arXiv subjects

Paul McFadden

Publications and source records attributed to Paul McFadden.

At least 19 recordsLinked to original sources

arXiv2026

Renormalised 3-point functions of stress tensors and conserved currents in CFT

We present a complete momentum-space prescription for the renormalisation of tensorial correlators in conformal field theories. Our discussion covers all 3-point functions of stress tensors and conserved currents in arbitrary spacetime dimensions. In dimensions three and four, we give explicit results for the renormalised correlators, the anomalous Ward identities they obey, and the conformal anomalies. For the stress tensor 3-point function in four dimensions, we identify the specific evanescent tensorial structure responsible for the type A Euler anomaly, and show this anomaly has the form of a double copy of the chiral anomaly.

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arXiv2024

Renormalisation of IR divergences and holography in de Sitter

We formulate a renormalisation procedure for IR divergences of tree-level in-in late-time de Sitter correlators. These divergences are due to the infinite volume of spacetime and are analogous to the divergences that appear in AdS dealt with by holographic renormalisation. Regulating the theory using dimensional regularisation, we show that one can remove all infinities by adding local counterterms at the future boundary of dS in the Schwinger-Keldysh path integral. The counterterms amount to renormalising the late-time bulk field. We frame the discussion in terms of bulk scalar fields in dS, using tree-level correlators of massless and conformal scalars for illustration. The relation to AdS via analytic continuation is discussed, and we show that different versions of the analytic continuation appearing in the literature are equivalent to each other. In AdS, one needs to add counterterms that are related to conformal anomalies, and also to renormalise the source part of the bulk field. The analytic continuation to dS projects out the traditional AdS counterterms, and links the renormalisation of the sources to the renormalisation of the late-time bulk field. We use these results to establish holographic formulae that relate tree-level dS in-in correlators to CFT correlators at up to four points, and we provide two proofs: one using the connection between the dS wavefunction and the partition function of the dual CFT, and a second by direct evaluation of the in-in correlators using the Schwinger-Keldysh formalism. The renormalisation of the bulk IR divergences is mapped by these formulae to UV renormalisation of the dual CFT via local counterterms, providing structural support for a possible duality. We also recast the regulated holographic formulae in terms of the AdS amplitudes of shadow fields, but show that this relation breaks down when renormalisation is required.

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arXiv2023

$\mathcal{A}$-hypergeometric functions and creation operators for Feynman and Witten diagrams

Both Feynman integrals and holographic Witten diagrams can be represented as multivariable hypergeometric functions of a class studied by Gel'fand, Kapranov & Zelevinsky known as GKZ or $\mathcal{A}$-hypergeometric functions. Among other advantages, this formalism enables the systematic construction of highly non-trivial weight-shifting operators known as 'creation' operators. We derive these operators from a physics perspective, highlighting their close relation to the spectral singularities of the integral as encoded by the facets of the Newton polytope. Many examples for Feynman and Witten diagrams are given, including novel weight-shifting operators for holographic contact diagrams. These in turn allow momentum-space exchange diagrams of different operator dimensions to be related while keeping the spacetime dimension fixed. In contrast to previous constructions, only non-derivative vertices are involved.

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arXiv2023

Shift operators from the simplex representation in momentum-space CFT

We derive parametric integral representations for the general $n$-point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman integral with the topology of an $(n-1)$-simplex, featuring an arbitrary function of momentum-space cross ratios. Here, we show all graph polynomials for this integral can be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of weight-shifting operators, expressible as determinants, that connect $n$-point functions in spacetime dimensions differing by two. Moreover, the action of all previously known weight-shifting operators preserving the spacetime dimension is manifest. Finally, the new parametric representations enable the validity of the conformal Ward identities to be established directly, without recourse to recursion in the number of points.

