Three-loop pentagonal Wilson loop with Lagrangian insertion

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.