For decades, theoretical particle physicists have struggled with complex computational problems known as Feynman integrals. They are central to all calculations, from forecasting How magnetic should a particle called a muon be?to calculate the exchange rate Higgs bosons should appear at the Large Hadron Collider (LHC). Now theorists have found a way to solve the integrals numerically by reducing them to linear algebra. This approach will allow for faster and more accurate theoretical calculations, which are essential for searching for hints of new particles and forces.

“Sometimes people have a deep mathematical understanding of these Feynman integrals, but they don’t help you calculate anything,” says Ayres Freitas, a theoretical physicist at the University of Pittsburgh. “This method will help.”

“It’s surprising [the method] works very well,” says Stefan Weinzierl, a theoretical physicist at the Johannes Gutenberg University in Mainz. 800 page book on integrals. “In principle, it’s so general that you can treat any Feynman integral.”

Feynman integrals have plagued particle theorists since the emergence of quantum field theory in the mid-20th century. Each integral corresponds to one of the strange diagrams Richard Feynman devised in 1948 to quickly figure out what to account for in certain particle interactions. For example, when two photons are exchanged, one electron can deflect the other, so the simplest Feynman diagram of the process consists of two lines representing the electron and a wave connecting the wave representing the photon. The doodle has a serious purpose: Each row represents a component in the integral that gives the probability of interaction with each electron’s initial and final momentum.

Even a graduate student can handle this integral. However, the interaction of electrons is more complicated. For example, an exchanged photon can become an electron-positron pair, which can become a photon, or a single electron can emit a particle called a Z boson and be reabsorbed. Such “functions” correspond to Feynman diagrams with one or more closed inner loops, and accurate computations must account for at least simple loop diagrams.

But the integral of the loop diagram is a disorder. With many variables, most cannot be done manually and require numerical methods. However, integrals tend to explode, forcing theorists to use many tricks and techniques to make the problem solvable. Most of the time, Freitas says, theorists struggle to obtain two-loop integrals.

This is not just an academic problem, notes Johannes Henn, a theoretical physicist at the Max Planck Institute for Physics. Every 2 years, particle physicists meet in a resort town in France and a would like a list of improved theoretical calculations They want to compare the results of the LHC experiment as they search for holes in the Standard Model, physicists’ dominant theory of fundamental particles. To fulfill these desires, Henn says, “the bottleneck is the ability to actually evaluate Feynman diagrams and integrals at the iterative level.”

Now, theoretical physicists Ji-Feng Liu and Yang-Ching Ma of Peking University have developed a method. This reduces such integrals in linear algebra to simpler problemsthey reported last week Physical Review Letters. Liu Ma et al. started with the old observation that a Feynman integral with a given number of loops can be written as a linear combination of other specific “master” integrals with the same number of loops. (A linear combination is a sum with coefficients: a Manhattan cocktail is a linear combination of two parts whiskey and one part vermouth.) So, for a given number of loops, theorists can solve any Feynman integral if they can solve the master integrals. .

Liu and Ma then invoke a second old theorem that allows the derivative of each master integral to be written as a linear combination of the same master integrals. This results in a set of differential equations involving all principal integrals. At this point, instead of directly factoring out the integrals, theorists can derive them implicitly by solving differential equations numerically. That’s something a computer can easily do, Ma said.

However, there is a catch. To solve differential equations, theorists need initial values or “boundary conditions” for the master integral. This usually requires finding symmetries to simplify the boundary conditions or integration for some fixed values of the input variables. But Liu Ma and others approach here A 2017 discovery by Ma’s team. They modify the basic integrals by introducing new parameters and then perform related derivatives. By setting the additional parameter to infinity, they obtain a set of differential equations that are easier to determine the boundary conditions. Once this is done, they set the new parameter to zero to recover the differential equation of the underlying integrals with the required boundary conditions.

Here’s how Liu and Ma define the boundary conditions: Setting their new parameter to infinity, the task is to evaluate the integrals of a “vacuum diagram” with no outer legs and only loops. But each of them can be equated to a corresponding diagram with one less loop. Repeating this motion eventually gives the vacuum integrals as linear combinations of integrals with no loops, which are trivial. Therefore, the boundary condition problem reduces to pure linear algebra.

Liu and Ma used literature results to prove their technique to obtain correct results for five-loop integrals. In January, the team made the technique available through a downloadable software package. In recent months, 80% of the papers published on the arXiv preprint server related to the calculation of Feynman integrals have used the package, Ma said.

Of course, theorists don’t get anything for nothing. This technique ignores numerical integration but requires more algebra. “The main computational time of our method is not to solve the differential equation, but to obtain the differential equation,” Ma said. Some calculations require a computer to solve half a billion linear equations, he said.

But even if it’s a clever combination of previous results, Weinzierl said, the fast new technique is likely to have broad utility. “Ingredients like gurl, salt and sugar are readily available, but you appreciate it when the chef makes a really tasty dish with it,” he says.