Symmetry properties of Wilson loops with a Lagrangian insertion

Abstract

Null Wilson loops in $$ \mathcal{N} $$ N = 4 super Yang-Mills are dual to planar scattering amplitudes. This duality implies hidden symmetries for both objects. We consider closely related infrared finite observables, defined as the Wilson loop with a Lagrangian insertion, normalized by the Wilson loop itself. Unlike ratio and remainder functions studied in the literature, this observable is non-trivial already for four scattered particles and bears close resemblance to (finite parts of) scattering processes in non-supersymmetric Yang-Mills theory. Moreover, by integrating over the insertion point, one can recover information on the amplitude, as was recently done to compute the full four-loop cusp anomalous dimension. We study the general structure of the Wilson loop with a Lagrangian insertion, focusing in particular on its leading singularities and their (hidden) symmetry properties. Thanks to the close connection of the observable to integrands of MHV amplitudes, it is natural to expect that its leading singularities can be written as certain Grassmannian integrals. The latter are manifestly dual conformal. They also have a conformal symmetry, up to total derivatives. We find that, surprisingly, the conformal symmetry becomes an invariance in the frame where the Lagrangian insertion point is sent to infinity. Furthermore, we use integrability methods to study how higher Yangian charges act on the Grassmannian integral. We evaluate the n-particle observable both at tree- and at one-loop level, finding compact analytic formulas. These results are explicitly written in the form of conformal leading singularities, multiplied by transcendental functions. We then compare these formulas to known expressions for all-plus amplitudes in pure Yang-Mills theory. We find a remarkable new connection: the Wilson loop with Lagrangian insertion in $$ \mathcal{N} $$ N = 4 super Yang-Mills appears to predict the maximal weight terms of the planar pure Yang-Mills all-plus amplitude. We test this relationship for the two-loop n-point Yang-Mills amplitude, as well as for the three-loop four-point amplitude.

Anomalous Ward identities for on-shell amplitudes at the conformal fixed point

“…[17,19]). For example, conformally-invariant expressions have been found to play an important role in all-plus scattering amplitudes [20][21][22], but also appear in more general helicity configurations.…”

Integrated negative geometries in ABJM

“…This is the same number of kinematic variables as for QCD n-point amplitudes. This similarity, together with a conjectured duality with pure Yang-Mills all-plus helicity-amplitudes [24], motivates further studies of these finite observables. We will focus on the four-particle case, for which one gets a function F(z) of a single cross-ratio.…”

“…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.…”