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arxiv: 2509.04974 · v3 · submitted 2025-09-05 · ✦ hep-th

Minkowski Space holography and Radon transform

Samrat Bhowmick , Koushik Ray This is my paper

Pith reviewed 2026-05-18 19:09 UTC · model grok-4.3

classification ✦ hep-th
keywords Minkowski spaceRadon transformholographybulk reconstructionscalar fieldde SitterMellin modeshypergeometric functions
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The pith

A free scalar field in Minkowski spacetime maps to a scalar field on a codimension-two sphere through Radon transform and bulk reconstruction.

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

The paper shows how a free scalar field living in flat Minkowski space can be connected to a field on a sphere that has two fewer dimensions. The connection works by taking the Radon transform of the Minkowski field, which produces a field on a hyperplane equivalent to a de Sitter or Euclidean anti-de Sitter space. That result is then matched to a field reconstructed from the codimension-two sphere using standard holographic bulk reconstruction methods. The authors also provide expressions for the Mellin modes of the original field as generalized hypergeometric functions. If correct, this gives a concrete dictionary between bulk flat-space data and lower-dimensional holographic data.

Core claim

We relate a free scalar field in the Minkowski spacetime with a scalar field with a certain scaling dimension on a sphere of codimension two. This is realised by first performing a Radon transform of the bulk field on the Minkowski space to a field on a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice. The Radon transform is identified in turn with a scalar field obtained from a sphere of one further lower dimension through the so-called bulk reconstruction programme. We write down the Mellin modes of the bulk field as generalised hypergeometric functions using the Lee-Pomeransky method developed for evaluation of Feynman loop diagrams.

What carries the argument

The Radon transform applied to the Minkowski bulk field, which maps it to a hyperplane field identified with the output of bulk reconstruction from the codimension-two sphere.

If this is right

  • This establishes a holographic-type relation for fields in flat spacetime.
  • The scaling dimension of the sphere field is determined by the matching to the transformed Minkowski field.
  • Computational tools from Feynman diagram evaluation can be used to find Mellin modes in this setup.
  • Similar mappings might apply to other fields or curved backgrounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could offer an alternative to AdS/CFT for understanding flat-space holography.
  • Extending this to gauge fields or gravity might reveal more about asymptotic symmetries in Minkowski space.
  • Testing the exact scaling dimension match numerically in low dimensions could verify the claim.

Load-bearing premise

That the field obtained after the Radon transform precisely equals the one from bulk reconstruction with the required scaling dimension.

What would settle it

Computing the Radon transform of a simple Minkowski scalar mode and comparing it to the reconstructed field from the sphere for the proposed scaling dimension; any mismatch would disprove the identification.

Figures

Figures reproduced from arXiv: 2509.04974 by Koushik Ray, Samrat Bhowmick.

Figure 1
Figure 1. Figure 1: Different regions of M1,d The inner product of two vectors X and Y in M1,d is X · Y = ηµνX µY ν = −X 0Y 0 + X 1Y 1 + · · · + X d−1Y d−1 + X dY d . (2) The spacetime is partitioned into different regions or patches, M+ = {X ∈ M1,d; |X| 2 > 0}, (3) M↑ − = {X ∈ M1,d; |X| 2 < 0, X0 > 0, }, M↓ − = {X ∈ M1,d; |X| 2 < 0, X0 < 0, }, M− = M↑ − ∪ M↓ −, (4) M↑ 0 = {X ∈ M1,d; |X| 2 = 0, X0 > 0}, M↓ 0 = {X ∈ M1,d; |X| … view at source ↗
read the original abstract

We relate a free scalar field in the Minkowski spacetime with a scalar field with a certain scaling dimension on a sphere of codimension two. This is realised by first performing a Radon transform of the ``bulk" field on the Minkowski space to a field on a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice. The Radon transform is identified in turn with a scalar field obtained from a sphere of one further lower dimension through the so-called bulk reconstruction programme. We write down the Mellin modes of the bulk field as generalised hypergeometric functions using the Lee-Pomeransky method developed for evaluation of Feynman loop diagrams.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to relate a free scalar field in Minkowski spacetime to a scalar field with a specific scaling dimension on a codimension-two sphere. This is realized by applying a Radon transform to map the Minkowski bulk field onto a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice, which is then identified with the scalar obtained from the sphere via the bulk reconstruction programme. The authors express the Mellin modes of the bulk field as generalized hypergeometric functions using the Lee-Pomeransky method.

