Pith. sign in

REVIEW 4 major objections 4 minor 90 references

Soft limits of gluon amplitudes in holography and cosmology

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read In AdS, soft gluon limits factorize into transition amplitudes

desk verdict A plausible and transparent first pass at soft-limit factorization in AdS, but the n=4 statement is regulator-dependent as written and the higher-n conjecture rests on unproven connectors; worth refereeing, not worth taking as settled. read the letter →

arxiv 2411.13652 v1 pith:VR6MMEV6 submitted 2024-11-20 hep-th gr-qc

Soner Albayrak , Savan Kharel This is my paper · ORCID
classification hep-thgr-qc
keywords softlimitsAdS/CFTgluonamplitudesYang-MillstheorytransitionWittendiagramsmomentumspaceunitarydecomposition
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

The paper asks what happens to the classic flat-space soft-gluon factorization when amplitudes live in anti-de Sitter space instead of flat space. In flat space a low-energy gluon factors out through a universal soft factor, but in AdS no propagator is singular in the soft limit, so the old argument fails. The authors show, using a unitary decomposition of the AdS propagator, that the soft limit of an $(n+1)$-point gluon amplitude is a sum of integrals of products of lower-point transition amplitudes, plus contributions intrinsic to curved space. The four-point case is worked out explicitly in $\mathrm{AdS}_{d+1}$, and a schematic all-$n$ relation is conjectured to hold in all dimensions.

What carries the argument

The central object is the 'unitary decomposition' of the AdS bulk-to-bulk propagator: it is split into a longitudinal piece (which vanishes in the flat-space limit and is thus intrinsically curved) and a sum over physical helicities of products of off-shell normalizable modes, integrated against a measure. Two new integration kernels—the boundary-to-boundary connector (4.2) and the boundary-to-bulk connector (4.3)—then perform the bulk-point integration at leading order in the soft momentum. These connectors convert Witten diagrams into transition amplitudes and are what allows the $(n+1)$-point soft limit to be expressed as products of lower-point transition amplitudes.

What would settle it

Compute the soft limit of a specific five-point Witten diagram in $\mathrm{AdS}_4$ by direct integration of the full momentum-space expression and compare term-by-term with the conjectured formula built from the connectors. If the kernels (4.2) and (4.3) do not reproduce the integrated result at leading order in the soft momentum, the conjecture collapses; a simpler check is to evaluate the boundary-to-boundary connector directly in $d=3$ against the known soft limit of a four-point diagram.

Watch

Extended reading notes

Core claim

The central claim is that the soft limit of a gluon amplitude in AdS does not factorize into a lower-point vacuum amplitude, but instead into lower-point transition amplitudes—correlators between coherent states—together with curved-space terms. For the four-point amplitude the soft limit is computed in three independent ways, and the 'unitary' decomposition (splitting the propagator into longitudinal and physical-helicity parts) shows that the result is the integral of a 3-point transition amplitude times a hypergeometric kernel, plus an intrinsically AdS piece. For arbitrary $(n+1)$-point diagrams, the paper conjectures that the leading soft behavior is, up to $O(1/n)$ edge-case corrections, a sum over $m$ of integrals of an $m$-point transition amplitude with an $(n-m+1)$-point transition amplitude, plus curved-space contributions, valid in all dimensions.

Load-bearing premise

The two connector kernels in equations (4.2) and (4.3) are presented without derivation and are taken to give the correct leading soft behavior after the bulk-point integration; the entire higher-point soft-limit conjecture rests on these kernels.

Editorial extensions

If this is right

  • The four-point soft limit can be written as a 3-point transition amplitude integrated against a kernel, with the intrinsically AdS contribution cleanly separated.
  • The unitary decomposition scales to higher points, so soft limits of higher-point gluon amplitudes can be computed from lower-point data without resolving all bulk integrals.
  • All contributions to the soft limit are of order $k_0^0$, confirming that no single diagram dominates in AdS and that a flat-space-style soft theorem cannot exist.
  • The proposed schematic relation is conjectured to hold in all dimensions, with explicit support in $\mathrm{AdS}_{d+1}$ for $n=4$ and in $\mathrm{AdS}_4$ for the four-gluon case.

