Claude pushes a particle-physics amplitude to nine loops

Anthropic’s Claude computed the six-particle scattering amplitude in planar N=4 super Yang-Mills theory through nine loops, surpassing the previous eight-loop record held by SLAC physicist Lance Dixon and collaborators. Running inside the Claude Science harness, the model worked for days with periodic continuation prompts and no mid-run scientific corrections. Dixon later checked the result using his group’s verification tools.

Physicist and science writer Matt von Hippel proposed the task on his 4gravitons blog just over a month earlier. He challenged AI companies to solve either an N=8 supergravity amplitude at seven loops or an N=4 super Yang-Mills amplitude at nine loops using resources available to an academic researcher. Anthropic chose the second problem.

Why each loop explodes

Scattering amplitudes predict the probabilities of particle interactions. Physicists calculate them as perturbative expansions, with each loop adding a higher-order quantum correction. The approximation can improve with each order, but the number and complexity of intermediate terms often grow exponentially or factorially.

• Most amplitudes used in phenomenology have reached two loops.
• A smaller set has reached three loops.
• The pure quantum-electrodynamics contribution to the electron’s anomalous magnetic moment has reached five loops.
• Dixon’s group had taken the six-particle amplitude in N=4 super Yang-Mills theory to eight loops.

Researchers use N=4 super Yang-Mills as an idealized laboratory for amplitude methods. Its extensive supersymmetry creates cancellations that make unusually deep calculations tractable. The planar limit simplifies the problem further by retaining the leading contributions when the number of color charges becomes large. Techniques developed there can inform work on quantum chromodynamics, the theory of the strong force, even though QCD calculations remain substantially harder.

Two routes through the bootstrap

The bootstrap method starts with a constrained space of functions that could represent the amplitude. Symmetries, boundary behavior, known limits and relationships to simpler quantities progressively fix the allowed coefficients. Von Hippel compares the process to Sudoku: each constraint eliminates possibilities until a unique answer remains, ideally with unused conditions available as checks.

Anthropic researchers Liam Fitzpatrick and Siddharth Mishra-Sharma gave Claude a one-line request for the nine-loop, six-particle hexagon amplitude. Subsequent prompts mainly instructed the model to continue and report its progress. Claude generated code, ran symbolic and numerical tools, diagnosed failures and completed the calculation through two related routes:

1. Direct bootstrap: Claude constructed the nine-loop amplitude by imposing the known constraints on its candidate-function space.
2. Form-factor route: Claude calculated a related quantity describing how a local operator couples to particle states, then mapped it back to the amplitude using antipodal duality, a symmetry identified by Dixon and Andy Liu in 2023.

A four-figure run

Anthropic reported a total end-user cost of roughly $1,000 to $2,000 for each run, with model inference accounting for most of the bill. The underlying numerical work remained comparatively inexpensive:

• The numerical bootstrap consumed about $100 of the budget.
• Anthropic compared that workload with running 96 CPUs for one week.
• The implementation used Python and the SymPy computer-algebra library.
• The agent operated for several days without a researcher correcting scientific mistakes during execution.

A result built to be checked

Dixon independently checked the output by translating the amplitude into the form factor his team had studied for years. His companion note describes the bootstrap as unusually fragile: an incorrect assumption or constraint can invalidate the entire construction. Much of the workflow’s practical knowledge had never been consolidated in a paper, so Claude had to recover the necessary scaffolding from the literature, code and intermediate results.

A simultaneous finish

A team led by Song He at the Chinese Academy of Sciences reported the symbol of the same nine-loop amplitude shortly after Anthropic contacted von Hippel. The symbol captures the amplitude’s iterated-integral structure while omitting constants and other terms invisible at that level. He’s group used GPT-6 to derive some constraints while researchers directed the overall calculation. The published Claude calculation and the concurrent calculation are available online.

The near-simultaneous results show that specialists were already close to nine loops. Claude applied established bootstrap methods, integrated them into a durable software workflow and executed that workflow quickly. Von Hippel’s assessment is that several amplitude problems may have tractable higher-loop extensions that researchers have lacked the time or tooling to pursue.

The harness behind the run

Claude Science is a paid platform that combines Claude with structured prompts, tool access, persistent task state and long-horizon execution. Those components allow the model to write and run code, inspect results, recover from failed approaches and continue across sessions. Anthropic has used a similar agent setup for projects including biomolecular modeling work. In this experiment, the initial instruction occupied one line, while brief continuation prompts kept the multi-day process moving.

Where scientific agents fit now

Developers and researchers evaluating agents for technical work can draw several practical lessons from the run:

1. Choose problems with formal checks. The amplitude had a constrained answer space, known symmetries and independent verification machinery.
2. Build verification before autonomy. Dixon could assess the output because his group already had tools capable of testing any candidate result.
3. Track inference and compute separately. Model usage dominated the reported cost, while the numerical calculation used modest conventional resources.
4. Define the human role precisely. Researchers selected the problem, supplied continuation prompts and validated the result; the agent handled the extended computational workflow.
5. Use redundant derivations when possible. The direct bootstrap and form-factor routes provided stronger evidence than a single successful run.
6. Limit generalization to comparable tasks. Performance on QCD calculations and open-ended theory discovery remains untested. This problem’s structured mathematics and machine-checkable constraints made it especially suitable for an agent.

The evidence supports a specific claim: an LLM agent can carry an established, delicate scientific method through a multi-day computation and produce a frontier result that experts can verify. A model-originated physical principle or mathematical method that survives independent review would support a broader claim about scientific discovery.