Unitarity approaches to two-loop all-plus amplitudes

Sebastian Pogel


With the continued advances in experimental measurements at the Large Hadron Collider (LHC), new high precision theoretical predictions are required to further test the validity of the Standard Model of particle physics (SM). An essential ingredient for such searches are predictions for processes in Quantum Chromodynamics (QCD). These make up the majority of the understood background in LHC measurements. A precise understanding of these processes is therefore vital for the search of physics beyond the SM.

High precision theory predictions require next-to-next-to-leading order virtual corrections, typically based on two-loop scattering amplitudes. As the computation of such amplitudes for experiments is generally a complex task, it is useful to study simplified cases, in order to develop computational techniques and search for new structures.

In this thesis I discuss the computation of such a simpler class of amplitudes, called all-plus amplitudes. They are highly symmetric, as they describe the interaction of gluons which all have the same helicity. The properties of the all-plus configuration lead to particularly compact forms, making all-plus amplitudes convenient objects to study.

A striking feature of all-plus amplitudes found so far for up to two loops is a reduction in computational complexity. Their tree amplitudes vanish, while their one-loop amplitudes can be obtained from techniques, which resemble those used at tree-level. In the two-loop case, many parts of these amplitudes have been shown to be computable using only one-loop techniques.

A part of two-loop all-plus amplitudes for which such a construction from one-loop techniques is presently not known in general are their rational parts. These are the parts of the amplitude that are free of polylogarithms and poles in dimensional regularization. Based on a previous conjecture, I present in this thesis an approach for the computation of the rational part of two-loop all-plus amplitudes based solely on one-loop generalized unitarity techniques. This approach is not limited to leading color, but appears to extend to the full-color amplitude. I show that this method reproduces all known results for such rational parts, including the non-planar ones, for up to seven gluons. It also matches the rational part of the seven gluon subleading single-trace amplitude, whose form is presently only known as part of an all-n conjecture.

Furthermore, I show that the rational parts can be determined not only from one-loop techniques, but also from a nested one-loop generalized unitarity computation. Here, one of the loops appears as the rational part of a one-loop amplitude. This mirrors a similar derivation found for the leading-color polylogarithmic parts of the amplitude.

Finally, I present new relations between the two-loop partial amplitudes of the all-plus, which involve powers of Mandelstam invariants. As such they have a striking similarity to BCJ relations, which relate tree-level amplitudes and loop-level integrands via powers of such invariants.

Funded in part by the European Union’s Horizon 2020 Framework Program under the Marie Skłodowska-Curie grant agreement No. 764850 (SAGEX).

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