In this note we study tree-level scattering amplitudes of gravitons under a natural deformation which in the large $\textit{z}$ limit can be interpreted either as a $\textit{k}$-hard-particle limit or as a ($\textit{n}$ − $\textit{k}$)-soft-particle limit. When $\textit{k}$ = 2 this becomes the standard BCFW deformation while for $\textit{k}$ = 3 it leads to the Risager deformation. The hard- to soft-limit map we define motivates a way of computing the leading order behavior of amplitudes for large $\textit{z}$ directly from soft limits. We check the proposal by applying the $\textit{k}$ = 3 and $\textit{k}$ = 4 versions to NMHV and N MHV gravity amplitudes respectively. The former reproduces in a few lines the result recently obtained by using CHY-like techniques in [1]. The N MHV formula is also remarkably simple and we give support for it using a CHY-like computation. In the $\textit{k}$ = 2 case applied to any gravity amplitude, the multiple soft-limit analysis reproduces the correct 𝒪($\textit{z}$ ) behavior while explicitly showing the source of the mysterious cancellation among Feynman diagrams that tames the behavior from the 𝒪($\textit{z}$ ) of individual Feynman diagrams down to the 𝒪($\textit{z}$ ) of the amplitude.

Canada

Freddy Cachazo

Pablo Leon

Connecting infinity to soft factors

<p>In the Cachazo-He-Yuan (CHY) representation, many tree-level scattering amplitudes are written as integrals localized by the scattering equations, where the CHY integrand encodes the theory and must reproduce the physical factorization pattern of Mandelstam singularities. The inverse CHY problem therefore asks for a rational integrand whose physical poles match a prescribed set of factorization channels, a task that becomes rapidly combinatorial at higher multiplicity. We present an algorithmic approach that makes the subset structure of pole data explicit by assigning each particle subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math> a generalized pole degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></math>, realized as a signed internal-edge count in a colored integrand graph. Additivity under integrand multiplication and an elementary face recursion on the subset lattice imply that all higher-channel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></math> are linear functions of the two-particle integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">{</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>s</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math>, so the inverse step becomes a mixed-integer linear feasibility problem. The same subset lattice provides a fixed computation graph for exact constraint satisfaction: pole requirements and recursion identities are treated as hard constraints, updates are restricted to those that preserve feasibility, and whenever a proposed change would enter a fixed node its constraint residual is rerouted and redistributed through unfixed degrees of freedom so that all constraints remain satisfied throughout. After a factorial rescaling, every local propagation update is purely integral, keeping the entire procedure exact in integer arithmetic and avoiding <i>a posteriori</i> reconstruction from numerical evaluations. Since physical tree-level amplitudes must contain only simple factorization poles as dictated by unitarity and locality, we further organize generalized integrands by an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>-regular grading under multiplication, where degree-zero factors act as Möbius-invariant insertions admitting a decomposition into products of four-point cross ratios, which guides efficient higher-order pole reduction. We illustrate the construction at six and eight points, including pole selection and higher-order pole reduction.</p>

China

Simeng Li

Yaobo Zhang

Inverse Cachazo-He-Yuan integrand construction from factorization channels with exact integer constraint propagation

The biadjoint scalar partial amplitude, $$ {m}_n\left(\mathbbm{I},\mathbbm{I}\right) $$ , can be expressed as a single integral over the positive tropical Grassmannian thus producing a $\textit{Global Schwinger Parameterization}$. The first result in this work is an extension to all partial amplitudes $\textit{m}$ ($\textit{α}$, $\textit{β}$) using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of $\textit{ϕ}$ amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into C regions, with C the $\textit{q}$ -Catalan number. The contribution from each region is identified with a $\textit{m}$ ($\textit{α}$, $$ \mathbbm{I} $$ ) amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for $\textit{ϕ}$ amplitudes using the Lagrange inversion construction. We start the exploration of $\textit{ϕ}$ theories, finding that their regions are encoded in non-crossing ($\textit{p}$ – 2)-chord diagrams. The structure of the expansion of $\textit{ϕ}$ amplitudes in terms of $\textit{ϕ}$ amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of Φ matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.

