Among the most significant results obtained through AMPLITUDES are:
1. An all-loop, non-planar, non-supersymmetric generalization of the prescriptive unitarity program. This culminated in the recent work “Building Bases of Loop Integrands”, co-authored with Enrico Herrmann, Cameron Langer, and Jaroslav Trnka, which included an all-orders proposal for a power-counting stratification of loop integrands (suitable for unitarity-based reconstruction of amplitudes) in any renormalizable quantum field theory in any number of dimensions, and an explicit enumeration of a complete basis suitable for the leading “transcendental-weight” part of any—involving arbitrary numbers and species of external states—three-loop scattering amplitude in any four-dimensional quantum field theory.
This work built upon a series of earlier results which were also completed as part of this project. In particular, this included the first closed formula for all-multiplicity amplitudes in any four-dimensional quantum field theory at two loops, which was published in the Physical Review Letters (PRL) in the article “All-Multiplicity Nonplanar Amplitude Integrands in sYM at Two Loops”, which followed very shortly after the result for six particles, published in JHEP, “Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops”, and the previous work on non-planar integrands “Amplitudes at Infinity”, published in PRL.
2. Development, refinement, and exploitation of new technology for loop integration. This included publications “Rationalizing Loop Integration”, “Manifestly Dual-Conformal Loop Integration”, “Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory”, “Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder”, and “All-Mass n-gon Integrals in n Dimensions”, all of which were published in JHEP.
3. The discovery and exploration of novel structures in the analytic form of non-polylogarithmic Feynman integrals. Specifically, my collaborators and I uncovered several infinite classes of finite Feynman integrals (defined in massless phi^4 theory in 4 dimensions) that involve period integrals Calabi-Yau manifolds of dimension increasing with loop order. These results were published in “Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms” and “A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries”, both published in PRL, and “Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space”, “Positive Geometries and Differential Forms with Non-Logarithmic Singularities, Part 1”, and “Traintrack Calabi-Yaus from Twistor Geometry”, published in JHEP.
4. The discovery of non-algebraic Yangian-invariants associated with on-shell functions integrated over elliptic (or higher-dimensional) contour integrals, described in "An Elliptic Yangian-Invariant `Leading Singularity' in PRL; the connection between this generalization of leading singularities and coefficients of loop integrands appearing in generalized unitarity was made clear in the subsequent work "Prescriptive Unitarity with Elliptic Leading Singularities".
5. The exploration of how integrand bases can be organized (`stratified') according to particular physical structure, including rigidity (or non-polylogarithmicity), IR/UV finiteness, etc, with applications to non-supersymmetric theories at one and two loops. In particular, in the work "The Stratification of Rigidity" published in JHEP, we showed that all leading-weight contributions to planar scattering amplitudes in four dimensions integrand bases could be represented in terms of integrals that were individually pure and polylogarithmic or pure and elliptic-polylogarithmic, with each of the later depending on a single elliptic curve geometry.
