AdS/CFT 4-point functions: How to succeed at z-integrals without really trying

Daniel Z. Freedman; Eric D'Hoker; Leonardo Rastelli

REVIEW

AdS/CFT 4-point functions: How to succeed at z-integrals without really trying

Eric D'Hoker, Daniel Z. Freedman, Leonardo Rastelli

High Energy Physics - Theory · Submitted 6 May 1999 · 19 references

Pith's verdict: A genuinely useful computational trick, cleanly executed for scalars and gravitons, with the vector uniqueness step and the n-point section left thinner than the rest.

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arXiv IDhep-th/9905049 (version 1)

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CitationDaniel Z. Freedman, Eric D'Hoker, Leonardo Rastelli. "AdS/CFT 4-point functions: How to succeed at z-integrals without really trying." arXiv:hep-th/9905049 (1999). https://pith.science/paper/6UKA2E3S

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The thing worth knowing: this replaces the ugly part of holographic four-point exchange calculations. Instead of expanding the bulk-to-bulk propagator in hypergeometric series and resumming, you hit the z-integral with the propagator's own wave equation, get an inhomogeneous ODE in t = w0²/w², and solve it by a downward recursion that terminates for exactly the trilinear couplings that show up in AdS5×S5. The propagator never has to be written down. That is a real simplification, not a cosmetic one, and it is the kind of paper people will actually use rather than cite and forget. The scalar case is done properly: ODE (2.20), recursion (2.22–2.24), and a genuine uniqueness proof via the two homogeneous solutions (2.26–2.27), with f1 killed by the t→1 singularity and f2 by the wrong t→0 scaling. The graviton section is the nicest part — an overdetermined four-equation system that collapses consistently to (4.15) and (4.19), which is a real check rather than a hand-wave. Reproducing [6] and [9] by a different route is a legitimate benchmark even though the author list overlaps; the method is independent, so the agreement means something. The observation that termination happens precisely for the supergravity couplings, with no fundamental explanation offered, is stated honestly as an open question and I'd leave it at that. Soft spots, in order of how much they matter. The vector uniqueness step (Sec. 3, after Eq. 3.15) is asserted by analogy — 'it is easy to check' — with no homogeneous solutions exhibited. For massless vectors the [6] cross-check covers it. For the massive-vector results, which are the new physics here, nothing external covers it. The gap is narrow (same hypergeometric structure, same x=1/t substitution should work) but it is exactly where the new claims live, so it should be filled in. Second, Section 5 is thinner than the rest: the recursion coefficients (5.12–5.14) are given, but the Feynman parameter integrals are not carried out and the claim that the polynomiality conditions are 'precisely obeyed' via R-symmetry is asserted rather than shown. Third, the d=2 massless cases are explicitly deferred, which the authors flag themselves. None of this threatens the central method. It's a good, compact methods paper for people doing explicit AdS/CFT correlator computations; the derivations are checkable and the results are usable. Send it out.

Referee report

Significance

If correct, the method removes the most cumbersome step in a well-studied class of AdS/CFT calculations and makes higher-point and massive-exchange amplitudes accessible. The paper reproduces the known scalar, massless gauge boson and graviton results by an independent route, which is a meaningful check, and the observation that the supergravity couplings are exactly those for which the recursion terminates is a genuinely interesting structural fact. The derivations are explicit and checkable by hand; the scalar case is complete, and the graviton case includes a nontrivial compatibility check of an overdetermined system (Eqs. 4.15-4.19). The main caveats are that the vector uniqueness step is asserted rather than proved and that the headline 'finite sum of quartic graphs' statement is written out explicitly only for scalars.

