The paper constructs and analyses two families of difficult two‑loop Feynman integrals needed for non‑factorisable quantum chromodynamics (QCD) corrections to Higgs‑boson production by vector boson fusion (VBF). These families, called the penta‑box (PB) and the hexa‑box (HB), are five‑point two‑loop topologies with one internal mass and several external energy scales. The authors produce a set of “master integrals” that obey a particularly simple form of differential equation, which makes them much easier to evaluate systematically.

What the researchers did: they reduced the many integrals that appear in the PB and HB families to a basis of master integrals using integration‑by‑parts (IBP) reduction and modern computer algebra tools (Reduze2 and a finite‑field implementation called Finred). They then found a change of basis so that the differential equations in the kinematic variables have an epsilon‑factorised or “canonical” form. Here epsilon (ε) is the small parameter used in dimensional regularisation, a standard trick in loop calculations. The kinematics involve seven independent variables: the internal mass squared and six independent combinations of the external momenta (commonly written as Mandelstam invariants), plus the sign of a parity‑odd invariant.

How the method works at a high level: master integrals satisfy differential equations in the external variables. If one picks a basis where the equations take the canonical form dI = ε A I, with A built from simple one‑forms (so‑called dlog letters), then the integrals can be expanded order‑by‑order in ε and written in terms of iterated integrals. This turns the hard problem of two‑loop integration into a controlled algebraic and numerical task. One technical hurdle the authors faced was a subsection of the PB family that introduces nested square roots in intermediate steps. Nested square roots often complicate the algebra and the identification of the dlog building blocks, but the authors describe strategies to derive the correct letters, to write the differential equations entirely in dlog form for their case, and to handle analytic continuation.