Significance. If the central identification holds, the work would provide a concrete integral-transform realization of flat-space holography linking Minkowski fields to lower-dimensional sphere data. The explicit use of the Lee-Pomeransky method to write Mellin modes as generalized hypergeometric functions is a strength, supplying concrete expressions that could be checked for consistency with the claimed scaling dimension.

major comments (2)
  1. The section on the Radon transform and subsequent identification: the central claim requires that the Radon transform of Minkowski plane-wave (or Mellin) modes yields a field on the hyperplane that exactly matches the bulk-reconstructed scalar with the stated scaling dimension. The manuscript does not demonstrate convergence of the transform for non-decaying modes or specify an iε prescription/cutoff that preserves the mode correspondence; this is load-bearing for the exact match asserted in the abstract.
  2. The part deriving the scaling dimension from bulk reconstruction: it is unclear whether the dimension is fixed independently by the reconstruction programme or is adjusted to fit the Radon-transformed field; an explicit check that the dimension emerges without post-hoc choice would strengthen the result.
minor comments (2)
  1. The abstract and introduction could state the numerical value of the scaling dimension obtained, rather than referring to it only as 'a certain scaling dimension'.
  2. Notation for the hyperplane identification with dS/EAdS slices should be introduced with a clear diagram or coordinate chart to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: The section on the Radon transform and subsequent identification: the central claim requires that the Radon transform of Minkowski plane-wave (or Mellin) modes yields a field on the hyperplane that exactly matches the bulk-reconstructed scalar with the stated scaling dimension. The manuscript does not demonstrate convergence of the transform for non-decaying modes or specify an iε prescription/cutoff that preserves the mode correspondence; this is load-bearing for the exact match asserted in the abstract.

    Authors: We agree that an explicit treatment of convergence is necessary for the claimed exact identification. In the revised manuscript we will add a dedicated paragraph (or short subsection) in the Radon-transform section that introduces a standard iε prescription for the integration contour, demonstrates distributional convergence for the Mellin modes, and verifies that the resulting hyperplane field coincides with the bulk-reconstructed scalar. The same regularization will be used consistently when expressing the modes via the Lee-Pomeransky representation. revision: yes

  2. Referee: The part deriving the scaling dimension from bulk reconstruction: it is unclear whether the dimension is fixed independently by the reconstruction programme or is adjusted to fit the Radon-transformed field; an explicit check that the dimension emerges without post-hoc choice would strengthen the result.

    Authors: The scaling dimension is fixed by the bulk-reconstruction programme itself: it is the unique value for which the reconstructed scalar on the dS/EAdS slice satisfies the correct wave equation and reproduces the known conformal dimension on the codimension-two sphere. This determination precedes and is independent of the subsequent Radon-transform step. To remove any ambiguity we will insert an explicit derivation (using the standard reconstruction formula) that obtains the dimension directly from the sphere data, followed by a consistency check confirming that the Radon-transformed Minkowski modes match this same dimension without adjustment. revision: yes

Circularity Check

0 steps flagged

No circularity: relation constructed via external transforms and standard bulk reconstruction

full rationale

The derivation applies the Radon transform (a standard integral transform) to map the Minkowski free scalar to a field on a hyperplane, then identifies this with the output of the bulk reconstruction programme on the codimension-two sphere. The Lee-Pomeransky method is invoked only for explicit mode computation and is an independent technique from Feynman integral evaluation. No equation reduces the target scaling dimension or the identification to a fitted parameter, self-definition, or prior self-citation chain. The central claim is an explicit construction relating two fields through these operations rather than a tautological renaming or forced prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard properties of the Radon transform and the bulk reconstruction framework from prior literature without introducing new free parameters or postulated entities.

axioms (2)
  • standard math The Radon transform maps the Minkowski scalar field to a well-defined field on a hyperplane that can be identified with a de Sitter or Euclidean anti-de Sitter slice.
    This identification is the first step stated in the abstract.
  • domain assumption Bulk reconstruction relates the hyperplane field to a scalar field on a codimension-two sphere with a definite scaling dimension.
    Invoked to complete the chain from Minkowski space to the sphere.

pith-pipeline@v0.9.0 · 5627 in / 1558 out tokens · 38352 ms · 2026-05-18T19:09:33.147111+00:00 · methodology

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 38 internal anchors

  1. [1]