Major objections (3)

[Section 3, paragraph following Eq. (3.15)] The text states: 'It is easy to check, in analogy with the scalar case, that if this condition is obeyed, (3.12-3.15) define the unique particular solution of (3.11) with the correct asymptotic properties.' For scalars this is proved: the two homogeneous solutions f1 (Eq. 2.26) and f2 (Eq. 2.27) are exhibited, f1 is rejected for its singularity at t=1 and f2 for its t->0 scaling. No analogous analysis is given for (3.11). The massless-vector comparison with [6] mitigates this, but the paper's new results are for massive vectors (m^2=(l-1)(l+1), l odd), for which no independent cross-check exists. Since the ODE is of the same hypergeometric type, the substitution x=1/t should reduce (3.11) to an equation whose two solutions can be analyzed as in Eqs. (2.25)-(2.27); please supply that short analysis, or state explicitly that the massive-vector results rest on the analogy.
[Abstract (s3) and Eq. (2.28); Sections 3-4] The claim that 'the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices' is demonstrated explicitly only for scalar exchange, where Eq. (2.28) exhibits the sum. For vector and graviton exchange, Sections 3 and 4 establish that the scalar coefficient functions f(t) and phi(t) are terminating polynomials, but the final step - assembling the w-integral into a finite sum of quartic scalar graphs - is not written out. Since this is the paper's headline structural claim, please either give the explicit reduction for at least one of the vector or graviton cases, or add a sentence explaining how the polynomial form of f(t) and phi(t) plus the structure of the w-integral produces the quartic-graph sum.
[Section 5 and abstract] The abstract states that 'the analogous integrals in n-point scalar diagrams for n>4 are also evaluated,' but Section 5 stops at S(w;delta;mu) in Eqs. (5.11)-(5.14) and leaves the Feynman-parameter integrals over alpha_i undone ('the Feynman integrals to be calculated are of a standard type'). In addition, the assertion that the polynomial condition delta-Delta-2 = 2 l0 is 'precisely obeyed thanks to the R-symmetry selection rules of AdS5 x S5 supergravity' is not demonstrated for n>4. Please either carry out one explicit n>4 example or soften the abstract to say that the z-integral is reduced to standard Feynman integrals.

Minor objections (8)

[Section 1 and Sections 3-4] The d=2 subtleties are acknowledged ('left to future work'), and reappear as the divergence of a_0 for m^2=0, d=2 (Section 3) and the ln(t-1) in Eq. (4.24). Since the abstract does not restrict d, a sentence in the abstract or conclusion stating that the results apply to d>2 would prevent misapplication.
[Section 2, after Eq. (2.28); Section 4, after Eq. (4.25)] 'wich' should be 'which', and 'coincise' should be 'concise'.
[Eqs. (2.21)-(2.24)] The text says k = Delta3 + l with l integer, but the recursion is used downward from a_{Delta3-1}; it would help to state explicitly that the sum runs over k = Delta3-1-j, j=0,...,(Delta3-1-k_min), and that Delta3 need not be an integer.
[Abstract and Section 2] 'Precisely for the cases that occur' suggests an iff. The paper shows that supergravity couplings satisfy the termination condition and that generic couplings give infinite sums, but does not show that every non-supergravity coupling fails. Suggest 'for the cases that occur ... the series terminates' rather than 'precisely'.
[Section 4, Eq. (4.7)] The pure-diffeomorphism terms D_mu D_nu X and D_{(mu}(delta_{nu)0}/w0 Y) are stated to drop out of the final w-integral; one more sentence making the argument explicit (the second conserved stress tensor) would help the reader.
[Section 4, comparison with [9]] 'Agreement is perfect after different normalizations are taken into account' - please give the relative normalization factor, since the reader cannot easily reconstruct it.
[Reference [8]] Reference [8] is listed as 'to appear in Nucl. Phys. B'; please update the citation status.
[Section 1] 'Left to future work (hopefully by someone else)' is informal for a journal; suggest rephrasing.

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The paper

The paper claims

A single differential equation replaces the hard half of every AdS exchange diagram, and for the supergravity couplings the answer is a finite polynomial.

