Four-point Correlation Research Articles

The Cosmological Bootstrap: Spinning correlators from symmetries and factorization

We extend the cosmological bootstrap to correlators involving massless spinning particles, focusing on spin-1 and spin-2. In de Sitter space, these correlators are constrained both by symmetries and by locality. In particular, the de Sitter isometries become conformal symmetries on the future boundary of the spacetime, which are reflected in a set of Ward identities that the boundary correlators must satisfy. We solve these Ward identities by acting with weight-shifting operators on scalar seed solutions. Using this weight-shifting approach, we derive three- and four-point correlators of massless spin-1 and spin-2 fields with conformally coupled scalars. Four-point functions arising from tree-level exchange are singular in particular kinematic configurations, and the coefficients of these singularities satisfy certain factorization properties. We show that in many cases these factorization limits fix the structure of the correlators uniquely, without having to solve the conformal Ward identities. The additional constraint of locality for massless spinning particles manifests itself as current conservation on the boundary. We find that the four-point functions only satisfy current conservation if the s, t, and u-channels are related to each other, leading to nontrivial constraints on the couplings between the conserved currents and other operators in the theory. For spin-1 currents this implies charge conservation, while for spin-2 currents we recover the equivalence principle from a purely boundary perspective. For multiple spin-1 fields, we recover the structure of Yang--Mills theory. Finally, we apply our methods to slow-roll inflation and derive a few phenomenologically relevant scalar-tensor three-point functions.

Towards charged hadron polarizabilities from four-point functions in lattice QCD

We show how to compute electromagnetic polarizabilities of charged hadrons using four-point functions in lattice QCD. The low-energy behavior of Compton scattering amplitude is matched to matrix elements of current-current correlation functions on the lattice. Working in momentum space, formulas for electric polarizability ($\alpha_E$) and magnetic polarizability ($\beta_M$) are derived for both charged pion and proton. Lattice four-point correlation functions are constructed from quark and gluon fields to be used in Monte-Carlo simulations. The content of the functions is assessed in detail and specific prescriptions are given to isolate the polarizabilities. The connected quark-line diagrams can be done today as a small lattice project. The disconnected diagrams are more challenging but are within reach of dedicated resources for medium to large lattice projects. We also draw attention to the potential of four-point functions as a multi-purpose tool for hadron structure.

Heterogeneous dynamics in [BMIM][PF6] + Cosolvent binary Mixtures: Does It depend upon cosolvent Polarity?

It is known that dilution of ionic liquid with appropriate cosolvents modifies a number of physicochemical properties that include surface tension, viscosity, ion diffusivity and many others. The cosolvents, particularly when present in low concentrations, also undergo substantial changes in their inherent dynamics. The work reported here explores, via molecular dynamics simulations, the composition dependence of modifications on motional features of both the neutral cosolvent molecules and the charged molecular ions constituting the ionic liquid, 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM] [PF6]). The cosolvents have been carefully chosen so that the polarities differ widely; they are acetonitrile (dipolar), 1,4-dioxane (quadrupolar) and n-hexane (non-polar). Mixture compositions studied are, FIL = 1.00, 0.90, 0.75, 0.50, 0.25, 0.10 and 0, FIL being the IL mole fraction. Signature of heterogeneous relaxations have been searched via examining several dynamical markers, such as, four-point density-time correlation functions, overlap functions, self-intermediate scattering functions, and other motional features arising from the deviations of displacements from Gaussian distributions. The presence of cosolvents affects the diffusivity of both the ions and the cosolvents, with dipolar acetonitrile causing the fastest ion diffusion in the mixture. Interestingly, all the dynamical heterogeneity features studied here have been found to respond significantly to both the cosolvent polarity and the mixture composition.

Notes on gravity multiplet correlators in AdS3 \xd7 S3

We present a compact formula in Mellin space for the four-point tree-level holographic correlators of chiral primary operators of arbitrary conformal weights in (2, 0) supergravity on AdS3× S3, with two operators in tensor multiplet and the other two in gravity multiplet. This is achieved by solving the recursion relation arising from a hidden six-dimensional conformal symmetry. We note the compact expression is obtained after carefully analysing the analytic structures of the correlators. Various limits of the correlators are studied, including the maximally R-symmetry violating limit and flat-space limit.

