Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

From bulk loops to boundary large-N expansion

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The singular part of a one-loop Witten diagram with a two-particle cut is fixed entirely by on-shell tree-level subdiagrams, through a factorization identity that matches large-N CFT expectations.

desk verdict A solid bulk-side derivation of the double-trace cut factorization for one-loop Witten diagrams, with the higher-spin application still resting on a flagged locality assumption. read the letter →

arxiv 1908.03974 v2 pith:YZ77FRJM submitted 2019-08-11 hep-th

Dmitry Ponomarev This is my paper · ORCID
classification hep-th
keywords AdS/CFTWittendiagramsconformalpartialwavesblocksloopamplitudeslarge-NexpansionCutkoskyrulesdouble-traceoperators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that loop-level scattering amplitudes in anti-de Sitter space are not new data: cutting a one-loop four-point diagram along two internal lines leaves a product of tree-level amplitudes evaluated on shell, integrated over the cut particles' phase space. If true, AdS loop amplitudes inherit their singular structure from simpler tree-level building blocks, exactly as flat-space unitarity dictates. The paper derives the factorization identity in the conformal partial wave representation and checks that it reproduces the relations the boundary large-N expansion predicts.

What carries the argument

The machinery is the conformal partial wave expansion of bulk amplitudes combined with the split representation of bulk-to-bulk propagators and the bubble integral formula for integrated products of conformal partial waves. The split representation turns a cut propagator into an integral over boundary points and spectral parameters, so cutting two lines expresses the loop as an integrated product of tree-level subamplitude expansions. The bubble integral then collapses the pair of integrated boundary points back to a single partial wave, leaving a double spectral integral whose contour-pinching singularities are evaluated by residues; those residues are the factorization identity.

What would settle it

Evaluate a one-loop four-point amplitude in a bulk theory with a non-local vertex, e.g. an exponential of a d'Alembertian acting on the cut lines, and compare the residue at the double-trace location with the right side of (4.12). If the derivative factors produce additional contour pinches in $\nu_1,\nu_2$, the residue will differ and the factorization fails.

Watch

Extended reading notes

Core claim

The central claim is that the singular part of a one-loop four-point scalar amplitude in Euclidean AdS, associated with a double-particle cut, factorizes: it equals the product of the on-shell tree-level subamplitudes obtained by cutting the two propagators, with the cut particles integrated over their AdS phase space. In conformal block language this reads $a^{(0)}_O = a^{(0)}_L a^{(0)}_R / a_M$ for generic external dimensions, and in the degenerate (identical-operator) case the second-derivative coefficient satisfies $a^{(2)}_O = a^{(1)}_L a^{(1)}_R / a_M$, with $a_M$ the mean-field-theory coefficient. The paper proves this by representing the loop diagram through its conformal partial wave expansion, analyzing the spectral integrals' pinching singularities, and computing residues; the same analysis yields AdS Cutkosky rules in which a propagator is replaced by $\Pi_\Delta - \Pi_{d-\Delta}$.

Load-bearing premise

The tree-level coefficient functions $I_L$ and $I_R$ have all their $\nu_1,\nu_2$ singularities coming from three-point $b$-factors, so derivative corrections merely multiply by polynomials and create no new spectral poles; the paper notes this can fail for non-local theories.

Editorial extensions

If this is right

  • The double-cut singularity of any one-loop four-point scalar amplitude can be computed from tree-level data alone, without evaluating the full loop integral.
  • Generic-dimension double-trace coefficients obey $a^{(0)}_O = a^{(0)}_L a^{(0)}_R / a_M$; for identical external operators the second-derivative coefficient obeys $a^{(2)}_O = a^{(1)}_L a^{(1)}_R / a_M$.
  • The AdS cutting rule $\Pi_\Delta \to \Pi_\Delta - \Pi_{d-\Delta}$ gives the singular part of a diagram, with shadow artifacts that still need projection out.
  • By iterating the bubble reduction, higher-loop and higher-point cuts reduce to multiple-trace singularities whose coefficients are built from tree-level data in the same factorized way.
  • On the CFT side, order-$1/N^2$ double-trace data determines the leading singular (double-trace) part of the four-point function at order $1/N^4$, consistent with large-N bootstrap relations.

