Conformal Correlation Functions Research Articles

On correlation functions of higher-spin currents in arbitrary dimensions d > 3

We revisit the problem of classification and explicit construction of the conformal three-point correlation functions of currents of arbitrary integer spin in arbitrary dimensions. For the conserved currents, we set up the equations for the conservation conditions and solve them completely for some values of spins, confirming the earlier counting of the number of independent structures matching them with the higher-spin cubic vertices in one higher dimension. The general solution for the correlators of conserved currents we delegate to a follow-up work.

Symmetries of conformal correlation functions

A program of wide interest in modern conformal bootstrap studies is to numerically solve general conformal field theories, based on a critical assumption that the dynamics is encoded in the conformal four-point crossing equations and positivity condition. In this paper we propose and verify a novel algebraic property of the crossing equations which provides strong restriction for this program. We show for various types of symmetries $\mathcal{G}$, the crossing equations can be linearly converted into the $SO(N)$ vector crossing equations associated with the $SO(N)\ensuremath{\rightarrow}\mathcal{G}$ branching rules and the transformations satisfy positivity condition. The dynamics constrained by the $\mathcal{G}$-symmetric crossing equations combined with the positivity condition degenerates to the $SO(N)$ symmetric cases, while the non-$SO(N)$ symmetric theories are not directly solvable without introducing the $SO(N)$ symmetry breaking assumptions on the spectrum.

The Grassmannian for celestial superamplitudes

Recently, scattering amplitudes in four-dimensional Minkowski spacetime have been interpreted as conformal correlation functions on the two-dimensional celestial sphere, the so-called celestial amplitudes. In this note we consider tree-level scattering amplitudes in mathcal{N} = 4 super Yang-Mills theory and present a Grassmannian formulation of their celestial counterparts. This result paves the way towards a geometric picture for celestial superamplitudes, in the spirit of positive geometries.

Celestial superamplitude in mathcal{N} = 4 SYM theory

Celestial amplitude is a new reformulation of momentum space scattering amplitudes and offers a promising way for flat holography. In this paper, we study the celestial amplitudes in mathcal{N} = 4 Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of mathcal{N} = 4 SYM theory. We also compute the three-point and four-point celestial super-amplitudes explicitly. They can be identified as the conformal correlation functions of the celestial superfields living at the celestial sphere. We further study the soft and collinear limits which give rise to the super-Ward identity and super-OPE on the celestial sphere, respectively. Our results initiate a new perspective of understanding the well-studied mathcal{N} = 4 SYM amplitudes via 2D celestial conformal field theory.

Conformal correlation functions in four dimensions from quaternionic Lauricella system

Correlation functions in Euclidean conformal field theories in four dimensions are expressed as representations of the conformal group $SL(2,\H)$, $\H$ being the field of quaternions, on the configuration space of points. The representations are obtained in terms of Lauricella system for quaternions. It generalizes the two-dimensional case, wherein the $N$-point correlation function is expressed in terms of solutions of Lauricella system on the configuration space of $N$ points on the complex plane, furnishing representation of the conformal group $SL(2,\C)$.

Loop corrections to celestial amplitudes

We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity. We first analyze finite one-loop celestial amplitudes in pure Yang-Mills theory and Einstein gravity. We then turn to our main focus: infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We compute the celestial one-loop amplitude in dimensional regularization and show that it can be recast as an operator acting on the celestial tree-level amplitude. This extends to any loop order and the re-summation of all planar loops enables us to write down an expression for the all-loop celestial amplitude. Finally, we show that the exponentiated all-loop expression given by the BDS formula gets promoted on the celestial sphere to an operator acting on the tree-level conformal correlation function, thus yielding, the celestial BDS formula.

Landau diagrams in AdS and S-matrices from conformal correlators

Quantum field theories in AdS generate conformal correlation functions on the boundary, and in the limit where AdS is nearly flat one should be able to extract an S-matrix from such correlators. We discuss a particularly simple position-space procedure to do so. It features a direct map from boundary positions to (on-shell) momenta and thereby relates cross ratios to Mandelstam invariants. This recipe succeeds in several examples, includes the momentum-conserving delta functions, and can be shown to imply the two proposals in [1] based on Mellin space and on the OPE data. Interestingly the procedure does not always work: the Landau singularities of a Feynman diagram are shown to be part of larger regions, to be called ‘bad regions’, where the flat-space limit of the Witten diagram diverges. To capture these divergences we introduce the notion of Landau diagrams in AdS. As in flat space, these describe on-shell particles propagating over large distances in a complexified space, with a form of momentum conservation holding at each bulk vertex. As an application we recover the anomalous threshold of the four-point triangle diagram at the boundary of a bad region.