    Dimensional Reduction in Quantum Gravity

    G. ’t Hooft,Dimensional reduction in quantum gravity, in *Salamfestschrift*, pp. 284–296, arXiv:gr-qc/9310026

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    The World as a Hologram

    L. Susskind,The World as a hologram, J. Math. Phys.36, 6377–6396 (1995), arXiv:hep-th/9409089

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  3. [3]

    J. M. Maldacena,The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2, 231–252 (1998), arXiv:hep-th/9711200

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  4. [4]

    Anti De Sitter Space And Holography

    E. Witten,Anti de Sitter space and holography, Adv. Theor. Math. Phys.2, 253–291 (1998), arXiv:hep-th/9802150

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  5. [5]

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov,Gauge theory correlators from noncritical string theory, Phys. Lett. B428, 105–114 (1998), arXiv:hep-th/9802109

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  6. [6]

    Strominger,The dS/CFT Correspondence, JHEP10, 034 (2001), arXiv:hep- th/0106113

    A. Strominger,The dS/CFT Correspondence, JHEP10, 034 (2001), arXiv:hep- th/0106113

    work page arXiv 2001

  7. [7]

    Quantum Gravity In De Sitter Space

    E. Witten,Quantum gravity in de Sitter space, arXiv:hep-th/0106109

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    A holographic reduction of Minkowski space-time

    J. de Boer and S. N. Solodukhin,A Holographic reduction of Minkowski space-time, Nucl. Phys. B665, 545–593 (2003), arXiv:hep-th/0303006

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  9. [9]

    S. N. Solodukhin,Reconstructing Minkowski space-time, IRMA Lect. Math. Theor. Phys.8, 123–163 (2005), arXiv:hep-th/0405252. 19

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  10. [10]

    Arcioni and C

    G. Arcioni and C. Dappiaggi,Exploring the holographic principle in asymptotically flat space-times via the BMS group, Nucl. Phys. B674, 553–592 (2003), arXiv:hep- th/0306142

    work page arXiv 2003

  11. [11]

    Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited

    G. Barnich and C. Troessaert,Symmetries of asymptotically flat 4 dimensional space- time at null infinity revisited, Phys. Rev. Lett.105, 111103 (2010), arXiv:0909.2617 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  12. [12]

    Aspects of the BMS/CFT correspondence

    G. Barnich and C. Troessaert,Aspects of the BMS/CFT correspondence, JHEP05, 062 (2010), arXiv:1001.1541 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  13. [13]

    Lectures on the Infrared Structure of Gravity and Gauge Theory

    A. Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory, Princeton University Press (2018), press.princeton.edu, arXiv:1703.05448 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    Null Infinity and Unitary Representation of The Poincare Group

    S. Banerjee,Null Infinity and Unitary Representation of the Poincar´ e Group, JHEP 01(2019) 205, arXiv:1801.10171 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  15. [15]

    Laddha, S

    A. Laddha, S. G. Prabhu, S. Raju and P. Shrivastava,The Holographic Nature of Null Infinity, SciPost Phys.10, 041 (2021), arXiv:2002.02448 [hep-th]

    work page arXiv 2021

  16. [16]

    Iacobacci, C

    L. Iacobacci, C. Sleight and M. Taronna,From Celestial Correlators to AdS, and Back, JHEP06, 053 (2023), arXiv:2208.01629 [hep-th]

    work page arXiv 2023

  17. [17]

    Furugori, N

    H. Furugori, N. Ogawa, S. Sugishita, T. Waki,Celestial Holography meets dS/CFT, arXiv:2507.17558 [hep-th]

    work page arXiv

  18. [18]

    E. Have, K. Nguyen, S. Prohazka, J. Salzer,Massive Carrollian Fields at Timelike Infinity, JHEP07(2024) 054, arXiv:2402.05190 [hep-th]

    work page arXiv 2024

  19. [19]

    Laddha, S

    A. Laddha, S. G. Prabhu, S. Raju and P. Shrivastava,Squinting at Massive Fields from Infinity, arXiv:2207.06406 [hep-th]

    work page arXiv

  20. [20]

    Sleight and M

    C. Sleight and M. Taronna,Celestial Holography Revisited, Phys. Rev. Lett.133, no.24, 241601 (2024), arXiv:2301.01810 [hep-th]

    work page arXiv 2024

  21. [21]