Reading between the lines

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

  • A testable extension: apply the same pinching analysis to a theory with an infinite-derivative (non-local) vertex; the paper's caveat predicts the factorization (4.12) can break exactly where derivative factors cease to be polynomial in the spectral parameters.
  • If the reconstruction-from-singularities program succeeds, bulk loop amplitudes could be assembled from their cut data plus Regge bounds, making direct two-loop integration unnecessary in practice.
  • For spinning fields, the mean-field sewing shortcut suggests $a^{(0)}_O = a^{(0)}_L a^{(0)}_R / a_M$ holds with spinning conformal block coefficients and the same inverse mean-field coefficient, but only for tensor structures present in mean field theory; explicit bubble integrals would settle the general case.
  • The shadow poles produced by the AdS Cutkosky replacement might be removed by continuing to Lorentzian signature or by monodromy projection; if they cannot be cleanly projected, the cutting rule needs refinement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper analyzes singularities of one-loop four-point Witten diagrams in Euclidean AdS using the conformal partial wave expansion. After expressing the loop diagram as a double spectral integral over tree-level subdiagrams via the split representation and the bubble integral (Eqs. (3.1)-(3.10)), the author studies pinch singularities of the integral and extracts the leading double-trace residues. The main result (Eqs. (4.7)-(4.9), translated to (4.12)-(4.14)) states that the singular part of the loop coefficient function factorizes into on-shell tree-level coefficient functions divided by the mean-field conformal block coefficient. The paper then compares this with the large-N OPE analysis on the boundary and discusses extensions to higher loops, spinning fields, AdS Cutkosky rules, and higher-spin theories.

Significance. If correct, this provides a purely bulk derivation of factorization relations that have been used extensively in the large-N bootstrap and holographic loop computations (e.g., [7,17,22]). The derivation is not circular: it uses analytic structure of spectral integrals rather than the large-N CFT relations as input, and a_M is a kinematical quantity, not a fitted parameter. The paper is careful to state what is captured ("less singular terms") and what is not (single-trace contributions, regular terms). The main limitation is that the proof relies on a locality assumption that is not verified for the non-local higher-spin theories discussed in Sec. 6.5; this conditionality is acknowledged in the text but not resolved.

major comments (2)
  1. [Section 4.1, Eqs. (4.1)-(4.7); Section 6.5] The factorization in (4.7)-(4.14) is derived under the assumption that all ν1,ν2-singularities of I_L and I_R are those of the three-point b-factors. The text correctly observes that derivative interactions are harmless in local theories, but explicitly allows that in non-local theories polynomial factors may sum to a singularity. Section 6.5 applies the formalism to higher-spin theories, which are non-local in the conventional sense (refs. [32,63]), and merely assumes that the off-shell extension has "the same analytic structure in ν1 and ν2". If this assumption fails, the pinching analysis in Section 4.1 and the residue formula (4.7) must be modified. The manuscript should either prove the required analytic structure for the non-local cases it claims to cover or restrict the statement of the main result to local bulk theories and present the higher-spin application as conditional. As written, the abstract's general claim of factorization goes beyond what is demonstrated.
  2. [Section 4.2, Eq. (4.7); Section 6.2, Eq. (6.4)] The status of shadow singularities in the proposed AdS Cutkosky rule is not fully specified. The text says that the explicit term also has shadow double-trace singularities that "should be ignored", and Appendix A shows for one toy integral that the deformed contour reproduces the physical residues despite extra shadow poles. However, no general prescription is given for identifying and removing the shadow contributions when (4.7)/(4.9) is used as an input to a loop computation, and Section 6.2 explicitly lists this removal as an open problem. Because the paper advertises (6.4) as the AdS version of Cutkosky rules, the rule is not yet a complete algorithm for computing the singular part of a general amplitude; this should be stated as a limitation rather than as a finished result.
minor comments (5)
  1. [Eq. (4.13)] Equation (4.13) appears to contain a typo: the right-hand side shows a^[0]_L times a^[0]_L, but consistency with (4.12) requires a^[0]_L times a^[0]_R.
  2. [Section 3.1, Eqs. (3.7)-(3.8)] The step from (3.7) to (3.8) is not shown in the text ("This is a straightforward computation"). Since this is a central formula, providing the intermediate Symanzik-star evaluation would improve reproducibility.
  3. [Section 4.4] The phrase "exact formula" for (4.13) is potentially overstatement; the derivation includes "less singular terms", so the text should say explicitly that the formula is exact for the coefficient of the leading double-trace singularity in the non-degenerate case.
  4. [Section 6.1] The higher-loop and higher-point extension is presented schematically without a complete pinching analysis for the multi-integral case. The text should state explicitly that this is an extrapolation of the one-loop argument rather than a fully worked proof.
  5. [Section 6.5] The "reasonable assumptions" under which (4.13) applies to higher-spin theories should be enumerated; the footnote about non-locality and the phrase "same analytic structure" are too brief for a reader wanting to apply the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bulk spectral-integral derivation is self-contained, and the CFT large-N relations are consistency checks rather than inputs.