Theory of two-dimensional spectroscopy with intense laser fields.

Two-dimensional vibrational and electronic spectroscopic observables of isotropically oriented molecular samples in solution are sensitive to laser field intensities and polarization. The third-order response function formalism predicts a signal that grows linearly with the field strength of each laser pulse, thus lacking a way of accounting for non-trivial intensity-dependent effects, such as saturation and finite bleaching. An analytical expression to describe the orientational part of the molecular response, which, in the weak-field limit, becomes equivalent to a four-point correlation function, is presented. Such an expression is evaluated for Liouville-space pathways accounting for diagonal and cross peaks for all-parallel and cross-polarized pulse sequences, in both the weak- and strong-field conditions, via truncation of a Taylor series expansion at different orders. The results obtained in the strong-field conditions suggest how a careful analysis of two-dimensional spectroscopic experimental data should include laser pulse intensity considerations when determining molecular internal coordinates.

Heterogeneous Orientational Relaxations and Translation-Rotation Decoupling in (Choline Chloride + Urea) Deep Eutectic Solvents: Investigation through Molecular Dynamics Simulations and Dielectric Relaxation Measurements.

Molecular dynamics simulations and dielectric relaxation (DR) measurements in the frequency window, 0.2 ≤ ν/GHz ≤ 50, have been performed to explore the heterogeneous reorientation dynamics in [f choline chloride + (1 - f) urea] deep eutectic solvents (DESs) at f = 0.33 and 0.40 in the temperature range 293 ≤ T/K ≤ 333. The solution viscosity varies by more than an order of magnitude. DR measurements in these DESs reveal multiple relaxation timescales-τ1 ∼ 500 ps, τ2 ∼ 100 ps, τ3 ∼ 30 ps, and τ4 ∼ 5 ps. Simulated rank-dependent collective single-particle reorientational (Cl(t), l being the rank) and structural H-bond [CHB(t)] relaxations can explain the microscopic origin of all these DR timescales. The average DR times, ⟨τDR⟩, exhibit a pronounced fractional viscosity dependence, ⟨τDR⟩ ∝ (η/T)p, with p = 0.1. This experimental evidence of pronounced heterogeneous reorientation dynamics in these DESs is supported by a strong translation-rotation decoupling and a significant deviation of the average reorientational correlation times (⟨τl⟩) from Debye's l(l + 1) law. The simulated ratios between the average rotation and translation timescales for both urea and choline correctly reduce to the appropriate hydrodynamic limit at high temperatures. The stretched exponential relaxations of the simulated self-dynamic structure factors and the non-Gaussian single-particle displacement distributions further support strong temporal heterogeneity in these DESs. Dynamic susceptibilities from the simulated four-point correlations exhibit long correlated timescales. Moreover, simulated activation energies estimated from the temperature-dependent C1(t) decays and the translational diffusion coefficients from the velocity autocorrelation functions agree favorably with those from the corresponding DR and the pulsed field gradient nuclear magnetic resonance measurements.

Octagon with finite bridge: free fermions and determinant identities

We continue the study of the octagon form factor which helps to evaluate a class of four-point correlation functions in mathcal{N} = 4 SYM theory. The octagon is characterised, besides the kinematical parameters, by a “bridge” of ℓ propagators connecting two nonadjacent operators. In this paper we construct an operator representation of the octagon with finite bridge as an expectation value in the Fock space of free complex fermions. The bridge ℓ appears as the level of filling of the Dirac sea. We obtain determinant identities relating octagons with different bridges, which we derive from the expression of the octagon in terms of discrete fermionic oscillators. The derivation is based on the existence of a previously conjectured similarity transformation, which we find here explicitly.

Conformal blocks from celestial gluon amplitudes

In celestial conformal field theory, gluons are represented by primary fields with dimensions ∆ = 1 + iλ, λ ∈ ℝ and spin J = ±1, in the adjoint representation of the gauge group. All two- and three-point correlation functions of these fields are zero as a consequence of four-dimensional kinematic constraints. Four-point correlation functions contain delta-function singularities enforcing planarity of four-particle scattering events. We relax these constraints by taking a shadow transform of one field and perform conformal block decomposition of the corresponding correlators. We compute the conformal block coefficients. When decomposed in channels that are “compatible” in two and four dimensions, such four-point correlators contain conformal blocks of primary fields with dimensions ∆ = 2 + M + iλ, where M ≥ 0 is an integer, with integer spin J = −M, −M + 2, …, M − 2, M. They appear in all gauge group representations obtained from a tensor product of two adjoint representations. When decomposed in incompatible channels, they also contain primary fields with continuous complex spin, but with positive integer dimensions.