The reading

Computing a four-point function in the AdS/CFT correspondence from an exchange graph requires two integrals over the bulk Anti-de Sitter space. The paper's target is the first of these, the $z$-integral, which previously had to be done by expanding the bulk-to-bulk propagator in a hypergeometric series, integrating term by term with Feynman parameters, and then recognising and resumming the resulting series. The authors show that this integral, as a function of the remaining bulk point $w$, is fixed by an inhomogeneous second-order ordinary differential equation in the variable $t = w_0^2/w^2$, and that a particular solution with the right behaviour at $t=0$ and $t=1$ can be built by a downward recursion on the coefficients of a power series in $t$. What makes this useful rather than merely tidier is that for the trilinear couplings that actually occur in the $AdS_5$ times $S^5$ compactification of Type IIB supergravity, the recursion terminates: the $z$-integral becomes a finite polynomial, and the whole exchange graph collapses into a finite sum of ordinary quartic scalar graphs whose asymptotics were already known.

What carries the argument

The carrier of the argument is the conversion of a bulk integral into an ODE by hitting it with the propagator's own wave operator. Because the bulk-to-bulk propagator satisfies $(-2k + m^2)G = 2k(z,w)$, applying that operator to $I(w) = 2k G(u) K(z,x_1) K(z,x_3)$ replaces the propagator by a delta function and leaves the product of the two bulk-to-boundary propagators as an explicit source. Scale and Poincar'e invariance fix the ansatz $I(w) = w_0^{2k} f(t)$ with $t = w_0^2/w^2$, so the integral is captured by one scalar function of one variable. The boundary conditions that single out the physical solution are regularity at $t=1$ (the bulk point is not at the boundary) and the power-law falloff $f(t) 2k t^{(2k-2k)/2}$ as $t 2k 0$, which follows from the small-$w_0$ behaviour of the propagator. For the scalar case the authors prove uniqueness by exhibiting the two homogeneous solutions and showing that one is singular at $t=1$ while the other has the wrong small-$t$ power. For vectors the same reduction is applied to $I_2k(w) = 2k w_2k/w^2 f(t)$, with a divergence argument used first to eliminate the second allowed tensor structure; for gravitons the ansatz carries $g_{2k2k} h(t)$ and $2k_{2k0}2k_{2k0} w_0^2 2k(t)$, and compatibility of the overdetermined system of four tensor equations reduces to a single first-order equation for $2k(t)$.

Extended reading notes

Core claim

The central claim is that the $z$-integral of an exchange diagram obeys a simple inhomogeneous differential equation that can be solved recursively, without ever writing down the bulk-to-bulk propagator in closed form. Applying the bulk wave operator to the integral and using the propagator's defining equation turns the problem into an ODE: for scalar exchange, $4t^2(t-1)f'' + 4t[(1+2k)t - 2k + d/2 - 1]f' + [2k(d-2k) + m^2]f = t^{2k}$-type equations with $k = (2k)/(2)$, and analogous equations for vector and graviton exchange. The authors set $f(t) = 2k t^k$ and run the recursion downward from a top coefficient, finding that the series terminates at a positive lower power precisely when the dimensions satisfy a condition: for scalars, that $2k$ be a positive even integer, and for vectors and gravitons, that a corresponding difference of dimensions be a non-negative integer. The authors then observe that these conditions are met for every trilinear coupling appearing in the $AdS_5$ times $S^5$ supergravity, which they attribute to $SU(4)$ selection rules and the standard triangle inequality. The consequence is the headline result: exchange diagrams in that theory reduce to a finite sum of graphs with quartic scalar vertices. The method reproduces earlier scalar, massless gauge boson and graviton results, gives new results for massive vector exchange, and extends to $n$-point scalar diagrams, where the source is Feynman-parametrised and the same recursion yields a closed-form coefficient $C_2k(2k)$.

Load-bearing premise

The load-bearing premise is that the terminating power series produced by the downward recursion is the unique solution with the required behaviour at $t=0$ and $t=1$; the authors prove this explicitly for scalar exchange by exhibiting the two homogeneous solutions, but for vector exchange they assert it by analogy with the scalar case rather than demonstrating it.