full rationale

The paper's central claim (4.12)-(4.14) is derived from the conformal partial wave representation of the one-loop amplitude, not from the large-N boundary relations. The coefficient function (3.10) is an exact integral over tree-level coefficient functions, and Section 4 obtains the double-trace singularities by a standard pinching analysis of the spectral integrals, using the external bubble-integral result of [20]. The factorization (4.7)-(4.9) follows from residues at the propagator poles, not from any CFT input. The mean-field coefficient a_M appearing in (4.12) is a fixed kinematic quantity taken from [22,53]; it is not fitted to the loop amplitude or renamed as a prediction. Section 5 explicitly frames its CFT analysis as consistency: "we will review the CFT dual counterpart of this analysis... demonstrate that relations such as (4.13) and (4.14) are consistent with the standard large-N considerations." Thus the CFT relations are not used to derive the bulk result. The only assumption that could affect the generality of the result is the locality statement (4.1), which the paper itself flags: "unless we are dealing with a non-local theory, in which such polynomial terms may sum up to a singularity." This is an explicit condition, not a circular reduction. The self-references in Section 6.5, to [17] and [32], are used to identify the higher-spin application and to flag the non-locality caveat, not as evidence for the bulk factorization. No parameter of the target result is determined by the target result itself, and no equation is equivalent to its input by construction. The derivation is therefore self-contained, with no circular step identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or new entities. It rests on standard conformal partial wave and AdS/CFT machinery; the main non-trivial input is the local-theory assumption for tree-level singularities and the standard contour-pinching technology.

assumptions (6)
  • standard math Conformal partial waves form a complete basis and every relevant amplitude admits an expansion (2.15) on the principal series.
    Used throughout Section 2.3 and in the master formula (3.9).
  • standard math The split representation (2.2)-(2.6) for delta-function and bulk-to-bulk propagator is valid in Euclidean AdS.
    Used in Section 2.4 and as the starting point of Section 3.
  • standard math The bubble integral formula (3.6) from [34,35] correctly evaluates the integrated product of two conformal three-point structures.
    Quoted and used in Section 3.1; the second term is claimed to give the same result but the computation is left to the reader.
  • domain assumption Singularities of the spectral integrals arise only from pinching configurations of poles, and residues can be computed by deforming contours.
    Used in Sections 4.1-4.2; the general pinching analysis is quoted from [20] and [26,36].
  • domain assumption For local bulk theories, the ν1,ν2 singularities of tree-level coefficient functions come only from b-factors; derivative vertices give polynomial factors.
    Section 4.1, equations (4.1)-(4.2). This is the weakest assumption and is explicitly qualified for non-local theories.
  • domain assumption The bulk loop expansion corresponds to the 1/N expansion on the boundary with g3 ~ 1/N and g4 ~ 1/N^2.
    Section 5, footnote 8, used to compare with large-N CFT.

how reviews work

0 comments
Cite this review

Pith. "Pith review of From bulk loops to boundary large-N expansion." pith.science (2026). https://pith.science/paper/YZ77FRJM

@misc{pith2026190803974,
  author       = {Pith},
  title        = {Pith review of: From bulk loops to boundary large-N expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YZ77FRJM}},
  note         = {Machine review of arXiv:1908.03974}
}
read the original abstract

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.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bulk-to-bulk photon propagator in AdS

    hep-th 2025-10 unverdicted novelty 6.0 of 10

    The bulk-to-bulk photon propagator in Euclidean AdS is derived in axial, Coulomb and covariant gauges, with the simplest position-space form in the Fried–Yennie gauge ξ=d/(d−2).