Conformal Regge theory at finite boost

The Operator Product Expansion is a useful tool to represent correlation functions. In this note we extend Conformal Regge theory to provide an exact OPE representation of Lorenzian four-point correlators in conformal field theory, valid even away from Regge limit. The representation extends convergence of the OPE by rewriting it as a double integral over continuous spins and dimensions, and features a novel “Regge block”. We test the formula in the conformal fishnet theory, where exact results involving nontrivial Regge trajectories are available.

Dispersive CFT sum rules

We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem.

More on holographic correlators: twisted and dimensionally reduced structures

Recently four-point holographic correlators with arbitrary external BPS operators were constructively derived in [1, 2] at tree-level for maximally superconformal theories. In this paper, we capitalize on these theoretical data, and perform a detailed study of their analytic properties. We point out that these maximally supersymmetric holographic correlators exhibit a hidden dimensional reduction structure à la Parisi and Sourlas. This emergent structure allows the correlators to be compactly expressed in terms of only scalar exchange diagrams in a dimensionally reduced spacetime, where formally both the AdS and the sphere factors have four dimensions less. We also demonstrate the superconformal properties of holographic correlators under the chiral algebra and topological twistings. For AdS5× S5 and AdS7× S4, we obtain closed form expressions for the meromorphic twisted correlators from the maximally R-symmetry violating limit of the holographic correlators. The results are compared with independent field theory computations in 4d mathcal{N} = 4 SYM and the 6d (2, 0) theory, finding perfect agreement. For AdS4× S7, we focus on an infinite family of near-extremal four-point correlators, and extract various protected OPE coefficients from supergravity. These OPE coefficients provide new holographic predictions to be matched by future supersymmetric localization calculations. In deriving these results, we also develop many technical tools which should have broader applicability beyond studying holographic correlators.

Toverline{T} Deformation of stress-tensor correlators from random geometry

We study stress-tensor correlators in the Toverline{T} -deformed conformal field theories in two dimensions. Using the random geometry approach to the Toverline{T} deformation, we develop a geometrical method to compute stress-tensor correlators. More specifically, we derive the Toverline{T} deformation to the Polyakov-Liouville conformal anomaly action and calculate three and four-point correlators to the first-order in the Toverline{T} deformation from the deformed Polyakov-Liouville action. The results are checked against the standard conformal perturbation theory computation and we further check consistency with the Toverline{T} -deformed operator product expansions of the stress tensor. A salient feature of the Toverline{T} -deformed stress-tensor correlators is a logarithmic correction that is absent in two and three-point functions but starts appearing in a four-point function.

The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop

We derive an optical theorem for perturbative CFTs which computes the double discontinuity of conformal correlators from the single discontinuities of lower order correlators, in analogy with the optical theorem for flat space scattering amplitudes. The theorem takes a purely multiplicative form in the CFT impact parameter representation used to describe high-energy scattering in the dual AdS theory. We use this result to study four-point correlation functions that are dominated in the Regge limit by the exchange of the graviton Regge trajectory (Pomeron) in the dual theory. At one-loop the scattering is dominated by double Pomeron exchange and receives contributions from tidal excitations of the scattering states which are efficiently described by an AdS vertex function, in close analogy with the known Regge limit result for one-loop string scattering in flat space at finite string tension. We compare the flat space limit of the conformal correlator to the flat space results and thus derive constraints on the one-loop vertex function for type IIB strings in AdS and also on general spinning tree level type IIB amplitudes in AdS.

Constraining momentum space correlators using slightly broken higher spin symmetry

In this work, building up on [1] we present momentum space Ward identities related to broken higher spin symmetry as an alternate approach to computing correlators of spinning operators in interacting theories such as the quasi-fermionic and quasi-bosonic theories. The direct Feynman diagram approach to computing correlation functions is intricate and in general has been performed only in specific kinematic regimes. We use higher spin equations to obtain the parity even and parity odd contributions to two-, three- and four-point correlators involving spinning and scalar operators in a general kinematic regime, and match our results with existing results in the literature for cases where they are available.One of the interesting facts about higher spin equations is that one can use them away from the conformal fixed point. We illustrate this by considering mass deformed free boson theory and solving for two-point functions of spinning operators using higher spin equations.

