CAS team uses AI to complete ten-loop calculation in maximally supersymmetric Yang–Mills theory

Recently, He Song, a researcher at the Institute of Theoretical Physics of the Chinese Academy of Sciences, led two students, Li Xiang and Jing Jirong, in using a newly developed joint method to complete, with the assistance of artificial intelligence (AI), the nine-loop and ten-loop "symbol" calculations of two types of six-particle scattering amplitudes (MHV and NMHV) in the maximally supersymmetric Yang–Mills theory. On the basis of extensive prior preparation, the actual runtime of the nine-loop calculation was no more than half a day; after further improving efficiency, the ten-loop calculations of the two types of amplitudes were completed within one day. Using the duality relation between the MHV amplitude and the three-point form factor, the team also obtained the ten-loop "symbol" result for the three-point form factor.

At almost the same time, the American company Anthropic also announced a nine-loop calculation completed with the help of Claude, but obtained the function result for only one of the MHV amplitudes, whereas He Song's team completed the "symbol" calculations for both types of amplitudes, that is, determined the key mathematical structure of the answer; obtaining the complete function still requires supplementing other information.

In an interview with China Science Daily, He Song emphasized that the two students did not merely use AI and ready-made tools, but jointly developed a new method that runs symmetry through every step of the calculation. In his view, the faster AI advances, the more attention should be paid to cultivating people, so that students can form physical intuition and research judgment as early as possible; being able to pose new questions and create new methods is far more important than demonstrating how difficult a calculation AI can complete.

Symbol results for the ten-loop six-point amplitude and three-point form factor in the maximally supersymmetric Yang-Mills theory.

The "loops" that trapped theoretical physicists

In high-energy physics, scattering amplitudes are the core quantities for calculating the probabilities of various outcomes after particle collisions, and they are also the "bridge" between theoretical predictions and collider experiments. Agreement between theory and experiment within the error range shows that the theory has passed the corresponding test; significant deviations deserve further study and may also point to new physics.

The maximally supersymmetric Yang–Mills theory is called the "hydrogen atom of the 21st century" and is an "ideal laboratory" for studying scattering amplitudes. The structurally simple hydrogen atom once helped people understand the quantum world; today, scientists hope to use this model, which shares a common foundation with realistic particle theories and possesses more symmetries, to see clearly the laws behind particle interactions, and then bring applicable methods to particle collision problems closer to reality.

But even in such an idealized model, calculation is not easy. Researchers usually start from simpler approximations and incorporate more complex quantum effects layer by layer. "Loop number" indicates the level of these quantum corrections; it does not mean that particles actually go around in circles.

The higher the loop number, the larger the expressions and conditions that usually need to be handled, and the greater the computation time, memory requirements, and implementation difficulty. Complex calculations are also interlocking, and a local error may affect subsequent results, so repeated testing and cross-checking are necessary.

Over the past 10-plus years, bootstrap methods have helped researchers push such amplitude calculations to higher loops. It is somewhat like solving Sudoku: instead of calculating all intermediate processes one by one, one starts from the rules the answer must obey and the existing clues, and gradually eliminates impossible options. When the conditions are sufficient, there is a chance to determine the answer.

In 2023, theoretical physicist Lance Dixon and collaborators used existing three-point form factor results and duality relations to push the calculation of one type of six-particle amplitude to eight loops. To go further, knowing which rules the answer should obey is not enough; how to efficiently organize and execute these rules is itself a difficult problem.

More than a record

This progress was built on continuous algorithmic accumulation. Previously, Li Zhenjie, who pursued a doctorate in He Song's team, developed algorithms to improve the efficiency of related bootstrap calculations. Subsequently, doctoral student Li Xiang and Jiang Xuhang, who had conducted postdoctoral research in the team, advanced the implementation of automated programs, organizing a series of calculation steps into reusable tools.

On this basis, Li Xiang and Jing Jirong jointly developed a new method that applies symmetry at every step. They did not leave symmetry only for the final filtering or checking, but from the very beginning used the symmetry relations among different mathematical objects to split the huge problem into more manageable parts, reducing repetitive work and memory requirements.

