arXiv subjects

Nima Arkani-Hamed

Publications and source records attributed to Nima Arkani-Hamed.

At least 55 records · Page 3Linked to original sources

arXiv2020

Binary Geometries, Generalized Particles and Strings, and Cluster Algebras

We introduce the notion of "binary" positive and complex geometries, giving a completely rigid geometric realization of the combinatorics of generalized associahedra attached to any Dynkin diagram. We also define open and closed "cluster string integrals" associated with these "cluster configuration spaces". The binary geometry of type ${\cal A}$ gives a gauge-invariant description of the usual open and closed string moduli spaces for tree scattering, making no explicit reference to a worldsheet. The binary geometries and cluster string integrals for other Dynkin types provide a generalization of particle and string scattering amplitudes. Both the binary geometries and cluster string integrals enjoy remarkable factorization properties at finite $α'$, obtained simply by removing nodes of the Dynkin diagram. As $α'\to 0$ these cluster string integrals reduce to the canonical forms of the ABHY generalized associahedron polytopes. For classical Dynkin types these are associated with $n$-particle scattering in the bi-adjoint $ϕ^3$ theory through one-loop order.

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arXiv2020

Kerr Black Holes as Elementary Particles

Long ago, Newman and Janis showed that a complex deformation $z\rightarrow z+i a$ of the Schwarzschild solution produces the Kerr solution. The underlying explanation for this relationship has remained obscure. The complex deformation has an electromagnetic counterpart: by shifting the Coloumb potential, we obtain the EM field of a certain rotating charge distribution which we term $\sqrt{\rm Kerr}$. In this note, we identify the origin of this shift as arising from the exponentiation of spin operators for the recently defined "minimally coupled" three-particle amplitudes of spinning particles coupled to gravity, in the large-spin limit. We demonstrate this by studying the impulse imparted to a test particle in the background of the heavy spinning particle. We first consider the electromagnetic case, where the impulse due to $\sqrt{\rm Kerr}$ is reproduced by a charged spinning particle; the shift of the Coloumb potential is matched to the exponentiated spin-factor appearing in the amplitude. The known impulse due to the Kerr black hole is then trivially derived from the gravitationally coupled spinning particle via the double copy.

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arXiv2019

Non-perturbative geometries for planar $\mathcal{N}=4$ SYM amplitudes

There is a remarkable well-known connection between the G$(4,n)$ cluster algebra and $n$-particle amplitudes in $\mathcal{N}=4$ SYM theory. For $n \ge 8$ two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of "stringy canonical forms" to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Conf${}_n(\mathbb{P}^{k-1}) \cong {\rm G}(k,n)/T$ that are manifestly finite for all $k$ and $n$. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig's canonical basis. For $(k,n) = (4,8)$ the latter include precisely the expected square roots, revealing them to be related to certain "overpositive" functions of the kinematical invariants.

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arXiv2019

The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities

Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and particle content. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlators are controlled by an approximate conformal symmetry on the boundary of the spacetime. After classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities completely fixes this correlator. A spin-raising operator relates it to the correlators associated with the exchange of particles with spin, while weight-shifting operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. We reproduce many classic results in the literature and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. The inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta.

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arXiv2019

On the Positive Geometry of Conformal Field Theory

It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2,R) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope U determined by the operator spectrum, while crossing demands they lie on a plane X. The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of U and X. We find that the conformal blocks enjoy a surprising positive determinant property. This implies that U is an example of a famous polytope -- the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection U and X by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.

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arXiv2018

On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes

The concepts of Lorentz invariance of local (flat space) physics, and unitarity of time evolution and the S-matrix, are famously rigid and robust, admitting no obvious consistent theoretical deformations, and confirmed to incredible accuracy by experiments. But neither of these notions seem to appear directly in describing the spatial correlation functions at future infinity characterizing the "boundary" observables in cosmology. How then can we see them emerge as {\it exact} concepts from a possible ab-initio theory for the late-time wavefunction of the universe? In this letter we examine this question in a simple but concrete setting, for the perturbative wavefunction in a class of scalar field models where an ab-initio description of the wavefunction has been given by "cosmological polytopes". Singularities of the wavefunction are associated with facets of the polytope. One of the singularities -- corresponding to the "total energy pole" -- is well known to be associated with the flat-space scattering amplitude. We show how the combinatorics and geometry of this {\it scattering facet} of the cosmological polytope straightforwardly leads to the emergence of Lorentz invariance and unitarity for the S-matrix. Unitarity follows from the way boundaries of the scattering facet factorize into products of lower-dimensional polytopes, while Lorentz invariance follows from a contour integral representation of the canonical form, which exists for any polytope, specialized to cosmological polytopes.

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arXiv2018

Cutting Deep Into The Amplituhedron

In this letter we compute a canonical set of cuts of the integrand for MHV amplitudes in planar ${\cal N}=4$ SYM, where all internal propagators are put on-shell. These "deepest cuts" probe the most complicated Feynman diagrams and on-shell processes that can possibly contribute to the amplitude, but are also naturally associated with remarkably simple geometric facets of the amplituhedron. The recent reformulation of the amplituhedron in terms of combinatorial geometry directly in the kinematic (momentum-twistor) space plays a crucial role in understanding this geometry and determining the cut. This provides us with the first non-trivial results on scattering amplitudes in the theory valid for arbitrarily many loops and external particle multiplicities.

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arXiv2018

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.

