⚛️ high-energy theory

Local vertices, quadratic propagators and double-copy structure of one-loop integrands from forward limits

This paper develops a systematic method to transform one-loop integrands from a non-local form into a form featuring quadratic propagators and local vertices by utilizing the forward limit of bi-adjoint scalar amplitudes, ultimately demonstrating that these integrands for gravity and Einstein-Yang-Mills theories inherit a double-copy structure.

Original authors: Chongsi Xie, Yi-Jian Du

Published 2026-02-11

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Chongsi Xie, Yi-Jian Du

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to build a massive, complex skyscraper (this is the Scattering Amplitude, which describes how subatomic particles collide).

Usually, architects use standard blueprints called Feynman Diagrams. These blueprints are great, but for the most complex buildings—like those representing Gravity or high-energy physics—the blueprints become so messy, cluttered, and "non-local" (meaning a change in a window on the 1st floor somehow instantly affects a door on the 100th floor in a way that doesn't make sense) that they become impossible to read.

This paper is essentially a new, revolutionary way to redraw those blueprints. Here is the breakdown:

1. The Problem: The "Messy Blueprint" (Non-locality)

In current physics, when we try to calculate how particles like gravitons (the theoretical particles of gravity) interact, our math produces "linear propagators."

Think of this like a plumbing system where the pipes are made of stretchy rubber bands. If you pull a pipe in one corner, the entire system vibrates unpredictably. In physics, we call this non-locality. It’s mathematically "ugly" because, in the real world, things should only affect what they are directly touching. We want "quadratic propagators"—which are like solid, rigid steel pipes. They are predictable, local, and much easier to work with.

2. The Solution: The "Lego-Block" Method (Double-Copy)

The researchers use a trick called the Double-Copy.

Imagine you have a set of instructions for building a simple Lego car (this is Yang-Mills theory, which describes forces like electromagnetism). The researchers discovered that if you take two identical sets of those car instructions and "multiply" them together in a very specific way, you suddenly have the instructions for a complex jet engine (this is Gravity).

Instead of trying to write the jet engine instructions from scratch—which would be a nightmare—they use the simpler car instructions to build it piece by piece.

3. The Innovation: The "Local Vertex" (The Magic Glue)

The core of this paper is how they handle the "glue" that holds these pieces together.

When they tried to combine the two sets of instructions, the "glue" was still messy and stretchy (the non-locality problem). The authors developed a systematic way to "localize" the math.

They found that by using specific mathematical patterns (which they call X-patterns and BCJ-patterns), they could cancel out the "stretchy rubber band" parts of the math. By canceling the mess, they reveal hidden, solid "joints" or Local Vertices.

The Analogy: Imagine you are trying to connect two complex machines. Instead of using messy, tangled wires that stretch everywhere, the researchers found a way to use standardized, solid "plug-and-play" connectors. Once you have these connectors, you can snap together any number of parts—whether you have 3 parts, 5 parts, or 10 parts—and the whole structure remains stable and easy to understand.

4. Why does this matter?

By turning these "stretchy" equations into "solid" ones, the researchers have provided a roadmap for:

Simplifying the Impossible: Calculations that used to take years of supercomputer time might now be done much faster.
Understanding Gravity: It brings us one step closer to a "Theory of Everything" by making the math of gravity look much more like the math of the other forces we already understand.
A Universal Toolkit: They didn't just solve one problem; they created a "General Formula." It’s like they didn't just build one specific bridge; they invented a new type of universal connector that can be used to build any bridge.

In short: They took the messy, tangled "rubber band" math of the universe and turned it into a clean, modular "Lego" system.

This paper, titled "Local vertices, quadratic propagators and double-copy structure of one-loop integrands from forward limits," presents a systematic framework for constructing one-loop scattering integrands for gravity and Einstein-Yang-Mills (EYM) theories using the worldsheet/forward-limit approach.

Below is a detailed technical summary.

