Conformal Partial Wave Research Articles

N-point functions in conformal quantum mechanics: a momentum space odyssey

In this paper, we study the implications of conformal invariance in momentum space for correlation functions in quantum mechanics. We find that three point functions of arbitrary operators can be written in terms of the 2F1 hypergeometric function. We then show that generic four-point functions can be expressed in terms of Appell’s generalized hypergeometric function F2 with one undetermined parameter that plays the role of the conformal cross ratio in momentum space. We also construct momentum space conformal partial waves, which we compare with the Appell F2 representation. We test our expressions against free theory and DFF model correlators, finding an exact agreement. We then analyze five, six, and all higher point functions. We find, quite remarkably, that n-point functions can be expressed in terms of the Lauricella generalized hypergeometric function, EA, with n − 3 undetermined parameters, which is in one-to-one correspondence with the number of conformal cross ratios. This analysis provides the first instance of a closed form for generic momentum space conformal correlators in contrast to the situation in higher dimensions. Further, we show that the existence of multiple solutions to the momentum space Ward identities can be attributed to the Fourier transforms of the various possible time orderings. Finally, we extend our analysis to theories with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, 2 supersymmetry, where we find that the constraints due to the superconformal ward identities are identical to identities involving hypergeometric functions.

Superconformal blocks for stress-tensor and chiral operators for 4D N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}=2$$\\end{document} superconformal field theories

We have computed the superconformal partial wave for the mixed correlators involving J, ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}, and ϕ†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^\\dagger $$\\end{document}, where J is the superconformal primary of 4D N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {N}}} = 2$$\\end{document} stress-tensor multiplet, ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document}, and ϕ†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^\\dagger $$\\end{document} are chiral and anti-chiral scalars, respectively in the channel ⟨JJϕϕ†⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle {J J}\\phi \\phi ^{\\dagger }\\rangle $$\\end{document}. We have used the superembedding formalism and computed the conformal partial wave corresponding to the exchange of long multiplet using the supershadow approach. This study provides part of the ingredients necessary for the study of conformal bootstrap for mixed correlator in 4D N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {N}}} = 2$$\\end{document} superconformal field theories.

Global conformal blocks via shadow formalism

We study \U0001d530\U0001d5292 and \U0001d530\U0001d5293 global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and \U0001d4b23 conformal field theories in the large central charge limit. In the \U0001d530\U0001d5292 case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the \U0001d530\U0001d5293 case and show that it allows to build simple integral representations for \U0001d530\U0001d5293 global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.

NLO corrections to the deeply virtual meson production revisited: impact on the extraction of generalized parton distributions

We revisit the next-to-leading order (NLO) perturbative QCD corrections for the deeply virtual meson production (DVMP) process, exploring its phenomenology both in isolation and in a multichannel fit combined with deeply virtual Compton scattering (DVCS). Our approach involves the conformal partial wave (CPaW) formalism, which allows for the straightforward inclusion of higher-order contributions and evolutionary effects. Our findings indicate that a description of the longitudinal component of the vector meson DVMP cross-section at high energies is achievable only at NLO within the standard collinear approach. Furthermore, we demonstrate a simultaneous description of DIS, DVCS, and DVMP processes, providing insights into the proton structure described at NLO by unique universal generalized parton distribution (GPD) functions.

A Celestial route to AdS bulk locality

We prove a precise form of AdS bulk locality by deriving analytical two-sided bounds on bulk Wilson coefficients. Our bounds are on the Wilson coefficients themselves, rather than their ratios, as is typically found in the literature. Inspired by the Celestial amplitudes program in flat space, we perform a Celestial transform of the CFT Mellin amplitude of four identical scalars. Using the crossing symmetric dispersion relation (CSDR), we express the resulting amplitude in terms of crossing symmetric conformal partial waves. The partial waves satisy remarkable positivity properties which along with the unitarity of the CFT prove sufficient to derive the bounds. We then employ our methods in the limit of large AdS radius and recover known bounds on flat space Wilson coefficients and new bounds on their large radius corrections. We check that the planar Mellin amplitude of four stress-tensor multiplets in mathcal{N} = 4 SYM satisfies our bounds. Finally, using null constraints, we derive a form of low-spin dominance in AdS EFTs. We find that the low-spin dominance is strongest in flat space and weakens as we move away from flat space towards higher AdS curvature.

