On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the $$ \mathcal{N} $$ = 4 Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $\textit{G}$($\textit{k}$, $\textit{n}$). Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $\textit{N}$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $\textit{G}$ ($\textit{k}$, $\textit{n}$) which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $\textit{N}$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $\textit{G}$(2, $\textit{n}$). These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are $\textit{internally planar}$ there always exists a strongly connected geometry, and for those that are $\textit{irreducible}$, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

United States

Artyom Lisitsyn

Umut Oktem

Melissa Sherman-Bennett

Jaroslav Trnka

Grassmannian geometries for non-planar on-shell diagrams

We propose a loop-level generalization of the inverse string theory Kawai-Lewellen-Tye (KLT) kernel: the planar $\textit{inverse KLT integrand}$. The integrand is defined constructively via a novel Berends-Giele-like recursion that exposes the inverse KLT kernel as the simplest toy model of a “stringy amplitude”. We show that, to all loop orders, the inverse KLT integrand is structurally equivalent to integrands in the cubic scalar tr$\textit{ϕ}$ theory. This simplicity is obscured in the conventional Feynman diagram approach, where the inverse KLT integrand receives contributions from an infinity of infinite towers of contact interactions. The inverse KLT integrand is a rational function of stringified kinematic variables and is naturally defined on the kinematic surface proposed by Arkani-Hamed et al. [1, 2]. It provides an elementary analogue of the surfacehedron integrand for the tr$\textit{ϕ}$ theory involving only scalar resonances and unifies the scattering of cubic scalars and pions in the non-linear sigma model (NLSM) to all loop orders via kinematic $\textit{α}$ -shifts.

Czechia

Christoph Bartsch

Karol Kampf

Jiří Novotný

A surface integrand for the inverse KLT kernel

We study correlators of null, $\textit{n}$-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.

Taro Brown

Germany

Johannes Henn

Elia Mazzucchelli

All-loop leading singularities of Wilson loops

We investigate how the positive geometry framework for loop integrands in $$ \mathcal{N}=4 $$ super Yang-Mills theory constrains the structure of the integrated answers. This is done in the context of a geometric expansion of Wilson loops with a Lagrangian insertion, called negative geometries, extending ideas previously used for scattering amplitudes related to the Amplituhedron. The procedure we adopt combines the knowledge of all maximal codimension boundaries of the geometry, which characterize all possible leading singularities of the integral, with a geometrically informed Landau analysis. The interplay between geometry and Landau analysis arises from associating Landau diagrams to geometric boundaries. The boundary structure of the geometry then determines which solutions to the Landau equations are spurious and which ones are physical, that is, which singularities are actually present in the integral. This method allows us to efficiently determine the symbol alphabet of the associated integral, and serves as a starting point for the symbol bootstrap. We successfully implement this procedure and compute the six-point two-loop and five-point three-loop ladder negative geometries at the symbol level. We also present the conjectural alphabet for ladder negative geometries at two loops for all multiplicities. These are finite integrals that serve as building blocks for the Wilson loop with Lagrangian insertion, and therefore provide insights into the function space of the latter.

France

Dmitry Chicherin

Qinglin Yang

Shun-Qing Zhang

Geometric Landau analysis and symbol bootstrap

A new perspective on the inverse string theory Kawai-Lewellen-Tye (KLT) kernel is provided which establishes the universality of scattering amplitudes in the biadjoint scalar theory, pions in the nonlinear sigma model (NLSM), and mixed amplitudes (NLSM+<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ϕ</mi><mn>3</mn></msup></math>) recently studied in the literature. We show that all these amplitudes can be viewed as equivalent, arising from a single function, the inverse string theory KLT kernel, evaluated at different kinematic points. In this way cubic colored scalars and pions become interchangeable through a procedure we call the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>α</mi><mo>′</mo></msup></math> shift. The latter complements the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>δ</mi></math> shift proposed by Arkani-Hamed et al., and demonstrates an inherent equivalence of scattering amplitudes in different quantum field theories by embedding them in a common stringy framework.

