arXiv subjects

Nima Arkani-Hamed

Publications and source records attributed to Nima Arkani-Hamed.

At least 37 records · Page 2Linked to original sources

arXiv2024

Tropical Amplitudes For Colored Lagrangians

Recently a new formulation for scattering amplitudes in Tr($Φ^3$) theory has been given based on simple combinatorial ideas in the space of kinematic data. This allows all-loop integrated amplitudes to be expressed as ''curve integrals'' defined using tropical building blocks - the ''headlight functions''. This paper shows how the formulation extends to the amplitudes of more general Lagrangians. We will present a number of different ways of introducing tropical ''numerator functions'' that allow us to describe general Lagrangian interactions. The simplest family of these ''tropical numerators'' computes the amplitudes of interesting Lagrangians with infinitely many interactions. We also describe methods for tropically formulating the amplitudes for general Lagrangians. One uses a variant of ''Wick contraction'' to glue together numerator factors for general interaction vertices. Another uses a natural characterization of polygons on surfaces to give a novel combinatorial description of all possible diagrams associated with arbitrary valence interactions.

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arXiv2023

Kinematic Flow and the Emergence of Time

Perhaps the most basic question we can ask about cosmological correlations is how their strength changes as we smoothly vary kinematic parameters. The answer is encoded in differential equations that govern this evolution in kinematic space. In this Letter, we introduce a new perspective on these differential equations. We show that, in the simplified setting of conformally coupled scalars in a general FRW spacetime, the equations for arbitrary tree-level processes can be obtained from a small number of simple combinatorial rules. While this "kinematic flow" is defined purely in terms of boundary data, it reflects the physics of bulk time evolution. The unexpected regularity of the equations suggests the existence of an autonomously defined mathematical structure from which cosmological correlations, and the time evolution of the associated spacetime, emerge.

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arXiv2023

Coulomb Branch Amplitudes from a Deformed Amplituhedron Geometry

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.

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arXiv2022

Positive configuration space

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang-Mills scattering amplitudes, which will be discussed in a sequel.

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arXiv2022

Causal Diamonds, Cluster Polytopes and Scattering Amplitudes

The "amplituhedron" for tree-level scattering amplitudes in the bi-adjoint $ϕ^3$ theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1+1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic "spacetime" with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain "walk", associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The ${\cal A}_{n{-}3},{\cal B}_{n{-}1}/{\cal C}_{n{-}1}$ and ${\cal D}_n$ polytopes are the amplituhedra for $n$-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope $\bar{\cal D}_n$, which chops the ${\cal D}_n$ polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.

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arXiv2022

Solving Scattering in N=4 Super-Yang-Mills Theory

As part of the Snowmass community planning exercise, we highlight an ongoing program of research into the structure of scattering amplitudes in N=4 super-Yang-Mills theory, particularly in the planar limit of a large number of colors. This theory sits at the nexus of a number of exciting topics in high-energy particle physics, including the AdS/CFT correspondence, conformal field theory, integrability, and string theory, and is believed to be exactly solvable in four dimensions. In many ways, planar N=4 super-Yang-Mills theory is the "hydrogen atom" of relativistic scattering: It has proven indispensable for learning about new geometrical formulations of quantum field theory, for exploring mathematical properties at high perturbative orders, and for developing powerful new computational methods that have found applicability in precision collider physics.

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arXiv2022

Feynman Polytopes and the Tropical Geometry of UV and IR Divergences

We introduce a class of polytopes that concisely capture the structure of UV and IR divergences of general Feynman integrals in Schwinger parameter space, treating them in a unified way as worldline segments shrinking and expanding at different relative rates. While these polytopes conventionally arise as convex hulls - via Newton polytopes of Symanzik polynomials - we show that they also have a remarkably simple dual description as cut out by linear inequalities defining the facets. It is this dual definition that makes it possible to transparently understand and efficiently compute leading UV and IR divergences for any Feynman integral. In the case of the UV, this provides a transparent geometric understanding of the familiar nested and overlapping divergences. In the IR, the polytope exposes a new perspective on soft/collinear singularities and their intricate generalizations. Tropical geometry furnishes a simple framework for calculating the leading UV/IR divergences of any Feynman integral, associating them with the volumes of certain dual cones. As concrete applications, we generalize Weinberg's theorem to include a characterization of IR divergences, and classify space-time dimensions in which general IR divergences (logarithmic as well as power-law) can occur. We also compute the leading IR divergence of rectangular fishnet diagrams at all loop orders, which turn out to have a surprisingly simple combinatorial description.