Editorial extensions

If this is right

Every previously published scalar, massless gauge boson and graviton exchange four-point amplitude in the AdS/CFT correspondence is recovered by a uniform short computation, with the bulk-to-bulk propagator never needed explicitly.
New massive vector exchange amplitudes follow, with the allowed Kaluza-Klein masses $m^2 = (\ell-1)(\ell+1)$ of the $AdS_5 \times S^5$ spectrum, and the terminating condition forces $\ell$ odd, matching the $SU(4)$ quadrality selection rule.
The exchange graph in the supergravity theory is exactly a finite sum of quartic scalar contact graphs, so its conformal-block asymptotics can be read off from already-derived quartic-graph expansions rather than computed afresh.
The same recursion handles $n$-point scalar diagrams with $n > 4$, giving a closed-form coefficient $C_\ell(\mu)$ in a finite sum of powers of $w_0/(w^2+\mu^2)$, which is what higher-order $1/N$ corrections require.
The method is stated to be general in $d$ and in the scaling dimensions, so it should apply beyond the specific $d=4$, $\mathcal{N}=4$ case that motivates the paper.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the terminating condition is really a consequence of $SU(4)$ selection rules rather than a coincidence of the cases checked, then the same collapse should occur in other supergravity compactifications with enough R-symmetry, which would give a cheap diagnostic for which holographic theories have polynomial exchange amplitudes.
The fact that generic couplings give an infinite series while supergravity couplings give a finite sum suggests the termination is a representation-theoretic statement about which conformal partial waves can appear in a three-point coupling, not merely an algebraic accident of the recursion.
Because the reduced answer is a finite sum of quartic graphs, numerical evaluation of exchange contributions to four-point functions in $\mathcal{N}=4$ super Yang-Mills at strong coupling becomes a sum of already-tabulated integrals, which lowers the cost of comparing strong-coupling predictions with weak-coupling perturbation theory.
The paper explicitly leaves the $d=2$ massless vector and graviton cases aside, where the recursion coefficient diverges and a logarithm appears; those cases are where a two-dimensional dual CFT would test the method most sharply.

Original abstract

A new method is discussed which vastly simplifies one of the two integrals over AdS(d+1) required to compute exchange graphs for 4-point functions of scalars in the AdS/CFT correspondence. The explicit form of the bulk-to-bulk propagator is not required. Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors. It is found that precisely for the cases that occur in the AdS(5) X S(5) compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices. The analogous integrals in n-point scalar diagrams for n>4 are also evaluated.

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What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

A new method is discussed which vastly simplifies one of the two integrals over AdS(d+1) required to compute exchange graphs for 4-point functions of scalars in the AdS/CFT correspondence. The explicit form of the bulk-to-bulk propagator is not required. Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors. It is found that precisely for the cases that occur in the AdS(5) X S(5) compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices. The analogous integrals in n-point scalar diagrams for n>4 are also evaluated.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A new method is discussed which vastly simplifies one of the two integrals over AdS(d+1) required to compute exchange graphs for 4-point functions of scalars in the AdS/CFT correspondence.	contribution 0.92	0.85	0.47	asserted 1.00
s1	The explicit form of the bulk-to-bulk propagator is not required.	method 0.64	0.67	0.08	asserted 1.00
s2	Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors.	result 0.74	0.88	0.12	asserted 1.00
s3	It is found that precisely for the cases that occur in the AdS(5) X S(5) compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices.	result 0.99	0.92	0.37	asserted 1.00
s4	The analogous integrals in n-point scalar diagrams for n>4 are also evaluated.	contribution 0.66	0.91	0.07	asserted 0.99

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 1.00.