Reading between the lines

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

  • If the $O(1/n)$ corrections are as mild as stated, the soft limit of a high-point holographic correlator approaches an exact convolution of transition amplitudes, which could be used to bootstrap higher-point correlators from lower-point ones.
  • The connectors (4.2) and (4.3) may be expressible as known AdS propagators or as integral transforms of them; if so, the schematic all-$n$ relation becomes a closed computational formula.
  • A direct test in $\mathrm{AdS}_4$ for the five-point amplitude—where the Bessel integrals reduce to exponentials and the edge cases are manageable—would confirm or refute the conjecture without needing arbitrary-dimension technology.
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

4 major / 4 minor

Summary. This paper studies the soft limit of tree-level Yang-Mills Witten diagrams in AdS_{d+1} using the momentum-space bulk-to-boundary and propagator formalism. The flat-space soft theorem is reviewed and contrasted with AdS, where no propagator develops a dominant singularity. The paper's main technical device is a 'unitary' decomposition of the bulk-to-bulk propagator into transverse modes plus a longitudinal piece, which splits the four-point s-channel diagram into a product of three-point transition amplitudes plus an intrinsically AdS term. The soft limit k1→0 is then evaluated with Bessel-function technology, leading to Eq. (3.19). Section IV extends the construction diagrammatically to (n+1)-point amplitudes, introducing boundary-to-boundary and boundary-to-bulk connectors (Eqs. (4.2)-(4.3)), and conjectures a schematic factorization into products of lower-point transition amplitudes up to O(1/n) edge corrections.

Significance. If made completely rigorous, the proposed relation between (n+1)-point amplitudes and transition amplitudes would be a useful organizing principle for holographic and cosmological correlators, complementing known AdS recursion relations and the growing literature on soft theorems in dS. The explicit four-point computation is the first concrete evidence in this direction, and the separation of transverse and longitudinal degrees of freedom is conceptually clean. The paper is also unusually candid about its limitations: the overlap with [50] is disclosed, the regularization dependence is acknowledged in footnotes 17-18, and the higher-point statement is presented as a conjecture with O(1/n) corrections rather than as a theorem. Those limitations, however, sit at the center of the claimed results and cannot be treated as peripheral.

major comments (4)
  1. [§III.C, Eqs. (3.17)-(3.19)] The 'explicit' four-point result is not yet a well-defined statement. In Eq. (3.17), the soft limit of the three-point transition amplitude carries the proportionality factor in footnote 18, which contains Γ(-1) and is therefore infinite; the authors themselves state that the result is 'regularization-dependent.' Consequently, Eq. (3.19) splits the soft limit into a divergent transition-amplitude factor times a 2F1 function plus an 'intrinsically AdS' remainder, but different regulators can move finite pieces between these two terms. The same ambiguity propagates into the higher-point conjecture in Section IV, because that conjecture is built by iterating the same construction. The paper needs a canonical regulator and subtraction scheme, together with a demonstration that the schematic relation is invariant under the choice of scheme.
  2. [§IV, Eqs. (4.2)-(4.3)] The boundary-to-boundary connector B(p1,p2) and the boundary-to-bulk connector B_i(k,p,z) are introduced as the result of 'carrying out the bulk-point integration at leading order in k0', but no derivation of these kernels is given. The entire higher-point expression in Eq. (4.4) and the conjecture that follows depend on these kernels being the correct leading-order integrals. The authors even note in footnote 24 that an in-depth analysis of these connectors is left to a future work. This is a load-bearing gap: either supply the integral identities used, or state explicitly which step is an assumption, and test the connectors by at least recovering Eq. (3.19) from the general formula in the n=4 case.
  3. [§III.B, footnote 17 and Eq. (3.16)] In several steps, including the derivation of Eq. (3.16), the soft limit is commuted with the radial integrations. Footnote 17 acknowledges that this interchange is 'not necessarily warranted' and that the relevant integrals are non-convergent in some cases. This is more than a technical caveat: if the limit and the integration do not commute, the coefficient of each O(k0^0) term can change. The paper should justify the interchange in a kinematic regime where the integrals converge and then specify the analytic continuation, or otherwise show that the regulator-independent part of the result is unaffected by the exchange.
  4. [§IV, unnumbered conjecture and Eq. (4.6)] The higher-point statement is a conjecture with an unspecified O(1/n) correction. The edge-case diagrams in Eq. (4.6) are said to modify the result, and the fraction of such diagrams is argued to be suppressed by 1/n, but no argument is given that the sum of the edge contributions is not enhanced by the connector kernels or by kinematic factors. As written, the claimed relation cannot be tested quantitatively because the error term is not defined. Either provide a counting argument for the full edge contribution, or weaken the claim to an explicit set of diagrams for which the factorization is exact.
minor comments (4)
  1. [§III.C, Eq. (3.15)] The notation KK^{(1)} and KK^{(2)} is used in Eq. (3.15) before the definitions in Appendix C; please add a forward reference to Eqs. (C.1e) and (C.1f) at first use.
  2. [Title and §V] The title advertises cosmology, but the explicit analysis in the paper is entirely in AdS; the dS connection is limited to the analytic-continuation discussion in Appendix A. Consider making the title or the introduction state the intended dS scope more precisely.
  3. [References] There are several formatting artifacts in the reference list, such as 'E. Witten„' in [67] and the publisher fields in [78], [79], and [81]; these should be normalized.
  4. [§IV, Eq. (4.1)] The diagrammatic notation in Eq. (4.1) is hard to parse, especially the labels h1p1 and h2p2; a short sentence explaining the placement of helicity and momentum labels on the internal lines would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the soft-limit factorization is computed for n=4 and conjectured for higher n, with the unproved connectors and regularization-dependence being gaps rather than circles.