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arXiv2022

On Unitarity of Tree-Level String Amplitudes

Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded in partial waves with all positive coefficients. While this follows (rather indirectly) from the no-ghost theorem, the simplicity of the statement and its fundamental importance for the physical consistency of string theory begs for a more direct and elementary understanding. In this note we take a step in this direction by presenting a new expression for the partial wave coefficients of string amplitudes, given by surprisingly simple double/triple contour integrals for open/closed strings. This representation allows us to directly prove unitarity of all superstring theories in $D \leq 6$ spacetime dimensions, and can also be used to determine various asymptotics of the partial waves at large mass levels.

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arXiv2021

Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron

The amplituhedron determines scattering amplitudes in planar ${\cal N}=4$ super Yang-Mills by a single "positive geometry" in the space of kinematic and loop variables. We study a closely related definition of the amplituhedron for the simplest case of four-particle scattering, given as a sum over complementary "negative geometries", which provides a natural geometric understanding of the exponentiation of infrared (IR) divergences, as well as a new geometric definition of an IR finite observable ${\cal F}(g,z)$ - dually interpreted as the expectation value of the null polygonal Wilson loop with a single Lagrangian insertion - which is directly determined by these negative geometries. This provides a long-sought direct link between canonical forms for positive (negative) geometries, and a completely IR finite post-loop-integration observable depending on a single kinematical variable $z$, from which the cusp anomalous dimension $Γ_{\rm cusp}(g)$ can also be straightforwardly obtained. We study an especially simple class of negative geometries at all loop orders, associated with a "tree" structure in the negativity conditions, for which the contributions to ${\cal F}(g,z)$ and $Γ_{\rm cusp}$ can easily be determined by an interesting non-linear differential equation immediately following from the combinatorics of negative geometries. This lets us compute these "tree" contributions to ${\cal F}(g,z)$ and $Γ_{\rm cusp}$ for all values of the 't Hooft coupling. The result for $Γ_{\rm cusp}$ remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.

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arXiv2021

Causality, Unitarity, and the Weak Gravity Conjecture

We consider the shift of charge-to-mass ratio for extremal black holes in the context of effective field theory, motivated by the Weak Gravity Conjecture. We constrain extremality corrections in different regimes subject to unitarity and causality constraints. In the asymptotic IR, we demonstrate that for any supersymmetric theory in flat space, and for all minimally coupled theories, logarithmic running at one loop pushes the Wilson coefficient of certain four-derivative operators to be larger at lower energies, guaranteeing the existence of sufficiently large black holes with $Q>M$. We identify two exceptional cases of nonsupersymmetric theories involving large numbers of light states and Planck-scale nonminimal couplings, in which the sign of the running is reversed, leading to black holes with negative corrections to $Q/M$ in the deep IR, but argue that these do not rule out extremal black holes as the requisite charged states for the WGC. We separately show that causality and unitarity imply that the leading threshold corrections to the effective action from integrating out massive states, in any weakly coupled theory, can be written as a sum of squares and is manifestly positive for black hole backgrounds. Quite beautifully, the shift in the extremal $Q/M$ ratio is directly proportional to the shift in the on-shell action, guaranteeing that these threshold corrections push $Q>M$ in compliance with the WGC. Our results apply for black holes with or without dilatonic coupling and charged under any number of ${\rm U}(1)$s.

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arXiv2021

Scattering Amplitudes For All Masses and Spins

We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called "spinning polynomials". Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be "elementary". The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can't be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. "Off-shell" observables like correlation functions and form-factors can be thought of as scattering amplitudes with external "probe" particles of general mass and spin, so all these objects--amplitudes, form factors and correlators, can be studied from a common on-shell perspective.

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arXiv2021

Cluster Configuration Spaces of Finite Type

For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and a partial compactification ${\widetilde {\mathcal{M}}}_D$. For $D = A_{n-3}$, we have ${\mathcal{M}}_{A_{n-3}} = {\mathcal{M}}_{0,n}$, the configuration space of $n$ points on ${\mathbb P}^1$, and the partial compactification ${\widetilde {\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The space ${\widetilde {\mathcal{M}}}_D$ is a smooth affine algebraic variety with a stratification in bijection with the faces of the Chapoton-Fomin-Zelevinsky generalized associahedron. The regular functions on ${\widetilde {\mathcal{M}}}_D$ are generated by coordinates $u_γ$, in bijection with the cluster variables of type $D$, and the relations are described completely in terms of the compatibility degree function of the cluster algebra. As an application, we define and study cluster algebra analogues of tree-level open string amplitudes.