    Iacobacci, C

    L. Iacobacci, C. Sleight and M. Taronna,Celestial Holography Revisited. Part II. Correlators and K¨ all´ en-Lehmann, JHEP08, 033 (2024), arXiv:2401.16591 [hep-th]

    work page arXiv 2024

  22. [22]

    Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity

    G. Barnich and C. Troessaert,Comments on holographic current algebras and asymp- totically flat four dimensional spacetime at null infinity, JHEP11(2013) 003, arXiv:1309.0794 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  23. [23]

    Asymptotic Symmetries of Yang-Mills Theory

    A. Strominger,Asymptotic Symmetries of Yang-Mills Theory, JHEP07(2014) 151, arXiv:1308.0589 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  24. [24]

    T. He, P. Mitra, A. P. Porfyriadis and A. Strominger,New Symmetries of Massless QED, JHEP10(2014) 112, arXiv:1407.3789 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  25. [25]

    Asymptotic symmetries of QED and Weinberg's soft photon theorem

    M. Campiglia and A. Laddha,Asymptotic symmetries of QED and Weinberg’s soft photon theorem, JHEP07(2015) 115, arXiv:1505.05346 [hep-th]. 20

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  26. [26]

    New Symmetries of QED

    D. Kapec, M. Pate and A. Strominger,New Symmetries of QED, arXiv:1506.02906 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  27. [27]

    T. He, P. Mitra and A. Strominger,2D Kac-Moody Symmetry of 4D Yang-Mills Theory, JHEP10(2016) 137, arXiv:1503.02663 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  28. [28]

    Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere

    S. Pasterski, S. H. Shao and A. Strominger,Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D96, 065026 (2017), arXiv:1701.00049 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  29. [29]

    A Conformal Basis for Flat Space Amplitudes

    S. Pasterski and S. H. Shao,Conformal basis for flat space amplitudes, Phys. Rev. D 96, 065022 (2017), arXiv:1705.01027 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  30. [30]

    Pasterski,Lectures on celestial amplitudes, arXiv:2108.04801 [hep-th]

    S. Pasterski,Lectures on celestial amplitudes, arXiv:2108.04801 [hep-th]

    work page arXiv

  31. [31]

    Neuenfeld,The Flat-Space Limit of AdS Coupled to a Bath, arXiv:2508.03798 [hep-th]

    D. Neuenfeld,The Flat-Space Limit of AdS Coupled to a Bath, arXiv:2508.03798 [hep-th]

    work page arXiv

  32. [32]

    P.-X. Hao, N. Ogawa, T. Takayanagi and T. Waki,Flat Space Holography via AdS/BCFT, arXiv:2509.00652 [hep-th]

    work page arXiv

  33. [33]

    What Do CFTs Tell Us About Anti-de Sitter Spacetimes?

    V. Balasubramanian, S. B. Giddings and A. E. Lawrence,What do CFTs tell us about Anti-de Sitter space-times?, JHEP03(1999) 001, hep-th/9902052

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  34. [34]

    On the construction of local fields in the bulk of AdS_5 and other spaces

    I. Bena,On the construction of local fields in the bulk of AdS(5) and other spaces, Phys. Rev. D62(2000) 066007, hep-th/9905186

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  35. [35]

    Local bulk operators in AdS/CFT: a boundary view of horizons and locality

    A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe,Local bulk operators in AdS/CFT: A Boundary view of horizons and locality, Phys. Rev. D73(2006) 086003, hep-th/0506118

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  36. [36]

    Holographic representation of local bulk operators

    A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe,Holographic representation of local bulk operators, Phys. Rev. D74(2006) 066009, hep-th/0606141

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  37. [37]

    Local bulk operators in AdS/CFT: A holographic description of the black hole interior

    A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe,Local bulk operators in AdS/CFT: A Holographic description of the black hole interior, Phys. Rev. D75 (2007) 106001, hep-th/0612053

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  38. [38]

    Holographic representation of bulk fields with spin in AdS/CFT

    D. Kabat, G. Lifschytz, S. Roy and D. Sarkar,Holographic representation of bulk fields with spin in AdS/CFT, Phys. Rev. D86(2012) 026004, arXiv:1204.0126 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  39. [39]

    Decoding the hologram: Scalar fields interacting with gravity

    D. Kabat and G. Lifschytz,Decoding the hologram: Scalar fields interacting with gravity, Phys. Rev. D89(2014) 066010, arXiv:1311.3020 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  40. [40]