full rationale

The four-point soft-limit relation is obtained by explicit computation: eq. (3.15) follows from the propagator decomposition, eq. (3.16) is the soft limit of that expression, and eq. (3.17) computes the soft limit of the three-point transition amplitude. The schematic form (3.19) is therefore a derived identity, not a restatement of the input. The higher-point generalization in Section IV is explicitly labelled a conjecture, and it relies on the connectors (4.2)-(4.3), which are presented without derivation; that is an unsupported or correctness gap, not circularity. The divergent, regularization-dependent prefactor in (3.17)-(3.18) means the n=4 decomposition is ambiguous until a scheme is fixed, but ambiguity is a well-definedness issue rather than a reduction of the conclusion to the premise. The paper does cite the authors' prior work for the AdS momentum-space formalism, the differential representation, and Bessel integral technology, but these are computational tools, many of which are re-derived in the appendices (e.g., eqs. (C.1)-(C.10)), and the soft-limit factorization itself is not imported from those references. No fitted parameters are introduced, the overlap with [50] is disclosed, and no equation or claim reduces by construction to its own input. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

No free parameters are fitted. The central claim rests on the propagator decomposition (domain assumption), the validity of exchanging soft limit and integrations (flagged in footnote 17), and two connector kernels presented without derivation (ad hoc). The Bessel identities in Appendix C are standard special-function results.

assumptions (5)
  • domain assumption Axial gauge bulk-to-boundary propagator form A_i^h_a(k,z) = epsilon_i^h k^nu z^nu K_nu(kz)
    Stated in eq (2.4); dictated by bulk regularity and Ward identity, taken from the authors' prior formalism.
  • domain assumption Unitary decomposition of the bulk-to-bulk propagator into a longitudinal piece plus a sum over physical helicity states, eqs (2.9)-(2.10)
    Central structural identity used throughout; asserted with 'One can show', building on the split representation from prior work.
  • domain assumption Commutation of the soft limit with the integrations over p and z in Witten diagrams
    Explicitly flagged in footnote 17 as not necessarily warranted, especially for non-convergent integrals; used to derive the n=4 soft limit eq (3.16).
  • standard math Flat-space Feynman propagator equals the sum over physical polarization states divided by the pole (unitarity), Appendix B
    Standard QFT result, used as the motivation for the AdS decomposition; referenced to Schwartz and prior literature.
  • ad hoc to paper Connectors in eqs (4.2) and (4.3) are the correct kernels after leading-order bulk-point integration
    Stated without derivation in Section IV; the paper says the result is obtained 'by carrying out the bulk-point integration' but does not show the steps.
invented entities (2)
  • Boundary-to-boundary connector B(p1,p2)
    purpose: Kernel for gluing two off-shell transition amplitudes in the soft limit of higher-point diagrams
    Defined in eq (4.2); a computational kernel without external confirmation or derivation.
  • Boundary-to-bulk connector B_i(k,p,z)
    purpose: Replaces a bulk-to-boundary propagator in the intrinsically AdS contributions to soft limits
    Defined in eq (4.3); stated to be an inherently curved-space contribution that vanishes in flat space; no derivation provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Soft limits of gluon amplitudes in holography and cosmology." pith.science (2026). https://pith.science/paper/VR6MMEV6

