Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Steinmann Violation and Minimal Cuts

Holmfridur S. Hannesdottir, Luke Lippstreu, Andrew J. McLeod, Maria Polackova

Abstract

The Steinmann relations are known to be violated with respect to some -- but not all -- two-particle momentum channels in massless Feynman integrals. We trace the source of this Steinmann violation to a special class of singularities, which arise from partially-overlapping minimal cuts. This allows us to propose an efficient graphical test for predicting which Steinmann relations will be violated by massless Feynman integrals of a given topology, which can be applied at any loop order. We provide evidence for this test by correctly predicting all instances of Steinmann violation in the complete set of known two-loop integrals that contribute to five-particle scattering with one or two external masses.

Steinmann Violation and Minimal Cuts

Abstract

The Steinmann relations are known to be violated with respect to some -- but not all -- two-particle momentum channels in massless Feynman integrals. We trace the source of this Steinmann violation to a special class of singularities, which arise from partially-overlapping minimal cuts. This allows us to propose an efficient graphical test for predicting which Steinmann relations will be violated by massless Feynman integrals of a given topology, which can be applied at any loop order. We provide evidence for this test by correctly predicting all instances of Steinmann violation in the complete set of known two-loop integrals that contribute to five-particle scattering with one or two external masses.
Paper Structure (14 sections, 1 theorem, 33 equations, 8 figures, 1 table)

This paper contains 14 sections, 1 theorem, 33 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Feynman Integrals and Landau Analysis
  3. The Landau Equations and Landau Diagrams
  4. Hierarchical Constraints and Minimal Cuts
  5. Steinmann Violation at One Loop
  6. Momentum Space Analysis
  7. Feynman Parameter Space Analysis
  8. A Simple Graphical Test
  9. Steinmann-Violating Minimal Cuts to All Loop Orders
  10. Landau Analysis of the Acnode
  11. Landau Analysis of the Envelope
  12. Landau Singularities at Higher Loops
  13. Steinmann Violation at Two Loops
  14. Conclusions

Key Result

Theorem 1

Any Feynman diagram with a partially-overlapping minimal cut $\cancel{G}^{\text{min}}_{s_I, s_J}$ supports a pair of solutions to the Landau equations that allow for Steinmann violation in the $s_I$ and $s_J$ channels whenever $\cancel{G}^{\text{min}}_{s_I, s_J}$ involves exactly four vertices and

Figures (8)

  • Figure 1: Examples of minimal cuts, for the two-mass hard box and the massless four-point double-box. In the box diagram, there exists a unique minimal cut for the threshold singularity in each momentum channel. Conversely, the $s_{12}$ channel of the double box diagram has four minimal cuts.
  • Figure 2: The two singular configurations that lead to Steinmann violation in the two-mass hard box. All momenta are taken to be incoming, and all propagators are on shell. (a) The solution in \ref{['eq:momentumspace2mhardsol2']} that gives rise to a singularity at $s_{234}=0$. Here, the momentum between the two massless corners is the zero vector and all Feynman parameters are zero except $\alpha_1$. (b) The solution in \ref{['eq:momentumspace2mhardsol']} that gives rise to a singularity at $s_{12}=0$. Only $\alpha_3$ is zero, and the momenta flowing through the edges associated with $\alpha_1$, $\alpha_2$, and $\alpha_4$ are all collinear for real external kinematics.
  • Figure 3: The two-mass hard acnode graph. Arrows indicate the orientations of the loop momenta $\ell_1^\mu$ and $\ell_5^\mu$.
  • Figure 4: The two-mass hard envelope graph. Arrows indicate the orientations of the loop momenta $\ell_1^\mu$, $\ell_5^\mu$, and $\ell_6^\mu$.
  • Figure 5: A one-mass pentabox integral, which supports three cuts in the $s_{345}$ channel and four cuts in the $s_{234}$ channel. When combined in all possible ways, these cuts give rise to the twelve partially-overlapping cuts shown on the right.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1