Three-loop pentagonal Wilson loop with Lagrangian insertion

TL;DR

This work computes the three-loop pentagonal Wilson loop with a Lagrangian insertion in planar N=4 sYM by combining a symbol bootstrap constrained by known leading singularities and near-collinear OPE data with an independent integral-reduction verification. The result uniquely fixes the weight-6 symbol for $F^{(3)}$, leveraging a reduced 30-letter alphabet and duality to all-plus amplitudes, and is independently validated by reducing the integrand to a canonical basis. The study provides the first physical observable in the three-loop five-point planar function space, offering a bridge between supersymmetric Wilson-loop techniques and non-supersymmetric Yang-Mills amplitudes, with potential extensions to finite coupling and higher loops.

Abstract

Employing a cutting-edge bootstrap method, we analytically compute the three-loop pentagonal Wilson loop with Lagrangian insertion in planar $\mathcal{N}=4$ super-Yang-Mills theory. This object is conjectured to coincide with the maximally transcendental part of the four-loop five-point all-plus amplitude in pure Yang-Mills theory. Our starting point is an ansatz that encodes the known leading singularities of this object, as well as the relevant function space. The latter has become available only recently, thanks to an analytic computation of all three-loop five-point planar massless Feynman integrals. We determine the coefficients in the ansatz by imposing physical constraints. This includes a near-collinear expansion, which so far has not been applied to this observable. Taken together, the constraints allow us to uniquely determine the symbol of the answer. We verify the symbol result by an independent integral reduction calculation.

Figures (2)

• Figure 1: Quadrilateral and pentagonal Wilson loop contours with cusps $x_i = Z_{i-1} \wedge Z_i$ and $x_{4'} = Z_3 \wedge Z_{4'}$, and the Lagrangian coordinate $x_0 = Z_A \wedge Z_B$. In the collinear limit, the polygon flattens, i.e. $Z_4, Z_5 \to Z_{4'}$, $x_{4} \to x_{4'}$ and $x_5 \in [x_1;x_{4'}]$.
• Figure 2: Feynman integrals that give rise to the letters $w_{21}$ and $w_{26}$, respectively.