⚛️ high-energy theory

Expansion formula of one-loop Einstein-Yang-Mills integrand

This paper establishes an expansion formula for one-loop Einstein-Yang-Mills integrands with a gluon loop in terms of conventional one-loop Yang-Mills integrands by utilizing a two-step expansion strategy through tree-level amplitudes, demonstrating that the resulting kinematic coefficients match those of Yang-Mills-scalar theories and thereby providing a foundation for constructing one-loop BCJ numerators.

Original authors: Yi-Jian Du, Chongsi Xie

Published 2026-06-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi-Jian Du, Chongsi Xie

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 the universe is built from tiny, invisible Lego bricks. Physicists have spent decades trying to figure out the exact instructions for how these bricks snap together to create everything from light (photons) to gravity.

This paper is a new set of instructions that helps physicists translate the "language" of gravity into the "language" of light. Here is the story of what they did, explained without the heavy math.

The Big Picture: Two Languages, One Translation

Physicists have two main ways of describing how particles interact:

The "Light" Language (Yang-Mills): This describes forces like electromagnetism and the strong nuclear force. It's like a complex, colorful puzzle where every piece has a specific color and shape.
The "Gravity" Language (Einstein-Yang-Mills): This describes gravity. It's like a much heavier, more complex version of the same puzzle.

For a long time, scientists knew there was a secret link between these two. They realized that if you take the "Light" puzzle, remove the colors, and swap the pieces around in a specific way, you get the "Gravity" puzzle. This is called Color-Kinematics Duality.

However, there was a problem. This trick worked perfectly for single interactions (like two particles bumping into each other once). But when particles loop around in a circle (a "one-loop" interaction, which is common in quantum physics), the instructions got messy. The "Gravity" instructions didn't quite match the "Light" instructions anymore.

The Problem: The "Linear" vs. "Quadratic" Trap

To understand the authors' solution, imagine you are trying to build a tower.

The Old Way (Linear Propagators): Imagine you are building the tower using a blueprint that says, "Put a block here, then move 1 inch to the right, then put another block." This is a straight line. It's easy to draw, but it's hard to build a stable, round tower with it. In physics, this is called a "linear propagator." It works for simple calculations but breaks down when you try to build the full, complex structure of gravity.
The Goal (Quadratic Propagators): The real world (and standard Feynman diagrams) works like a curved path. You need to know the distance from the center in all directions at once. This is a "quadratic propagator." It's the standard, correct way to describe the tower.

The authors found that the "Light" instructions they had were written in the "Linear" language. They needed to translate them into the "Quadratic" language so they could build the "Gravity" tower correctly.

The Solution: The "Two-Step" Translator

The authors, Yi-Jian Du and Chongsi Xie, developed a rigorous proof that acts as a translator. They showed that you can take the messy "Gravity" instructions and break them down into two steps:

Step 1: The "Scalar" Middleman. They introduced a third, simpler language called "Bi-adjoint Scalar" (BS). Think of this as a universal translator that speaks both "Light" and "Gravity." They showed that the "Gravity" instructions could be written as a sum of these simple "Scalar" instructions.
Step 2: The Consistency Check. Here is the magic trick. The authors proved that if the "Light" instructions follow a specific rule (called a consistency condition), they can be rearranged from the messy "Linear" format into the clean "Quadratic" format.

The Analogy:
Imagine you have a recipe for a complex cake (Gravity) written in a strange code.

You know that a simple cookie recipe (Scalar) can be made from the cake recipe.
You also know that a standard cake recipe (Light) can be made from the cookie recipe.
The authors proved that if you follow the cookie recipe exactly and ensure the ingredients are measured in a specific way (the consistency condition), you can automatically rewrite the complex cake recipe using the standard cake ingredients.

The Key Discovery

The most exciting part of their finding is that the coefficients (the numbers and multipliers in the recipe) are identical.

The numbers used to turn "Light" into "Scalar" are the exact same numbers needed to turn "Gravity" into "Light."

This means that if physicists figure out how to write the "Light" instructions correctly for these loops, they automatically know how to write the "Gravity" instructions. They don't need to do the hard work twice.

Why This Matters

This paper doesn't invent new particles or predict new forces. Instead, it provides the mathematical bridge that was missing.

Before this, trying to calculate gravity loops was like trying to solve a Rubik's cube while blindfolded.
Now, the authors have shown that if you solve the "Light" version of the puzzle correctly, the "Gravity" version solves itself automatically.

They used a method called the Forward Limit, which is like taking a movie of a particle interaction, running it forward, and then running it backward to see how the pieces fit together in a loop. By proving that this method works consistently, they have given physicists a reliable tool to calculate how gravity behaves at the smallest scales, using the simpler rules of light as a guide.

