arXiv subjects

Freddy Cachazo

Publications and source records attributed to Freddy Cachazo.

At least 37 records · Page 2Linked to original sources

arXiv2017

Combinatorial Factorization

The simplest integrands in the CHY formulation of scattering amplitudes are constructed using the so-called Parke-Taylor functions. Parke-Taylor functions also turn out to belong to a large class of rational functions known as MHV leading singularities. In fact, Parke-Taylor functions correspond to planar MHV leading singularities. In this note we study the behavior of CHY integrands constructed using non-planar MHV leading singularities under collinear and multi-particle factorization limits. General $n$-particle MHV leading singularities are completely characterized by a set of $(n-2)$ triples of particle labels. We give a simple operation on this combinatorial data which "factors" the list into two sets of triples defining two lower point MHV leading singularities. The fact that general MHV leading singularities form a closed set under "multi-particle factorizations" is surprising from their gauge theoretic origin.

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arXiv2017

Leading Singularities and Classical Gravitational Scattering

In this work we propose to use leading singularities to obtain the classical pieces of amplitudes of two massive particles whose only interaction is gravitational. Leading singularities are generalizations of unitarity cuts. At one-loop we find that leading singularities obtained by multiple discontinuities in the t-channel contain all the classical information. As the main example, we show how to obtain a compact formula for the fully relativistic classical one-loop contribution to the scattering of two particles with different masses. The non-relativistic limit of the leading singularity agrees with known results in the post-Newtonian expansion. We also compute a variety of higher loop leading singularities including some all-loop families and study some of their properties.

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arXiv2016

Scattering Equations: Real Solutions and Particles on a Line

We find $n(n-3)/2$-dimensional regions of the space of kinematic invariants, where all the solutions to the scattering equations (the core of the CHY formulation of amplitudes) for $n$ massless particles are real. On these regions, the scattering equations are equivalent to the problem of finding stationary points of $n-3$ mutually repelling particles on a finite real interval with appropriate boundary conditions. This identification directly implies that for each of the $(n-3)!$ possible orderings of the $n-3$ particles on the interval, there exists one stable stationary point. Furthermore, restricting to four dimensions, we find that the separation of the solutions into $k\in \{2,3,\ldots ,n-2\}$ sectors naturally matches that of permutations of $n-3$ labels into those with $k-2$ descents. This leads to a physical realization of the combinatorial meaning of the Eulerian numbers.

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arXiv2016

Extensions of Theories from Soft Limits

We study a variety of field theories with vanishing single soft limits. In all cases, the structure of the soft limit is controlled by a larger theory, which provides an extension of the original one by adding more fields and interactions. Our main example is the $U(N)$ non-linear sigma model in its CHY representation. Its extension is a theory in which the NLSM Goldstone bosons interact with a cubic biadjoint scalar. Other theories we study and extend are the special Galileon and Born-Infeld theory, including its maximally supersymmetric version in four dimensions, the DBI-Volkov-Akulov theory. In all the cases, we propose the CHY representation of the complete tree-level S-matrix of the extended theories. In fact, CHY formulas are the key technique for studying the single soft limit behavior of the original theories. As a byproduct, we show that the tree-level S-matrix of the extended NLSM theory can be constructed using a very compact BCFW-like recursion relation, where physical poles are at most linear in the deformation parameter.

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arXiv2016

Minimal Basis in Four Dimensions and Scalar Blocks

We find a construction that expresses any tree-level $n$-particle ${\rm N^{k-2}MHV}$ color-ordered partial amplitude in gauge theory as a linear combination of a basis of dimension $\eulerian{n-3}{k-2}$. Here $\eulerian{p}{q}$ denotes the $(p,q)$ Eulerian number. The coefficients of the expansion are independent of the helicities of the particles. This basis is a four-dimensional refinement of the $(n-3)!$-element BCJ basis which is valid in any number of dimensions. The construction uses a new kind of objects which we call {\it scalar blocks}. Here we initiate the study of these objects. Scalar blocks provide an "${\rm N^{k-2}MHV}$ sector" decomposition of a bi-adjoint scalar amplitude in four dimensions. As byproducts of the construction, we also find an intrinsically four-dimensional version of KLT relations for gravity amplitudes.