From celestial correlators to AdS, and back

We present a general relation between celestial correlation functions in d-dimensions and Witten diagrams in (d + 1)-dimensional Euclidean anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as a continuum of particle exchanges in EAdS where the exchanged particles carrying unitary Principal Series representations of SO(d + 1, 1). One can then try to import familiar EAdS techniques to study the properties of celestial correlators. In this work we use this relation to infer the analytic structure of the spectral density in the conformal partial wave expansion of celestial correlators which, at least perturbatively, should be a meromorphic function of the spectral parameter. We also discuss non-perturbative constraints from unitarity in Euclidean Conformal Field Theory, which requires positivity of the spectral density. This extends similar relations recently uncovered between boundary correlation functions in de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a central role in efforts towards holography for all lambdas.

The shadow formalism of Galilean CFT2

In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (GCFT2). We define the principal series representation of Galilean conformal symmetry group and find its relation with the Wigner classification, then we determine the shadow transform of local operators. Using this formalism we derive the OPE blocks, Clebsch-Gordan kernels, conformal blocks and conformal partial waves. A new feature is that the conformal block admits additional branch points, which would destroy the convergence of OPE for certain parameters. We establish another inversion formula different from the previous one, but get the same result when decomposing the four-point functions in the mean field theory (MFT). We also construct a continuous series of bilocal actions of MFT, and an exceptional series of local actions, one of which is the BMS free scalar model. We notice that there is an outer automorphism of the Galilean conformal symmetry, and the GCFT2 can be regarded as null defect in higher dimensional CFTs.

Towards the non-perturbative cosmological bootstrap

We study quantum field theory on a de Sitter spacetime dSd+1 background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group SO(d + 1, 1). As the first application of the Hilbert space formalism, we recover the Källen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity proper- ties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS2.

Bulk locality from the celestial amplitude

In this paper, we study the implications of bulk locality on the celestial amplitude. In the context of the four-point amplitude, the fact that the bulk S-matrix factorizes locally in poles of Mandelstam variables is reflected in the imaginary part of the celestial amplitude. In particular, on the real axis in the complex plane of the boost weight, the imaginary part of the celestial amplitude can be given as a positive expansion on the Poincaré partial waves, which are nothing but the projection of flat-space spinning polynomials onto the celestial sphere. Furthermore, we derive the celestial dispersion relation, which relates the imaginary part to the residue of the celestial amplitude for negative even integer boost weight. The latter is precisely the projection of low energy EFT coefficients onto the celestial sphere. We demonstrate these properties explicitly on the open and closed string celestial amplitudes. Finally, we give an explicit expansion of the Poincaré partial waves in terms of 2D conformal partial waves.

Analyticity and unitarity for cosmological correlators

We study the fundamentals of quantum field theory on a rigid de Sitter space. We show that the perturbative expansion of late-time correlation functions to all orders can be equivalently generated by a non-unitary Lagrangian on a Euclidean AdS geometry. This finding simplifies dramatically perturbative computations, as well as allows us to establish basic properties of these correlators, which comprise a Euclidean CFT. We use this to infer the analytic structure of the spectral density that captures the conformal partial wave expansion of a late-time four-point function, to derive an OPE expansion, and to constrain the operator spectrum. Generically, dimensions and OPE coefficients do not obey the usual CFT notion of unitarity. Instead, unitarity of the de Sitter theory manifests itself as the positivity of the spectral density. This statement does not rely on the use of Euclidean AdS Lagrangians and holds non-perturbatively. We illustrate and check these properties by explicit calculations in a scalar theory by computing first tree-level, and then full one- loop-resummed exchange diagrams. An exchanged particle appears as a resonant feature in the spectral density which can be potentially useful in experimental searches.

The F-theorem in the melonic limit

The F-theorem states that in three dimensions the sphere free energy of a field theory must decrease between ultraviolet and infrared fixed points of the renormalization group flow, and it has been proven for unitary conformal field theories (CFTs).We consider here the long-range bosonic O(N)3 model on a spherical background, at next-to-next-to-leading order of the 1/N expansion. The model displays four large-N fixed points and we test and confirm the F-theorem holds in this case. This is non-trivial as one of the couplings is imaginary, and therefore the model is non-unitary at finite N. Despite this, several tests indicating that the large-N CFTs are in fact unitary have been performed: for instance all the OPE coefficients computed so far in the large-N limit are real, and the spectrum of bilinear operators is real and above unitarity bounds. Our result, namely that the F theorem holds at large N, can be viewed as further indication that such theories are unitary.As an added bonus, we show how conformal partial waves expansions in conformal field theory can be used to resum infinite classes of vacuum diagrams. Non-perturbatively, the jump in the value of the free energy has the interpretation of the inclusion at the ultraviolet fixed point of an extra non-normalizable contribution in the conformal partial wave expansion. This can be seen in perturbation theory as the reversal of the sign of an infinite class of diagrams due to the flow of a coupling constant.