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not examined

full text
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relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** A general t–channel exchange diagram. 2 Scalar exchange As in most past work, we calculate on the Euclidean continuation of AdSd+1, which is modelled as the upper half space zµ ∈ Rd+1, with z0 > 0, and metric gµν of constant negative curvature R = −d(d + 1), given by ds2 = X d µ,ν=0 gµνdzµdzν = 1 z 2 0 (dz2 0 + X d i=1 dz2 i ). (2.1) The Christoffel symbols are Γ κ µν = 1 z0 (δ κ 0 δµν − δµ0δ κ ν − δν0δ κ µ ). (2.2) It is well-known that AdS–invariant functions, such as scalar propagators, are simply expressed [13] as functions of the chordal distance u,

Figures reproduced from arXiv:hep-th/9905049 by the authors.

References

Works cited (19)

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D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, “ Comments on 4 point functions in the CFT/AdS correspondence,” Phys. Lett. B452, 61 (1999), hep-th/9808006

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H. Liu, ‘Scattering in Anti–de Sitter space and operator product expansion’, hep– th/9811152. 16

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E. D’Hoker and D. Z. Freedman, ‘General scalar exchange i n AdSd+1’, hep–th/9811257, to appear in Nucl. Phys. B

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P. Breitenlohner and D.Z. Freedman, “Positive Energy In Anti-De Sitter Backgrounds And Gauged Extended Supergravity,”Phys. Lett. 115B, 197 (1982); “Stability In Gauged Ex- tended Supergravity,” Ann. Phys.144, 249 (1982). L. Mezincescu and P.K. Townsend, ‘Stability at a Local Maxim um in Higher Dimensional anti–de Sitter Space and Applications to Supergrav...

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All Holographic Four-Point Functions in All Maximally Supersymmetric CFTs
hep-th 2020-06 conditional novelty 6.0 of 10

A new 'maximally R-symmetry violating' limit of holographic Mellin amplitudes determines all tree-level four-point functions of half-BPS operators in maximally supersymmetric CFTs in 3, 4, and 6 dimensions.
Towards the higher point holographic momentum space amplitudes II: Gravitons
hep-th 2019-08 conditional novelty 6.0 of 10

A method that expresses AdS4 graviton Witten diagrams as a differential operator acting on one scalar factor, with explicit three, four, and five point results.
Holographic correlators in AdS$_3$
hep-th 2018-12 conditional novelty 6.0 of 10

The authors derive the first holographic four-point correlators of single-trace operators in AdS3/CFT2, for half-BPS dimension-(1/2,1/2) operators of the D1D5 CFT, by extrapolating heavy-heavy-light-light results and ...
Recursion Relations in Witten Diagrams and Conformal Partial Waves
hep-th 2018-12 accept novelty 6.0 of 10

An infinite set of linear recursion relations among crossed-channel conformal block coefficients of exchange Witten diagrams yields a recursive algorithm that fixes all subleading double-trace OPE coefficients from le...
Shockwaves from the Operator Product Expansion
hep-th 2017-09 conditional novelty 6.0 of 10

The Regge-limit OPE of two scalars is the AdS shockwave operator, and the CFT chaos bound reproduces the Engelhardt-Fischetti bulk causality condition.
Spinning AdS Loop Diagrams: Two Point Functions
hep-th 2017-08 accept novelty 6.0 of 10

A spectral decomposition of spinning propagators reduces AdS one-loop two-point Witten diagrams to Mellin-Barnes spectral integrals times boundary conformal integrals, and shows that bubble diagrams alone miss the ano...
Holographic 4-point correlators with heavy states
hep-th 2017-05 conditional novelty 6.0 of 10

A holographic four-point correlator with two heavy and two light states in AdS3/CFT2 is computed at strong coupling and shown to differ from the free orbifold result, establishing that it is not protected.
The Mellin Formalism for Boundary CFT$_d$
hep-th 2017-05 accept novelty 6.0 of 10

The Mellin representation of conformal correlators is extended to boundary and interface CFTs, and geodesic Witten diagrams in a sliced AdS geometry are shown to reproduce the corresponding conformal blocks.
Four Point Functions of the Stress Tensor and Conserved Currents in AdS_4/CFT_3
hep-th 2012-01 conditional novelty 6.0 of 10

