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Uniformly transcendental bases for protected two-point functions
High Energy Physics - Theory · Submitted 12 Dec 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Inspired by this, we construct an explicit basis of uniformly transcendental master integrals for these correlators, through four loops in four and three loops in three dimensions.
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The perturbative expansion of two-point functions of lowest dimension supersymmetric operators in $\mathcal{N}=4$ SYM and ABJM theory exhibits uniform transcendental weight. Inspired by this, we construct an explicit basis of uniformly transcendental master integrals for these correlators, through four loops in four and three loops in three dimensions. In terms of these bases, the two-point functions simplify to rational linear combinations. Conversely, such explicit bases of uniformly transcendental integrals can be useful for other applications.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The perturbative expansion of two-point functions of lowest dimension supersymmetric operators in $\mathcal{N}=4$ SYM and ABJM theory exhibits uniform transcendental weight. | background 0.95 | 0.18 | 0.15 | asserted 1.00 |
| s1 | Inspired by this, we construct an explicit basis of uniformly transcendental master integrals for these correlators, through four loops in four and three loops in three dimensions. | contribution 0.99 | 0.96 | 0.44 | asserted 1.00 |
| s2 | In terms of these bases, the two-point functions simplify to rational linear combinations. | result 0.97 | 0.84 | 0.44 | asserted 1.00 |
| s3 | Conversely, such explicit bases of uniformly transcendental integrals can be useful for other applications. | interpretation 0.84 | 0.13 | 0.07 | hedged possibility 0.89 |
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The paper
Original abstract
The perturbative expansion of two-point functions of lowest dimension supersymmetric operators in $\mathcal{N}=4$ SYM and ABJM theory exhibits uniform transcendental weight. Inspired by this, we construct an explicit basis of uniformly transcendental master integrals for these correlators, through four loops in four and three loops in three dimensions. In terms of these bases, the two-point functions simplify to rational linear combinations. Conversely, such explicit bases of uniformly transcendental integrals can be useful for other applications.
Human review
References
How the literature uses it
3 arXiv papers cite this in 10 sentences
Papers citing it · 6 sentences without a confident label
Contests it 1 paper
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In the conventions of [ref], the two-loop coefficient is therefore[Equations (3.18) and (4.9) of the published version of [ref] contain typos, which have been corrected in the revised arXiv submission.
Universal and Transcendental Structures in Protected ABJM Two-Point Functions · Dimension-one correlator · contest 0.80
Uses its method or tool 1 paper
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To make uniform transcendentality manifest, we switch to the uniformly transcendental basis proposed in [ref] and obtain [equation] As above, solid dots in the integral diagrams denote squared propagators, corresponding to $(k^2)^-2$ for momentum $k$, while empty dots denote numerator factors $k^2$.
Tracing Transcendentality in Protected Correlators of N=4 SYM · Two loops · 0.99
Uses its data or values 1 paper
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The full expressions up to transcendental order 12, taken from [ref] and based on [ref] , are stored in the ancillary file , both in the following form and after Fourier transformation to position space and multiplication by $1-4ϵ/1-2ϵ$ to preserve uniform transcendentality and contribute directly to the reduced correlators $f_n,i,j$. 01 = -1/ϵ^6 +2 ζ_2/ϵ^4 +184 ζ_3/3 ϵ^3 +86 ζ_4/ϵ^2 +4144 ζ_5/5-368 ζ_2 ζ_3/3/ϵ +(1608 ζ_6-16928 ζ_3^2/9) +O(ϵ) 12 = -1/ϵ^6 +2 ζ_2/ϵ^4 +64 ζ_3/3 ϵ^3 +26 ζ_4/ϵ^2 +544 ζ_5/5-128 ζ_2 ζ_3/3/ϵ +(118 ζ_6-2048 ζ_3^2/9) +O(ϵ) 13 = -1/ϵ^6 +2 ζ_2/ϵ^4 +136 ζ_3/3 ϵ^3 +62 ζ_4/ϵ^2 +3184 ζ_5/5-272 ζ_2 ζ_3/3/ϵ +(1252 ζ_6-9248 ζ_3^2/9) +O(ϵ) 14 = -1/ϵ^6 +2 ζ_2/ϵ^4 +166 ζ_3/3 ϵ^3 +77 ζ_4/ϵ^2 +3934 ζ_5/5-332 ζ_2 ζ_3/3/ϵ +(3099 ζ_6/2-13778 ζ_3^2/9) +O(ϵ) 22 = 6 ζ_3/ϵ^3 +9 ζ_4/ϵ^2
Tracing Transcendentality in Protected Correlators of N=4 SYM · Master integral expansions · 0.99
Cites it as background 1 paper
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Recently, Bianchi has considered two-point functions of BPS operators in $d=4-2 ε$ dimensions in $𝒩=4$ SYM [ref] and also in $d=3-2 ε$ in ABJM [ref] .