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arXiv2017

One-Loop Integrals from Spherical Projections of Planes and Quadrics

We initiate a systematic study of one-loop integrals by investigating the connection between their singularity structures and geometric configurations in the projective space associated to their Feynman parametrization. We analyze these integrals by two recursive methods, which leads to two independent algebraic algorithms that determine the symbols of any one-loop integrals in arbitrary spacetime dimensions. The discontinuities of Feynman diagrams are shown to arise from taking certain "spherical contour" residues in Feynman parameter space, which is geometrically interpreted as a projection of the quadric surface (associated to the Symanzik polynomial at one loop) through faces of the integration region (which is a simplex). This geometry also leads to a manifestly Lorentz-invariant understanding for perturbative unitarity at one loop.

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arXiv2017

Cosmological Polytopes and the Wavefunction of the Universe

We present a connection between the physics of cosmological time evolution and the mathematics of positive geometries, roughly analogous to similar connections seen in the context of scattering amplitudes. We consider the wavefunction of the universe in a class of toy models of conformally coupled scalars (with non-conformal interactions) in FRW cosmologies. The contribution of each Feynman diagram to the wavefunction of the universe is associated with a certain universal rational integrand, which we identify as the canonical form of a "cosmological polytope", which have an independent, intrinsic definition, making no reference to physics. The singularity structure of the wavefunction for this model of scalars is common to all theories, and is geometrized by the cosmological polytope. Natural triangulations of the polytope reproduce the path-integral and "old-fashioned perturbation theory" representations of the wavefunction, and we also find new representations of the wavefunction with no extant physical interpretation. We show in suitable examples how symmetries of the cosmological polytope descend to symmetries of the wavefunction, (such as conformal invariance). In cases such as $ϕ^3$ theory in $dS_4$, the final wavefunction obtained from integration of the rational functions gives rise to polylogarithms associated with every graph. We give an explicit expression for the symbol of these polylogs, which record the geometry of sequential projections of the cosmological polytope.

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arXiv2017

Positive Geometries and Canonical Forms

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

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arXiv2017

Nnaturalness

We present a new solution to the electroweak hierarchy problem. We introduce $N$ copies of the Standard Model with varying values of the Higgs mass parameter. This generically yields a sector whose weak scale is parametrically removed from the cutoff by a factor of $1/\sqrt{N}$. Ensuring that reheating deposits a majority of the total energy density into this lightest sector requires a modification of the standard cosmological history, providing a powerful probe of the mechanism. Current and near-future experiments will explore much of the natural parameter space. Furthermore, supersymmetric completions which preserve grand unification predict superpartners with mass below $m_W \times M_{\text{pl}} / M_{\text{GUT}} \sim 10$ TeV.

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arXiv2017

Unwinding the Amplituhedron in Binary

We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized 'winding number'. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of 'sign flips' of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural 'dual' of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar N = 4 SYM as a differential form on the space of physical kinematical data.

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arXiv2016

Locality and Unitarity from Singularities and Gauge Invariance

We conjecture that the leading two-derivative tree-level amplitudes for gluons and gravitons can be derived from gauge invariance together with mild assumptions on their singularity structure. Assuming locality (that the singularities are associated with the poles of cubic graphs), we prove that gauge-invariance in just (n-1) particles together with minimal power-counting uniquely fixes the amplitude. Unitarity in the form of factorization then follows from locality and gauge invariance. We also give evidence for a stronger conjecture: assuming only that singularities occur when the sum of a subset of external momenta go on-shell, we show in non-trivial examples that gauge-invariance and power-counting demand a graph structure for singularities. Thus both locality and unitarity emerge from singularities and gauge invariance. Similar statements hold for theories of Goldstone bosons like the non-linear sigma model and Dirac-Born-Infeld, by replacing the condition of gauge invariance with an appropriate degree of vanishing in soft limits.

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arXiv2016

Unification and New Particles at the LHC

Precision gauge coupling unification is one of the primary quantitative successes of low energy or split supersymmetry. Preserving this success puts severe restrictions on possible matter and gauge sectors that might appear at collider-accessible energies. In this work we enumerate new gauge sectors which are compatible with unification, consisting of horizontal gauge groups acting on vector-like matter charged under the Standard Model. Interestingly, almost all of these theories are in the supersymmetric conformal window at high energies and confine quickly after the superpartners are decoupled. For a range of scalar masses compatible with both moderately tuned and minimally split supersymmetry, the confining dynamics happen at the multi-TeV scale, leading to a spectrum of multiple spin-0 and spin-1 resonances accessible to the LHC, with unusual quantum numbers and striking decay patterns.

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arXiv2015

Physics Opportunities of a 100 TeV Proton-Proton Collider

The discovery of the Higgs boson at the LHC exposes some of the most profound mysteries fundamental physics has encountered in decades, opening the door to the next phase of experimental exploration. More than ever, this will necessitate new machines to push us deeper into the energy frontier. In this article, we discuss the physics motivation and present the physics potential of a proton-proton collider running at an energy significantly beyond that of the LHC and a luminosity comparable to that of the LHC. 100 TeV is used as a benchmark of the center of mass energy, with integrated luminosities from 3 inverse ab to 30 inverse ab.

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arXiv2015

Cosmological Collider Physics

We study the imprint of new particles on the primordial cosmological fluctuations. New particles with masses comparable to the Hubble scale produce a distinctive signature on the non-gaussianities. This feature arises in the squeezed limit of the correlation functions of primordial fluctuations. It consists of particular power law, or oscillatory, behavior that contains information about the masses of new particles. There is an angular dependence that gives information about the spin. We also have a relative phase that crucially depends on the quantum mechanical nature of the fluctuations and can be viewed as arising from the interference between two processes. While some of these features were noted before in the context of specific inflationary scenarios, here we give a general description emphasizing the role of symmetries in determining the final result.

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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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