Reference graph

Works this paper leans on

65 extracted references · 13 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. M. Maldacena,The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200]

    arXiv 1999

  2. [2]

    Witten,Anti-de Sitter space and holography, Adv

    E. Witten,Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150]

    arXiv 1998

  3. [3]

    S. S. Gubser, I. R. Klebanov and A. M. Polyakov,Gauge theory correlators from noncritical string theory, Phys. Lett. B428 (1998) 105 [hep-th/9802109]

    arXiv 1998

  4. [4]

    Heemskerk, J

    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully,Holography from Conformal Field Theory, JHEP 10 (2009) 079 [0907.0151]

    arXiv 2009

  5. [5]

    L. F. Alday, A. Bissi and E. Perlmutter,Holographic Reconstruction of AdS Exchanges from Crossing Symmetry, JHEP 08 (2017) 147 [1705.02318]. – 30 –

    arXiv 2017

  6. [6]

    L. F. Alday,Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [1611.01500]

    arXiv 2017

  7. [7]

    Aharony, L

    O. Aharony, L. F. Alday, A. Bissi and E. Perlmutter,Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036 [1612.03891]

    arXiv 2017

  8. [8]

    Caron-Huot,Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [1703.00278]

    S. Caron-Huot,Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [1703.00278]

    arXiv 2017

Show all 65 references
  1. [9]

    L. F. Alday and A. Bissi,Loop Corrections to Supergravity onAdS5×S5, Phys. Rev. Lett. 119 (2017) 171601 [1706.02388]

    2017 arXiv

  2. [10]

    Aprile, J

    F. Aprile, J. M. Drummond, P. Heslop and H. Paul,Quantum Gravity from Conformal Field Theory, JHEP 01 (2018) 035 [1706.02822]

    2018 arXiv

  3. [11]

    L. F. Alday and S. Caron-Huot,Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [1711.02031]

    2018 arXiv

  4. [12]

    Aprile, J

    F. Aprile, J. M. Drummond, P. Heslop and H. Paul,Loop corrections for Kaluza-Klein AdS amplitudes, JHEP 05 (2018) 056 [1711.03903]

    2018 arXiv

  5. [13]

    Aharony, L

    O. Aharony, L. F. Alday, A. Bissi and R. Yacoby,The Analytic Bootstrap for LargeN Chern-Simons Vector Models, JHEP 08 (2018) 166 [1805.04377]

    2018 arXiv

  6. [14]

    L. F. Alday, A. Bissi and E. Perlmutter,Genus-One String Amplitudes from Conformal Field Theory, JHEP 06 (2019) 010 [1809.10670]

    2019 arXiv

  7. [15]

    Ghosh,Polyakov-Mellin Bootstrap for AdS loops, 1811.00504

    K. Ghosh,Polyakov-Mellin Bootstrap for AdS loops, 1811.00504

    arXiv

  8. [16]

    L. F. Alday,On Genus-one String Amplitudes onAdS5×S5, 1812.11783

    arXiv

  9. [17]

    Ponomarev, E

    D. Ponomarev, E. Sezgin and E. Skvortsov,On one loop corrections in higher spin gravity, 1904.01042

    1904 arXiv

  10. [18]

    L. F. Alday and E. Perlmutter,Growing Extra Dimensions in AdS/CFT, 1906.01477

    1906 arXiv

  11. [19]

    L. F. Alday, J. Henriksson and M. van Loon,An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order1/N2, 1907.02445

    1907 arXiv

  12. [20]

    Penedones,Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [1011.1485]

    J. Penedones,Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [1011.1485]

    2011 arXiv

  13. [21]

    A. L. Fitzpatrick and J. Kaplan,Analyticity and the Holographic S-Matrix, JHEP 10 (2012) 127 [1111.6972]

    2012 arXiv

  14. [22]

    A. L. Fitzpatrick and J. Kaplan,Unitarity and the Holographic S-Matrix, JHEP 10 (2012) 032 [1112.4845]

    2012 arXiv

  15. [23]

    Cardona,Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS, 1708.06339

    C. Cardona,Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS, 1708.06339

    arXiv

  16. [24]