Conformal block expansion in celestial CFT

The 4D 4-point scattering amplitude of massless scalars via a massive exchange is expressed in a basis of conformal primary particle wavefunctions. This celestial amplitude is expanded in a basis of 2D conformal partial waves on the unitary principal series, and then rewritten as a sum over 2D conformal blocks via contour deformation. 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.

From dS to AdS and back

We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

Conformal partial waves in momentum space

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension d \geq 3d≥3 (including non-integer dd). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in d = 4d=4 dimensions.

Dimensional reduction of higher-point conformal blocks

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d − 2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms of certain graphical Feynman-like rules and the associated finite set of vertex and edge factors. Notably, these rules can be fully determined by considering the explicit power-series representation of just three particular conformal blocks: the four-point block, the five-point block and the six-point block of the so-called OPE/snowflake topology. In principle, this method can be applied to obtain the arbitrary-point dimensional reduction of conformal blocks with spinning exchanges as well. We also show how to systematically extend the dimensional reduction relation of conformal partial waves to higher-points.

Scattering equations in AdS: scalar correlators in arbitrary dimensions

We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two cur- rent algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

Unitarity methods in AdS/CFT

We develop a systematic unitarity method for loop-level AdS scattering amplitudes, dual to non-planar CFT correlators, from both bulk and boundary perspectives. We identify cut operators acting on bulk amplitudes that put virtual lines on shell, and show how the conformal partial wave decomposition of the amplitudes may be efficiently computed by gluing lower-loop amplitudes. A central role is played by the double discontinuity of the amplitude, which has a direct relation to these cuts. We then exhibit a precise, intuitive map between the diagrammatic approach in the bulk using cutting and gluing, and the algebraic, holographic unitarity method of [1] that constructs the non-planar correlator from planar CFT data. Our analysis focuses mostly on four-point, one-loop diagrams — we compute cuts of the scalar bubble, triangle and box, as well as some one-particle reducible diagrams — in addition to the five-point tree and four-point double-ladder. Analogies with S-matrix unitarity methods are drawn throughout.

Bootstrapping inflationary correlators in Mellin space

We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdSd+1 and dSd+1 is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dSd+1 from known results for Witten diagrams in AdSd+1 by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdSd+1 and dSd+1 in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-ℓ. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general d and generic scaling dimensions, in particular massive, light and (partially-)massless fields. It moreover naturally identifies boundary correlation functions for generic fields with multi-variable Meijer-G functions. When d = 3 we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin- ℓ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.

Energy correlations in the end-point region

The energy-energy correlation (EEC) measures the angular distribution of the energy that flows through two calorimeters separated by some relative angle in the final state created by a source. We study this observable in the limit of small and large angles when it describes the correlation between particles belonging, respectively, to the same jet and to two almost back-to-back jets. We present a new approach to resumming large logarithmically enhanced corrections in both limits that exploits the relation between the energy correlations and four-point correlation functions of conserved currents. At large angle, we derive the EEC from the behaviour of the correlation function in the limit when four operators are light-like separated in a sequential manner. At small angle, in a conformal theory, we obtain the EEC from resummation of the conformal partial wave expansion of the correlation function at short-distance separation between the calorimeters. In both cases, we obtain a concise representation of the EEC in terms of the conformal data of twist-two operators and verify it by comparing with the results of explicit calculation at next-to-next-to-leading order in maximally supersymmetric Yang-Mills theory. As a byproduct of our analysis, we predict the maximal weight part of the analogous QCD expression in the back-to-back limit.

From bulk loops to boundary large-N expansion

We study the analytic structure of loop Witten diagrams in Euclidean AdS represented by their conformal partial wave expansions. We show that, as in flat space, amplitude’s singularities are associated with non-trivial cuts of the diagram and factorize into products of the coefficient functions for the subdiagrams resulting from these cuts. We consider an example of a one-loop four-point diagram in detail and then briefly discuss how the procedure can be extended to more general diagrams. Finally, we show that this analysis reproduces simple relations that follow from the large-N considerations on the boundary.