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arXiv2016

One-Loop Corrections from Higher Dimensional Tree Amplitudes

We show how one-loop corrections to scattering amplitudes of scalars and gauge bosons can be obtained from tree amplitudes in one higher dimension. Starting with a complete tree-level scattering amplitude of n+2 particles in five dimensions, one assumes that two of them cannot be "detected" and therefore an integration over their LIPS is carried out. The resulting object, function of the remaining n particles, is taken to be four-dimensional by restricting the corresponding momenta. We perform this procedure in the context of the tree-level CHY formulation of amplitudes. The scattering equations obtained in the procedure coincide with those derived by Geyer et al from ambitwistor constructions and recently studied by two of the authors for bi-adjoint scalars. They have two sectors of solutions: regular and singular. We prove that the contribution from regular solutions generically gives rise to unphysical poles. However, using a BCFW argument we prove that the unphysical contributions are always homogeneous functions of the loop momentum and can be discarded. We also show that the contribution from singular solutions turns out to be homogeneous as well.

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arXiv2015

Computation of Contour Integrals on ${\cal M}_{0,n}$

Contour integrals of rational functions over ${\cal M}_{0,n}$, the moduli space of $n$-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined by the critical points of a certain Morse function on ${\cal M}_{0,n}$. The integrand is a general rational function of the puncture locations with poles of arbitrary order as two punctures coincide. In this note we provide an algorithm for the analytic computation of any such integral. The algorithm uses three ingredients: an operation we call general KLT, Petersen's theorem applied to the existence of a 2-factor in any 4-regular graph and Hamiltonian decompositions of certain 4-regular graphs. The procedure is iterative and reduces the computation of a general integral to that of simple building blocks. These are integrals which compute double-color-ordered partial amplitudes in a bi-adjoint cubic scalar theory.

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arXiv2015

New Double Soft Emission Theorems

We study the behavior of the tree-level S-matrix of a variety of theories as two particles become soft. By analogy with the recently found subleading soft theorems for gravitons and gluons, we explore subleading terms in double soft emissions. We first consider double soft scalar emissions and find subleading terms that are controlled by the angular momentum operator acting on hard particles. The order of the subleading theorems depends on the presence or not of color structures. Next we obtain a compact formula for the leading term in a double soft photon emission. The theories studied are a special Galileon, DBI, Einstein-Maxwell-Scalar, NLSM and Yang-Mills-Scalar. We use the recently found CHY representation of these theories in order to give a simple proof of the leading order part of all these theorems

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arXiv2015

Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM

The tree-level S-matrix of Einstein's theory is known to have a representation as an integral over the moduli space of punctured spheres localized to the solutions of the scattering equations. In this paper we introduce three operations that can be applied on the integrand in order to produce other theories. Starting in $d+M$ dimensions we use dimensional reduction to construct Einstein-Maxwell with gauge group $U(1)^M$. The second operation turns gravitons into gluons and we call it "squeezing". This gives rise to a formula for all multi-trace mixed amplitudes in Einstein-Yang-Mills. Dimensionally reducing Yang-Mills we find the S-matrix of a special Yang-Mills-Scalar (YMS) theory, and by the squeezing operation we find that of a YMS theory with an additional cubic scalar vertex. A corollary of the YMS formula gives one for a single massless scalar with a $ϕ^4$ interaction. Starting again from Einstein's theory but in $d+d$ dimensions we introduce a "generalized dimensional reduction" that produces the Born-Infeld theory or a special Galileon theory in $d$ dimensions depending on how it is applied. An extension of Born-Infeld formula leads to one for the Dirac-Born-Infeld (DBI) theory. By applying the same operation to Yang-Mills we obtain the $U(N)$ non-linear sigma model (NLSM). Finally, we show how the Kawai-Lewellen-Tye relations naturally follow from our formulation and provide additional connections among these theories. One such relation constructs DBI from YMS and NLSM.

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arXiv2014

On-Shell Structures of MHV Amplitudes Beyond the Planar Limit

We initiate an exploration of on-shell functions in $\mathcal{N}=4$ SYM beyond the planar limit by providing compact, combinatorial expressions for all leading singularities of MHV amplitudes and showing that they can always be expressed as a positive sum of differently ordered Parke-Taylor tree amplitudes. This is understood in terms of an extended notion of positivity in $G(2,n)$, the Grassmannian of 2-planes in $n$ dimensions: a single on-shell diagram can be associated with many different "positive" regions, of which the familiar positive region associated with planar diagrams is just one example. The decomposition into Parke-Taylor factors is simply a "triangulation" of these extended positive regions. The $U(1)$ decoupling and Kleiss-Kuijf (KK) relations satisfied by the Parke-Taylor amplitudes also follow naturally from this geometric picture. These results suggest that non-planar MHV amplitudes in $\mathcal{N}=4$ SYM at all loop orders can be expressed as a sum of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes.

