Past Amplitude Workshop

We study the celestial description of the O(N) sigma model in the large N limit. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. This is a joint work with D. García-Sepúlveda, A. Guevara, J. Kulp.

The scattering equations connect two modern descriptions of scattering amplitudes: the CHY formalism and the framework of positive geometries. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry in the CHY moduli space. In this talk, I consider the general problem of pushing forward rational differential forms via the scattering equations. I will present some recent results (2206.14196) for achieving this without ever needing to explicitly solve any scattering equations. These results use techniques from computational algebraic geometry, and they extend the application of similar results for rational functions to rational differential forms.

Detectors in collider experiments are modeled by light-ray operators in Quantum Field Theory. For example, energy detectors are certain null integrals of the stress-energy tensor, localized at an angle on the celestial sphere, where they collect quanta that escape in their direction. In this talk, I will discuss a series of work developing a nonperturbative, convergent operator product expansion (OPE) for light-ray operators in Conformal Field Theories (CFTs). Objects appearing in the expansion are more general light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. An important application is to event shapes in collider physics, which correspond to correlation functions of light-ray operators within the state created by the incoming particles. I will discuss some applications of the light-ray OPE in CFT, and mention some extensions to QCD which make contact with measurements at the LHC. Talk based primarily on [1905.01311] and [2010.04726].

The on-shell superspace formulation of N=4 SYM allows the writing of all possible scattering processes in one compact object called the super-amplitude.   Famously, the super-amplitude integrand can be extracted from generalized polyhedra called the amplituhedron. In this talk, I will review this construction and present a natural generalization of the amplituhedron that we proved at tree level and conjectured at loop level to correspond to the product of two parity conjugate superamplitudes. The sum of all parity conjugate amplitudes corresponds to a particular limit of the supercorrelator through the Wilson Loop/Amplitude duality.  I will conclude by discussing this connection from a geometrical point of view. This talk is based on the reference arXiv:2106.09372 .

The AdS/CFT correspondence provides a rich setup to study the properties of gauge theories and the dual theories of gravity, in particular their thermodynamic properties. On RxS3, the maximally supersymmetric Yang-Mills theory with gauge group U(N) exhibits a phase transition that resembles the confinement-deconfinement transition of QCD. For infinite N, this transition is characterized by Hagedorn behavior. We show how the corresponding Hagedorn temperature can be calculated at any value of the ’t Hooft coupling via integrability. For large but finite N, we show how the Hagedorn behavior is replaced by Lee-Yang behavior.

This will be a zoom seminar with communal viewing in L4

I will present work in progress with Erik Panzer, Matteo Parisi and Ömer Gürdoğan on the analytic structure of Feynman(esque) integrals: We consider integrals of meromorphic differential forms over relative cycles in a compact complex manifold, the underlying geometry encoded in a certain (parameter dependant) subspace arrangement (e.g. Feynman integrals in their parametric representation). I will explain how the analytic struture of such integrals can be studied via methods from differential topology; this is the seminal work by Pham et al (using tools and methods developed by Leray, Thom, Picard-Lefschetz etc.). Although their work covers a very general setup, the case we need for Feynman integrals has never been worked out in full detail. I will comment on the gaps that have to be filled to make the theory work, then discuss how much information about the analytic structure of integrals can be derived from a careful study of the corresponding subspace arrangement.

Event shape observables describe how energy is distributed in the final state in scattering processes. Recent years have seen increasing interest from different physics areas in event shapes, in particular the energy correlators. They define a class of observable quantities which admit a simple and unified formulation in quantum  field theory.

Three-point energy correlators (EEEC) measure the energy flow through three detectors as a function of the three angles between them. We analytically compute the one-loop EEEC in maximally supersymmetric Yang-Mills theory. The result is a linear combination of logarithms and dilogarithms, decomposed onto a basis of single-valued transcendental functions. Its symbol contains 16 alphabet letters, revealing a dihedral symmetry of the three-point event shape.  Our results represent the first perturbative computation of a three-parameter event-shape observable, providing information on the function space at higher-loop order, and valuable input to the study of conformal light-ray OPE.

