Tropical field theory could tame the rising cost of collider predictions

Theoretical predictions for particle collisions become far more costly to compute as precision rises, because the workload grows factorially with each extra digit. Michael Borinsky, a research faculty member at the Perimeter Institute, proposes a way around this: a simplified version of quantum field theory called tropical field theory, which turns factorial growth into polynomial growth. The work appears in Annales Henri Poincaré.

Why precision is expensive

Interpreting collider data depends on quantum field theory (QFT), the framework that describes how particles behave and interact. A theorist begins with a model that encodes local interaction rules and works towards predictions that can be compared with measurements on larger scales. Borinsky describes this as a pipeline from simple local rules to observable quantities.

The pipeline is slow, and it slows further as accuracy improves. The reason is mathematical. The computational demand rises factorially, which outpaces even exponential growth. With five steps, an exponential process of the form 2n gives 32, whereas 5! gives 120, and the gap widens rapidly as n increases. In practice, Borinsky says, a few digits can be computed with effort, but the next digit then demands far more time, so researchers meet a wall. As experiments probe ever shorter length scales, they need predictions of higher precision, and so the wall matters more.

What 'tropical' means here

The name comes from tropical geometry, a branch of mathematics that simplifies a geometric object by breaking it into smaller pieces, straightening them and reassembling the result. Borinsky compares this with taking a complicated, many-jointed pretzel, smoothing its bumpy outline and cutting it into nearly straight segments. The shape remains recognisable while becoming much easier to handle.

Applied to QFT, the idea is to replace the original theory with a deformed, cartoon-like version that keeps the essential details but is far easier to compute. Borinsky aims for polynomial rather than merely exponential growth in cost, since polynomial scaling is much more manageable for computers. According to the source, the tropical approach achieves this shift. He says calculations in the tropical setting that would normally take years can sometimes be done on paper.

Current use and open questions

The method is not yet ready for particle accelerator work. The source states that more development is needed before tropical QFTs can be applied effectively to collider experiments. It is, however, already being used to improve the accuracy of predictions for condensed matter experiments that study exotic phases of matter.

A central open question is what the tropical simplification preserves. Borinsky notes that it is still unclear which information is lost and which survives, and that understanding this is his current focus. Without that knowledge, it is hard to say how closely tropical results can stand in for those of the full theory in a given calculation.

Why it matters

Better algorithms for QFT predictions are a concern at Perimeter and in many other groups, since computing time is a real constraint on how precisely theory can be matched to data. A mathematical shortcut that reduced the scaling of the calculations could make high-precision predictions cheaper and help researchers interpret collider measurements. Borinsky expects that a few new fundamental insights will be required, after which particle predictions could be accelerated considerably.

For now, the result is a promising formal development with demonstrated use in condensed matter physics. Whether it can be adapted to the demands of collider phenomenology will depend on clarifying what the tropical approximation retains.

Source: How a 'tropical' shortcut could help particle collider experiments (Phys.org, 31 October 2026).

Image: AI-generated illustration.