@misc{pith2026241113652,
  author       = {Pith},
  title        = {Pith review of: Soft limits of gluon amplitudes in holography and cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VR6MMEV6}},
  note         = {Machine review of arXiv:2411.13652}
}
abstract

In this work, we extend the study of soft limits to (Anti) de Sitter spaces, investigating the analytic structure of holographic gluon correlators as part of a broader effort to reveal new symmetries and fundamental structures in gauge theories. By reorganizing perturbation theory in AdS to align with flat space unitarity, we analyze the contributions intrinsic to curved spacetime and their behavior in the soft limit. Our analysis uncovers schematic relations between $(n+1)$-point amplitude and $n$-point transition amplitudes in arbitrary-dimensional AdS, with explicit results derived for $n=4$ in AdS$_{d+1}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 73 canonical work pages

  1. [50]

    Bzowski, P

    A. Bzowski, P. McFadden, and K. Skenderis, Phys. Rev. Lett. 124 no. 13, (2020) 131602

    work page 2020

  2. [1]

    The soft limit of 𝑛+1 point diagram contains a piece (first term in right hand side) which is simply integra- tion of lower point diagrams (with a leg made off-shell) against a kernel. In fact, since we sum over all diagrams in the𝑛+1 point amplitude, we might expect that these terms add up to the integration of products of lower- point transition amplitu...

  3. [2]

    Using the differential representation in eqn. (2.3)

  4. [3]

    (2.10) We will carry out all of these approaches to investigate their pros and cons

    Utilizing the “unitary” decomposition of the propaga- tor in eqn. (2.10) We will carry out all of these approaches to investigate their pros and cons. For concreteness, we will consider the 𝑠−channel computation. A. Direct approach The expressions for the respective Witten diagram reads 𝑊 = ∞ ∫ 0 𝑑𝑧𝐿 𝑧𝑑−3 𝐿 𝑑𝑧𝑅 𝑧𝑑−3 𝑅 𝐴ℎ1 𝑖 (𝐤1,𝑧𝐿)𝐴ℎ2 𝑗 (𝐤2,𝑧𝐿)𝐴ℎ3 𝑘 (𝐤3,𝑧...

  5. [4]

    The resemblance of the above-mentioned term with unitary-cuts23 is natural: by using the unitary de- composition of the propagator, we managed to relate higher point amplitudes to products of lower point ones, in an analogous fashion to the Cutkosky cuts

  6. [5]

    The inherently AdS pieces in the soft limit of𝑛+1 point diagram can not be rewritten as products of lower point amplitudes, at least in the traditional sense since some of the bulk-to-boundary propagators are replaced by the bulk-to-boundary connectors.24

  7. [6]

    all of them go as (𝐤0)0

    The suppression of different terms in the soft limit are of the same severity, i.e. all of them go as (𝐤0)0. This is consistent with our main point in the introduction, i.e. unlike the flat space where particular diagrams are dominant in the soft limit (making Weinberg’s soft the- orem conceptually easier to grasp), and a similar anal- ogy is missing in A...

  8. [7]

    negative frequency

    The analysis of eqn. (4.4) gets corrections as there are also the edge cases, where the soft leg is not connected 22 For instance, the first term on the right hand side of eqn. (4.4) reads as lim 𝐤0→0 𝑊(𝐤ℎ0 0 , 𝐤ℎ1 1 ,…, 𝐤ℎℎ𝑛 ) ⊃ −𝑖2 √ 2𝐤𝐿⋅𝜖ℎ0(𝐤0)𝑘(𝑑−2)/2 0 ∑ ℎ ×∫ 𝑝𝐿𝑝𝑅𝑑𝑝𝐿𝑑𝑝𝑅 4(𝑘2 𝐿+𝑝2 𝐿+𝑖𝜖)(𝑘2 𝑅+𝑝2 𝑅+𝑖𝜖) (𝑝𝐿,𝑝𝑅) ×𝑊(𝐤ℎ∗ 𝐿 ,𝑝𝐿; 𝐤ℎ1 1 ,…, 𝐤ℎ𝑚 𝑚 ) ×𝑊(𝐤∗ 𝑅,𝑝𝑅...