TripleK: A Mathematica package for evaluating triple-K integrals and conformal correlation functions

I present a Mathematica package designed for manipulations and evaluations of triple-K integrals and conformal correlation functions in momentum space. Additionally, the program provides tools for evaluation of a large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators. The package is accompanied by five Mathematica notebooks containing detailed calculations of numerous conformal 3-point functions in momentum space. Program Title: TripleK CPC Library link to program files: http://dx.doi.org/10.17632/5sz4bt28vr.1 Developer’s repository link: https://triplek.hepforge.org/ Licensing provisions: GNU General Public License v3.0 Programming language: Wolfram Language [1] (Mathematica 10.0 or higher) Supplementary material: The package includes five Mathematica notebooks containing bulk of the results regarding the structure of conformal 3-point functions. Nature of problem: Triple-K integrals were introduced in [2] as a convenient tool for the analysis of conformal 3-point functions in momentum space. All 3-point functions of scalar operators, conserved currents and stress tensor can be expressed in terms of triple-K integrals. Furthermore, a large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators is expressible in terms of triple-K integrals as well. Since the expressions are usually long and unwieldy, an automated tool is essential for efficient manipulations. Solution method: In [3] an effective reduction algorithm was provided for expressing a large class of triple-K integrals in terms of master integrals. The presented package implements this reduction scheme. As far as the multi-loop Feynman integrals are concerned, the conversion to multiple-K integrals proceeds by means of Schwinger parameterization. Additional comments including restrictions and unusual features: Despite extensive testing, this package is a one man job, therefore bugs are unavoidable. Please, report all issues at [email protected] or [email protected] [1] Wolfram Research Inc., Mathematica, Version 11.2, 12.0, Champaign, IL, 2020 [2] A. Bzowski, P. McFadden, K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111. http://arxiv.org/abs/1304.7760 arXiv:1304.7760, https://doi.org/10.1007/JHEP03(2014)111 doi:10.1007/JHEP03(2014)111 [3] A. Bzowski, P. McFadden, K. Skenderis, Evaluation of conformal integrals, JHEP 02 (2016) 068. http://arxiv.org/abs/1511.02357 arXiv:1511.02357, https://doi.org/10.1007/JHEP02(2016)068 doi:10.1007/JHEP02(2016)068

Conformal four-point correlation functions from the operator product expansion

We show how to compute conformal blocks of operators in arbitrary Lorentz representations using the formalism described in [1, 2] and present several explicit examples of blocks derived via this method. The procedure for obtaining the blocks has been reduced to (1) determining the relevant group theoretic structures and (2) applying appropriate predetermined substitution rules. The most transparent expressions for the blocks we find are expressed in terms of specific substitutions on the Gegenbauer polynomials. In our examples, we study operators which transform as scalars, symmetric tensors, two-index antisymmetric tensors, as well as mixed representations of the Lorentz group.

New methods for conformal correlation functions

The most general operator product expansion in conformal field theory is obtained using the embedding space formalism and a new uplift for general quasi-primary operators. The uplift introduced here, based on quasi-primary operators with spinor in- dices only and standard projection operators, allows a unified treatment of all quasi-primary operators irrespective of their Lorentz group irreducible representations. This unified treatment works at the level of the operator product expansion and hence applies to all correlation functions. A very useful differential operator appearing in the operator product expansion is established and its action on appropriate products of embedding space coordinates is explicitly computed. This computation leads to tensorial generalizations of the usual Exton function for all correlation functions. Several important identities and contiguous relations are also demonstrated for these new tensorial functions. From the operator product expansion all correlation functions for all quasi-primary operators, irrespective of their Lorentz group irreducible representations, can be computed recursively in a systematic way. The resulting answer can be expressed in terms of tensor structures that carry all the Lorentz group information and linear combinations of the new tensorial functions. Finally, a summary of the well-defined rules allowing the computation of all correlation functions constructively is presented.

Conformal two-point correlation functions from the operator product expansion

We compute the most general embedding space two-point function in arbitrary Lorentz representations in the context of the recently introduced formalism in [1, 2]. This work provides a first explicit application of this approach and furnishes a number of checks of the formalism. We project the general embedding space two-point function to position space and find a form consistent with conformal covariance. Several concrete examples are worked out in detail. We also derive constraints on the OPE coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations.

Correlation functions at the bulk point singularity from the gravitational eikonal S-matrix

The bulk point singularity limit of conformal correlation functions in Lorentzian signature acts as a microscope to look into local bulk physics in AdS. From it we can extract flat space scattering processes localized in AdS that ultimate should be related to corresponding observables on the conformal field theory at the boundary. In this paper we use this interesting property to propose a map from flat space s-matrix to conformal correlation functions and try it on perturbative gravitational scattering. In particular, we show that the eikonal limit of gravitation scattering maps to a correlation function of the expected form at the bulk point singularity. We also compute the inverse map recovering a previous proposal in the literature.