In short: They found the universal key that unlocks the door between the physics of light and the physics of gravity, proving that if you get the light right, the gravity follows automatically.

Technical Summary: Expansion Formula of One-Loop Einstein-Yang-Mills Integrand

Problem Statement
The construction of Bern-Carrasco-Johansson (BCJ) numerators satisfying color-kinematics duality is a central challenge in understanding the relationship between Yang-Mills (YM) theory and General Relativity (GR). While systematic constructions exist for tree-level amplitudes, a general construction for one-loop BCJ numerators remains an open problem. A specific difficulty arises in the "forward limit" approach, where $n$ -point one-loop integrands are expressed as $(n+2)$ -point tree amplitudes. In this representation, loop propagators appear as linear functions of the loop momentum ( $\ell \cdot k$ ), whereas conventional Feynman diagrams utilize quadratic propagators ( $( \ell + k)^2$ ). Consequently, expansion coefficients derived from the forward limit cannot be naively identified as one-loop BCJ numerators. Recent work established that if expansion coefficients satisfy specific "consistency conditions" (invariance under cyclic permutations combined with loop momentum shifts), they can generate valid one-loop numerators. However, an explicit expansion formula for Einstein-Yang-Mills (EYM) integrands in terms of conventional one-loop YM integrands (with quadratic propagators) was lacking, particularly one that inherits the coefficients from the known scalar-loop Yang-Mills-scalar (YMS) expansion.

Methodology
The authors employ a two-step expansion strategy based on the forward limit approach and explicit analysis of Bi-Adjoint Scalar (BS) Feynman diagrams:

Forward Limit Representation: The one-loop EYM integrand is first expressed as a sum of tree-level amplitudes where two particles carry loop momenta $\pm \ell$ . Using the tree-level expansion formula, these are decomposed into doubly color-ordered BS amplitudes.
Linear to Quadratic Propagator Conversion: The paper analyzes the structure of these BS amplitudes, which initially contain linear propagators. By grouping terms related by cyclic permutations of the external particles and applying the consistency conditions on the kinematic coefficients, the authors demonstrate how sums of linear-propagator diagrams can be reorganized into single diagrams with quadratic propagators.
Coefficient Inheritance: The study assumes the existence of consistent coefficients for the YMS expansion (which have been proven in prior literature). It then investigates whether the EYM integrand can be expanded into YM integrands using the same coefficients as the YMS expansion.

Key Contributions and Results

Derivation of the Expansion Formula: The paper provides a rigorous proof that the one-loop EYM integrand (with a gluon loop) can be expanded as a linear combination of conventional one-loop YM integrands with quadratic propagators. The formula is given by:
$I_{EYM}(\ell; 1, \dots, r || H) = \sum_{\sigma} C(\ell; 1, \sigma_2, \dots, \sigma_n) I_{YM}(\ell; 1, \sigma_2, \dots, \sigma_n)$
where $H$ represents the set of gravitons and the coefficients $C$ are identical to those found in the scalar-loop YMS expansion.
Explicit Example: Section 4 details a specific case involving two gluons and two gravitons. The authors explicitly demonstrate the reorganization of linear-propagator terms into quadratic-propagator YM integrands, verifying that the consistency conditions allow the coefficients to factor out and the remaining diagrammatic sum to reconstruct the YM integrand.
General Proof: Section 5 extends the example to a general proof for arbitrary numbers of gluons and gravitons. The proof relies on the identity that the sum of cyclic permutations of BS subcurrents with linear propagators equals a single BS integrand with quadratic propagators.
Coefficient Correspondence: The results confirm that the kinematic coefficients required for the EYM expansion are exactly those derived for the YMS expansion. This implies that the known explicit formulas for YMS coefficients (up to three gluons) directly provide the coefficients for EYM expansions (up to three gravitons).

Significance and Claims
The paper claims to provide the "proper foundation for constructing BCJ numerators in both YMS and EYM theories at one loop." By establishing that the expansion coefficients for EYM are inherited from YMS, the work highlights a shared kinematic structure between these theories. The authors state that this correspondence allows for the construction of GR integrands (via the double copy of EYM) in terms of YM integrands with the same coefficients used in the YM/YMS sector. The work does not propose new experimental applications or future implications beyond offering a "rigorous proof" and "further insights for understanding the BCJ duality at the one-loop level." The significance lies in bridging the gap between the forward limit representation (linear propagators) and the conventional Feynman representation (quadratic propagators) for EYM theories, thereby facilitating the systematic construction of one-loop BCJ numerators.

The Big Picture: Two Languages, One Translation

The Problem: The "Linear" vs. "Quadratic" Trap

The Solution: The "Two-Step" Translator

The Key Discovery

Why This Matters

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