pith. sign in

arxiv: 2606.13665 · v1 · pith:B2I63WVEnew · submitted 2026-06-11 · ✦ hep-th

One-loop five-point gluing analytically

B. Eden , M. Gottwald This is my paper

Pith reviewed 2026-06-27 05:44 UTC · model grok-4.3

classification ✦ hep-th
keywords one-loop five-point functionN=4 super Yang-Millsstress tensor multipletsgluing correctionsEuler integralsintersection theoryanalytic evaluationgeneralised hypergeometric functions
0
0 comments X

The pith

The one-loop five-point function of stress-tensor multiplets in N=4 SYM receives a complete analytic evaluation for every process.

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

The paper shows how to convert the series of residues that arise from integrability-based gluing corrections into Euler integrals. These integrals are then evaluated directly when possible or recovered through intersection theory on generalised hypergeometric functions. The procedure yields explicit analytic expressions for all contributions to the five-point correlator. A reader would care because the result replaces earlier partial or numerical matches with closed-form answers that can be inspected or used in further calculations.

Core claim

For the first time, we present a full analytic evaluation of all processes. The starting point is the re-summation of series of residues into Euler integrals. These are directly integrated where possible or otherwise recovered from intersection theory for generalised hypergeometric functions.

What carries the argument

Re-summation of residue series into Euler integrals, followed by direct integration or intersection-theory recovery of generalised hypergeometric functions.

If this is right

  • Every process contributing to the one-loop five-point function now possesses an explicit analytic formula.
  • The integrability gluing corrections can be matched to field-theory results without intermediate numerical checks.
  • The same residue-to-Euler-integral conversion supplies the analytic building blocks needed for any further manipulation of the correlator.

Where Pith is reading between the lines

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

  • The same conversion technique may extend to six-point or higher correlators once the corresponding residue series are written down.
  • Closed analytic forms make it feasible to extract OPE coefficients or study light-cone limits without numerical fitting.
  • Intersection-theory recovery offers a template for handling other hypergeometric integrals that appear in higher-loop calculations.

Load-bearing premise

The re-summation that turns every series of residues from the gluing corrections into Euler integrals remains valid and exhaustive for all five-point processes.

What would settle it

Direct numerical evaluation of one or more five-point correlators at a generic kinematic point that differs from the closed-form expression obtained after the Euler-integral step.

read the original abstract

The one-loop five-point function of stress tensor multiplets in N=4 super Yang-Mills theory in four dimensions has previously been studied by integrability methods. The finite-size viz gluing corrections defining it were originally matched on the known field theory result - establishing many features of the formalism on the way - and later to a good extent also analytically evaluated. For the first time, we present a full analytic evaluation of all processes. The starting point is the re-summation of series of residues into Euler integrals. These are directly integrated where possible or otherwise recovered from intersection theory for generalised hypergeometric functions.

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

1 major / 0 minor

Summary. The paper claims to deliver the first complete analytic evaluation of all processes in the one-loop five-point function of stress-tensor multiplets in N=4 SYM. It starts from the re-summation of residue series arising in the integrability gluing corrections, converts them into Euler integrals, and evaluates those either by direct integration or by intersection theory applied to generalised hypergeometric functions.

Significance. If the re-summation step is exhaustive and the resulting Euler integrals are correctly identified, the work would supply the first fully analytic expressions for the five-point correlator, extending earlier integrability-based results that were only partially analytic or numerically matched to field theory. The systematic use of intersection theory for the hypergeometric sector would constitute a concrete technical advance.

major comments (1)
  1. The central claim rests on the assertion that the re-summation of residue series into Euler integrals is complete for every five-point process. No explicit derivation, completeness proof, or cross-check against all channels is supplied in the available text, leaving the mapping from integrability corrections to the claimed Euler integrals unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for greater transparency on the re-summation step. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim rests on the assertion that the re-summation of residue series into Euler integrals is complete for every five-point process. No explicit derivation, completeness proof, or cross-check against all channels is supplied in the available text, leaving the mapping from integrability corrections to the claimed Euler integrals unverified.