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arXiv2014

Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations

We present the building blocks that can be combined to produce tree-level S-matrix elements of a variety of theories with various spins mixed in arbitrary dimensions. The new formulas for the scattering of $n$ massless particles are given by integrals over the positions of $n$ points on a sphere restricted to satisfy the scattering equations. As applications, we obtain all single-trace amplitudes in Einstein--Yang--Mills (EYM) theory, and generalizations to include scalars. Also in EYM but extended by a B-field and a dilaton, we present all double-trace gluon amplitudes. The building blocks are made of Pfaffians and Parke--Taylor-like factors of subsets of particle labels.

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arXiv2014

On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes

We present evidence that loop amplitudes in maximally supersymmetric $\mathcal{N}=4$ Yang-Mills (SYM) beyond the planar limit share some of the remarkable structures of the planar theory. In particular, we show that through two loops, the four-particle amplitude in full $\mathcal{N}=4$ SYM has only logarithmic singularities and is free of any poles at infinity---properties closely related to uniform transcendentality and the UV-finiteness of the theory. We also briefly comment on implications for maximal ($\mathcal{N}=8$) supergravity.

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arXiv2014

Are Soft Theorems Renormalized?

We show that the distributional nature of soft theorems requires the soft limit expansion to take priority over the regulator expansion of Feynman loop integrals. We start the study of soft graviton theorems at loop level from this perspective by considering a five-particle one-loop amplitude in ${\cal N}=8$ supergravity. Surprisingly, we find that a soft theorem recently introduced by one of the authors and Strominger is not renormalized in this case. Computations are done in $4-2ε$ dimensions and for terms of order $ε^{-2}$, $ε^{-1}$ and $ε^{0}$.

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arXiv2014

Evidence for a New Soft Graviton Theorem

The single-soft-graviton limit of any quantum gravity scattering amplitude is given at leading order by the universal Weinberg pole formula. Gauge invariance of the formula follows from global energy-momentum conservation. In this paper evidence is given for a conjectured universal formula for the finite subleading term in the expansion about the soft limit, whose gauge invariance follows from global angular momentum conservation. The conjecture is non-trivially verified for all tree-level graviton scattering amplitudes using a BCFW recursion relation. One hopes to understand this infinity of new soft relations as a Ward identity for a new superrotation Virasoro symmetry of the quantum gravity S-matrix.

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arXiv2014

Gravity in Twistor Space and its Grassmannian Formulation

We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that physical amplitudes are known to satisfy, with the same initial conditions. As part of the proof, the behavior of the new formula under large BCFW deformations is studied. An unexpected bonus of the analysis is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity. In addition, we provide a description of gravity amplitudes as a multidimensional contour integral over a Grassmannian. The Grassmannian formulation has a very simple structure; in the N$^{k-2}$MHV sector the integrand is essentially the product of that of an MHV and an $\overline{\rm MHV}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.

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arXiv2014

Scattering Amplitudes and the Positive Grassmannian

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations; in particular, BCFW deformations correspond to adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with positive coordinates and an invariant measure which determines the on-shell function associated with the diagram. This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. Yangian invariance of scattering amplitudes is transparently represented by diffeomorphisms of G(k,n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop integrand for scattering amplitudes, presenting all integrands in a novel dLog form which directly reflects the underlying positive structure.

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arXiv2013

Scattering of Massless Particles: Scalars, Gluons and Gravitons

In a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U(N) color structures while the second is a Pfaffian. The S-matrix of a U(N)xU(N') cubic scalar theory is obtained by simply replacing the Pfaffian with a U(N') version of the previous U(N) factor. Given that gravity amplitudes are obtained by replacing the U(N) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. We find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials.

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arXiv2013

Scattering of Massless Particles in Arbitrary Dimension

We present a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. The new formula for the scattering of n particles is given by an integral over the position of n points on a sphere restricted to satisfy a dimension-independent set of equations. The integrand is constructed using the reduced Pfaffian of a 2n by 2n matrix that depends on momenta and polarization vectors. In its simplest form, the gravity integrand is a reduced determinant which is the square of the Pfaffian in the Yang-Mills integrand. Gauge invariance is completely manifest as it follows from a simple property of the Pfaffian.

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