1. The Problem: Nonlocality and Linear Propagators

In the modern scattering amplitude program, the forward-limit approach (derived from the CHY formula and ambitwistor strings) is a powerful tool to construct one-loop integrands. However, integrands constructed this way suffer from two major technical drawbacks compared to traditional Feynman diagrams:

Linear Propagators: The loop propagators in the resulting integrands are linear functions of the loop momentum ( $1/l$ ), whereas conventional Feynman diagrams use quadratic propagators ( $1/l^2$ ).
Nonlocality: The expansion formulas for these integrands do not manifest locality. Lorentz contractions (e.g., $\epsilon \cdot \epsilon$ or $\epsilon \cdot k$ ) often occur between particles that are separated by loop propagators, meaning the "vertices" are not local.

These issues prevent a direct application of the Bern-Carrasco-Johansson (BCJ) double-copy construction to one-loop integrands, as the BCJ numerators are typically defined with respect to quadratic propagators.

2. Methodology: The dYMS Approach and Localization

The authors propose a systematic method to transform these nonlocal, linear-propagator integrands into local, quadratic-propagator integrands. The methodology follows these steps:

A. Introduction of Double Yang-Mills-Scalar (dYMS) Integrands

The authors observe that EYM and Gravity (GR) integrands can be expanded into double Yang-Mills-Scalar (dYMS) integrands. A dYMS integrand is constructed by taking two copies of the left half-integrands of a Yang-Mills-Scalar (YMS) theory. This allows the problem of gravity to be reduced to the study of the simpler YMS theory.

B. Cancellation of Nonlocalities (Localization)

The core of the paper is the "localization" procedure. The authors demonstrate that the nonlocalities in the dYMS integrand arise from two sources:

Left Coefficients: These are handled by using the known local expression of the YMS integrand.
Right Coefficients: These are more complex and involve X-patterns (momentum contractions like $k \cdot X$ ) and BCJ-patterns (subcurrents satisfying off-shell BCJ relations).

The authors prove that through the systematic cancellation of these X- and BCJ-patterns, the nonlocal terms vanish, leaving behind a structure where all Lorentz contractions are contained within local multi-point vertices.

C. Conversion to Quadratic Propagators

Once locality is achieved, the authors use the partial fraction identity to convert the sum of linear propagators into a single quadratic propagator ( $1/l^2$ ). This results in a final expression where tree-level "effective subcurrents" are attached to a quadratic loop propagator via local vertices.

3. Key Contributions and Results

A. Systematic Extraction of Local Vertices

The paper provides a rigorous classification and extraction of local vertices based on the number of particles carrying two copies of polarization vectors (denoted by $|W|$ ):

$|W|=0$ : Derives the general local formula for dYMS, YM, and YMS.
$|W|=1$ : Identifies new 3-point, 4-point, and 5-point vertices (e.g., $2x-1y-1z-1w$ vertices).
$|W|=2$ : Uncovers new 6-point vertices and complex 5-point vertices.
$|W|=3$ : Provides an algorithm to extract even higher-order contact terms (e.s., $2x-3w$ vertices).

B. Double-Copy Structure of Effective Currents

A major result is the proof that the effective subcurrents used in these local vertices possess an explicit double-copy structure. Specifically, a tree-level effective current is expressed as a combination of tree-level Bi-Adjoint Scalar (BS) currents associated with two copies of kinematic coefficients.

C. Extension to EYM and GR

By applying the dYMS results, the authors derive the quadratic propagator formulas for:

Einstein-Yang-Mills (EYM) integrands.
General Relativity (GR) integrands.
They also extend these results to tree-level amplitudes, showing that the localization procedure is consistent across loop levels.

4. Significance

This work is significant for several reasons:

Bridge to BCJ Numerators: It provides an intermediate step between the "forward-limit" (linear) approach and the "expected" one-loop BCJ double-copy (quadratic) approach. It offers a path toward the general construction of BCJ numerators for one-loop amplitudes.
Computational Efficiency: By providing local vertex expressions, the paper moves the worldsheet approach closer to the computational utility of Feynman diagrams while retaining the elegance of the amplitude program.
Theoretical Unification: It demonstrates how gauge invariance (via BCJ relations) and the worldsheet approach (via the forward limit) work together to ensure the locality of physical theories like gravity.