Johannes M. Henn

Employing a cutting-edge bootstrap method, we analytically compute the three-loop pentagonal Wilson loop with Lagrangian insertion in planar N = 4 super-Yang-Mills theory. Via a conjectured duality, this provides the first prediction for 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 quantity. 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, which moreover yields the full function-level answer.

A bstract: We study correlators of null, n -sided polygonal Wilson loops with a Lagrangian insertion in the planar limit of the 𝒩 = 4 supersymmetric Yang-Mills theory. This finite observable is closely related to loop integrands of maximally-helicity-violating amplitudes in the same theory, and, conjecturally, to all-plus helicity amplitudes in pure Yang-Mills theory. The resulting function has been observed to have an expansion in terms of functions of uniform transcendental weight, multiplied by certain rational prefactors, called leading singularities. In this work we prove several conjectures about the leading singularities: we classify and compute them at any loop order and for any number of edges of the Wilson loop, and show that they have a hidden conformal symmetry. This is achieved by leveraging the geometric definition of the loop integrand via the Amplituhedron. The leading singularities can be seen as maximal codimension residues of the integrand, and the boundary structure of the Amplituhedron geometry restricts which iterative residues are accessible. Combining this idea with a further geometric decomposition of the Amplituhedron in terms of so-called negative geometries allows us to identify the complete set of leading singularities.

A bstract: We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives have a fixed sign, in the Euclidean kinematic region. This imposes strong constraints on the function space. Simultaneously, these integrals obey systems of linear differential equations with respect to kinematic parameters. By imposing that the solutions to these differential equations satisfy complete monotonicity across the Euclidean region, we develop an efficient and highly constraining numerical bootstrap method. We provide a proof of principle of the power of our approach by applying it to a class of multi-loop Feynman integrals with internal masses. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, such as dimension and propagator exponents, are not only CM but in fact Stieltjes functions. The latter can be described efficiently by Padé approximants that are known to converge in the cut complex plane. This means that these representations are valid also in analytically continued kinematics, such as physical scattering regions. These insights allow us to obtain rational approximations to Feynman integrals from minimal information, such as a Taylor expansion about a soft limit. We demonstrate the effectiveness of this method by applying it to a 20-loop banana-type Feynman integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.

We present a symbol-level bootstrap construction of the planar, two-loop six-gluon scattering amplitude for the − − + + + + helicity configuration in quantum chromodynamics (QCD), focusing on the maximal weight pieces—the “most complicated terms” in the sense of Lipatov Building on recent advances in the understanding of the relevant function space, we incorporate as a crucial new ingredient the complete set of leading singularities, obtained from an analysis of on-shell diagrams. The resulting expressions are manifestly conformally invariant and clarify the structure of previous five-particle results. Combining this with the symbol bootstrap, we show that constraints from physical limits are sufficient to determine the answer. We thus obtain the first concrete characterization of two-loop six-gluon amplitudes at symbol level and at highest weight. Remarkably, we find that the effective function space involves only 137 symbol letters—significantly fewer than the full set of 167 possible letters, suggesting a yet-unexplained underlying structure akin to that seen in maximally supersymmetric Yang-Mills theory. From the novel amplitude results we extract previously unknown symbol-level results describing two-loop triple collinear and double soft limits.

The study of collinear behavior for gauge theories in the spacelike region is of great phenomenological and theoretical importance. We analytically calculate the two-loop spacelike splitting amplitude for the full color N = 4 super-Yang-Mills theory. The result is derived by two complementary methods starting from the known amplitude: one is based on a discontinuity analysis, while the other one is based on calculating Feynman integrals analytically in the collinear limit. Our result explicitly shows terms that violate naive factorization. However we show that factorization is restored at the level of color-summed unpolarized squared amplitudes at next-to-next-to-next-to leading order. We conjecture that the two-loop tripole terms in the generalized splitting amplitudes in quantum chromodynamics are identical to what we obtain in N = 4 super Yang-Mills theory.