    2023

Show all 90 references
  1. [8]

    Once one sticks to a partic- ular dimension, many of the formulae actually simplify dras- tically

    AdS 4 computations The discussion in the body of the paper is purely agnos- tic to the spacetime dimension, making expressions rather technical and lengthy as a result. Once one sticks to a partic- ular dimension, many of the formulae actually simplify dras- tically. We demons...

  2. [9]

    Three point amplitudes Even though we have the technology to compute any 𝑛−point amplitude, the computation is tedious with lengthy results that provide little to no physical insight. The excep- tion is the tree point amplitudes, and in this section, we are going to provide so...

  3. [10]

    S. J. Parke and T. R. Taylor, Phys. Rev. Lett. 56 (1986) 2459

    1986

  4. [11]

    Cachazo, P

    F. Cachazo, P. Svrcek, and E. Witten, JHEP 09 (2004) 006

    2004

  5. [12]

    Britto, F

    R. Britto, F. Cachazo, and B. Feng, Nucl. Phys. B 715 (2005) 499–522

    2005

  6. [13]

    Britto, F

    R. Britto, F. Cachazo, B. Feng, and E. Witten, Phys. Rev. Lett. 94 (2005) 181602

    2005

  7. [14]

    Arkani-Hamed, J

    N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot, and J. Trnka, JHEP 01 (2011) 041

    2011

  8. [15]

    Bloch and A

    F. Bloch and A. Nordsieck, Phys. Rev. 52 (1937) 54–59

    1937

  9. [16]

    F. E. Low, Phys. Rev. 96 (1954) 1428–1432

    1954

  10. [17]

    Gell-Mann and M

    M. Gell-Mann and M. L. Goldberger, Phys. Rev. 96 (1954) 1433–1438

    1954

  11. [18]

    F. E. Low, Phys. Rev. 110 (1958) 974–977

    1958

  12. [19]

    Kazes, Il Nuovo Cimento (1955-1965) 13 (1959) 1226–1239

    E. Kazes, Il Nuovo Cimento (1955-1965) 13 (1959) 1226–1239

    1959

  13. [20]

    D. R. Yennie, S. C. Frautschi, and H. Suura, Annals Phys. 13 (1961) 379–452

    1961

  14. [21]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. 140 (1965) B516–B524

    1965

  15. [22]

    Strominger, JHEP 07 (2014) 152

    A. Strominger, JHEP 07 (2014) 152

    2014

  16. [23]

    Cachazo and A

    F. Cachazo and A. Strominger, arXiv:1404.4091 [hep-th]

    arXiv

  17. [24]

    Casali, JHEP 08 (2014) 077

    E. Casali, JHEP 08 (2014) 077

    2014

  18. [25]

    Albayrak, S

    S. Albayrak, S. Kharel, and D. Meltzer, JHEP 03 (2021) 249

    2021

  19. [26]

    Armstrong, A

    C. Armstrong, A. E. Lipstein, and J. Mei, JHEP 02 (2021) 194

    2021

  20. [27]

    Zhou, Phys

    X. Zhou, Phys. Rev. Lett. 127 no. 14, (2021) 141601

    2021

  21. [28]

    Cheung, J

    C. Cheung, J. Parra-Martinez, and A. Sivaramakrishnan, arXiv:2201.05147 [hep-th]

    arXiv

  22. [29]

    J. M. Maldacena, JHEP 05 (2003) 013

    2003

  23. [30]

    Creminelli and M

    P. Creminelli and M. Zaldarriaga, JCAP 10 (2004) 006

    2004

  24. [31]

    Creminelli, J

    P. Creminelli, J. Noreña, and M. Simonović, JCAP 07 (2012) 052

    2012

  25. [32]

    Assassi, D

    V. Assassi, D. Baumann, and D. Green, JCAP 11 (2012) 047

    2012

  26. [33]

    Hinterbichler, L

    K. Hinterbichler, L. Hui, and J. Khoury, JCAP 08 (2012) 017

    2012

  27. [34]

