Symmetry properties of Wilson loops with a Lagrangian insertion

Abstract

Null Wilson loops in 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 N = 4 super Yang-Mills appears to predict the maximal weight terms of the planar pure Yang-Mills allplus amplitude. We test this relationship for the two-loop n-point Yang-Mills amplitude, as well as for the three-loop four-point amplitude.

A Unitarity Approach to Two-Loop All-Plus Rational Terms

“…These amplitudes may also be a portal to exploring hidden connections between different theories: at one loop, there is an intriguing connection [7,8] between the all-plus amplitude and the simplest N = 4 super-Yang-Mills amplitude, scattering all gluons but two of like helicity (MHV). Furthermore, for four and five gluons in the planar limit, the leading transcendental weight parts of all-plus amplitudes at two loops (and three loops for four gluons) have been shown to be dual to those of N = 4 Wilson loops with Lagrangian insertions at one lower loop order [9]. This duality is conjectured to hold for any loop order.…”

“…14: An example of a one-loop-squared cut contributing to the A 12;6,9 partial amplitude. As explained in the text, it contributions to the coefficient of the triple-trace structure ITr(1, 2, 3, 4, 5, 6) ITr (7,8,9) ITr (10,11,12).…”

“…We wrote a set of Mathematica packages to automate the evaluation of unitarity cuts on specified (rational) kinematic points 9 . These packages are extensions of the SpinorHelicity6D package of ref.…”

“…(b)shows the six different representations, for which we construct unitarity cuts. The insertions on the edges correspond to a color structure ITr(1, 2, 3) ITr(4, 5, 6) ITr(7,8,9)…”

All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron

“…In N = 4 SYM, it is beneficial to decompose n = 4 amplituhedron into building blocks called negative geometries [38], and at each loop, the non-trivial geometries combine to give the integrand for an infrared-finite observable closely related to logarithm of amplitudes (or equivalently Wilson loops with a single insertion) [39][40][41][42][43][44]. Analogous decomposition of projected amplituhedron reveals enormous simplifications from D = 4 to D = 3, since we only have a tiny fraction of negative geometries, namely those correspond to bipartite graphs, contribute to the integrand, with very simple pole structures.…”