    Authors: We agree that the manuscript states the re-summation of residue series into Euler integrals as its starting point but does not supply an explicit, channel-by-channel derivation or a formal completeness argument in the main text. This omission was unintentional and weakens the central claim. In the revised version we will add a dedicated subsection (or appendix) that (i) derives the re-summation procedure from the integrability gluing corrections for a representative set of five-point processes, (ii) states the structural reason why the same mapping applies to all remaining channels, and (iii) provides explicit cross-checks against previously known partial analytic or numerical results for at least two independent processes. We therefore accept the referee’s criticism and will revise accordingly. revision: yes

Circularity Check

0 steps flagged

Minor reliance on prior matched integrability formalism; new re-summation step presented as independent

full rationale

The abstract notes that gluing corrections were previously matched to field theory results to establish features of the formalism, indicating some self-citation in the foundational setup. However, the central claim of a full analytic evaluation via re-summation of residue series into Euler integrals (followed by direct integration or intersection theory) is introduced as a new starting point without any quoted reduction showing that this re-summation is defined in terms of the target results, fitted to data by construction, or justified solely by overlapping-author citations that themselves lack independent verification. No equations or steps in the provided text exhibit self-definitional loops, fitted inputs renamed as predictions, or ansatz smuggling. The derivation chain for the new results therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities; all such elements remain unknown.

pith-pipeline@v0.9.1-grok · 5615 in / 1175 out tokens · 29563 ms · 2026-06-27T05:44:22.983521+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 4 linked inside Pith

  1. [1]

    The large N limit of superconformal field theories and supergravity,

    J.Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv.Theor. Math.Phys.2(1998) 231; S.Gubser, I.Klebanov, A.Polyakov, “Gauge theory correlators from non- critical string theory,” Phys.Lett.B428(1998) 105; E.Witten, “Anti de Sitter space and holography,” Adv.Theor.Math.Phys.2(1998) 253

    1998

  2. [2]

    The Bethe-ansatz forN= 4 Super Yang-Mills,

    J. Minahan and K. Zarembo, “The Bethe-ansatz forN= 4 Super Yang-Mills,” JHEP03(2003) 013; N. Beisert, C. Kristjansen and M. Staudacher, “The dilatation operator of conformalN= 4 super Yang- Mills theory,” Nucl. Phys.B664(2003) 131; N. Beisert and M. Staudacher, “Long-rangeP SU(2,2|4) Bethe ansaetze for gauge theory and strings,” Nucl. Phys.B727(2005) 1

    2003

  3. [3]

    Integrability and transcendentality,

    B. Eden and M. Staudacher, “Integrability and transcendentality,” J. Stat. Mech.11(2006) P11014

    2006

  4. [4]

    Transcendentality and crossing,

    B. Eden, N. Beisert and M. Staudacher, “Transcendentality and crossing,” J.Stat. Mech.01(2007) P01021

    2007

  5. [5]

    Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling,

    B. Basso, G. Korchemsky and J. Kotanski, “Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling,” Phys. Rev. Lett.100(2008) 091601

    2008

  6. [6]

    A Test of the AdS/CFT correspondence using high-spin operators,

    M. Benna, S. Benvenuti, I. Klebanov and A. Scardicchio, “A Test of the AdS/CFT correspondence using high-spin operators,” Phys. Rev. Lett.98(2007), 131603

    2007

  7. [7]

    Quantum spectral curve for planarN= 4 super- Yang-Mills theory,

    N. Gromov, V. Kazakov, S. Leurent and D. Volin, “Quantum spectral curve for planarN= 4 super- Yang-Mills theory,” Phy. Rev. Lett.112(2014) no.1, 011602; C. Marboe and D. Volin, “Quantum spectral curve as a tool for a perturbative quantum field theory,” Nucl. Phys.B899(2015) 810

    2014

  8. [8]

    Structure constants and integrable bootstrap in planarN= 4 SYM Theory,

    B. Basso, S. Komatsu and P. Vieira, “Structure constants and integrable bootstrap in planarN= 4 SYM Theory,”hep-th/1505.06745

    Pith/arXiv arXiv

  9. [9]

    Tessellating cushions: four-point functions inN= 4 SYM,

    B. Eden and A. Sfondrini, “Tessellating cushions: four-point functions inN= 4 SYM,” JHEP10(2017) 098

    2017

  10. [10]

