Three-point Functions Research Articles

A new twist on spinning (A)dS correlators

Massless spinning correlators in cosmology are extremely complicated. In contrast, the scattering amplitudes of massless particles with spin are very simple. We propose that the reason for the unreasonable complexity of these correlators lies in the use of inconvenient kinematic variables. For example, in de Sitter space, consistency with unitarity and the background isometries imply that the correlators must be conformally covariant and also conserved. However, the commonly used kinematic variables for correlators do not make all of these properties manifest. In this paper, we introduce twistor space as a powerful way to satisfy all kinematic constraints. We show that conformal correlators of conserved currents can be written as twistor integrals, where the conservation condition translates into holomorphicity of the integrand. The functional form of the twistor-space correlators is very simple and easily bootstrapped. For the case of two- and three-point functions, we verify explicitly that this reproduces known results in embedding space. We also perform a half-Fourier transform of the twistor-space correlators to obtain their counterparts in momentum space. We conclude that twistors provide a promising new avenue to study conformal correlation functions that exposes their hidden simplicity.

Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics. Part 2. Quartic interactions and consistency relations

We prove explicitly the absence of one-loop corrections to large scales from small scales in transient non-slow-roll dynamics. Specifically, we address loop corrections to the primordial power spectrum, relative to tree-level, that are independent of the ratio between the two scales. We review all the necessary components, adapted to our context, to express one-loop diagrams as three-point functions, emphasizing the crucial role played by quartic interactions. Notably, we include the quartic Hamiltonian induced by the cubic Lagrangian and quartic interactions that are ensured by diffeomorphism invariance. We then explicitly prove consistency relations for an arbitrary transient non-slow-roll phase involving operators with (time) derivatives. Finally, we calculate one-loop corrections by including contributions from the relevant cubic and quartic interactions, and express the final result as a total derivative term over comoving momenta, utilizing the consistency relations we established. This leads us to conclude that one-loop corrections to long-wavelength modes are unaffected by the physics of short and enhanced modes in non-slow-roll dynamics.

Discrete Lehmann representation of three-point functions

Discrete Lehmann representation of three-point functions

The streaming model for the three-point correlation function and its connection to standard perturbation theory

Redshift-space distortions (RSDs) present a significant challenge in building models for the three-point correlation function (3PCF). We compare two possible lines of attack: the streaming model and standard perturbation theory (SPT). The two approaches differ in their treatment of the non-linear mapping from real to redshift space: SPT expands this mapping perturbatively, while the streaming model retains its non-linear form but relies on simplifying assumptions about the probability density function (PDF) of line-of-sight velocity differences between pairs or triplets of tracers. To assess the quality of the predictions and the validity of the assumptions of these models, we measure the monopole of the matter 3PCF and the first two moments of the pair- and triplewise velocity PDF from a suite of N-body simulations. We also evaluate the large-scale limit of the streaming model and determine under which conditions it aligns to SPT. On scales larger than 10 h -1 Mpc, we find that the streaming model for the 3PCF monopole is dominated by the first two velocity moments, making the exact shape of the PDF irrelevant. This model can match the accuracy of a Stage-IV galaxy survey, if the velocity moments are measured directly from the simulations. However, replacing the measurements with perturbative expressions to leading order generates large errors already on scales of 60–70 h -1 Mpc. This is the primary drawback of the streaming model. On the other hand, the SPT model for the 3PCF cannot account for the significant velocity dispersion that is present at all scales, and consequently provides predictions with limited accuracy. We demonstrate that this issue can be approximately addressed by isolating the large-scale limit of the dispersion, which leads to typical Fingers-of-God damping functions. Overall, the SPT model with a damping function provides the best compromise in terms of accuracy and computing time.

Correlators for pseudo Hermitian systems

Pseudo-Hermitian system is a class of non-Hermitian system with Hamiltonian satisfying the condition η−1H†η = H. We develop the in-in and Schwinger Keldysh formalism to calculate cosmological correlators for pseudo-Hermitian systems. We study a model consists of massive symplectic fermions coupled to the primordial curvature perturbation. The three-point function for the primordial curvature perturbation is computed up to one-loop and compared to earlier work where the loop correction comes from a massive scalar boson. The two results differ by a minus sign. Therefore, the one loop correction to the three-point function cannot be used to distinguished scalar bosons and symplectic fermions. To conclude, we discuss possibilities where the scalar bosons and symplectic fermions may be distinguished.