    Bulk equations of motion from CFT correlators

    D. Kabat and G. Lifschytz,Bulk equations of motion from CFT correlators, JHEP 09(2015) 059, arXiv:1505.03755 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  41. [41]

    S. R. Roy and D. Sarkar,Hologram of a pure state black hole, Phys. Rev. D92(2015) 126003, arXiv:1505.03895 [hep-th]. 21

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  42. [42]

    Local bulk physics from intersecting modular Hamiltonians

    D. Kabat and G. Lifschytz,Local bulk physics from intersecting modular Hamiltoni- ans, JHEP06(2017) 120, arXiv:1703.06523 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  43. [43]

    The Boundary Dual of Bulk Local Operators

    F. Sanches and S. J. Weinberg,The Boundary Dual of Bulk Local Operators, Phys. Rev. D96(2017) 026004, arXiv:1703.07780 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  44. [44]

    CFT descriptions of bulk local states in the AdS black holes

    K. Goto and T. Takayanagi,CFT Descriptions of Bulk Local States in the AdS Black Holes, JHEP10(2017) 153, arXiv:1704.00053 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  45. [45]

    S. R. Roy and D. Sarkar,Holographic Bulk Reconstruction Beyond (Super)gravity, arXiv:1704.06294 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  46. [46]

    Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields

    G. S´ arosi and T. Ugajin,Modular Hamiltonians of Excited States, OPE Blocks and Emergent Bulk Fields, JHEP01(2018) 012, arXiv:1705.01486 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [47]

    Bulk reconstruction in AdS and Gel'fand-Graev-Radon Transform

    S. Bhowmick, K. Ray and S. Sen,Bulk reconstruction in AdS and Gel’fand–Graev–Radon Transform, JHEP10(2017) 082, arXiv:1705.06985 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  48. [48]

    Bhowmick, K

    S. Bhowmick, K. Ray and S. Sen,Holography in de Sitter and anti-de Sitter Spaces and Gel’fand Graev Radon transform, Phys. Lett. B798(2019) 134977, arXiv:1903.07336 [hep-th]

    work page arXiv 2019

  49. [49]

    Holography on local fields via Radon Transform

    S. Bhowmick and K. Ray,Holography on local fields via Radon Transform, JHEP09 (2018) 126, arXiv:1805.07189 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  50. [50]

    Aoki and J

    S. Aoki and J. Balog,Extension of the HKLL bulk reconstruction for small∆, JHEP 05(2023) 034, arXiv:2302.11693 [hep-th]

    work page arXiv 2023

  51. [51]

    I. M. Gel’fand, M. I. Graev and N. Ya. Vilenkin,Generalized Functions, Vol. 5, Academic Press, 1966, AMS Chelsea Publishing

    work page 1966

  52. [52]

    Helgason,Integral Geometry and Radon Transforms, Springer, 2011

    S. Helgason,Integral Geometry and Radon Transforms, Springer, 2011

    work page 2011

  53. [53]

    Kumahara and M

    K. Kumahara and M. Wakayama,On Radon transform for Minkowski space, J. Fac. Gen. Ed. Tottori Univ.27(1993) 139

    work page 1993

  54. [54]

    R. N. Lee and A. A. Pomeransky,Critical Points and Number of Master Integrals, JHEP11(2013) 165, arXiv:1308.6676 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  55. [55]

    R. P. Klausen,Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems, JHEP04(2020) 121, arXiv:1910.08651 [hep-th]

    work page arXiv 2020

  56. [56]

    Weinzierl,Feynman Integrals, Springer, 2022, arXiv:2201.03593 [hep-th]

    S. Weinzierl,Feynman Integrals, Springer, 2022, arXiv:2201.03593 [hep-th]

    work page arXiv 2022

  57. [57]

    Laplace, Residue, and Euler integral representations of GKZ hypergeometric functions

    S.-J. Matsubara-Heo,Laplace, residue, and Euler integral representations of GKZ hypergeometric functions, arXiv:1801.04075 [math.CA]

    work page internal anchor Pith review Pith/arXiv arXiv

  58. [58]

    Pal and K

    A. Pal and K. Ray,Conformal integrals in all dimensions as generalised hypergeomet- ric functions and Clifford groups, J. Math. Phys.66, 043504 (2025), arXiv:2303.17326 [hep-th]. 22

    work page arXiv 2025