    Flauger, D

    R. Flauger, D. Green, and R. A. Porto, JCAP 08 (2013) 032

    2013

  28. [35]

    G. L. Pimentel, JHEP 02 (2014) 124

    2014

  29. [36]

    W. D. Goldberger, L. Hui, and A. Nicolis, Phys. Rev. D 87 no. 10, (2013) 103520

    2013

  30. [37]

    Hinterbichler, L

    K. Hinterbichler, L. Hui, and J. Khoury, JCAP 01 (2014) 039

    2014

  31. [38]

    Joyce, J

    A. Joyce, J. Khoury, and M. Simonović, JCAP 01 (2015) 012

    2015

  32. [39]

    Raju, Phys

    S. Raju, Phys. Rev. D 85 (2012) 126009

    2012

  33. [40]

    Albayrak, C

    S. Albayrak, C. Chowdhury, and S. Kharel, JHEP 10 (2019) 274

    2019

  34. [41]

    Albayrak and S

    S. Albayrak and S. Kharel, Phys. Rev. D 103 no. 2, (2021) 026004

    2021

  35. [42]

    Albayrak, C

    S. Albayrak, C. Chowdhury, and S. Kharel, Phys. Rev. D 101 no. 12, (2020) 124043

    2020

  36. [43]

    Albayrak, S

    S. Albayrak, S. Kharel, and X. Wang, JHEP 07 (2024) 281

    2024

  37. [44]

    Baumann, C

    D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, JHEP 12 (2020) 204

    2020

  38. [45]

    Baumann, C

    D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, SciPost Phys. 11 (2021) 071

    2021

  39. [46]

    Baumann, W.-M

    D. Baumann, W.-M. Chen, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, JHEP 09 (2022) 010

    2022

  40. [47]

    Bzowski, P

    A. Bzowski, P. McFadden, and K. Skenderis, JHEP 03 (2014) 111. pressions are valid for higher spinning particles as well; for instance, the scalar factor of a spin−3 conserved current in AdS4 is same as the gravi- ton expression in AdS6: we refer reader to [34] for a discussio...

    2014

  41. [48]

    Bzowski, P

    A. Bzowski, P. McFadden, and K. Skenderis, JHEP 03 (2016) 066

    2016

  42. [49]

    Bzowski, P

    A. Bzowski, P. McFadden, and K. Skenderis, JHEP 11 (2018) 159

    2018

  43. [51]

    Bzowski, P

    A. Bzowski, P. McFadden, and K. Skenderis, JHEP 01 (2021) 192

    2021

  44. [52]

    Bzowski, P

    A. Bzowski, P. McFadden, and K. Skenderis, JHEP 12 (2022) 039

    2022

  45. [53]

    Bzowski, JHEP 04 (2024) 082

    A. Bzowski, JHEP 04 (2024) 082

    2024

  46. [54]

    S. Jain, R. R. John, and V. Malvimat, JHEP 11 (2020) 049

    2020

  47. [55]

    S. Jain, R. R. John, A. Mehta, A. A. Nizami, and A. Suresh, JHEP 09 (2021) 041

    2021

  48. [56]

    S. Jain, D. K. S, D. Mazumdar, and S. Yadav, JHEP 09 (2024) 027

    2024

  49. [57]

    Isono, T

    H. Isono, T. Noumi, and G. Shiu, JHEP 07 (2018) 136

    2018

  50. [58]

    Isono, T

    H. Isono, T. Noumi, and G. Shiu, JHEP 10 (2019) 183

    2019

  51. [59]

    Chowdhury, A

    C. Chowdhury, A. Lipstein, J. Mei, and Y. Mo, JHEP 10 (2024) 070

    2024

  52. [60]

    Albayrak and S

    S. Albayrak and S. Kharel, JHEP 02 (2019) 040

    2019

  53. [61]

    Albayrak and S

    S. Albayrak and S. Kharel, JHEP 12 (2019) 135

    2019

  54. [62]

    M. F. Paulos, JHEP 10 (2011) 074

    2011

  55. [63]

    Chu and S

    J. Chu and S. Kharel, Phys. Rev. D 109 no. 10, (2024) L101901

    2024

  56. [64]