Light-cone sum rules study on the purely nonfactorizable Λc+→Ξ0K+ decay

We investigate the purely nonfactorizable Λc+→Ξ0K+ decay using light-cone sum rules. To extract the decay amplitudes, a three-point correlation function is defined and calculated at hadron and quark-gluon level, respectively. Both the W-exchange and the W-inward emission diagrams are considered in the quark-gluon level calculation, where the two-particle light-cone distribution amplitudes (LCDAs) of kaon are used as nonperturbative input. We obtain the decay amplitudes contributed from the twist-2 and twist-3 kaon LCDAs, respectively. The obtained P-wave amplitude is consistent with those predicted in the literature, while the S-wave one is much smaller. This significant difference between the S- and P -wave amplitudes leads to small up-down spin asymmetry, close to the experimental measurement. Published by the American Physical Society 2024

Pair counting without binning - a new approach to correlation functions in clustering statistics

Abstract This paper presents a novel perspective on correlation functions in the clustering analysis of the large-scale structure of the universe. We begin with the recognition that pair counting in bins of radial separation is equivalent to evaluating counts-in-cells (CIC), which can be modelled using a filtered density field with a binning-window function. This insight leads to an in situ expression for the two-point correlation function (2PCF). Essentially, the core idea underlying our method is to introduce a window function to define the binning scheme, enabling pair-counting without binning. This approach develops an idea of generalised 2PCF, which extends beyond conventional discrete pair counting by accommodating non-sharp-edged window functions. In the context of multiresolution analysis, we can implement a fast algorithm to estimate the generalised 2PCF. To extend this framework to N-point correlation functions (NPCF) using current optimal edge-corrected estimators, we developed a binning scheme that is independent of the specific parameterisation of polyhedral configurations. In particular, we demonstrate a fast algorithm for the three-point correlation function (3PCF), where triplet counting is accomplished by assigning either a spherical tophat or a Gaussian filter to each vertex of triangles. Additionally, we derive analytical expressions for the 3PCF using a multipole expansion in Legendre polynomials, accounting for filtered field (binning) corrections. Our method provides an exact solution for quantifying binning effects in practical measurements and offers a high-speed algorithm, enabling high-order clustering analysis in extremely large datasets from ongoing and upcoming surveys such as Euclid, LSST, and DESI.

Higher-rank sectors in the hexagon formalism and marginal deformations

Abstract The hexagon approach provides an integrability framework for the computation of structure constants in N = 4 super Yang–Mills theory in four dimensions. Three-point functions are cut into two hexagonal patches, on which the excitations of the long-range Bethe ansatz of the spectrum problem scatter. To this end, the Bethe states representing the operators also need to be cut into two parts. In rank-one sectors such entangled states are fairly straightforward to construct so that most applications of the method have so far been restricted to this simplest case. In this article we construct entangled states for operators in p s u ( 1 , 1 | 2 ) sectors, importing a minimum of information from the nested Bethe ansatz. The idea is successfully tested against free field theory for a sample set of correlators with up to three higher-rank operators. Further, we take a look at the same correlators in the presence of marginal deformations of the theory. While a systematic modification of the hexagon procedure remains out of reach for now, in practical applications the undeformed amplitudes are surprisingly efficient, especially for a certain one-parameter deformation.

Conformal bootstrap equations from the embedding space operator product expansion

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the M-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-ℓ exchange, with finite ℓ-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars S and vectors V, namely ⟨SSSV⟩.

All-electron first-principles GWΓ simulations for accurately predicting core-electron binding energies considering first-order three-point vertex corrections.

In the conventional GW method, the three-point vertex function (Γ) is approximated to unity (Γ ∼ 1). Here, we developed an all-electron first-principles GWΓ method beyond a conventional GW method by considering a first-order three-point vertex function (Γ(1) = 1 + iGGW) in a one-electron self-energy operator. We applied the GWΓ method to simulate the binding energies (BEs) of B1s, C1s, N1s, O1s, and F1s for 19 small-sized molecules. Contrary to the one-shot GW method [or G0W0(LDA)], which underestimates the experimentally determined absolute BEs by about 3.7eV for B1s, 5.1eV for C1s, 6.9eV for N1s, 7.8eV for O1s, and 5.8eV for F1s, the GWΓ method successfully reduces these errors by approximately 1-2eV for all the elements studied here. Notably, the first-order three-point vertex corrections are more significant for heavier elements, following the order of F > O > N > C > B1s. Finally, the computational cost analysis revealed that one term in the GWΓ one-electron self-energy operator, despite being computationally intensive, contributes negligibly (<0.1eV) to the C1s, N1s, O1s, and F1s.

