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

Germany

Johannes Henn

Elia Mazzucchelli

United States

Jaroslav Trnka

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.

Czechia

Christoph Bartsch

Karol Kampf

Jiří Novotný

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

Umut Oktem

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.

All-loop soft theorem for pions

We study a novel geometric expansion for scattering amplitudes in the planar sector of <math> <mi>N</mi> </math> $$ \mathcal{N} $$ = 4 super Yang-Mills theory, in the context of the Amplituhedron which reproduces the all-loop integrand as a canonical differential form on the positive geometry. In a paper by Arkani-Hamed, Henn and one of the authors, it was shown that this result can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momenta, represented by lines in momentum twistor space. One can then calculate an all-loop order result in the approximation where only tree graphs in the space of all loops are considered. Furthermore, using differential equation methods, it is possible to calculate and resum integrated expressions and obtain strong coupling results. In this paper, we provide a more general framework for the ‘loops of loops’ expansion and outline a powerful method for the determination of differential forms for higher-order geometries. We solve the problem completely for graphs with one internal cycle, but the method can be used more generally for other geometries.

Taro Brown

Loops of loops expansion in the amplituhedron

The amplituhedron provides, via geometric means, the all-loop integrand of scattering amplitudes in maximally supersymmetric Yang-Mills theory. Unfortunately, dimensional regularization, used conventionally for integration, breaks the beautiful geometric picture. This motivates us to propose a “deformed” amplituhedron. Focusing on the four-particle amplitude, we introduce two deformation parameters, which can be interpreted as particle masses. We provide evidence that the mass pattern corresponds to a specific choice of vacuum expectation values on the Coulomb branch. The deformed amplitude is infrared finite, making the answer well defined in four dimensions. Leveraging four-dimensional integration techniques based on differential equations, we compute the amplitude up to two loops. In the limit where the deformation parameters are taken to zero, we recover the known Bern-Dixon-Smirnov amplitude. In the limit where only one deformation parameter is taken to zero, we find a connection to the angle-dependent cusp anomalous dimension.

Nima Arkani-Hamed

Johannes M. Henn

Coulomb Branch Amplitudes from a Deformed Amplituhedron Geometry

In this Letter, we derive new expressions for tree-level graviton amplitudes in <math><mi>N</mi><mo>=</mo><mn>8</mn></math> supergravity from Britto-Cachazo-Feng-Witten (BCFW) recursion relations combined with new types of bonus relations. These bonus relations go beyond the famous <math><mn>1</mn><mo>/</mo><msup><mi>z</mi><mn>2</mn></msup></math> behavior under a large BCFW shift and use knowledge about certain zeros of graviton amplitudes in collinear kinematics. This extra knowledge can be used in the context of global residue theorems by writing the amplitude in a special form using canonical building blocks. In the next-to-maximally-helicity-violating case, these building blocks are dressed one-loop leading singularities, the same objects that appear in the expansion of Yang-Mills amplitudes, where each term corresponds to an <math><mi>R</mi></math> invariant. Unlike other approaches, our formula is not an expansion in terms of cyclic objects and does not manifest color-kinematics duality but rather preserves the permutational symmetry of its building blocks. We also comment on the possible connection to Grassmannian geometry and give some nontrivial evidence of such structure for graviton amplitudes.