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arxiv: 2512.21947 · v2 · submitted 2025-12-26 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Notes on off-shell conformal integrals and correlation functions at five points

Chia-Kai Kuo , Qinglin Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-16 19:53 UTC · model grok-4.3

classification ✦ hep-th
keywords conformal integralsfive-point functionshalf-BPS correlatorsuniform transcendentalleading singularitiesN=4 SYMtwo-loopoff-shell integrals
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0 comments X

The pith

A basis of six uniform-transcendental integrals spans the five-point off-shell conformal cases at two loops.

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 construct a complete set of uniform-transcendental pure integrals for five-point conformal integrals by using conformal invariance to diagonalize leading singularities. This produces six distinct topologies that can be mapped to standard four-mass integral families. The method combines canonical differential equations with integration-by-parts reduction to obtain integrated results at the symbol level. These results are then applied to compute the two-loop five-point half-BPS correlation functions in maximally supersymmetric Yang-Mills theory, covering both maximal and non-maximal sectors. The approach provides a practical way to handle the complexity of higher-point conformal integrals in perturbation theory.

Core claim

By diagonalizing leading singularities subject to conformal invariance, the authors obtain a basis of uniform-transcendental pure integrals that spans six distinct topologies and covers all five-point off-shell conformal integrals appearing in the two-loop half-BPS correlators.

What carries the argument

Diagonalization of leading singularities under conformal invariance to generate a basis of UT pure integrals across six topologies.

If this is right

  • The integrated symbol-level results give explicit expressions for the two-loop five-point half-BPS correlators.
  • This basis can be used to simplify computations of similar integrals in related theories or at higher orders.
  • Mapping to known two-loop four-mass integral families allows reuse of existing computational tools.
  • Both maximal and non-maximal sectors of the correlators are now accessible at symbol level.

Where Pith is reading between the lines

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

  • If the basis is complete, it may extend to six-point or higher-point conformal integrals by similar diagonalization techniques.
  • The symbol results could be used to extract physical information like OPE coefficients or check consistency with known bootstrap constraints.
  • Generalizing the conformal frame fixing might reduce the number of independent integrals in other dimensions or signatures.

Load-bearing premise

The six topologies obtained by diagonalizing leading singularities under conformal invariance form a complete basis for all five-point off-shell conformal integrals that appear in the two-loop half-BPS correlators.

What would settle it

Finding a five-point off-shell conformal integral in the two-loop half-BPS correlator that cannot be expressed as a linear combination of the six topologies would show the basis is incomplete.

Figures

Figures reproduced from arXiv: 2512.21947 by Chia-Kai Kuo, Qinglin Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. Two-loop five-point relevant topologies. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Excluded two–loop five–point topologies. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fixing frame of conformal kinematics to get general [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. 2-loop diagram with 4 external massive legs [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We study five-point off-shell conformal integrals and the associated half-BPS correlation functions at two loops in the 't Hooft coupling expansion of maximally supersymmetric Yang-Mills theory. We construct a basis of uniform-transcendental (UT) pure integrals spanning six distinct topologies by diagonalizing leading singularities subject to conformal invariance. By fixing conformal frames, this basis can be mapped to known two-loop four-mass integral families. We then compute the integrated results by combining canonical differential equations with integration-by-parts reduction. As an application, we present symbol-level integrated results for the two-loop five-point half-BPS correlators, including both maximal and non-maximal sectors.

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. This paper develops a basis for five-point off-shell conformal integrals at two loops by constructing six uniform-transcendental pure integrals through the diagonalization of leading singularities under conformal invariance constraints. The basis is mapped to standard two-loop four-mass integral families, and the authors derive symbol-level integrated expressions for the two-loop five-point half-BPS correlation functions in maximally supersymmetric Yang-Mills theory using canonical differential equations and integration-by-parts methods.

Significance. Should the six-topology basis prove complete, this manuscript would provide an important technical tool for computing conformal integrals and correlation functions at higher points and loops. The explicit results for the half-BPS correlators, including non-maximal sectors, contribute to the growing body of knowledge on multi-point functions in N=4 SYM, potentially aiding in checks of integrability and AdS/CFT predictions.

major comments (1)
  1. [Basis construction (as described in the abstract and main text)] The completeness of the six-topology UT basis is assumed without an explicit proof via vector space dimension counting or by demonstrating that a generic IBP reduction of five-point topologies yields masters that all reduce into this basis. If integrals without leading singularities or additional topologies are missed, the canonical DE solutions and symbol results for the correlators would be incomplete. This is a load-bearing assumption for the central claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comment on the basis construction. We address the concern directly below and will revise the manuscript to strengthen the justification.

read point-by-point responses

  1. Referee: [Basis construction (as described in the abstract and main text)] The completeness of the six-topology UT basis is assumed without an explicit proof via vector space dimension counting or by demonstrating that a generic IBP reduction of five-point topologies yields masters that all reduce into this basis. If integrals without leading singularities or additional topologies are missed, the canonical DE solutions and symbol results for the correlators would be incomplete. This is a load-bearing assumption for the central claims.

    Authors: We agree that an explicit demonstration of completeness strengthens the central claims. In the revised manuscript we will add a new subsection that enumerates the dimension of the vector space of two-loop five-point off-shell conformal integrals of uniform transcendental weight. We show that the only leading-singularity structures compatible with conformal invariance are those spanned by the six topologies obtained via our diagonalization procedure. We further verify, by explicit IBP reduction on a representative set of five-point topologies, that all masters reduce to linear combinations within this basis and that no additional UT integrals appear. This establishes that the canonical differential equations and the symbol-level results for the half-BPS correlators are complete. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity detected; basis construction and results rely on external mappings and standard IBP

full rationale

The derivation constructs a six-topology UT basis via diagonalization of leading singularities under conformal invariance, then maps the basis to known external two-loop four-mass integral families and applies canonical differential equations combined with standard IBP reduction. No central quantity is defined in terms of a fitted parameter from the same paper, no self-citation forms the load-bearing justification for completeness, and no step reduces by construction to its own inputs. The completeness of the basis for appearing integrals is asserted rather than dimensionally proven in the provided text, but this is an assumption about coverage rather than a definitional or fitted-input circularity. The overall chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard conformal invariance and the existence of a complete UT basis obtained from leading singularities; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Conformal invariance constrains the allowed leading singularities of the five-point integrals.
    Invoked when diagonalizing leading singularities to obtain the UT basis.
  • domain assumption The six topologies span the full space of five-point off-shell conformal integrals relevant to the half-BPS correlators.
    Central completeness assumption required for the basis to be usable.

pith-pipeline@v0.9.0 · 5399 in / 1363 out tokens · 16800 ms · 2026-05-16T19:53:57.031275+00:00 · methodology

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Reference graph

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