    Hexagonalization of correlation functions,

    T. Fleury and S. Komatsu, “Hexagonalization of correlation functions,” JHEP01(2017) 130

    2017

  11. [11]

    Four point functions inN= 4 supersymmetric Yang-Mills theory at two loops,

    B. Eden, P. Howe, C. Schubert, E. Sokatchev and P. West, “Four point functions inN= 4 supersymmetric Yang-Mills theory at two loops,” Nucl. Phys.B557(1999) 355; B. Eden, C. Schubert and E. Sokatchev, “Three loop four point correlator inN= 4 SYM,” Phys. Lett.B482(2000) 309; M. Bianchi, S. Kovacs, G. Rossi and Y. Stanev, “Anomalous dimensions inN= 4 SYM the...

    1999

  12. [12]

    Octagons I: combinatorics and non-planar resummations,

    T.Bargheer, F.Coronado, P.Vieira, “Octagons I: combinatorics and non-planar resummations,” JH08 (2019) 162; I.Kostov, V.Petkova, D.Serban, “Determinant formula for the octagon form factor in N= 4 supersymmetric Yang-Mills theory,” Phys.RevLett.122(2019) no.23, 231601; A.Belitsky, G.Korchemsky, “Octagon at finite coupling,” JH07(2020) 219

    2019

  13. [13]

    Analytical result for dimensionally regularized massless on-shell double box,

    V. Smirnov, “Analytical result for dimensionally regularized massless on-shell double box,” Phys. Lett. B460(1999) 397

    1999

  14. [14]

    Hexagonalization of correlation functions II: two-particle contributions,

    T. Fleury and S. Komatsu, “Hexagonalization of correlation functions II: two-particle contributions,” JHEP02(2018) 177

    2018

  15. [15]

    Superprotected n-point correlation functions of local operators inN= 4 super Yang-Mills,

    N. Drukker, J. Plefka, “Superprotected n-point correlation functions of local operators inN= 4 super Yang-Mills,” JHEP04(2009) 052

    2009

  16. [16]

    Multi-particle finite-volume effects for hexagon tessellations,

    M. De Leeuw, B. Eden, D. le Plat, T. Meier and A. Sfondrini, “Multi-particle finite-volume effects for hexagon tessellations,” JHEP09(2020) 039

    2020

  17. [17]

    Thesu(2|2) dynamic S-matrix,

    N. Beisert, “Thesu(2|2) dynamic S-matrix,” Adv. Theor. Math. Phys.12(2008) 945

    2008

  18. [18]

    The bound state S-matrix for AdS(5) x S**5 superstring,

    G. Arutyunov, M. de Leeuw and A. Torrielli, “The bound state S-matrix for AdS(5) x S**5 superstring,” Nucl. Phys.B819(2009) 319

    2009

  19. [19]

    The Dressing Factor and Crossing Equations,

    G. Arutyunov and S. Frolov, “The Dressing Factor and Crossing Equations,” J. Phys.A42(2009) 425401. 32

    2009

  20. [20]

    Bound state scattering simplified,

    M. de Leeuw, B. Eden and A. Sfondrini, “Bound state scattering simplified,” Phys. Rev.D102(2020) no.12, 126001

    2020

  21. [21]

    Theory of hypergeometric functions,

    K. Aomoto and M. Kita, “Theory of hypergeometric functions,” Springer Monographs in Mathematics, Springer Japan (2011)

    2011

  22. [22]

    Feynman integrals and intersection theory,

    P. Mastrolia and S. Mizera, “Feynman integrals and intersection theory,” JHEP02(2019) 139; H. Frellesvig, F. Gasparotto, M. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, “Vector space of Feynman integrals and multivariate intersection numbers,” Phys. Rev. Lett.123(2019) no.20, 201602; H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastroli...