    Giombi, C

    S. Giombi, C. Sleight and M. Taronna,Spinning AdS Loop Diagrams: Two Point Functions, JHEP 06 (2018) 030 [1708.08404]

    2018 arXiv

  17. [25]

    E. Y. Yuan,Loops in the Bulk, 1710.01361

    arXiv

  18. [26]

    E. Y. Yuan,Simplicity in AdS Perturbative Dynamics, 1801.07283

    arXiv

  19. [27]

    Bertan and I

    I. Bertan and I. Sachs,Loops in Antide Sitter Space, Phys. Rev. Lett.121 (2018) 101601 [1804.01880]. – 31 –

    2018 arXiv

  20. [28]

    J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin,d-dimensional SYK, AdS Loops, and6j Symbols, JHEP 03 (2019) 052 [1808.00612]

    2019 arXiv

  21. [29]

    Bertan, I

    I. Bertan, I. Sachs and E. D. Skvortsov,Quantum φ4 Theory in AdS4 and its CFT Dual, JHEP 02 (2019) 099 [1810.00907]

    2019 arXiv

  22. [30]

    Taronna,Pseudo-local Theories: A Functional Class Proposal, inProceedings, International Workshop on Higher Spin Gauge Theories: Singapore, Singapore, November 4-6, 2015, pp

    M. Taronna,Pseudo-local Theories: A Functional Class Proposal, inProceedings, International Workshop on Higher Spin Gauge Theories: Singapore, Singapore, November 4-6, 2015, pp. 59–84, 2017,1602.08566, DOI

    2015 arXiv

  23. [31]

    Bekaert, J

    X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight,Bulk quartic vertices from boundary four-point correlators, inProceedings, International Workshop on Higher Spin Gauge Theories: Singapore, Singapore, November 4-6, 2015, pp. 291–303, 2017,1602.08570, DOI

    2015 arXiv

  24. [32]

    Ponomarev,A Note on (Non)-Locality in Holographic Higher Spin Theories, Universe 4 (2018) 2 [1710.00403]

    D. Ponomarev,A Note on (Non)-Locality in Holographic Higher Spin Theories, Universe 4 (2018) 2 [1710.00403]

    2018 arXiv

  25. [33]

    Rastelli and X

    L. Rastelli and X. Zhou,How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [1710.05923]

    2018 arXiv

  26. [34]

    V. K. Dobrev, G. Mack, I. T. Todorov, V. B. Petkova and S. G. Petrova,On the Clebsch-Gordan Expansion for the Lorentz Group in n Dimensions, Rept. Math. Phys.9 (1976) 219

    1976

  27. [35]

    Karateev, P

    D. Karateev, P. Kravchuk and D. Simmons-Duffin,Harmonic Analysis and Mean Field Theory, 1809.05111

    arXiv

  28. [36]

    R. J. Eden, P. V. Landshoff, O. D. I. and J. C. Polkinghorne,The Analytic S-Matrix. Cambridge University Press, 1966

    1966

  29. [37]

    M. F. Paulos,Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [1107.1504]

    2011 arXiv

  30. [38]

    M. S. Costa, V. Gonalves and J. Penedones,Spinning AdS Propagators, JHEP 09 (2014) 064 [1404.5625]

    2014 arXiv

  31. [39]

    Fronsdal,Elementary particles in a curved space

    C. Fronsdal,Elementary particles in a curved space. ii, Phys. Rev. D10 (1974) 589

    1974

  32. [40]

    V. K. Dobrev,Intertwining operator realization of the AdS / CFT correspondence, Nucl. Phys. B553 (1999) 559 [hep-th/9812194]

    1999 arXiv

  33. [41]

    Leonhardt, R

    T. Leonhardt, R. Manvelyan and W. Ruhl,The Group approach to AdS space propagators, Nucl. Phys. B667 (2003) 413 [hep-th/0305235]

    2003 arXiv

  34. [42]

    Leonhardt, W

    T. Leonhardt, W. Ruhl and R. Manvelyan,The Group approach to AdS space propagators: A Fast algorithm, J. Phys. A37 (2004) 7051 [hep-th/0310063]

    2004 arXiv

  35. [43]

    D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli,Correlation functions in the CFT(d) / AdS(d+1) correspondence, Nucl. Phys. B546 (1999) 96 [hep-th/9804058]

    1999 arXiv

  36. [44]

