arXiv subjects

Freddy Cachazo

Publications and source records attributed to Freddy Cachazo.

At least 19 recordsLinked to original sources

arXiv2024

Connecting Infinity to Soft Factors

In this note we study tree-level scattering amplitudes of gravitons under a natural deformation which in the large $z$ limit can be interpreted either as a $k$-hard-particle limit or as a $(n-k)$-soft-particle limit. When $k=2$ this becomes the standard BCFW deformation while for $k=3$ 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 $z$ directly from soft limits. We check the proposal by applying the $k=3$ and $k=4$ versions to NMHV and N$^2$MHV gravity amplitudes respectively. The former reproduces in a few lines the result recently obtained by using CHY-like techniques in \cite{BCL}. The N$^2$MHV formula is also remarkably simple and we give support for it using a CHY-like computation. In the $k=2$ case applied to any gravity amplitude, the multiple soft-limit analysis reproduces the correct ${\cal O}(z^{-2})$ behavior while explicitly showing the source of the mysterious cancellation among Feynman diagrams that tames the behavior from the ${\cal O}(z^{n-5})$ of individual Feynman diagrams down to the ${\cal O}(z^{-2})$ of the amplitude.

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arXiv2024

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are "color" decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of "local planarity" into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the "colorless" generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.

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arXiv2024

Computing NMHV Gravity Amplitudes at Infinity

In this note we show how the solutions to the scattering equations in the NMHV sector fully decompose into subsectors in the $z\to \infty$ 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.

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arXiv2023

Generalized Color Orderings: CEGM Integrands and Decoupling Identities

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 $X(k,n)$ of $n$ points on $\mathbb{CP}^{k-1}$ in generic position. Unlike the $k=2$ case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for $k>2$. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all $(3,n<9)$ integrands, all $(3,9)$ integrands up to four undetermined constants, and $95 \%$ of $(4,8)$ integrands up to 24 undetermined constants. $k=2$ partial amplitudes are known to satisfy identities. Among them, the so-called $U(1)$ decoupling identities are the simplest ones. These are characterized by a label $i$ and a color ordering in $X(2,|[n]\setminus \{i\}|)$. Here we introduce decoupling identities for $k>2$ determined combinatorially using GCOs. Moreover, we identify the natural analog of $U(1)$ identities as those characterized by a pair of labels $i\neq j$, and a pair of GCOs, one in $X(k,|[n]\setminus \{i\}|)$ and the other in $X(k-1,|[n]\setminus \{j\}|)$. We call them {\it 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).

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arXiv2023

Planar Matrices and Arrays of Feynman Diagrams

Very recently planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of $k=3$ biadjoint amplitudes. Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition. In this work we introduce planar matrices of Feynman diagrams as the objects that compute $k=4$ biadjoint amplitudes. These are symmetric matrices of metric trees satisfying compatibility conditions. We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections. As applications of the first, we find all $693$, $13\,612$, and $346\,710$ collections for $(k,n)=(3,7), (3,8),$ and $(3,9)$ respectively. As applications of the second kind, we find all $90\, 608$ and $30\,659\,424$ planar matrices that compute $(k,n)=(4,8)$ and $(4,9)$ biadjoint amplitudes respectively. As an example of the evaluation of matrices of Feynman diagrams, we present the complete form of the $(4,8)$ and $(4,9)$ biadjoint amplitudes. We also start the study of higher dimensional arrays of Feynman diagrams, including the combinatorial version of the duality between $(k,n)$ and $(n-k,n)$ objects.

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arXiv2023

Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes

In the Grassmannian formulation of the S-matrix for planar $\mathcal{N}=4$ Super Yang-Mills, $N^{k-2}MHV$ scattering amplitudes for $k$ negative and $n-k$ positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of $(k-2)(n-k-2)$-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian $G(k,n)$. In this paper, we replace the Grassmannian $G(3,n)$ with its torus quotient, the moduli space of $n$ points in the projective plane in general position, and planar $\mathcal{N}=4$ SYM with generalized biadjoint scalar amplitudes $m^{(3)}_n$ as introduced by Cachazo-Early-Guevara-Mizera (CEGM). Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such $(n-5)$-dimensional residue of $m^{(3)}_n$ is an entire biadjoint scalar partial amplitude $m^{(2)}_n$, that is a sum over all tree-level Feynman diagrams for a fixed planar order. We propose a generalization which would give rise to identifications of $m^{(2)}_n$ inside $m^{(k)}_n$ for $k\ge 4$, via $(k-2)(n-k-2)$-dimensional residues. Our proof for $k=3$ uses the CEGM formula for $m^{(3)}_n$; it predicts a new Minkowski sum realization of the associahedron in terms of certain positroid polytopes in the second hypersimplex $Δ_{2,n}$.