Three-point functions of higher-spin supercurrents in 4D N=1 SCFT: General formalism for arbitrary superspins

We analyze the general structure of the three-point functions involving conserved higher-spin “vectorlike” supercurrents Js(z)≔Jα(s)α˙(s)(z) in four-dimensional N=1 superconformal field theory. Using the constraints of superconformal symmetry and superfield conservation equations, we utilize a computational approach to analyze the general structure of the three-point function ⟨Js1(z1)Js2′(z2)Js3′′(z3)⟩ and provide a general classification of the results. We demonstrate that the three-point function is fixed up to 2 min(si)+2 independent conserved structures, which we propose to hold for arbitrary superspins. In addition, we show that the conserved structures can be classified as parity-even or parity-odd in superspace based on their transformation properties under superinversion. Published by the American Physical Society 2024

Evaluation of three-point correlation functions from structural images on CPU and GPU architectures: Accounting for anisotropy effects.

Structures, or spatial arrangements of matter and energy, including some fields (e.g., velocity or pressure) are ubiquitous in research applications and frequently require description for subsequent analysis, or stochastic reconstruction from limited data. The classical descriptors are two-point correlation functions (CFs), but the computation of three-point statistics is known to be advantageous in some cases as they can probe non-Gaussian signatures, not captured by their two-point counterparts. Moreover, n-point CFs with n≥3 are believed to possess larger information content and provide more information about studied structures. In this paper, we have developed algorithms and code to compute S_{3},C_{3},F_{sss},F_{ssv},and F_{svv} with a right-angle and arbitrary triangle pattern. The former was believed to be faster to compute, but with the help of precomputed regular positions we achieved the same speed for arbitrary pattern. In this work we also implement and demonstrate computations of directional three-point CFs-for this purpose right-triangular pattern seems to be superior due to explicit orientation and high coverage. Moreover, we assess the errors in CFs' evaluation due to image or pattern rotations and show that they have minor effect on accuracy of computations. The execution times of our algorithms for the same number of samples are orders of magnitude lower than in existing published counterparts. We show that the volume of data produced gets unwieldy very easily, especially if computations are performed in frequency domain. For these reasons until information content of different sets of correlation functions with different "n-pointness" is known, advantages of CFs with n>3 are not clear. Nonetheless, developed algorithms and code are universal enough to be easily extendable to any n with increasing computational and random access memory (RAM) burden. All results are available as part of open-source package correlationfunctions.jl [V.Postnicov et al., Comput. Phys. Commun. 299, 109134 (2024)10.1016/j.cpc.2024.109134.]. As described in this paper, three-point CFs computations can be immediately applied in a great number of research applications, for example: (1) flow and transport velocity fields analysis or any data with non-Gaussian signatures, (2) deep learning for structural and physical properties, and (3) structure taxonomy and categorization. In all these and numerous other potential cases the ability to compute directional three-point functions may be crucial. Notably, the organization of the code functions allows computation of crosscorrelation, i.e., one can compute three-point CFs for multiphase images (while binary structures were used in this paper for simplicity of explanations).

Towards testing the general bounce cosmology with the CMB B-mode auto-bispectrum

It has been shown that a three-point correlation function of tensor perturbations from a bounce model in general relativity with a minimally-coupled scalar field is highly suppressed, and the resultant three-point function of cosmic microwave background (CMB) B-mode polarizations is too small to be detected by CMB experiments. On the other hand, bounce models in a more general class with a non-minimal derivative coupling between a scalar field and gravity can predict the three-point correlation function of the tensor perturbations without any suppression, the amplitude of which is allowed to be much larger than that in general relativity. In this paper, we evaluate the three-point function of the B-mode polarizations from the general bounce cosmology with the non-minimal coupling and show that a signal-to-noise ratio of the B-mode auto-bispectrum in the general class can reach unity for ℓ max=100 in the full-sky case, with and without the lensing B-mode added to cosmic variance. Considering additionally the LiteBIRD experimental noise, we obtain a SNR smaller than unity.