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arXiv2022

A handbook of holographic 4-point functions

We present a comprehensive discussion of tree-level holographic $4$-point functions of scalar operators in momentum space. We show that each individual Witten diagram satisfies the conformal Ward identities on its own and is thus a valid conformal correlator. When the $β= Δ- d/2$ are half-integral, with $Δ$ the dimensions of the operators and $d$ the spacetime dimension, the Witten diagrams can be evaluated in closed form and we present explicit formulae for the case $d=3$ and $Δ=2,3$. These correlators require renormalization, which we carry out explicitly, and lead to new conformal anomalies and beta functions. Correlators of operators of different dimension may be linked via weight-shifting operators, which allow new correlators to be generated from given `seed' correlators. We present a new derivation of weight-shifting operators in momentum space and uncover several subtleties associated with their use: such operators map exchange diagrams to a linear combination of exchange and contact diagrams, and special care must be taken when renormalization is required.

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arXiv2021

Conformal correlators as simplex integrals in momentum space

We find the general solution of the conformal Ward identities for scalar $n$-point functions in momentum space and in general dimension. The solution is given in terms of integrals over $(n-1)$-simplices in momentum space. The $n$ operators are inserted at the $n$ vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where $n$-point functions are built in terms of $(n-1)$-point functions. To illustrate our discussion, we derive the simplex representation of $n$-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves $(n-2)$ integrations, which is an improvement (when $n>4$) relative to the Mellin representation that involves $n(n-3)/2$ integrations.

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arXiv2020

Conformal $n$-point functions in momentum space

We present a Feynman integral representation for the general momentum-space scalar $n$-point function in any conformal field theory. This representation solves the conformal Ward identities and features an arbitrary function of $n(n-3)/2$ variables which play the role of momentum-space conformal cross-ratios. It involves $(n-1)(n-2)/2$ integrations over momenta, with the momenta running over the edges of an $(n-1)$-simplex. We provide the details in the simplest non-trivial case (4-point functions), and for this case we identify values of the operator and spacetime dimensions for which singularities arise leading to anomalies and beta functions, and discuss several illustrative examples from perturbative quantum field theory and holography.

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arXiv2019

Double copy structure and the flat space limit of conformal correlators in even dimensions

We analyse the flat space limit of 3-point correlators in momentum space for general conformal field theories in even spacetime dimensions, and show they exhibit a double copy structure similar to that found in odd dimensions. In even dimensions, the situation is more complicated because correlators contain branch cuts and divergences which need to be renormalised. We describe the analytic continuation of momenta required to extract the flat space limit, and show that the flat space limit is encoded in the leading singularity of a 1-loop triangle integral which serves as a master integral for 3-point correlators in even dimensions. We then give a detailed analysis of the renormalised correlators in four dimensions where the flat space limit of stress tensor correlators are controlled by the coefficients in the trace anomaly.

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arXiv2019

Double copy structure of CFT correlators

We consider the momentum-space 3-point correlators of currents, stress tensors and marginal scalar operators in general odd-dimensional conformal field theories. We show that the flat space limit of these correlators is spanned by gauge and gravitational scattering amplitudes in one higher dimension which are related by a double copy. Moreover, we recast three-dimensional CFT correlators in terms of tree-level Feynman diagrams without energy conservation, suggesting double copy structure beyond the flat space limit.

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arXiv2018

Implications of conformal invariance in momentum space

We present a comprehensive analysis of the implications of conformal invariance for 3-point functions of the stress-energy tensor, conserved currents and scalar operators in general dimension and in momentum space. Our starting point is a novel and very effective decomposition of tensor correlators which reduces their computation to that of a number of scalar form factors. For example, the most general 3-point function of a conserved and traceless stress-energy tensor is determined by only five form factors. Dilatations and special conformal Ward identities then impose additional conditions on these form factors. The special conformal Ward identities become a set of first and second order differential equations, whose general solution is given in terms of integrals involving a product of three Bessel functions (`triple-K integrals'). All in all, the correlators are completely determined up to a number of constants, in agreement with well-known position space results. We develop systematic methods for explicitly evaluating the triple-K integrals. In odd dimensions they are given in terms of elementary functions while in even dimensions the results involve dilogarithms. In some cases, the triple-K integrals diverge and subtractions are necessary and we show how such subtractions are related to conformal anomalies. This paper contains two parts that can be read independently of each other. In the first part, we explain the method that leads to the solution for the correlators in terms of triple-K integrals and how to evaluate these integrals, while the second part contains a self-contained presentation of all results. Readers interested only in results may directly consult the second part of the paper.

