Prescriptive unitarity for non-planar six-particle amplitudes at two loops

Abstract

We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all softcollinear singularities in both theories. As a consequence, this integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as ancillary files to this work's submission to the arXiv. arXiv:1909.09131v1 [hep-th]

Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

“…The banana graphs have rigidity L−1, while the tardigrades have rigidity 2(L−1). 4 The two-dimensional massive banana graphs are required, for example, in the calculation of the electron self-energy in QED [103], while the massless two-loop tardigrades enter the integrand basis for massless two-loop amplitudes using prescriptive unitarity [63,90,91,104,105].…”

“…[62] and the examples discussed in ref. [63]). A general understanding of the types of integrals that can show up is currently lacking.…”

Rooting out letters: octagonal symbol alphabets and algebraic number theory

“…Note that it is, however, necessary to consider an entire amplitude, as it is well known that local integral representations can involve 'spurious' symbol letters (or even 'spurious' nonpolylogarithmic parts-see e.g. [50,51]) that cancel between terms. Surprisingly, in the component under study, this is precisely what happens: the local integrals that contribute to the amplitude individually involve quadratic roots, but these roots cancel.…”

Conformally-regulated direct integration of the two-loop heptagon remainder

“…At the integrand level, there are several new and extremely powerful frameworks for expressing perturbative scattering amplitudes of an increasingly general class of theories. These tools include all-loop recursion relations [7,8], bootstrap methods [9][10][11], Q-cuts [12], and the broad reach of generalized [13][14][15][16][17][18][19][20][21][22] and prescriptive [23][24][25][26][27][28][29] unitarity. It remains to be seen, however, how much of the simplicity of integrands can survive loop integration.…”