We compute the planar part of the two-loop five gluon amplitude with all helicities positive. To perform the calculation we develop a D-dimensional generalized unitarity procedure allowing us to reconstruct the amplitude by cutting into products of six-dimensional trees. We find a compact form for the integrand which only requires topologies with six or more propagators. We perform cross checks of the universal infra-red structure using numerical integration techniques.

The coherent interaction between quantum emitters and photonic modes in cavities underlies many of the current strategies aiming at generating and controlling photonic quantum states. A plasmonic nanocavity provides a powerful solution for reducing the effective mode volumes down to nanometre scale, but spatial control at the atomic scale of the coupling with a single molecular emitter is challenging. Here we demonstrate sub-nanometre spatial control over the coherent coupling between a single molecule and a plasmonic nanocavity in close proximity by monitoring the evolution of Fano lineshapes and photonic Lamb shifts in tunnelling electron-induced luminescence spectra. The evolution of the Fano dips allows the determination of the effective interaction distance of ∼1 nm, coupling strengths reaching ∼15 meV and a giant self-interaction induced photonic Lamb shift of up to ∼3 meV. These results open new pathways to control quantum interference and field–matter interaction at the nanoscale.

We evaluate analytically all previously unknown nonplanar master integrals for massless fiveparticle scattering at two loops, using the differential equations method. A canonical form of the differential equations is obtained by identifying integrals with constant leading singularities, in D space-time dimensions. These integrals evaluate to Q-linear combinations of multiple polylogarithms of uniform weight at each order in the expansion in the dimensional regularization parameter, and are in agreement with previous conjectures for nonplanar pentagon functions. Our results provide the complete set of two-loop Feynman integrals for any massless 2 → 3 scattering process, thereby opening up a new level of precision collider phenomenology.

We show that the integration-by-parts reductions of various two-loop integral topologies can be efficiently obtained by applying unitarity cuts to a specific set of subgraphs and solving associated polynomial (syzygy) equations.