The first explicit four-point stress-tensor correlator in AdS4/CFT3 is derived via recursion relations and shown to reproduce the flat-space MHV graviton amplitude in the appropriate limit.
Towards Feynman rules for Mellin amplitudes in AdS/CFT
hep-th 2011-07 conditional novelty 6.0 of 10

Tree-level Witten diagram correlators in AdS/CFT, rewritten in Mellin space, appear to obey Feynman-like rules whose vertices are Lauricella hypergeometric functions, checked through six-point scalar and four-current ...
Defect Conformal Field Theory and Locally Localized Gravity
hep-th 2003-03 accept novelty 6.0 of 10

Bulk mode expansion in AdS equals the boundary operator product expansion at a defect, and local localization of gravity corresponds to a reduced operator in that expansion acquiring a negative anomalous dimension.
Carrollian Amplitudes from Holographic Correlators
hep-th 2024-06 accept novelty 5.0 of 10

The flat-space limit of AdS boundary correlators, taken in Bondi coordinates, yields Carrollian scattering amplitudes, with the bulk flat limit matching a boundary Carrollian limit.
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hep-th 2020-01 conditional novelty 5.0 of 10

The Arkani-Hamed-Benincasa-Postnikov algorithm transfers to AdS transition amplitudes for conformally coupled scalars, giving closed-form tree-level results and a cutoff-regularized one-loop example.
New relation for AdS amplitudes
hep-th 2019-04 conditional novelty 5.0 of 10

Tree-level gauge-boson Witten diagrams in AdS4 momentum space can be computed algebraically by combining a known cutting algorithm for straight-line scalar graphs with a new k to 0 limit that generates crossed-line graphs.
Towards the higher point holographic momentum space amplitudes
hep-th 2018-10 conditional novelty 5.0 of 10

Explicit momentum-space expressions are given for tree-level four- and five-point vector Witten diagrams in AdS4, extending Raju's recursion-relation results to higher points.
S-matrix singularities and CFT correlation functions
hep-th 2017-02 conditional novelty 5.0 of 10

Imposing mass conservation at each cubic vertex makes the factorization singularities of flat-space S-matrix diagrams localize all AdS3 bulk points to a single point, reproducing contact Witten diagrams.
Dirichlet, Neumann, Mixed and self-dual holography: (self-dual) Yang--Mills theory II
hep-th 2026-06 unverdicted novelty 4.0 of 10

Derives bulk and boundary propagators and computes 3- and 4-point correlators for YM, CS and SDYM in AdS/CFT with multiple boundary conditions to relate their observables.
Large N Field Theories, String Theory and Gravity
hep-th 1999-05 accept novelty 3.0 of 10

The review argues and assembles the evidence that N=4 supersymmetric Yang-Mills theory in four dimensions is dual to type IIB string theory on AdS5 x S5, and surveys what the duality predicts.
Pre-Strings Lectures on Holographic Correlators and Analytic Bootstrap
hep-th 2026-09 unverdicted novelty 2.0 of 10

A pedagogical review and synthesis of analytic bootstrap techniques for holographic correlators, with extensions to higher-point functions, loops, string corrections, and defect CFTs.
Fuzzballs and Microstate Geometries: Black-Hole Structure in String Theory
hep-th 2022-04 conditional novelty 2.0 of 10

String theory replaces black holes with horizonless fuzzballs, and this, the authors argue, resolves the black-hole information paradox.

How later papers used it

86 later papers cite it

Evidence: substantial. 84 of the citing papers share no author with it, and 28 of those carry a confident label from Pith's citation classifier.

Pith read 29 of the 143 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

9 later papers by other authors are listed below: 2 built on it and 7 used its method or tool.

Builds on it 2 of 3 papers with this label

Extended this paper's method to calculate the integral for general conformal dimensions.

Aspects of higher-point functions in BCFT$_d$ (2023) · 0.93 · paraphrased: the licence does not allow quoting
Generalized this paper's calculation of the vector z-integral to include the case when the vector field couples to scalar fields of different mass.

