Pentagon Wilson loop with Lagrangian insertion at two loops in $$ \mathcal{N} $$ = 4 super Yang-Mills theory

Abstract

We compute the two-loop result for the null pentagonal Wilson loop with a Lagrangian insertion (normalized by the Wilson loop without insertion) in planar, maximally supersymmetric Yang-Mills theory. This finite observable is closely related to the Amplituhedron, and it is reminiscent of finite parts of planar two-loop five-particle scattering amplitudes. We verify that, up to this loop order, the leading singularities are given by the same conformally invariant expressions that appear in all-plus pure Yang-Mills amplitudes. The accompanying weight-four transcendental functions are expressed in terms of the pentagon functions space known from planar two-loop five-particle amplitudes, but interestingly only a subset of the functions appears. Being a function of four dimensionless variables, the observable has interesting asymptotic limits. We verify that our analytic result is consistent with soft and collinear limits, and find an intriguingly simple pattern in the multi-Regge limit. Thanks to the new result we can also conjecturally predict, for general kinematics, the maximal weight piece of the planar three-loop five-particle all-plus amplitude in pure Yang-Mills theory. Motivated by the Amplituhedron geometry, we investigate positivity properties of the integrated answer. Generalizing previous results at four particles, we find numerical evidence that the two-loop five-particle result has uniform sign in a kinematic region suggested by the loop Amplituhedron.

Integrated negative geometries in ABJM

“…Furthermore, we use our results to compute the first non-trivial contribution to Γ cusp , finding perfect agreement with the literature [51]. Finally, we discover that the leading singularities of the integrated results also possess a hidden conformal symmetry, in a similar manner to what was found in the four-dimensional case [24,25].…”

“…Also surprisingly, the leading singularities of these integrated negative geometries enjoy a (hidden) conformal symmetry [24,25]. Furthermore, identities relating F(z) to all-plus amplitudes in pure Yang-Mills theory have been found [24,25]. Finally, one can also note that the perturbative expansion of F(z) respects a uniform transcendentality principle [23].…”

“…Such results would allow a comparison with the non-perturbative derivation of Γ cusp coming from integrability [30,31]. Also surprisingly, the leading singularities of these integrated negative geometries enjoy a (hidden) conformal symmetry [24,25]. Furthermore, identities relating F(z) to all-plus amplitudes in pure Yang-Mills theory have been found [24,25].…”

“…It arises when one considers the integration of the L-loop negative geometry over L − 1 of the loop variables, i.e. leaving one of the loop variables unintegrated [16][17][18][19][20][21][22][23][24][25]. Remarkably, this object is infrared (IR) finite, and all the divergences concentrate on the L-th loop integral.…”

“…In this paper we focus on the ABJM theory, and we explicitly perform the (L − 1)-loop integrations of the four-particle negative geometries for the L ≤ 3 cases, showing that the integrated results are given by finite and uniform-transcendental polylogarithmic functions. In an analogous way to the five-particle case of N = 4 sYM [25], we find it convenient to organize the integrated results in parity-even and -odd terms, which are described by two functions F(z) and G(z), respectively. As we will see, it is straightforward to show that only the former contributes to the cusp anomalous dimension Γ cusp after the last loop integration.…”