How a “tropical” shortcut could help particle collider experiments

Particle accelerators are expensive. Not just to build or to run, but also to predict the outcome of measurements obtained from them.

Predicting the outcomes of particle collisions is essential to understanding the measured results seen in colliders. These predictions use quantum field theory (QFT), the language that describes particle behaviour, and they have a major problem, according to Perimeter Research Faculty member Michael Borinsky.

“We take a theoretical model of physics as input, translate it into the QFT language, and it’s very hard to produce predictions,” he says. “What you put in encodes a law of nature, how particles interact locally, and what you want is some prediction of what follows on a larger scale. It’s a pipeline from these local, very simple rules to things that can be measured.”

But predictions aren’t just intellectually difficult. They’re computationally difficult, more so with every digit calculated, which translates directly into expensive computer hours. And as scientists investigate phenomena at increasingly tiny length scales, the predictions need to be more precise.

Every digit counts, but each new digit requires significantly more compute. And significantly more money.

“This pipeline, this algorithm, has a flaw. It’s very difficult and it’s very slow,” Borinsky explains. “It’s just very hard to make these predictions, and they get more expensive if we want more accurate results. Getting more digits becomes much harder at each step.”

Beyond exponential growth

At its core, the QFT prediction problem is mathematical: predictions increase their computational demands factorially instead of exponentially, which outpaces exponential growth. That also means the cost to run these predictions increases factorially as well.

For example, let’s assume n=5 and compare exponential to factorial growth.

The exponential growth would look like this: 2⁵=2×2×2×2×2=32

But the factorial growth looks like this: 5!=5×4×3×2×1=120

“That’s a horrible growth rate,” says Borinsky. “You can maybe compute three digits, but then computing the fourth digit takes much, much more time. You really hit this wall.”

Borinsky’s goal is to move the QFT calculations away from factorial growth. But he doesn’t want to settle for exponential growth. He wants polynomial growth which, while conceptually more complicated, is significantly more efficient for computers.

For that, Michael Borinsky has a solution. It’s called “tropical field theory.”

A “tropical” efficiency for QFT predictions

Tropical field theory gets its name from tropical geometry, a mathematical concept used to systematically simplify a geometric object, break it into smaller pieces, and put it back together again.

The concept (like many things) is best explained through baked goods.

Imagine a pretzel with a very, very complicated shape with many joints. Your goal is to simplify it while preserving as much important information about its shape as possible.

One way to better understand its shape is to smooth out its exterior, to sacrifice the detail of its bumpy outside so you can better understand its overall shape. Then, you can cut the pretzel up into many small pieces, smoothing the curves into almost straight chunks. You now have many smooth, fairly straight objects that can be reconstructed into that pretzel. Overall, the pretzel is less complicated while you can still grasp its shape. This is the heart of tropical geometry.

Borinsky applied the same concept to QFT: a simpler version that will still hold the necessary details. “We have this original quantum field theory that is very complicated, hard to understand for both humans and computers,” he says. “But then we deform it into this cartoon version that the tropical theory provides, which makes the computation of the original theory much more efficient.”

And the results are promising, shifting the factorial growth to a much, much more efficient polynomial growth.

“We land on something that can be solved extremely easily compared to the standards from normal QFTs,” he says. “Even on pen and paper, one can make computations in this tropical world that would otherwise take years.”

As it stands, more work needs to be done before tropical QFTs will be effective for particle accelerator experiments. But the tropical solution is already being used to increase the accuracy of predictions for condensed matter experiments, which investigate exotic phases of matter.

Currently, Borinsky is interested in what specifically survives tropicalization. “So far, it’s not clear what information is lost and what survives,” he says.

Finding solutions to the growing demands of predictions and simulations for particle accelerators is a topic studied at Perimeter and around the world. A mathematical solution like tropical QFTs could help researchers better understand measured results while keeping the computational demand both efficient and inexpensive.

“There’s going to be a couple of new fundamental insights needed,” he says. “Then it would really accelerate particle predictions significantly.”

Some of Borinsky’s research on Tropical QFTs was recently published in Ann. Henri Poincaré.