Bruno Umbert

Connecting scalar amplitudes using the positive tropical Grassmannian

In this note we show how the solutions to the scattering equations in the NMHV sector fully decompose into subsectors in the z → ∞ limit of a Risager deformation. Each subsector is characterized by the punctures that coalesce in the limit. This naturally decomposes the E(n − 3, 1) solutions into sets characterized by partitions of n − 3 elements so that exactly one subset has more than one element. We present analytic expressions for the leading order of the solutions in an expansion around infinite z for any n. We also give a simple algorithm for numerically computing arbitrarily high orders in the same expansion. As a consequence, one has the ability to compute Yang-Mills and gravity amplitudes purely from this expansion around infinity. Moreover, we present a new analytic computation of the residue at infinity of the n = 12 NMHV tree-level gravity amplitude which agrees with the results of Conde and Rajabi. In fact, we present the analytic form of the leading order in 1/z of the Cachazo-Skinner-Mason/CHY formula for graviton amplitudes for each subsector and to all multiplicity. As a byproduct of the all-order algorithm, one has access to the numerical value of the residue at infinity for any n and hence to the corrected CSW (or MHV) expansion for NMHV gravity amplitudes.

Dawit Belayneh

Computing NMHV gravity amplitudes at infinity

In a recent paper, we defined generalized color orderings (GCO) and Feynman diagrams (GFD) to compute color-dressed generalized biadjoint amplitudes. In this work, we study the Cachazo-Early-Guevara-Mizera (CEGM) representation of generalized partial amplitudes and “decoupling” identities. This representation is a generalization of the Cachazo-He-Yuan (CHY) formulation as an integral over the configuration space <math><mi>X</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo></math> of n points on <math><msup><mrow><mi>CP</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math> in generic position.Unlike the <math><mi>k</mi><mo>=</mo><mn>2</mn></math> case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for <math><mi>k</mi><mo>&gt;</mo><mn>2</mn></math>. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all <math><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>&lt;</mo><mn>9</mn><mo>)</mo></math> integrands, all <math><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math> integrands up to four undetermined constants, and 95% of <math><mo>(</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>)</mo></math> integrands up to 24 undetermined constants.<math><mi>k</mi><mo>=</mo><mn>2</mn></math> partial amplitudes are known to satisfy identities. Among them, the so-called <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> decoupling identities are the simplest ones. These are characterized by a label i and a color ordering in <math><mi>X</mi><mo>(</mo><mn>2</mn><mo>,</mo><mo>|</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>∖</mo><mo>{</mo><mi>i</mi><mo>}</mo><mo>|</mo><mo>)</mo></math>. Here we introduce decoupling identities for <math><mi>k</mi><mo>&gt;</mo><mn>2</mn></math> determined combinatorially using GCOs. Moreover, we identify the natural analog of <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> identities as those characterized by a pair of labels <math><mi>i</mi><mo>≠</mo><mi>j</mi></math>, and a pair of GCOs, one in <math><mi>X</mi><mo>(</mo><mi>k</mi><mo>,</mo><mo>|</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>∖</mo><mo>{</mo><mi>i</mi><mo>}</mo><mo>|</mo><mo>)</mo></math> and the other in <math><mi>X</mi><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>|</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>∖</mo><mo>{</mo><mi>j</mi><mo>}</mo><mo>|</mo><mo>)</mo></math>. We call them double extension identities.We also provide explicit connections among different ways of representing GCOs, such as configurations of lines, configurations of points, and reorientation classes of uniform oriented matroids (chirotopes).

Germany

Nick Early

Yong Zhang

Generalized color orderings: CEGM integrands and decoupling identities

In the Grassmannian formulation of the S-matrix for planar $$ \mathcal{N} $$ = 4 Super Yang-Mills, $\textit{N}$ $\textit{MHV}$ scattering amplitudes for $\textit{k}$ negative and $\textit{n − k}$ positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of ($\textit{k −}$ 2)($\textit{n − k −}$ 2)-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian $\textit{G}$($\textit{k, n}$). In this paper, we replace the Grassmannian $\textit{G}$(3$\textit{, n}$) with its torus quotient, the moduli space of $\textit{n}$ points in the projective plane in general position, and planar $$ \mathcal{N} $$ = 4 SYM with generalized biadjoint scalar amplitudes $$ {m}_n^{(3)} $$ as introduced by Cachazo-Early-Guevara-Mizera (CEGM) [1]. Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such ($\textit{n −}$ 5)-dimensional residue of $$ {m}_n^{(3)} $$ an entire biadjoint scalar partial amplitude $$ {m}_n^{(2)} $$ , that is a sum over all tree-level Feynman diagrams for a fixed planar order. We make a proposal which would do the same for $\textit{k ≥}$ 4.

