tadpoles and to the elementary graph as well.
  • Figure 5: A triangle integral with one external massive leg and internal propagators of equal mass $m$.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Remark 1
  • proof
  • Remark 2