Crossing bridges with strong Szeg\u0151 limit theorem

We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar mathcal{N} = 4 SYM theory which admit factorization into a product of two octagon form factors with an arbitrary bridge length. We show that the octagon can be expressed as the Fredholm determinant of the integrable Bessel operator and demonstrate that this representation is very efficient in finding the octagons both at weak and strong coupling. At weak coupling, in the limit when the four half-BPS operators become null separated in a sequential manner, the octagon obeys the Toda lattice equations and can be found in a closed form. At strong coupling, we exploit the strong Szegő limit theorem to derive the leading asymptotic behavior of the octagon and, then, apply the method of differential equations to determine the remaining subleading terms of the strong coupling expansion to any order in the inverse coupling. To achieve this goal, we generalize results available in the literature for the asymptotic behavior of the determinant of the Bessel operator. As a byproduct of our analysis, we formulate a Szegő-Akhiezer-Kac formula for the determinant of the Bessel operator with a Fisher-Hartwig singularity and develop a systematic approach to account for subleading power suppressed contributions.

On genus-one string amplitudes on AdS5 \xd7 S5

We study non-planar correlators in mathcal{N} = 4 super-Yang-Mills in Mellin space. We focus in the stress tensor four-point correlator to order 1/N4 and in a strong coupling expansion. This can be regarded as the genus-one four-point graviton amplitude of type IIB string theory on AdS5× S5 in a low energy expansion. Both the loop supergravity result as well as the tower of stringy corrections have a remarkable simple structure in Mellin space, making manifest important properties such as the correct flat space limit and the structure of UV divergences.

Simulation of time-dependent resonant inelastic x-ray scattering using nonequilibrium dynamical mean-field theory

We develop a framework to evaluate the time-dependent resonant inelastic X-ray scattering (RIXS) signal with the use of non-equilibrium dynamical mean field theory simulations. The approach is based on the solution of a time-dependent impurity model which explicitly incorporates the probe pulse. It avoids the need to compute four-point correlation functions, and can in principle be combined with different impurity solvers. This opens a path to study time-resolved RIXS processes in multi-orbital systems. The approach is exemplified with a study of the RIXS signal of a melting Mott antiferromagnet.

Detection of photon statistics and multimode field correlations by Raman processes.

Glauber's g(2)-function provides a common measure of quantum field statistics through two-photon coincidence counting in Hanbury Brown-Twiss measurements. Here, we propose to use nonlinear optical signals as a tool for the characterization of quantum light. In particular, we show that Raman measurements provide an alternative direct probe for a different component of the four-point correlation function underlying the g(2)-function. We illustrate this capacity for a specific quantum state obtained from a frequency conversion process. Our work points out how the analysis of controlled optical nonlinear processes can provide an alternative window toward the analysis of quantum light sources.

One-loop string corrections to AdS amplitudes from CFT

We consider α′ corrections to the one-loop four-point correlator of the stress- tensor multiplets in mathcal{N} = 4 super Yang-Mills at order 1/N4. Holographically, this is dual to string corrections of the one-loop supergravity amplitude on AdS5 × S5. While this correlator has been considered in Mellin space before, we derive the corresponding position space results, gaining new insights into the analytic structure of AdS loop amplitudes. Most notably, the presence of a transcendental weight three function involving new singularities is required, which has not appeared in the context of AdS amplitudes before. We thereby confirm the structure of string corrected one-loop Mellin amplitudes, and also provide new explicit results at orders in α′ not considered before.

From correlation functions to event shapes in QCD

We present a method for calculating event shapes in QCD based on correlation functions of conserved currents. The method has been previously applied to the maximally supersymmetric Yang-Mills theory, but we demonstrate that supersymmetry is not essential. As a proof of concept, we consider the simplest example of a charge-charge correlation at one loop (leading order). We compute the correlation function of four electromagnetic currents and explain in detail the steps needed to extract the event shape from it. The result is compared to the standard amplitude calculation. The explicit four-point correlation function may also be of interest for the CFT community.