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arXiv2018

Renormalised CFT 3-point functions of scalars, currents and stress tensors

We discuss the renormalisation of mixed 3-point functions involving tensorial and scalar operators in conformal field theories of general dimension. In previous work we analysed correlators of either purely scalar or purely tensorial operators, in each case finding new features and new complications: for scalar correlators, renormalisation leads to beta functions, novel conformal anomalies of type B, and unexpected analytic structure in momentum space; for correlators of stress tensors and/or conserved currents, beta functions vanish but anomalies of both type B and type A (associated with a $0/0$ structure) are present. Mixed correlators combine all these features: beta functions and anomalies of type B, plus the possibility of new type A anomalies. Following a non-perturbative and general momentum-space analysis, we present explicit results in dimensions $d=3,4$ for all renormalised 3-point functions of stress tensors, conserved currents and scalars of dimensions $Δ=d$ and $Δ=d-2$. We identify all anomalies and beta functions, and explain the form of the anomalous conformal Ward identities. In $d=3$, we find a $0/0$ structure but the corresponding type A anomaly turns out to be trivial. In addition, the correlators of two currents and a scalar, and of two stress tensors and a scalar, both feature universal tensor structures that are independent of the scalar dimension and vanish for opposite helicities.

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arXiv2018

Evaluation of conformal integrals

We present a comprehensive method for the evaluation of a vast class of integrals representing 3-point functions of conformal field theories in momentum space. The method leads to analytic, closed-form expressions for all scalar and tensorial 3-point functions of operators with integer dimensions in any spacetime dimension. In particular, this encompasses all 3-point functions of the stress tensor, conserved currents and marginal scalar operators.

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arXiv2016

Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies

We present a comprehensive discussion of renormalisation of 3-point functions of scalar operators in conformal field theories in general dimension. We have previously shown that conformal symmetry uniquely determines the momentum-space 3-point functions in terms of certain integrals involving a product of three Bessel functions (triple-K integrals). The triple-K integrals diverge when the dimensions of operators satisfy certain relations and we discuss how to obtain renormalised 3-point functions in all cases. There are three different types of divergences: ultralocal, semilocal and nonlocal, and a given divergent triple-K integral may have any combination of them. Ultralocal divergences may be removed using local counterterms and this results in new conformal anomalies. Semilocal divergences may be removed by renormalising the sources, and this results in CFT correlators that satisfy Callan-Symanzik equations with beta functions. In the case of non-local divergences, it is the triple-K representation that is singular, not the 3-point function. Here, the CFT correlator is the coefficient of the leading nonlocal singularity, which satisfies all the expected conformal Ward identities. Such correlators exhibit enhanced symmetry: they are also invariant under dual conformal transformations where the momenta play the role of coordinates. When both anomalies and beta functions are present the correlators exhibit novel analytic structure containing products of logarithms of momenta. We illustrate our discussion with numerous examples, including free field realisations and AdS/CFT computations.

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arXiv2015

Soft limits in holographic cosmology

We study the soft limits of cosmological correlators from a holographic perspective, showing how the inflationary consistency relations arise from the diffeomorphism invariance of the dual quantum field theory. Starting from the corresponding Ward identity, by taking moments we derive the leading and subleading behaviour of the stress tensor 3-point function in the limit as one momentum vanishes. These results are non-perturbative and valid in quantum field theories of a very general nature. Exploiting the known mapping of correlators in the dual quantum field theory to those of the cosmology, we then obtain the leading and subleading soft behaviour of all cosmological 3-point correlators of curvature perturbations and gravitons. Our results thus provide a holographic derivation of all leading and subleading consistency relations for cosmological 3-point functions, and our method is easily generalised. We verify our results explicitly for slow-roll inflation and for strongly coupled holographic cosmologies with a perturbative dual description.

