Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

Freddy Cachazo, Nick Early, Yong Zhang

Abstract

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are "color" decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of "local planarity" into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the "colorless" generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

Abstract

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of . Amplitudes are "color" decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of "local planarity" into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on . We present results for generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the "colorless" generalized scalar amplitude. We also propose a notion of GCOs for arbitrary , discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.
Paper Structure (32 sections, 5 theorems, 101 equations, 5 figures, 5 tables)

This paper contains 32 sections, 5 theorems, 101 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Standard color decomposition
  3. Generalized color orderings
  4. Generalized Feynman diagrams
  5. Color dressed generalized biadjoint amplitudes
  6. Examples
  7. Properties of $\boldsymbol{k=3}$ color orderings and examples
  8. Arrangements of lines vs. arrangements of pseudo-lines
  9. Connecting color orderings using triangle flips
  10. Generalized decoupling identity
  11. Decoupling Identities for $\boldsymbol{(3,6)}$ color factors
  12. Decoupling identities for $\boldsymbol{(3,7)}$ color factors
  13. Bootstrapping GFDs via flips compatible with color orderings
  14. Flips in $\boldsymbol{k=2}$ Feynman diagrams
  15. Flips in $\boldsymbol{k=3}$ Feynman diagrams
  16. ...and 17 more sections

Key Result

Theorem 6.1

An $n$-tuple of standard $($or $k=2)$ color orderings such that the $i^{\rm th}$ one is defined on the set $[n]\setminus \{i\}$ can be represented as an arrangement of pseudo-lines on $\mathbb{RP}^2$ if the following holds:

Figures (5)

  • Figure 1: Top: Arrangement of lines corresponding to the generalized color ordering $\Sigma = ((2345),(1345),(1245),(1235),(1234))$. The dashed circle is at infinity, points on it with the same label are identified. Each line defines a $k=2$ color ordering by identifying its points on the boundary to make a circle. Bottom: Five $(k,n)=(2,4)$ color orderings obtained from $\Sigma$ by the order in which lines intersect a given one.
  • Figure 2: Representatives of arrangements of lines of different types for $(3,6)$.
  • Figure 3: Blue cycles represent the $k=2$ ordering in $\Sigma$, red cycles represent the $k=2$ ordering in $\tilde{\Sigma}$. Following the CHY diagrammatic description of biadjoint amplitudes, the dual to each red cycle gives rise to the possible Feynman diagrams compatible with both orderings. In this case, there is a single Feynman diagram in each entry and it is drawn in black. They together make up a collection of Feynman diagrams with a non-vanishing metric, i.e., a GFD.
  • Figure 4: Left: Lines $L_i$, $L_j$, $L_k$ bound a triangle with line $L_k$ at the bottom. Center: Line $L_k$ moves up until the triangle becomes a point where all three lines intercept. Right: The three lines bound a triangle again but with $L_k$ bounding the top.
  • Figure 5: A $(4,7)$ GFD which is present in a more redundant way by a symmetric matrix such that the Feynman diagram in the $i^{\rm th}$-row and $j^{\rm th}$-column has leaves $i$, $j$ pruned.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 4.1: herrmann2008draw
  • Definition 4.2
  • Definition 4.3
  • Definition 5.1
  • Theorem 6.1
  • Claim
  • ...and 18 more