Researchers calculated a new, precise correction to the top quark mass. They computed the three-loop mixed quantum chromodynamics (QCD) and electroweak (EW) contribution at order α α_s^2. When they translate between two common mass definitions — the pole mass and the MSbar (modified minimal subtraction) mass — they find a shift of −216 MeV for m_t(m_t) when the pole mass M_t is 172.3 GeV. That size is comparable to the current experimental uncertainty on the top mass.

What the paper is about is improving the relationship between two ways physicists quote the top quark mass. The pole mass is tied to the particle’s propagator pole and is close to an intuitive “physical” mass, but it is sensitive to low-energy (infrared) effects. The MSbar mass is a short-distance, scheme-dependent number commonly used in calculations. The authors compute the three-loop contribution that mixes QCD and electroweak forces so theorists can convert reliably between these two numbers at higher precision.

To get this result the authors evaluated the three-loop top-quark self-energy and the needed lower-loop pieces and derivatives. They worked in the t’Hooft–Feynman gauge and used a scheme due to Jegerlehner and Kalmykov that keeps tadpole diagrams (certain self-interaction pieces) included so the mass definition stays gauge invariant. The many Feynman diagrams were reduced to a set of master integrals. Those master integrals were then solved using differential equations written in an ε-factorized form, which lets the solution be built as iterated (nested) integrals.

The calculation encounters mathematical objects that are more complicated than ordinary logarithms and polylogarithms. In particular, some integrals involve periods of elliptic curves and a class of three-loop on-shell integrals known as “banana” integrals. Handling those required special iterated-integral functions (the paper labels some as F1 and F2) and modular-form structures. The authors note that while some of these special functions are needed to define intermediate integrals, the final physical self-energy is independent of one of these extra functions.