    Mack,Group Theoretical Approach to Conformal Invariant Quantum Field Theory, NATO Sci

    G. Mack,Group Theoretical Approach to Conformal Invariant Quantum Field Theory, NATO Sci. Ser. B5 (1974) 123

    1974

  37. [45]

    Mack,Osterwalder-Schrader Positivity in Conformal Invariant Quantum Field Theory, Lect

    G. Mack,Osterwalder-Schrader Positivity in Conformal Invariant Quantum Field Theory, Lect. Notes Phys.37 (1975) 66

    1975

  38. [46]

    V. K. Dobrev, V. B. Petkova, S. G. Petrova and I. T. Todorov,Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D13 (1976) 887. – 32 –

    1976

  39. [47]

    V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova and I. T. Todorov,Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys.63 (1977) 1

    1977

  40. [48]

    Simmons-Duffin, D

    D. Simmons-Duffin, D. Stanford and E. Witten,A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [1711.03816]

    2018 arXiv

  41. [49]

    Bekaert, J

    X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight,Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory, JHEP 11 (2015) 149 [1508.04292]

    2015 arXiv

  42. [50]

    Bekaert, J

    X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight,Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange, JHEP 03 (2015) 170 [1412.0016]

    2015 arXiv

  43. [51]

    Symanzik,On Calculations in conformal invariant field theories, Lett

    K. Symanzik,On Calculations in conformal invariant field theories, Lett. Nuovo Cim.3 (1972) 734

    1972

  44. [52]

    Mack,D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models

    G. Mack,D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, 0907.2407

    arXiv

  45. [53]

    F. A. Dolan and H. Osborn,Conformal four point functions and the operator product expansion, Nucl. Phys. B599 (2001) 459 [hep-th/0011040]

    2001 arXiv

  46. [54]

    R. E. Cutkosky,Singularities and discontinuities of Feynman amplitudes, J. Math. Phys.1 (1960) 429

    1960

  47. [55]

    Simmons-Duffin,Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146 [1204.3894]

    D. Simmons-Duffin,Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146 [1204.3894]

    2014 arXiv

  48. [56]

    Maldacena, S

    J. Maldacena, S. H. Shenker and D. Stanford,A bound on chaos, JHEP 08 (2016) 106 [1503.01409]

    2016 arXiv

  49. [57]

    G. J. Turiaci and A. Zhiboedov,Veneziano Amplitude of Vasiliev Theory, JHEP 10 (2018) 034 [1802.04390]

    2018 arXiv

  50. [58]

    Zhou,Recursion Relations in Witten Diagrams and Conformal Partial Waves, JHEP 05 (2019) 006 [1812.01006]

    X. Zhou,Recursion Relations in Witten Diagrams and Conformal Partial Waves, JHEP 05 (2019) 006 [1812.01006]

    2019 arXiv

  51. [59]

    Giombi and I

    S. Giombi and I. R. Klebanov,One Loop Tests of Higher Spin AdS/CFT, JHEP 12 (2013) 068 [1308.2337]

    2013 arXiv

  52. [60]

    Giombi, I

    S. Giombi, I. R. Klebanov and B. R. Safdi,Higher Spin AdSd+1/CFTd at One Loop, Phys. Rev. D89 (2014) 084004 [1401.0825]

    2014 arXiv

  53. [61]

    Giombi, I

    S. Giombi, I. R. Klebanov and A. A. Tseytlin,Partition Functions and Casimir Energies in Higher Spin AdSd+1/CFTd, Phys. Rev. D90 (2014) 024048 [1402.5396]

    2014 arXiv

  54. [62]

    E. D. Skvortsov and T. Tran,AdS/CFT in Fractional Dimension and Higher Spin Gravity at One Loop, Universe 3 (2017) 61 [1707.00758]

    2017 arXiv

  55. [63]

    Sleight and M

    C. Sleight and M. Taronna,Higher-Spin Gauge Theories and Bulk Locality, Phys. Rev. Lett. 121 (2018) 171604 [1704.07859]

    2018 arXiv

  56. [64]

    A. L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin,The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [1212.3616]

    2013 arXiv

  57. [65]

    Komargodski and A

    Z. Komargodski and A. Zhiboedov,Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [1212.4103]. – 33 –

    2013 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.