Event probabilities in particle collisions are calculated by squaring a scattering amplitude, itself computed using quantum field theory (QFT). However, calculating amplitudes is computationally expensive, and the upcoming High-Luminosity Large Hadron Collider (HL-LHC) upgrade will provide a plethora of high-precision measurements, outpacing scientists’ ability to make theoretical predictions.
Contemporary algorithms used to compute amplitudes dramatically improved the efficiency of these calculations compared with Feynman’s diagrammatic approach, but they took decades to develop. The HL-LHC should be ready in four years. As experimental uncertainties continue to drop, how can theorists keep up?
Yi Gu and Sven Krippendorf at the University of Cambridge think that large language models (LLMs) may help. In a recent arXiv preprint entitled Scattering Amplitudes as Programs: Self-Evolving Search for Theory and Event Generation, they provide a proof-of-concept that LLMs can be used as a search tool for new algorithms. (Note that as a preprint submitted to the arXiv server, this paper has not yet been peer-reviewed.)
Self-evolving search
The researchers did not set out to beat existing amplitude software. They focused on the simplest case: tree-level scattering amplitudes (in contrast, quantum corrections involve loops, which dramatically complicate calculations; think of them as higher-order Taylor series terms). Rather, they asked whether LLMs are a useful tool for such a goal, and if so, which objectives and evaluation metrics produce the best results.
The general principle is simple: OpenAI’s Codex proposes a candidate program to compute amplitudes, and a test program evaluates the candidate on correctness and efficiency. The LLM then sees the evaluation and proposes a new candidate. In this way, the search is self-evolving.
The expectation is that because LLMs have absorbed physics literature, they can propose changes to the underlying algorithmic representation that improve the efficiency of the calculation by more than simple code fixes. The search can thus take motivated steps through the enormous space of representations.
Search objectives and their outcomes
The unanswered question here is, what does efficiency mean? The search must optimize for a quantifiable objective. Rather than proposing a single definition, the researchers tested three different objectives: scaling, structure and complexity. The scaling test looked at computational efficiency. The structural test asked whether the search could find cancellations between unphysical singularities. Lastly, the complexity search studied a cruder definition of efficiency: the number of arithmetic operations.
In each case, correctness was used as a hard gate: an incorrect amplitude was rejected. Additionally, the LLM was provided with a seed algorithm from which to start its search, with the prompt explaining the test and selection criteria, as well as providing ideas for algorithmic improvement.
Each objective produced a champion algorithm faster than the seeded code. Interestingly, the LLM provided qualitatively different improvements for each objective. Optimizing for scaling, the search abandoned its starting formalism, finding one whose costs grow linearly with particle number (for a fixed helicity configuration), 805 times faster than the seed. Optimizing for structure, the search combined terms with spurious boundaries that would ordinarily cancel at the end of calculations. Lastly, the complexity test yielded an improvement using a variety of standard algebraic QFT tricks, cutting the number of operations by factor of 48.
Importantly, none of these techniques were novel. Instead, the key takeaway is that even when runtime is hidden from the LLM (as in the latter tests), different selection pressures nevertheless produce more time-efficient calculations through qualitatively different means.
Comparison with existing event generators
The researchers then compared the complexity test champion with existing event generators used in the field: Sherpa and MadGraph. For six external particles, the independent Python prototype was slower than both of these. They note, however, that the study’s aim was to test whether AI-driven search could improve a starting algorithm, rather than outperform established compiled generators.
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Gu and Krippendorf emphasize that their conclusion is not that existing matrix element calculation software should be replaced. Unlike Sherpa and MadGraph, the algorithms they developed do not cover the entire standard model: they don’t transfer to arbitrary processes, helicities or multiplicities. Rather, the researchers conclude that further use of self-evolving search as a tool is justified due to the significant performance increase provided by these tests.
Producing further efficiency increases and generalizing the algorithms will likely require the input and collaboration of amplitude theorists, phenomenologists, numerical experts and generator developers.
Krippendorf is candid about the limit. “When a problem goes beyond established results and there is no known baseline for comparison – for example in some higher-loop calculations – we currently have to rely on human verification,” he explains. The method works where a code-improvement task has a fast numerical check, such as an existing codebase. As such, he thinks that the community should try it there.