Four-point functions of different-weight operators in the AdS/CFT correspondence (2007) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 7 of 26 papers with this label

Used this paper's result to find that the cubic integral involving a graviton field reduces to a sum of contact vertices.

$20'$ Five-Point Function from $AdS_5\times S^5$ Supergravity (2019) · 0.80 · paraphrased: the licence does not allow quoting
Following this paper, wrote down the expression for the diagram in d=4.

Shockwaves from the Operator Product Expansion (2017) · 0.81 · paraphrased: the licence does not allow quoting
Following this paper, performed a translation followed by a conformal inversion in the three-point integral.

Recursion Relations in Witten Diagrams and Conformal Partial Waves (2018) · 0.99 · paraphrased: the licence does not allow quoting
From those without derivatives we have [equation] The quartic vertex with derivatives gives [equation] which, using the identity [ref] in the appendix [ref], becomes [equation] The exchange diagrams are [equation] These exchange diagrams can be related to simpler contact integrals through a differential operator [ref] (see [ref] for recent reviews) : [equation] where $C_(2)^ij$ is the quadratic Casimir acting on the two boundary points $i$ and $j$.

Bootstrapping line defects in AdS$_3$/CFT$_2$ (2024) · Correlators from fluctuations $ϕ_tr$ · 0.98
Using the results is [ref] for a scalar exchange we write (X, P_1, P_2) =⟨Φ(P_1) ϵ(X) Φ^*(P_2) ⟩∝1/(-2P_1·P_2)^Δf_ϵ( -P_1·P_2/2 X ·P_1 X ·P_2) , [] with f_ϵ(t) =Γ(Δ_ϵ-3/2+Δ)Γ(Δ-Δ_ϵ/2)/4Γ(Δ)^2[t^Δ_ϵ/2Γ(Δ_ϵ/2)^2_2F̃_1(Δ_ϵ/2,Δ_ϵ/2,Δ_ϵ-1/2,t) -t^ΔΓ(Δ)^2_3F̃_2 ( [ 1, Δ, Δ; Δ-Δ_ϵ/2+1, Δ+Δ_ϵ-1/2 ] ; t )] , [] where we have defined the regularized Hypergeometric function _pF̃_q ( [ a_1, ⋯, a_p; b_1, ⋯, b_q; ] ; t ) ≡1/Γ(b_1)⋯Γ(b_q)_pF_q ( [ a_1, ⋯, a_p; b_1, ⋯, b_q; ] ; t ) .

Holography and Regge Phases with $U(1)$ Charge (2024) · Quantitative Analysis at $Q=2, 3,4$ · 0.95
Used the formulae in this paper to get its result.

How to Succeed at Witten Diagram Recursions without Really Trying (2020) · 0.87 · paraphrased: the licence does not allow quoting
Expressed exchange and contact Witten diagrams in terms of D-functions using this paper's method.

One-loop Gluon Amplitudes in AdS (2021) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

80 arXiv papers cite this in 135 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 64 sentences without a confident label

Contests it 1 paper

The standard method of [ref] is not applicable for these diagrams.[If one naively applies the method of [ref] , one finds the answer is divergent.

$\mathbf{AdS_3\times S^3}$ Tree-Level Correlators: Hidden Six-Dimensional Conformal Symmetry · The position space algorithm · contest 0.78

Builds on it 3 papers

In this appendix we extend the method of [ref] to higher-point Witten diagrams with more than one internal line, and evaluate the various diagrams that appear in Section [ref].

$20'$ Five-Point Function from $AdS_5\times S^5$ Supergravity · Integrating out an internal line · 1.00
Here we extend the computation in [ref] using the method of [ref] for the massive symmetric tensor $z$-integral when the tensor field is coupled to scalar fields of different mass.