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arXiv2013

On the power spectrum of inflationary cosmologies dual to a deformed CFT

We analyse slow-roll inflationary cosmologies that are holographically dual to a three-dimensional conformal field theory deformed by a nearly marginal scalar operator. We show the cosmological power spectrum is inversely proportional to the spectral density associated with the 2-point function of the trace of the stress tensor in the deformed CFT. Computing this quantity using second-order conformal perturbation theory, we obtain a holographic power spectrum in exact agreement with the expected inflationary power spectrum to second order in slow roll.

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arXiv2012

Holography for inflation using conformal perturbation theory

We provide a precise and quantitative holographic description of a class of inflationary slow-roll models. The dual QFT is a deformation of a three-dimensional CFT by a nearly marginal operator, which, in the models we consider, generates an RG flow to a nearby IR fixed point. These models describe hilltop inflation, where the inflaton rolls from a local maximum of the potential in the infinite past (corresponding to the IR fixed point of the dual QFT) to reach a nearby local minimum in the infinite future (corresponding to the UV of the dual QFT). Through purely holographic means, we compute the spectra and bispectra of scalar and tensor cosmological perturbations. The QFT correlators to which these observables map holographically may be calculated using conformal perturbation theory, even when the dual QFT is strongly coupled. Both the spectra and the bispectra may be expressed this way in terms of CFT correlators that are fixed, up to a few constants, by conformal invariance. The form of slow-roll inflationary correlators is thus determined by the perturbative breaking of the de Sitter isometries away from the fixed point. Setting the constants to their values obtained by AdS/CFT at the fixed point, we find exact agreement with known expressions for the slow-roll power spectra and non-Gaussianities.

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arXiv2012

The relativistic fluid dual to vacuum Einstein gravity

We present a construction of a (d+2)-dimensional Ricci-flat metric corresponding to a (d+1)-dimensional relativistic fluid, representing holographically the hydrodynamic regime of a (putative) dual theory. We show how to obtain the metric to arbitrarily high order using a relativistic gradient expansion, and explicitly carry out the computation to second order. The fluid has zero energy density in equilibrium, which implies incompressibility at first order in gradients, and its stress tensor (both at and away from equilibrium) satisfies a quadratic constraint, which determines its energy density away from equilibrium. The entire dynamics to second order is encoded in one first order and six second order transport coefficients, which we compute. We classify entropy currents with non-negative divergence at second order in relativistic gradients. We then verify that the entropy current obtained by pulling back to the fluid surface the area form at the null horizon indeed has a non-negative divergence. We show that there are distinct near-horizon scaling limits that are equivalent either to the relativistic gradient expansion we discuss here, or to the non-relativistic expansion associated with the Navier-Stokes equations discussed in previous works. The latter expansion may be recovered from the present relativistic expansion upon taking a specific non-relativistic limit.

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arXiv2012

The holographic fluid dual to vacuum Einstein gravity

We present an algorithm for systematically reconstructing a solution of the (d+2)-dimensional vacuum Einstein equations from a (d+1)-dimensional fluid, extending the non-relativistic hydrodynamic expansion of Bredberg et al in arXiv:1101.2451 to arbitrary order. The fluid satisfies equations of motion which are the incompressible Navier-Stokes equations, corrected by specific higher derivative terms. The uniqueness and regularity of this solution is established to all orders and explicit results are given for the bulk metric and the stress tensor of the dual fluid through fifth order in the hydrodynamic expansion. We establish the validity of a relativistic hydrodynamic description for the dual fluid, which has the unusual property of having a vanishing equilibrium energy density. The gravitational results are used to identify transport coefficients of the dual fluid, which also obeys an interesting and exact constraint on its stress tensor. We propose novel Lagrangian models which realise key properties of the holographic fluid.

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