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Landau Analysis in the Grassmannian

Benjamin Hollering, Elia Mazzucchelli, Matteo Parisi, Bernd Sturmfels

Abstract

Momentum twistors for scattering amplitudes in particle physics are lines in three-space. We develop Landau analysis for Feynman integrals in this setting. The resulting discriminants and resultants are identified with Hurwitz and Chow forms of incidence varieties in products of Grassmannians. We study their degrees and factorizations, and the kinematic regimes in which the fibers of the Landau map are rational or real. Identifying this map with the amplituhedron map on positroid varieties, and the associated recursions with promotion maps, yields a geometric mechanism for the emergence of positivity and cluster structures in planar N=4 super Yang-Mills theory.

Landau Analysis in the Grassmannian

Abstract

Momentum twistors for scattering amplitudes in particle physics are lines in three-space. We develop Landau analysis for Feynman integrals in this setting. The resulting discriminants and resultants are identified with Hurwitz and Chow forms of incidence varieties in products of Grassmannians. We study their degrees and factorizations, and the kinematic regimes in which the fibers of the Landau map are rational or real. Identifying this map with the amplituhedron map on positroid varieties, and the associated recursions with promotion maps, yields a geometric mechanism for the emergence of positivity and cluster structures in planar N=4 super Yang-Mills theory.

Paper Structure

This paper contains 13 sections, 35 theorems, 73 equations, 9 figures.

Table of Contents

  1. Introduction
  2. From Feynman Graphs to Lines in 3-Space
  3. Incidence Varieties and the Landau Map
  4. Discriminants and Resultants
  5. Chow and Hurwitz
  6. NLS Geometry
  7. Recursive Formulas
  8. Rationality
  9. Positivity and Reality
  10. Positroid Varieties
  11. Promotion Maps
  12. Cluster Variables
  13. Future Directions

Key Result

Theorem 3.2

Let $G$ be an outerplanar graph. Then the incidence variety $V_G$ is a complete intersection. It has precisely $2^{\tau(G)}$ irreducible components, as described above, one for each bicoloring $\sigma$ of the triangles. In particular, $V_G$ is irreducible when the graph $G$ is triangle-free.

Figures (9)

  • Figure 1: The triple pentagon is a Feynman graph at $\ell\!=\!3$ loops for $d\!=\!9$ interacting particles.
  • Figure 2: Regions of the triple pentagon labeled by dual variables (left). The dual graph labeled by momentum twistors (right). These are lines in $3$-space with prescribed incidences.
  • Figure 3: A Landau diagram which is reducible with respect to $\mathcal{L}_1 = G_{1,u_1} \cup H_{u_1}$.
  • Figure 4: Plabic graphs $\mathcal{G}_{G,\sigma,u}$ for $\sigma={\bf b}$ on the left and for $\sigma = {\bf w}$ in the center. The perfect orientation for $\mathcal{G}_{G,{\bf w},u}$ on the right has the source set $\{1,3,5,7,9,13\}$. This is a basis of $\Pi_\mathcal{G}$.
  • Figure 5: Left: Vertex of a Grassmann graph of degree $n'$ and helicity $h(v)=n'-2$. Center: Plabic graphs for the positroid $U_{n'-2,n'}$. Right: Grassmann graph $\mathcal{G}_{G,\sigma,u}$ in Example \ref{['ex:positroid_Fibonacci']}, with $\sigma=\{{\bf bbww}\}$. The positroid variety $\Pi_{G,\sigma,u}$ in ${\rm Gr}(10,30)$ has dimension $40$ of intersection number $24$. Shown in orange is the bicoloring $\sigma$ of the subdivided hexagon $G_u$.
  • ...and 4 more figures

Theorems & Definitions (92)

  • Example 3.1: Triangle
  • Theorem 3.2
  • proof
  • Example 3.3
  • Example 4.1: $\ell=1$
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • ...and 82 more