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arXiv2022

Connecting Scalar Amplitudes using The Positive Tropical Grassmannian

The biadjoint scalar partial amplitude, $m_n(\mathbb{I},\mathbb{I})$, can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial amplitudes $m_n(α,β)$ using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of $ϕ^4$ amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into $\textrm{C}_{n/2-1}$ regions, with $\textrm{C}_q$ the $q^{\rm th}$-Catalan number. The contribution from each region is identified with a $m_{n/2+1}(α,\mathbb{I})$ amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for $ϕ^4$ amplitudes using the Lagrange inversion construction. We start the exploration of $ϕ^p$ theories, finding that their regions are encoded in non-crossing $(p-2)$-chord diagrams. The structure of the expansion of $ϕ^p$ amplitudes in terms of $ϕ^3$ amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of $Φ^{p-1}$ matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.

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arXiv2022

Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes

In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $G(n-k,n)$ and the generalized CHY potential on $X(k,n)$. We also check the recent conjecture that generalized biadjoint amplitudes evaluate to $k$-dimensional Catalan numbers on PK for several examples including $k=3$ and $n\leq 40$ and $(k,n)=(6,13)$. We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over ${\rm Trop}^+ G(k,n)$ and we use it to evaluate the amplitude on PK with the purpose of exhibiting how GFD's glue together. We initiate the study of two minimal lattice polytopal neighborhoods of the planar kinematics point. One of these, the rank-graded root polytope $\mathcal{R}_{k,n}$, in the case $k=2$, is a projection of the standard type A root polytope. The other, denoted $Π_{k,n}$, in the case $k=2$, is a degeneration of the associahedron. We check up to and including $\mathcal{R}_{3,9}$ and $\mathcal{R}_{4,9}$ that the relative volume of $\mathcal{R}_{k,n}$ is the multi-dimensional Catalan number $C^{(k)}_{n-k}$, hinting towards the possibility of deeper geometric and combinatorial interpretations of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ near the PK point.

math.CO↗

arXiv2022

Smoothly Splitting Amplitudes and Semi-Locality

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 $\textit{smooth splitting}$ and $\textit{semi-locality}$. 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 $\binom{n}{3}-n$ such 3-splits, one for each generic, interior triangle 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 ${\rm Tr}\, G(2,n)$ formulation to ${\rm Tr}\, G(k,n)$, where ${\rm Tr}\, G(k,n)$ 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.

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arXiv2021

Minimal Kinematics: An All $k$ and $n$ Peek into ${\rm Trop}^+{\rm G}(k,n)$

In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all $k$ generalization of the one recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we have checked analytically through $n=10$ for all $k$. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the $90\,608$ planar arrays for ${\rm Tr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for $k=2$ and uses the planar basis recently introduced by Early for all $k$. Based on numerical computations through $n=8$ for all $k$, we conjecture that on the planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For $k=2$ these are the standard Catalan numbers.

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arXiv2021

Diagonally Embedded Sets of ${\rm Trop}^+G(2,n)$'s in ${\rm Trop}\, G(2,n)$: Is There a Critical Value of $n$?

The tropical Grassmannian ${\rm Trop}\, G(2,n)$ is known to be the moduli space of unrooted metric trees with $n$ leaves. A positive part can be defined for each of the $(n-1)!/2$ possible planar orderings, $α$, and agrees with the corresponding planar trees in the moduli space, ${\rm Trop}^αG(2,n)$. Motivated by a physical application we study the way ${\rm Trop}^αG(2,n)$ and ${\rm Trop}^βG(2,n)$ intersect in ${\rm Trop}\, G(2,n)$. We define their intersection number as the number of unrooted binary trees that belong to both and construct a $(n-1)!/2\times (n-1)!/2$ intersection matrix. We are interested in finding the diagonal (up to permutations of rows and columns) submatrices of maximum possible rank for a given $n$. We prove that such diagonal matrices cannot have rank larger than $(n-3)!$ using the CHY formalism. We also prove that the bound is saturated for $n=5$ (the condition is trivial for $n=4$), that for $n=6$ the maximum rank is $4$, and that for $n=7$ the maximum rank is $\geq 14$. We also ask the following question: Is there a value $n_{\rm c}$ so that for any $n>n_{\rm c}$ the bound $(n-3)!$ is always saturated? We review and extend two relevant results in the literature. The first is the Kawai-Lewellen-Tye (KLT) choice of sets which leads to a $(n-3)!\times (n-3)!$ block diagonal submatrix with blocks of size $d\times d$ with $d = \lceil (n-3)/2\rceil !\lfloor (n-3)/2\rfloor !$. The second result is that the number of ${\rm Trop}^αG(2,n)$'s that intersect a given one grows as $\exp(n\log (3+\sqrt{8}))$ for large $n$ which implies that the density of the intersection matrix goes as $\exp(-n(\log(n)-2.76))$. We interpret this as an indication that the generic behavior is not seen until $n \approx \exp (2.76)$, i.e. $n = 16$. We also find an exact formula for the number of zeros in a KLT block.