math.AG↗

arXiv2021

Naturalness and the muon magnetic moment

We study a predictive model for explaining the apparent deviation of the muon anomalous magnetic moment from the Standard Model expectation. There are no new scalars and hence no new hierarchy puzzles beyond those associated with the Higgs; the only new particles at the TeV scale are vector-like singlet and doublet leptons. Interestingly, this simple model provides a calculable example violating the Wilsonian notion of naturalness: despite the absence of any symmetries prohibiting its generation, the coefficient of the naively leading dimension-six operator for $(g-2)$ vanishes at one-loop. While effective field theorists interpret this either as a surprising UV cancellation of power divergences, or as a delicate cancellation between matching UV and calculable IR corrections to $(g-2)$ from parametrically separated scales, there is a simple explanation in the full theory: the loop integrand is a total derivative of a function vanishing in both the deep UV and IR. The leading contribution to $(g-2)$ arises from dimension-eight operators, and thus the required masses of new fermions are lower than naively expected, with a sizeable portion of parameter space already covered by direct searches at the LHC. The viable parameter space free of fine-tuning for the muon mass will be fully covered by future direct LHC searches, and {\it all} of the parameter space can be probed by precision measurements at planned future lepton colliders.

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arXiv2021

The EFT-Hedron

We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden "totally positive" structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the "EFThedron". We initiate a systematic investigation of the boundary structure of the EFThedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFThedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

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arXiv2021

The Muon Smasher's Guide

We lay out a comprehensive physics case for a future high-energy muon collider, exploring a range of collision energies (from 1 to 100 TeV) and luminosities. We highlight the advantages of such a collider over proposed alternatives. We show how one can leverage both the point-like nature of the muons themselves as well as the cloud of electroweak radiation that surrounds the beam to blur the dichotomy between energy and precision in the search for new physics. The physics case is buttressed by a range of studies with applications to electroweak symmetry breaking, dark matter, and the naturalness of the weak scale. Furthermore, we make sharp connections with complementary experiments that are probing new physics effects using electric dipole moments, flavor violation, and gravitational waves. An extensive appendix provides cross section predictions as a function of the center-of-mass energy for many canonical simplified models.

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arXiv2021

The Weak Scale as a Trigger

Does the value of the Higgs mass parameter affect the expectation value of local operators in the Standard Model? For essentially all local operators the answer to this question is "no", and this is one of the avatars of the hierarchy problem: nothing is "triggered" when the Higgs mass parameter crosses zero. In this letter, we explore settings in which Higgs mass parameters $can$ act as a "trigger" for some local operators ${\cal O}_T$. In the Standard Model, this happens for ${\cal O}_T = {\rm Tr} (G \tilde G)$. We also introduce a "type-0" two Higgs doublet model, with a $Z_4$ symmetry, for which ${\cal O}_T = H_1 H_2$ is triggered by the Higgs masses, demanding the existence of new Higgs states necessarily comparable to or lighter than the weak scale, with no wiggle room to decouple them whatsoever. Surprisingly, this model is not yet entirely excluded by collider searches, and will be incisively probed by the high-luminosity run of the LHC, as well as future Higgs factories. We also discuss a possibility for using this trigger to explain the origin of the weak scale, invoking a landscape of extremely light, weakly interacting scalars $ϕ_i$, with a coupling to ${\cal O}_T$ needed to make it possible to find vacua with small enough cosmological constant. The weak scale trigger links the tuning of the Higgs mass to that of the cosmological constant, while coherent oscillations of the $ϕ_i$ can constitute dark matter.

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arXiv2021

Stringy Canonical Forms

Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce "stringy canonical forms", which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter $α'$. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As $α' \to 0$, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite $α'$, they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the $α' \to 0$ limit of tree-level string amplitudes, and scattering equations that appear when studying the $α' \to \infty$ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A for string amplitudes), and other natural integrals over the positive Grassmannian.

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arXiv2020

Celestial Amplitudes from UV to IR

Celestial amplitudes represent 4D scattering of particles in boost, rather than the usual energy-momentum, eigenstates and hence are sensitive to both UV and IR physics. We show that known UV and IR properties of quantum gravity translate into powerful constraints on the analytic structure of celestial amplitudes. For example the soft UV behavior of quantum gravity is shown to imply that the exact four-particle scattering amplitude is meromorphic in the complex boost weight plane with poles confined to even integers on the negative real axis. Would-be poles on the positive real axis from UV asymptotics are shown to be erased by a flat space analog of the AdS resolution of the bulk point singularity. The residues of the poles on the negative axis are identified with operator coefficients in the IR effective action. Far along the real positive axis, the scattering is argued to grow exponentially according to the black hole area law. Exclusive amplitudes are shown to simply factorize into conformally hard and conformally soft factors. The soft factor contains all IR divergences and is given by a celestial current algebra correlator of Goldstone bosons from spontaneously broken asymptotic symmetries. The hard factor describes the scattering of hard particles together with the boost-eigenstate clouds of soft photons or gravitons required by asymptotic symmetries. These provide an IR safe $\mathcal{S}$-matrix for the scattering of hard particles.

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