With this method, the team can already construct on a single workstation the huge candidate space used for high-loop calculations. They also plan to further incorporate the new algorithm into the automated program package AsymBoot, so that these improvements become tools that can be repeatedly used in subsequent research.

He Song explained that the tasks undertaken by AI were very specific: assisting with the tedious calculations when imposing physical conditions, and automating part of the workflow. Researchers propose algorithms and methods, and AI helps implement some of the complex, time-consuming calculation steps. When Jing Jirong first joined this research, he had just graduated from Ningbo University with a bachelor's degree. In September this year, he became a first-year graduate student at the University of the Chinese Academy of Sciences.

He Song specifically noted that computation runtime does not equal the time taken by the entire research project. Previously, the team had already carried out extensive theoretical research, algorithm development, program implementation, and data preparation. After the algorithms, programs, and required inputs were ready, the actual runtime of the nine-loop calculations for the two types of six-particle amplitudes was no more than half a day; after further improving the methods and increasing efficiency, the ten-loop "symbol" calculations for these two types of amplitudes were completed within one day.

Using the duality relation between the MHV amplitude and the three-point form factor, the team also obtained the ten-loop "symbol" result for the three-point form factor, with the number of terms reaching 538819472142, more than 500 billion terms.

“Connections between different physical quantities allow the answer to one problem to be used to solve another; and such a massive scale also makes efficient organization of computation key,” He Song said.

He Song said the significance of this work goes beyond setting a record; more importantly, it accumulates algorithms and computational steps into reusable tools, so that subsequent similar research does not have to start from scratch—it is like building new “infrastructure” for theoretical physics. The bootstrap idea is also used in research fields such as phase transitions and the early universe, while specific problems still require their own physical conditions and computational methods.

The computation is complete, but the research is not over. In addition to proposing algorithms and developing new methods, analyzing the physics results is also a core contribution of researchers. Scattering amplitudes, form factors, and Wilson loops have exact connections under specific conditions; complex formulas are also connected to mathematical structures such as cluster algebras, as if sharing a hidden “grammar.”

Higher-loop calculations are therefore also a kind of “theoretical experiment”: do the relations discovered in simple cases still hold in more complex cases? Can common patterns be found that run through different loop orders, or even arbitrary loop orders? Researchers hope to use these results to distinguish accidental coincidences from deeper physical and mathematical structures.

More important is cultivating people

In He Song’s view, in the AI era, the goal of training students should not stop at proficient use of tools, but should enable them to develop the ability to conduct research independently as early as possible. Li Xiang and Jing Jirong not only completed the higher-loop calculations but also jointly developed new methods. Enabling young people to move from executing calculations to posing questions and creating methods is the more important gain from this work than the record itself.

He Song explained that in the past, even if students encountered frontier research earlier, they often had to invest a great deal of energy in tedious technical calculations. Today, AI and automation tools can take on part of that work, allowing them to devote their energy earlier to understanding physics, forming intuition, developing methods, and analyzing results.

This is not about letting students learn less physics, but about having them face real research problems earlier. Why adopt these conditions? How can one judge whether an answer is trustworthy? Which phenomena in the results are worth investigating in depth? To answer these questions, solid training in physics and mathematics is still indispensable.

“What is scarce now is attention, scientific taste, and judgment,” He Song said. When tools can support increasingly complex calculations, it becomes even more important to choose which problems are worth studying, how to interpret results, and how to propose new questions from them.

He Song encourages students to use AI, but requires them to explain the research process clearly in their own understanding, including what conditions were used, why the calculation was done this way, what tests the results underwent, and what they show. In his view, tools can help complete calculations, but they cannot replace students’ own understanding and judgment. Students must learn to pose questions, test answers, and create methods in real research.

AI can help push calculations further, but for He Song, cultivating young people who can think independently is always the more important matter. Records will continue to be broken, and what remains should not be only a result, but also better methods and researchers capable of posing the next important question.