Universality of colored scalars from the stringy Kawai-Lewellen-Tye kernel

We explore scattering amplitudes on the Coulomb branch of maximally supersymmetric Yang-Mills theory. We introduce a particular pattern of scalar vacuum expectation values that allow us to define amplitudes with a different mass pattern compared to what was studied previously. This is motivated by an extension of the Amplituhedron that leads to infrared-finite four-particle amplitudes involving massive particles. We work out the Feynman rules on the Coulomb branch and use them, together with generalized unitarity techniques, to perform consistency checks on the Amplituhedron expectations for the one- and two-loop integrands for the four-particle amplitude. We present details of the computation of the required two-loop four-point integrals via a four-dimensional version of the differential equations method. Finally, we study the Regge limit of the four-point amplitude, including the first power suppressed terms. We find that when organized in terms of a suitable expansion parameter, the subleading power term exponentiates, with the exponent matching the anomalous dimension of a cusped Wilson line with a local operator insertion. The latter is known from integrability, which leads to a prediction at higher loop orders in the Regge limit.

Italy

Wojciech Flieger

Anders Schreiber

Two-loop four-point amplitudes on the Coulomb branch of $$ \mathcal{N} $$ = 4 super Yang-Mills

In this paper we discuss the geometric integrand expansion of the five-point Wilson loop with one Lagrangian insertion in maximally supersymmetric Yang-Mills theory. We construct the integrand corresponding to an all-loop class of ladder-type geometries. We then investigate the known two-loop observable from this geometric viewpoint. To do so, we evaluate analytically the new two-loop integrals corresponding to the negative geometry contribution, using the canonical differential equations method. Inspecting the analytic result, we present numerical evidence that in this decomposition, each piece has uniform sign properties, when evaluated in the Amplituhedron region. Finally, we present an alternative bootstrap approach for the ladder-type geometries. We find that certain minimal bootstrap assumptions can be satisfied at two loops, but lead to a contradiction at three loops. This suggests to us that novel alphabet letters are required at this loop order. Indeed studying planar three-loop Feynman integrals, we do identify novel pentagon alphabet letters.

Positivity properties of five-point two-loop Wilson loops with Lagrangian insertion

Recently, Arkani-Hamed et al. proposed the existence of zeros in scattering amplitudes in certain quantum field theories including the cubic adjoint scalar theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Tr</mi><mo stretchy="false">(</mo><msup><mi>ϕ</mi><mn>3</mn></msup><mo stretchy="false">)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> nonlinear sigma model and Yang-Mills theory. These hidden zeros are special kinematic points where the amplitude vanishes and factorizes into a product of lower-point amplitudes, similar to factorization near poles. In this paper, we show a close connection between the existence of such zeros and the double-copy map. In fact, compatibility with the Bern-Carrasco-Johansson relations requires the presence of these zeros. We also show that these zeros extend via the Kawai-Lewellen-Tye relations to special Galileon amplitudes and their corrections, evincing that these hidden zeros are also present in permutation-invariant amplitudes.

Taro V. Brown

Shruti Paranjape

Hidden amplitude zeros from the double-copy map

In this paper, we study the scattering amplitudes and soft theorems for the sigma models with two scalars. We show that if the particles are Goldstone bosons, then you necessarily get Adler zero with no possibility for non-trivial soft theorems. For non-Goldstone bosons, the soft behavior is generically captured by the geometric soft theorem studied by Cheung et al., and the right-hand side contains derivatives of lower-point amplitudes. Inspired by the recent work on the 2D sigma models, we study one special two-scalar sigma model, where the presence of symmetries in the target space translates into a special but non-trivial soft theorem without derivatives. We further generalize the construction to two larger classes of such models and derive certain soft theorem sum rules, again avoiding the derivatives of amplitudes. Our analysis provides an interesting hierarchy of two-scalar sigma models and soft theorems, ranging from Goldstone boson case to a generic target space, and showing that there are interesting theories in between.

Jiri Novotny

Mikhail Shifman

Soft theorems for two-scalar sigma models

In this paper we discuss a generalization of the Adler zero to loop integrands in the planar limit of the <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> nonlinear sigma model (NLSM). The Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a soft theorem satisfied by loop integrands with any number of loops and legs. This requires a generalization of NLSM integrands to an off shell framework with certain deformed kinematics. Defining an algebraic soft limit, we identify a simple nonvanishing soft behavior of integrands, which we call the algebraic soft theorem. We find that the proposed soft theorem is satisfied by the “surface” integrand of Arkani-Hamed and Figueiredo [arXiv:2403.04826], which is obtained from the shifted <math><mi>Tr</mi><msup><mi>φ</mi><mn>3</mn></msup></math> surfacehedron integrand. Finally, we derive an on shell version of the algebraic soft theorem that takes an interesting form in terms of self-energy factors and lower-loop integrands in a mixed theory of pions and scalars.