For decades, theoretical particle physicists have struggled with tricky computational problems called Feynman integrals. They are at the heart of all the calculations they perform, from predicting the degree of magnetism of a particle called a muon to estimating the speed at which Higgs bosons should emerge at the Large Hadron Collider (LHC). Now theorists have found a way to solve integrals numerically by reducing them to linear algebra. The method promises faster and more accurate theoretical calculations, which are essential for searching for clues of new particles and forces.
“Sometimes people come in with deep mathematical knowledge of these Feynman integrals, but they don’t actually help you calculate things,” says Ayres Freitas, a theoretical physicist at the University of Pittsburgh. “This method will help.”
“It is surprising that [the method] works so well,” says Stefan Weinzierl, a theoretical physicist at Johannes Gutenberg University Mainz, who has written an 800-page book on integrals. “In principle, it’s absolutely general, so you can treat any full Feynman with him.”
Feynman integrals have plagued particle theorists since the rise of quantum field theory in the mid-twentieth century. Each integral corresponds to one of the original diagrams concocted in 1948 by Richard Feynman to quickly determine what to calculate for a particular particle interaction. For example, one electron can deflect another when the two exchange a photon, so the simplest Feynman diagram for the process consists of two lines representing the electrons and a wavy line connecting them which represents the photon . The doodle has a serious purpose: each line represents a component in an integral giving the probability of interaction given the initial and final momentum of each electron.
Even a graduate student can handle this integral. However, electronic interactions are much more complex. For example, the exchanged photon can transform into an electron-positron pair and become a photon again, or an electron can emit and reabsorb a particle called a Z boson. Such “disturbances” correspond to Feynman diagrams with one or more internal loops closed, and accurate calculations should consider at least the simpler loop diagrams.
Integrals for loop diagrams are however pathological. With multiple variables, many are impossible to do by hand and require numerical methods. Even then, the integrals tend to blow up, forcing theorists to employ a myriad of tricks and techniques to solve the problems. In most cases, theorists struggle to get past two-loop integrals, says Freitas, of which any particular interaction can have thousands.
This is not just an academic question, notes Johannes Henn, theoretical physicist at the Max Planck Institute for Physics. Every 2 years, particle physicists gather at a seaside resort in France to compile a wish list of improved theoretical calculations which they wish to compare with experimental results from the LHC as they search for holes in the dominant theory of particles. physicists on fundamental particles, the standard model. In striving to achieve these wishes, Henn says, “the bottleneck is our ability to actually evaluate Feynman diagrams and integrals at the loop level.”
Now Zhi-Feng Liu and Yan-Qing Ma, theoretical physicists at Peking University, have developed a method that reduces such an integral to an easier problem in linear algebra, they reported last week in Physical examination letters. Liu and Ma start with an old observation that any Feynman integral with a given number of loops can be written as a linear combination of some other “master” integral 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, if theorists can solve the principal integrals, they can solve any Feynman integral.
Liu and Ma then invoke a second old theorem that allows them to write the derivative of each master integral as a linear combination of the same master integrals. This produces a set of differential equations relating all the master integrals. At this point, instead of directly calculating the integrals, theorists can derive them implicitly by solving the differential equations numerically. It’s something a computer can do easily, says Ma.
There is a catch, however. To solve differential equations, theorists need certain initial values or “boundary conditions” for the master integrals. This usually requires finding a symmetry that makes the boundary conditions simple or biting the bullet and doing the integration for some fixed values of the input variables. Here, however, Liu and Ma resort to a breakthrough made by Ma’s team in 2017. They modify the master integrals by inserting a new parameter and then performing the derivatives with respect to it. By fixing this additional parameter at infinity, they obtain a set of differential equations for which the determination of the boundary conditions is much easier. This done, they adjust the new parameter to zero to recover the differential equations of the principal integrals, with the necessary boundary conditions.
Here’s how Liu and Ma determine the boundary condition: With their new parameter set to infinity, this task is to evaluate integrals for “vacuum diagrams” that have no external branches and only loops. But each of these can be likened to a related diagram with one less loop. Iterating this motion finally gives the vacuum integrals as linear combinations of loop-free integrals, which is trivial. Thus, the problem of boundary conditions is reduced to pure linear algebra.
Using results from the literature, Liu and Ma confirmed that their technique reproduced correct results for five-loop integrals. In January, the team made the technique available via a downloadable software package. Over the past few months, according to Ma, 80% of articles published on the arXiv preprint server involving the calculation of Feynman integrals have used the package.
Of course, theorists don’t get something for nothing. The technique avoids brute-force numerical integration, but it requires a lot more algebra. “The main computation time in our method is not solving the differential equations, but getting the differential equations,” Ma says. Some calculations might require a computer to solve half a billion linear equations, says- he.
Still, the new rapid technique is likely to prove very useful, Weinzierl says, even if it’s largely a clever combination of earlier results. “Ingredients like flour, salt, sugar, etc. are all readily available,” he says, “but if a cook makes a very delicious meal out of it, you appreciate it.”
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