FeynFacet: AI agents for analytic NLO QCD

The universe is built from a handful of fundamental particles, but the rules governing how they interact are written in a language of staggering complexity. At the heart of this complexity lies quantum chromodynamics, the theory that describes how quarks and gluons—the building blocks of protons and neutrons—stick together to form the matter we see around us. To understand the behavior of these particles, physicists must calculate how they scatter when they collide at high speeds. These calculations are not simple arithmetic; they involve tracking countless possible paths a particle can take, summing up probabilities that include invisible, fleeting fluctuations in the vacuum. For decades, the most precise predictions required teams of experts to spend months or even years deriving the necessary mathematical formulas by hand, a process prone to human error and limited by the sheer volume of algebra involved.

Now, a new approach has emerged that changes how these calculations are performed. Researchers have developed a workflow where artificial intelligence agents act as specialized assistants, writing and running the computer programs needed to solve these intricate problems. Instead of a human deriving every step, the AI is given the physical rules and the specific question to answer, then it constructs the code to find the solution. This method has been used to produce a set of new, highly precise formulas describing how the spin of a particle is transferred during a collision. These results fill a critical gap in our understanding of how the internal spin of protons influences the particles they produce, providing the necessary tools to interpret data from the world's most powerful particle colliders.

The core achievement of this work is the derivation of new analytic formulas for a process called single-inclusive transverse spin transfer. In simple terms, this describes what happens when a spinning particle, such as a quark inside a proton, collides with another particle and produces a new, spinning particle that flies out in a specific direction. The "transverse" aspect refers to the direction of the spin, which is perpendicular to the direction of motion. While physicists have long known how to calculate the average behavior of these collisions, the specific details of how the spin orientation is preserved or altered in the final products have remained difficult to pin down with high precision. The new formulas provide a complete, exact description of this process for both collisions between two protons and collisions between a lepton and a proton.

To reach this result, the researchers employed a sophisticated workflow they named FeynFacet. In this system, human physicists define the physical scenario—specifying the particles involved, the type of collision, and the specific spin configurations they are interested in. An artificial intelligence agent then takes these instructions and automatically generates the computer code required to perform the calculation. The agent handles the heavy lifting of symbolic mathematics, which involves manipulating thousands of terms that represent the different ways particles can interact. It uses established mathematical techniques to simplify these terms, converting complex integrals into a manageable set of standard forms that can be solved exactly.

The process is rigorous and self-checking. The AI agents do not just produce a single answer; they run the same calculation using three different independent software tools to ensure the results match. They also verify the outcome by checking fundamental physical laws, such as whether the results remain consistent when the reference frame is changed or when the energy scale of the interaction is adjusted. If any inconsistency is found, the system is designed to diagnose the error and correct it, much like a human researcher would, but with the speed and thoroughness of a machine. This multi-layered verification ensures that the final formulas are not just plausible, but mathematically certain.

The new formulas cover all the possible ways quarks and antiquarks can interact to produce a spinning particle in a proton-proton collision. This includes seven distinct classes of interactions, ranging from the most common scenarios to those that only occur at a higher level of precision. The researchers also calculated the corresponding results for lepton-proton collisions, where a lepton, such as an electron, strikes a proton. In these cases, the calculation had to account for the fact that the scattered lepton is not always detected, requiring a specific adjustment to the math to handle the missing information. The resulting formulas are "analytic," meaning they are written as exact mathematical expressions rather than numerical approximations. This allows physicists to plug in different values for energy and momentum instantly, without having to re-run a massive computer simulation for every new data point.

These new results are immediately useful for interpreting data from current and future experiments. For instance, they are essential for studying the production of polarized hyperons, which are particles containing strange quarks that can reveal details about the spin structure of the original proton. They also improve the analysis of the Collins effect, a phenomenon where the spin of a quark influences the direction in which it fragments into a jet of particles. Furthermore, the formulas help refine measurements of energy correlations within jets, which are streams of particles created in high-energy collisions. By providing a precise theoretical baseline, these calculations allow experimentalists to extract the properties of the proton's internal spin with greater accuracy than ever before.

The work also serves as a validation of the AI-assisted approach itself. The researchers used the same workflow to reproduce a wide range of previously known results, including calculations for unpolarized collisions and those involving longitudinal spin, where the spin is aligned with the direction of motion. In every case, the AI-generated formulas matched the established results perfectly. This success demonstrates that the method is robust enough to handle the most demanding problems in quantum field theory. It also confirms that the AI can correctly navigate the subtle mathematical pitfalls that often trip up human calculations, such as the handling of terms that vanish in four dimensions but leave finite effects when the calculation is extended to higher dimensions.

Looking forward, this workflow opens the door to calculations that were previously considered too difficult or time-consuming to attempt. The ability to generate exact formulas for complex spin-dependent processes means that theorists can now keep pace with the increasing precision of experimental data. As the Large Hadron Collider continues to operate and the future Electron-Ion Collider comes online, the demand for these precise theoretical inputs will only grow. The new formulas provide the necessary foundation to explore the three-dimensional structure of the proton and to understand how the spin of its constituent quarks shapes the matter of the universe.

The researchers have made all the new formulas and the computer programs used to generate them publicly available. This transparency allows other scientists to use the results immediately in their own work and to build upon the methods developed here. By sharing the tools and the data, the team ensures that the benefits of this AI-assisted approach can be felt across the entire field of particle physics. The work stands as a testament to the potential of combining human insight with machine precision, creating a new standard for how the fundamental laws of nature are calculated and understood.