Primary fields in celestial CFT

The basic ingredient of CCFT holography is to regard four-dimensional amplitudes describing conformal wave packets as two-dimensional conformal correlation functions of the operators associated to external particles. By construction, these operators transform as quasi-primary fields under SL(2, ℂ) conformal symmetry group of the celestial sphere. We derive the OPE of the CCFT energy-momentum tensor with the operators representing gauge bosons and show that they transform as Virasoro primaries under diffeomorphisms of the celestial sphere.

Logarithmic correlation functions for critical dense polymers on the cylinder

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter nn. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty)α∈(0,∞). These correlators are defined as ratios Z(x)/Z_0Z(x)/Z0 of partition functions, where Z_0Z0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x)Z(x), the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 11 and xx where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite nn using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \inftyn→∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1)τ=x−1n∈(0,1). For small \tauτ, the leading behaviours are proportional to \tau^{1/4}τ1/4, \tau^{1/4}\log \tauτ1/4logτ, \log \taulogτ and \log^2 \taulog2τ. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge c=-2c=−2 and conformal dimensions \Delta = -\frac18Δ=−18 or \Delta = 0Δ=0. With these assumptions, we obtain differential equations of order two and three satisfied by the conformal correlation functions, solve these equations in terms of hypergeometric functions, and find a perfect agreement with the lattice results. We use the lattice results to compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

Stochastic interface dynamics in the Hele-Shaw cell.

A one-parametric stochastic regularized dynamics of the interface in the Hele-Shaw cell is introduced. The short-distance regularization suggested by the aggregation model stabilizes the growth by preventing the formation of cusps at the interface and makes the interface dynamics chaotic. The introduced stochastic growth process generates universal complex patterns with the well-developed fjords of oil separating the fingers of water. In a long time asymptotic, by coupling a conformal field theory to the stochastic growth process, we introduce a set of observables (the martingales), whose expectation values are constant in time. The martingales are closely connected to degenerate representations of the Virasoro algebra and can be written in terms of conformal correlation functions. A direct link between Laplacian growth and conformal Liouville field theory with the central charge c≥25 is proposed.

Symmetries of celestial amplitudes

Celestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional sphere describing Minkowski space at null infinity. We construct the generators of Poincare and conformal groups in the celestial representation and discuss how these symmetries are manifest in the amplitudes.

Towards spinning Mellin amplitudes

We construct the Mellin representation of four point conformal correlation function with external primary operators with arbitrary integer spacetime spins, and obtain a natural proposal for spinning Mellin amplitudes. By restricting to the exchange of symmetric traceless primaries, we generalize the Mellin transform for scalar case to introduce discrete Mellin variables for incorporating spin degrees of freedom. Based on the structures about spinning three and four point Witten diagrams, we also obtain a generalization of the Mack polynomial which can be regarded as a natural kinematical polynomial basis for computing spinning Mellin amplitudes using different choices of interaction vertices.

Conformal correlators and blocks with spinors in general dimensions

We compute conformal correlation functions with spinor, tensor, and spinor-tensor primary fields in general dimensions with Euclidean and Lorentzian metrics. The spinors are taken to be Dirac spinors, which exist for any dimensions. For this, the embedding space formalism is employed and the polarisation spinors are introduced to simplify the computations. Three-point functions are rewritten in terms of differential operators acting on scalar-scalar-tensor correlation functions. This enables us to determine conformal blocks for four-point functions with scalar and spinor fields by acting the differential operators on scalar conformal blocks, which will be useful in finding their geodesic Witten diagrams.

Holographic conformal blocks from interacting Wilson lines

We present a simple prescription for computing conformal blocks and correlation functions holographically in AdS$_3$ in terms of Wilson lines merging at a bulk vertex. This is shown to reproduce global conformal blocks and heavy-light Virasoro blocks. In the case of higher spin theories the space of vertices is in one-to-one correspondence with the space of ${\cal W}_N$ conformal blocks, and we show how the latter are obtained by explicit computations.

Conformal correlation functions in the Brownian loop soup

We define and study a set of operators that compute statistical properties of the Brownian Loop Soup, a conformally invariant gas of random Brownian loops (Brownian paths constrained to begin and end at the same point) in two dimensions. We prove that the correlation functions of these operators have many of the properties of conformal primaries in a conformal field theory, and compute their conformal dimension. The dimensions are real and positive, but have the novel feature that they vary continuously as a periodic function of a real parameter. We comment on the relation of the Brownian Loop Soup to the free field, and use this relation to establish that the central charge of the Loop Soup is twice its intensity.