    2019

  23. [23]

    Multiple series representations of n-fold Mellin- Barnes integrals,

    B. Ananthanarayan, S. Banik, S. Friot and S. Ghosh, “Multiple series representations of n-fold Mellin- Barnes integrals,” Physical. Rev. Lett.127(2021) no.15, 151601

    2021

  24. [24]

    Kotikov, L

    A. Kotikov, L. Lipatov and V. Velizhanin, Phys. Lett.B557(2003) 114

    2003

  25. [25]

    Critical points and number of master integrals,

    R. Lee, A. Pomeransky, “Critical points and number of master integrals,” JHEP11(2013) 165

    2013

  26. [26]

    Gluing via intersection theory,

    G. Crisanti, B. Eden, M. Gottwald, P. Mastrolia and T. Scherdin, “Gluing via intersection theory,” Phys. Rev. Lett.134(2025) no.25, 251603

    2025

  27. [27]

    Multiloop integrals in dimensional regularization made simple,

    J.Henn, “Multiloop integrals in dimensional regularization made simple,” Phys. Rev. Lett.110(2013) 251601

    2013

  28. [28]

    Multiple polylogarithms, cyclotomy and modular complexes,

    A.Goncharov, “Multiple polylogarithms, cyclotomy and modular complexes,” Math. Res. Lett.5(1998), 497; A. Goncharov, M. Spradlin, C. Vergu and A.Volovich, “Classical polylogarithms for amplitudes and Wilson loops,” Phys. Rev. Lett.105(2010) 151605

    1998

  29. [29]

    The off-shell one- and two-loop box recovered from intersection theory,

    B. Eden, “The off-shell one- and two-loop box recovered from intersection theory,” Nucl. Phys.B1024 (2026) 117336

    2026

  30. [30]

    Modern techniques of multiloop calculations,

    R. N. Lee, “Modern techniques of multiloop calculations,” Proceedings of the 49th Rencontres de Moriond- 2014 QCD and High Energy Interactions, (2014) 297,hep-ph/1405.5616; K. Larsen and Y. Zhang, “Integration-by-parts reductions from unitarity cuts and algebraic geometry,” Phys. Rev.D93(2016) no.4, 041701

    Pith/arXiv arXiv 2014

  31. [31]

    Five legs @ three loops: N=4 sYM amplitude near mass-shell,

    A. Belitsky, L. Bork, R. Lee and V. Smirnov, “Five legs @ three loops: N=4 sYM amplitude near mass-shell,”hep-th/2605.17056, “Five legs @ three loops: slightly off-shell dual conformal integrals,” hep-th/2605.17057

    Pith/arXiv arXiv

  32. [32]

    Space-time S-matrix and flux-tube S-matrix at finite coupling,

    B. Basso, A. Sever and P. Vieira, “Space-time S-matrix and flux-tube S-matrix at finite coupling,” Phys. Rev. Lett.111(2013) 091602; “Space-time S-matrix and flux tube S-matrix II. Extracting and matching data,” JHEP01(2014) 008; “Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions,” JHEP08(2014) 085; “Space-time S-matrix and fl...

    2013

  33. [33]

    Hexagon OPE resummation and multi-Regge kinematics,

    J. Drummond and G. Papathanasiou, “Hexagon OPE resummation and multi-Regge kinematics,” JHEP 02(2016) 185

    2016

  34. [34]

    The massless higher-loop two-point function,

    F. Brown, “The massless higher-loop two-point function,” Comm. Math. Phys.287(2009) 925; E. Panzer, “Algorithms for the symbolic integration of hyperlogarithms with applications to Feyn- man integrals,” Comput. Phys. Commun.188(2014) 148; “Feynman integrals and hyperlogarithms,” math-ph/1506.07243

    Pith/arXiv arXiv 2009

  35. [35]

    Cordova, JHEP1701(2017) 051; H

    L. Cordova, JHEP1701(2017) 051; H. Lam, M. von Hippel, JHEP1612(2016) 011

    2017

  36. [36]

    Thermodynamic Bethe ansatz in relativistic models. Scaling three state Potts and Lee-Yang models,

    A. B. Zamolodchikov, “Thermodynamic Bethe ansatz in relativistic models. Scaling three state Potts and Lee-Yang models,” Nuclear. Phys.B342(1990) 695; G. Arutyunov and S. Frolov, “On string S-matrix, bound states and TBA,” JHEP12(2007) 024, “The S-matrix of string bound states,” Nucl. Phys. B804(2008) 90; N. Gromov, V. Kazakov and P. Vieira, “Exact spectr...

    1990