math.CO↗

arXiv2020

Singular Solutions in Soft Limits

A generalization of the scattering equations on $X(2,n)$, the configuration space of $n$ points on $\mathbb{CP}^1$, to higher dimensional projective spaces was recently introduced by Early, Guevara, Mizera, and one of the authors. One of the new features in $X(k,n)$ with $k>2$ is the presence of both regular and singular solutions in a soft limit. In this work we study soft limits in $X(3,7)$, $X(4,7)$, $X(3,8)$ and $X(5,8)$, find all singular solutions, and show their geometrical configurations. More explicitly, for $X(3,7)$ and $X(4,7)$ we find $180$ and $120$ singular solutions which when added to the known number of regular solutions both give rise to $1\, 272$ solutions as it is expected since $X(3,7)\sim X(4,7)$. Likewise, for $X(3,8)$ and $X(5,8)$ we find $59\, 640$ and $58\, 800$ singular solutions which when added to the regular solutions both give rise to $188\, 112$ solutions. We also propose a classification of all configurations that can support singular solutions for general $X(k,n)$ and comment on their contribution to soft expansions of generalized biadjoint amplitudes.

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arXiv2020

Notes on Biadjoint Amplitudes, ${\rm Trop}\,G(3,7)$ and $X(3,7)$ Scattering Equations

In these notes we use the recently found relation between facets of tropical Grassmannians and generalizations of Feynman diagrams to compute all "biadjoint amplitudes" for $n=7$ and $k=3$. We also study scattering equations on $X(3,7)$, the configuration space of seven points on $\mathbb{CP}^2$. We prove that the number of solutions is $1272$ in a two-step process. In the first step we obtain $1162$ explicit solutions to high precision using near-soft kinematics. In the second step we compute the matrix of $360\times 360$ biadjoint amplitudes obtained by using the facets of ${\rm Trop}\, G(3,7)$, subtract the result from using the $1162$ solutions and compute the rank of the resulting matrix. The rank turns out to be $110$, which proves that the number of solutions in addition to the $1162$ explicit ones is exactly $110$.

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arXiv2019

Compatible Cycles and CHY Integrals

The CHY construction naturally associates a vector in $\mathbb{R}^{(n-3)!}$ to every 2-regular graph with $n$ vertices. Partial amplitudes in the biadjoint scalar theory are given by the inner product of vectors associated with a pair of cycles. In this work we study the problem of extending the computation to pairs of arbitrary 2-regular graphs. This requires the construction of compatible cycles, i.e. cycles such that their union with a 2-regular graph admits a Hamiltonian decomposition. We prove that there are at least $(n-2)!/4$ such cycles for any 2-regular graph. We also find a connection to breakpoint graphs when the graph only has double edges. We end with a comparison of the lower bound on the number of randomly selected cycles needed to generate a basis of $\mathbb{R}^{(n-3)!}$, using the super Catalan numbers, and our lower bound for compatible cycles.

math-ph↗

arXiv2019

Generalized Planar Feynman Diagrams: Collections

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 collections 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.

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arXiv2019

Scattering Equations: From Projective Spaces to Tropical Grassmannians

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ${\mathbb{CP}^1}$, to higher-dimensional projective spaces $\mathbb{CP}^{k-1}$. The standard, $k=2$ Mandelstam invariants, $s_{ab}$, are generalized to completely symmetric tensors $\textsf{s}_{a_1a_2\ldots a_k}$ subject to a `massless' condition $\textsf{s}_{a_1a_2\cdots a_{k-2}\,b\,b}=0$ and to `momentum conservation'. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the $k=3$ case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all `biadjoint amplitudes' for $(k,n)=(3,6)$ and find a direct connection to the tropical Grassmannian. This leads to the notion of $k=3$ Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for $k=2$, and provides analytic solutions analogous to the MHV ones.

hep-th↗

arXiv2018

$Δ$-Algebra and Scattering Amplitudes

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $Δ$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $Δ$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$.

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arXiv2018

The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps

We present new formulas for $n$-particle tree-level scattering amplitudes of six-dimensional $\mathcal{N}=(1,1)$ super Yang-Mills (SYM) and $\mathcal{N}=(2,2)$ supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the $(n-3)!$ solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-$n$ and odd-$n$ formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-$n$ amplitudes of $\mathcal{N}=(1,1)$ SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-$n$ amplitudes, we deduce the form of the rational maps and the integrand for $n$ odd. The odd-$n$ formulas obtained in this way have a new redundancy that is intertwined with the usual $\text{SL}(2, \mathbb{C})$ invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional $\mathcal{N}=(2,2)$ SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional $\mathcal{N}=4$ SYM on the Coulomb branch.

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