A road map to cosmological parameter analysis with third-order shear statistics

Context. Third-order lensing statistics contain a wealth of cosmological information that is not captured by second-order statistics. However, the computational effort it takes to estimate such statistics in forthcoming stage IV surveys is prohibitively expensive. Aims. We derive and validate an efficient estimation procedure for the three-point correlation function (3PCF) of polar fields such as weak lensing shear. We then use our approach to measure the shear 3PCF and the third-order aperture mass statistics on the KiDS-1000 survey. Methods We constructed an efficient estimator for third-order shear statistics that builds on the multipole decomposition of the 3PCF. We then validated our estimator on mock ellipticity catalogs obtained from N-body simulations. Finally, we applied our estimator to the KiDS-1000 data and presented a measurement of the third-order aperture statistics in a tomographic setup. Results. Our estimator provides a speedup of a factor of ∼100–1000 compared to the state-of-the-art estimation procedures. It is also able to provide accurate measurements for squeezed and folded triangle configurations without additional computational effort. We report a significant detection of tomographic third-order aperture mass statistics in the KiDS-1000 data (S/N = 6.69). Conclusions. Our estimator will make it computationally feasible to measure third-order shear statistics in forthcoming stage IV surveys. Furthermore, it can be used to construct empirical covariance matrices for such statistics.

Spectral flow and the conformal block expansion for strings in AdS3

We present a detailed study of spectrally flowed four-point functions in the SL(2,ℝ) WZW model, focusing on their conformal block decomposition. Dei and Eberhardt conjectured a general formula relating these observables to their unflowed counterparts. Although the latter are not known in closed form, their conformal block expansion has been formally established. By combining this information with the integral transform that encodes the effect of spectral flow, we show how to describe a considerable number of s-channel exchanges, including cases with both flowed and unflowed intermediate states. For all such processes, we compute the normalization of the corresponding conformal blocks in terms of products of the recently derived flowed three-point functions with arbitrary spectral flow charges. Our results constitute a highly non-trivial consistency check, thus strongly supporting the aforementioned conjecture, and establishing its computational power.

Roles of boundary and equation-of-motion terms in cosmological correlation functions

We revisit the properties of total time-derivative terms as well as terms proportional to the free equation of motion (EOM) in a Schwinger-Keldysh formalism. They are relevant to the correct calculation of correlation functions of curvature perturbations in the context of inflationary Universe. We show that these two contributions to the action play different roles in the operator or the path-integral formalism, but they give the same correlation functions as each other. As a concrete example, we confirm that the Maldacena's consistency relations for the three-point correlation function in the slow-roll inflationary scenario driven by a minimally coupled canonical scalar field hold in both the operator and path-integral formalisms. We also give some comments on loop calculations.

Simulating Holographic Conformal Field Theories on Hyperbolic Lattices.

We demonstrate how tabletop settings combining hyperbolic lattices with nonlinear dynamics universally encode aspects of the bulk-boundary correspondence between gravity in anti-de-Sitter (AdS) space and conformal field theory (CFT). Our concrete and broadly applicable holographic toy model simulates gravitational self-interactions in the bulk and features an emergent CFT with nontrivial correlations on the boundary. We measure the CFT data contained in the two- and three-point functions and clarify how a thermal CFT is simulated through an effective black hole geometry. As a concrete example, we propose and simulate an experimentally feasible protocol to measure the holographic CFT using electrical circuits.

Light-ray sum rules and the c-anomaly

In a four-dimensional quantum field theory that flows between two fixed points under the renormalization group, the change in the conformal anomaly ∆a has been related to the average null energy. We extend this result to derive a sum rule for the other anomaly coefficient, ∆c, in terms of the stress tensor three-point function. While the sum rule for ∆a is an expectation value of the averaged null energy operator, and therefore positive, the result for ∆c involves the off-diagonal matrix elements, so it does not have a fixed sign.

Boundary Liouville conformal field theory in four dimensions

Higher dimensional Euclidean Liouville conformal field theories (LCFTs) consist of a log-correlated real scalar field with a background charge and an exponential potential. We analyse the LCFT on a four-dimensional manifold with a boundary. We extend to the boundary, the conformally covariant GJMS operator and the Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{Q} $$\\end{document}-curvature term in the LCFT action and classify the conformal boundary conditions. Working on a flat space with plate boundary, we calculate the dimensions of the boundary conformal primary operators, the two- and three-point functions of the displacement operator and the boundary conformal anomaly coefficients.

Missing local operators, zeros, and twist-4 trajectories

The number of local operators in a CFT below a given twist grows with spin. Consistency with analyticity in spin then requires that at low spin, infinitely many Regge trajectories must decouple from local correlation functions, implying infinitely many vanishing conditions for OPE coefficients. In this paper we explain the mechanism behind this infinity of zeros. Specifically, the mechanism is related to the two-point function rather than the three-point function, explaining the vanishing of OPE coefficients in every correlator from a single condition. We illustrate our result by studying twist-4 Regge trajectories in the Wilson-Fisher CFT at one loop.