Biadjoint scalars and associahedra from residues of generalized amplitudes

In this paper, we study a novel behavior developed by certain tree-level scalar scattering amplitudes, including the biadjoint, NLSM, and special Galileon, when a subset of kinematic invariants vanishes without producing a singularity. This behavior exhibits properties which we call smooth splitting and semi-locality. The former means that an amplitude becomes the product of exactly three amputated Berends-Giele currents, while the latter means that any two currents share one external particle. We call these smooth splittings 3-splits. In fact, there are exactly   <math> <mfenced> <mtable> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mfenced> <mo>−</mo> <mi>n</mi> </math>  $$ \left(\begin{array}{c}n\\ {}3\end{array}\right)-n $$  such 3-splits of an n-particle amplitude, one for each tripod in a polygon; as they cannot be obtained from standard factorization, they are a new phenomenon in Quantum Field Theory. In fact, the resulting splitting is analogous to the one first seen in Cachazo-Early-Guevara-Mizera (CEGM) amplitudes which generalize standard cubic scalar amplitudes from their Tr G(2, n) formulation to Tr G(k, n), where Tr G(k, n) is the tropical Grassmannian. Along the way, we show how smooth splittings naturally lead to the discovery of mixed amplitudes in the NLSM and special Galileon theories and to novel BCFW-like recursion relations for NLSM amplitudes.

Smoothly splitting amplitudes and semi-locality

Tree-level Feynman diagrams in a cubic scalar theory can be given a metric such that each edge has a length. The space of metric trees is made out of orthants joined where a tree degenerates. Here we restrict to planar trees since each degeneration of a tree leads to a single planar neighbor. Amplitudes are computed as an integral over the space of metrics where edge lengths are Schwinger parameters. In this work we propose that a natural generalization of Feynman diagrams is provided by what are known as metric tree arrangements. These are collections of metric trees subject to a compatibility condition on the metrics. We introduce the notion of planar col lections of Feynman diagrams and argue that using planarity one can generate all planar collections starting from any one. Moreover, we identify a canonical initial collection for all n. Generalized k = 3 biadjoint amplitudes, introduced by Early, Guevara, Mizera, and one of the authors, are easily computed as an integral over the space of metrics of planar collections of Feynman diagrams.

Francisco Borges

Generalized planar Feynman diagrams: collections

We extend on-shell bootstrap methods from spacetime amplitudes to the world sheet objects of the Cachazo-He-Yuan formalism. We observe that the integrands corresponding to tree-level nonlinear sigma model, Yang-Mills and <math><mo>(</mo><mi>D</mi><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup></math> theory are determined by demanding enhanced UV scaling under Britto-Cachazo-Feng-Witten shifts. Back in spacetime, we also find that <math><mo>(</mo><mi>D</mi><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup></math> theory is fixed by gauge invariance/UV scaling and simple locality assumptions.

France

Laurentiu Rodina

UV consistency conditions for Cachazo-He-Yuan integrands

The Cachazo-He-Yuan formula was originally proposed to describe on-shell scattering of particles from a single massless field. We present a method to modify it to include several interacting scalar fields, all possessing different masses and possibly off-shell momenta. The method is applied to Yukawa interactions between a number of scalar nucleons and pions, and to the <math><msub><mi>ϕ</mi><mn>1</mn></msub><msub><mi>ϕ</mi><mn>2</mn></msub><msub><mi>ϕ</mi><mn>3</mn></msub></math> coupling of three different scalar fields. Composite models constructed from existing theories can be used to broaden the scope of the method. The modification is applied to describe Compton scattering from a massive particle and to photon bremsstrahlung. It is also employed to generalize the disk function <math><mi>Z</mi></math> and the sphere function <math><mi>J</mi></math>.