    Chu and S

    J. Chu and S. Kharel, Phys. Rev. D 109 no. 10, (2024) 106003

    2024

  57. [65]

    Raju, Phys

    S. Raju, Phys. Rev. D 85 (2012) 126008

    2012

  58. [66]

    Raju, Phys

    S. Raju, Phys. Rev. D 83 (2011) 126002

    2011

  59. [67]

    Meltzer and A

    D. Meltzer and A. Sivaramakrishnan, JHEP 11 (2020) 073

    2020

  60. [68]

    Albayrak and S

    S. Albayrak and S. Kharel, JHEP 05 (2023) 151

    2023

  61. [69]

    Cheung, K

    C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, Phys. Rev. Lett. 116 no. 4, (2016) 041601

    2016

  62. [70]

    Bittermann and A

    N. Bittermann and A. Joyce, JHEP 03 (2023) 092

    2023

  63. [71]

    Boucher-Veronneau and A

    C. Boucher-Veronneau and A. J. Larkoski, JHEP 09 (2011) 130

    2011

  64. [72]

    Witten, Adv

    E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253–291

    1998

  65. [73]

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428 (1998) 105–114

    1998

  66. [74]

    Banks, M

    T. Banks, M. R. Douglas, G. T. Horowitz, and E. J. Martinec, arXiv:hep-th/9808016

    arXiv

  67. [75]

    Harlow and D

    D. Harlow and D. Stanford, arXiv:1104.2621 [hep-th]

    arXiv

  68. [76]

    Quantum gravity in de Sitter space,

    E. Witten„ “Quantum gravity in de Sitter space, ” inStrings 2001: International Conference . 6, 2001. arXiv:hep-th/0106109

    2001 arXiv

  69. [77]

    Strominger, JHEP 10 (2001) 034

    A. Strominger, JHEP 10 (2001) 034

    2001

  70. [78]

    J. B. Hartle and S. W. Hawking, Phys. Rev. D 28 (1983) 2960–2975

    1983

  71. [79]

    Witten, JHEP 12 (1998) 012

    E. Witten, JHEP 12 (1998) 012

    1998

  72. [80]

    Lifschytz and V

    G. Lifschytz and V. Periwal, JHEP 04 (2000) 026

    2000

  73. [81]

    de Boer, E

    J. de Boer, E. P. Verlinde, and H. L. Verlinde, JHEP 08 (2000) 003

    2000

  74. [82]

    Ghosh, N

    A. Ghosh, N. Kundu, S. Raju, and S. P. Trivedi, JHEP 07 (2014) 011

    2014

  75. [83]

    Heemskerk and J

    I. Heemskerk and J. Polchinski, JHEP 06 (2011) 031

    2011

  76. [84]

    M. D. Schwartz„ Quantum Field Theory and the Standard Model. Cambridge University Press, 3, 2014

    2014

  77. [85]

    Meltzer, E

    D. Meltzer, E. Perlmutter, and A. Sivaramakrishnan, JHEP 03 (2020) 061

    2020

  78. [86]

    Albayrak, P

    S. Albayrak, P. Benincasa, and C. Duaso Pueyo, SciPost Phys. 16 no. 6, (2024) 157

    2024

  79. [87]

    B. M. Project, H. Bateman, A. Erdélyi, and U. S. O. of Naval Research„ Higher transcendental functions. No. v. 1 in 15 Higher Transcendental Functions. McGraw-Hill, 1953. https://books.google.com/books?id= AA6AvgEACAAJ

    1953

  80. [88]

    Appell and J

    P. Appell and J. de Fériet„ Fonctions hypergéométriques et hypersphériques: polynomes d’Hermite . Gauthier-Villars, 1926. https://books.google.com.tw/books?id= znr7fqnNPyoC

    1926

  81. [89]

    W. N. Bailey, Proceedings of the London Mathematical Society s2-40 no. 1, (1936) 37–48. https: //londmathsoc.onlinelibrary.wiley.com/ doi/abs/10.1112/plms/s2-40.1.37

    1936 doi

  82. [90]

    Zwillinger„ Table of Integrals, Series, and Products

    D. Zwillinger„ Table of Integrals, Series, and Products . Elsevier Science, 2014. https://books.google.com.tr/ books?id=NjnLAwAAQBAJ

    2014

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.