In this talk I will discuss a striking duality, T-duality, we discovered between two seemingly unrelated objects: the hypersimplex and the m=2 amplituhedron. We draw novel connections between them and prove many new properties. We exploit T-duality to relate their triangulations and generalised triangles (maximal cells in a triangulation). We subdivide the amplituhedron into chambers as the hypersimplex can be subdivided into simplices - both enumerated by Eulerian numbers. Along the way, we prove several conjectures on the amplituhedron and find novel cluster-algebraic structures, e.g. a generalisation of cluster adjacency.

This is based on the joint work with Lauren Williams and Melissa Sherman-Bennett https://arxiv.org/abs/2104.08254.

The 4D 4-point scattering amplitude of massless scalars via a massive exchange can be expressed in a basis of conformal primary particle wavefunctions. In this talk I will show that the resulting celestial amplitude admits a decomposition as a sum over 2D conformal blocks. This decomposition is obtained by contour deformation upon expanding the celestial amplitude in a basis of conformal partial waves. The conformal blocks include intermediate exchanges of spinning light-ray states, as well as scalar states with positive integer conformal weights. The conformal block prefactors are found as expected to be quadratic in the celestial OPE coefficients. Finally, I will comment on implications of this result for celestial holography and discuss some open questions.

In this talk, I will discuss AdS super gluon scattering amplitudes in various spacetime dimensions. These amplitudes are dual to correlation functions in a variety of non-maximally supersymmetric CFTs, such as the 6d E-string theory, 5d Seiberg exceptional theories, etc. I will introduce a powerful method based on symmetries and consistency conditions, and show that it fixes all the infinitely many four-point amplitudes at tree level. I will also point out many interesting properties and structures of these amplitudes, which include the flat space limit, Parisi-Sourlas-like dimensional reduction, hidden conformal symmetry, and a color-kinematic duality in AdS. Along the way, I will also review some earlier progress and the relation with this work. I will conclude with a brief discussion of various open problems. 

In this talk I review the recent discovery of Yangian symmetry for massive Feynman integrals and how it can be used to set up a Yangian Bootstrap. I will provide elementary proofs of the symmetry at one and two loops, whereas at generic loop order I conjecture that all graphs cut from regular tilings of the plane with massive propagators on the boundary enjoy the symmetry. After demonstrating how the symmetry may be used to constrain the functional form of Feynman integrals on explicit examples, I comment on how a subset of the diagrams for which the symmetry is conjectured to hold is naturally embedded in a Massive Fishnet theory that descends from gamma-deformed Coulomb branch N=4 Super-Yang-Mills theory in a particular double scaling limit.

Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this talk we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C.Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-point amplitude from four plabic graphs. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).

In this talk I will review the work that has been done by me, N. Gromov, V. Kazakov, G. Korchemsky and G. Sizov on the analysis of fishnet Feynman graphs in a particular scaling limit of $\mathcal N=4$ SYM, a theory dubbed $\chi$FT$_4$. After introducing said theory, in which the Feynman graphs take a very simple fishnet form — in the planar limit — I will review how to exploit integrable techniques to compute these graphs and, consequently, extract the anomalous dimensions of a simple class of operators.

I will review recent works on geometries underlying scattering amplitudes of (certain generalizations of) particles and strings  Tree amplitudes of a cubic scalar theory are given by "canonical forms" of the so-called ABHY associahedra defined in kinematic space. The latter can be naturally extended to generalized associahedra for finite-type cluster algebra, and for classical types their canonical forms give scalar amplitudes through one-loop order. We then consider vast generalizations of string amplitudes dubbed “stringy canonical forms”, and in particular "cluster string integrals" for any Dynkin diagram, which for type A reduces to usual string amplitudes. These integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram; as $\alpha'\rightarrow 0$ they reduce to canonical forms of generalized associahedra, or the aforementioned tree and one-loop scalar amplitudes.

In this talk I will discuss relations between the low-energy expansions  of open- and closed string amplitudes. At genus zero, it has been shown that the single-valued map of MZVs maps open-string amplitudes to their closed-string counterparts. After reviewing this story, I will discuss recent work at genus one which aims to define a similar mapping from the open to the closed string. Our construction is driven by the differential equations and degeneration limits of certain generating functions of string integrals and suggests a pairing of integration cycles and forms at genus one - analogous to the duality between Parke-Taylor factors and disk boundaries at genus zero. Finally, I will discuss the impact of said mapping on the elliptic MZVs and modular graph forms which arise naturally upon solving these differential equations.