Four-point functions of different-weight operators in the AdS/CFT correspondence · Symmetric Tensor Exchange · 0.99
In Section [ref] we defined the integral $A^μ(w,x_2,x_3)$ [equation] We extend the method of [ref] to calculate the integral for general conformal dimensions.

Aspects of higher-point functions in BCFT$_d$ · Integrated vertex identity for vector exchange · 0.93

Uses its method or tool 25 papers

Let $Δ_1=Δ_2=Δ_ext$, the cubic integral is [equation] Using the result of [ref] , we find that this integral reduces to the following sum of contact vertices [equation] where [equation]

$20'$ Five-Point Function from $AdS_5\times S^5$ Supergravity · Graviton exchange · 0.80
Following the technique of Refs. [ref] , we isolate the part of BasicDiag which depends on $P_1,P_2$ and $P_3$: Fdef F_Δ_L,Δ_C,Δ_R (P_1,P_2,P_3,Z^') ≡∫_Z∈γ_12 K̂_Δ_1(P_1,Z) K̂_Δ_2(P_2,Z) K̂_Δ_L(P_3,Z) (ξ(Z,Z^') 2)^Δ_C K̂_Δ_R(P_3,Z^') , and use its invariance under simultaneous $SO(n+1,1)$ rotations of $P_1,P_2,P_3$ and $Z^'$ to conclude Finvariance ( L^(1)_AB + L^(2)_AB+ L^(3)_AB ) F_Δ_L,Δ_C,Δ_R(P_1,P_2,P_3,Z^') = -L^(Z^')_AB F_Δ_L,Δ_C,Δ_R(P_1,P_2,P_3,Z^') , which immediately leads to FCasimir ( L^(1) + L^(2) + L^(3) )^2 F_Δ_L,Δ_C,Δ_R(P_1,P_2,P_3,Z^') = ( L^(Z^') )^2 F_Δ_L,Δ_C,Δ_R(P_1,P_2,P_3,Z^') .

Holographic dual of the five-point conformal block · Proof by conformal Casimir equation · 0.95
This is in contrast to the connected tree-diagram given in Fig. [ref]b, which can be reduced to a finite sum of contact diagrams following the method of [ref] .

Holographic correlators with multi-particle states · Correlators with multi-trace operators: the setup · 0.94

Compares with it 3 papers

The extra factor of $i$ arises due to Wick rotation and the extra factor of $(-1)$ is due to the different convention for the Green function (compare [ref] and (2.7) of [ref] ) ] [equation] In the limit $χ_12 →∞$, corresponding to the flat limit (see below [ref]), the bulk-to-bulk propagator behaves as [equation] We identified the chordal distance [ref] in the last equality.

Carrollian Amplitudes from Holographic Correlators · Bulk-to-bulk propagator · 0.86
This result is in perfect agreement with what was found by D'Hoker, Freedman and Rastelli in [ref] and also independently by Dolan and Osborn in [ref] .

Quantum Scalar Theories in the AdS/CFT Correspondence · 4-Point Function · 1.00
It would also be interesting to contrast this with the analysis of four-point in [ref] .

Revisiting Extremal Couplings in AdS/CFT · Summary and Discussion · 0.80

Cites it as background 24 papers

The $1/N$ expansion of the CFT correlators maps onto the perturbative expansion of the AdS QFT amplitudes [ref] ,

Dressing in AdS and a Conformal Bethe-Salpeter Equation · Introduction · 0.95
For example, the propagator for $ϕ$ is [equation] The solution $G_Δ$ is called bulk-to-bulk propagator, and it can be obtained in closed form [ref] [equation] Similarly, the propagator for $ϕ$ is called brane-to-brane propagator.

The Witten Diagram Bootstrap for Holographic Defects · Effective action and Witten diagrams · 0.81
Heavy scalar exchange contributions to CFT four-point functions have been studied using a variety of techniques (see e.g. [ref] ), but extracting information about anomalous dimensions has proven to be relatively difficult using the standard methods.

Effective Conformal Theory and the Flat-Space Limit of AdS · Dimensional Analysis with $n$ · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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