Two-point functions in $4-2\,\varepsilon$ dimensions from localization · Introduction · 0.91
What it cites · 78 sentences without a confident label
Builds on 2 papers
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By searching for such alternative representatives, we also construct uniformly transcendental versions of the three most complicated topologies of [ref] .
Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve · Four dimensions · 0.94
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We perform an analogous analysis in three dimensions, building on our previous work [ref] , and obtain several novel results.
Transcendentality of ABJM two-point functions · Uniformly transcendental basis for two-point functions · 1.00
Uses the method or tool of 5 papers
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For these reductions we employed Forcer [ref] scripts executed in Form [ref] , which are tailored to this class of propagator-type integrals.
The Multithreaded version of FORM · Two-point functions of short protected operators · 1.00
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For these reductions we employed Forcer [ref] scripts executed in Form [ref] , which are tailored to this class of propagator-type integrals.
Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams · Two-point functions of short protected operators · 1.00
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For these reductions we employed Forcer [ref] scripts executed in Form [ref] , which are tailored to this class of propagator-type integrals.
FORM version 4.2 · Two-point functions of short protected operators · 1.00
Uses data or values from 1 paper
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We work up to transcendental weight twelve, for which explicit $ϵ$-expansions are available from [ref] through four loops.
Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve · Four dimensions · 0.94
Compares with 2 papers
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For comparison, the expansion in the master-integral basis of Forcer reads [ref] \begin{dmath*} ⟨O(p) O̅(-p) ⟩^(3)= -4(3 ϵ-1) (4 ϵ-1) (5 ϵ-1) (14484 ϵ^3+10379 ϵ^2+2433 ϵ+190)9 ϵ^3 (3 ϵ+1) (4 ϵ+1) (5 ϵ+1) -2.5mm < g r a p h i c s > + +16 (4 ϵ-1) (324 ϵ^2+157 ϵ+18)(2 ϵ-1)^23 ϵ^3 (4 ϵ+1) (5 ϵ+1) -1.6mm < g r a p h i c s > -20 (5 ϵ-1) (2 ϵ-1)^2ϵ^3 -2.5mm < g r a p h i c s > +16 (3 ϵ-1) (4 ϵ-1) (1257 ϵ^3+833 ϵ^2+165 ϵ+10) (2 ϵ-1)9 ϵ^3 (3 ϵ+1) (4 ϵ+1) (5 ϵ+1) -1.6mm < g r a p h i c s > +8 (2 ϵ-1)^3ϵ^3 -0.8mm < g r a p h i c s > -4 (3 ϵ-1) (408 ϵ^2+187 ϵ+21) (2 ϵ-1)^2ϵ^3 (4 ϵ+1) (5 ϵ+1) -0.8mm < g r a p h i c s > -4(3 ϵ-1) (5 ϵ-1) (3288 ϵ^3+2629 ϵ^2+579 ϵ+38) (2 ϵ-1)9 ϵ^3 (3 ϵ+1) (4 ϵ+1) (5 ϵ+1) -2.5mm < g r a p h i c s > +4(4 ϵ-1) (5 ϵ-1) (3592 ϵ^3+2938 ϵ^2+761 ϵ+63) (2 ϵ-1)3 ϵ^3 (3 ϵ+1) (4 ϵ+1) (5 ϵ+1) -2.5mm < g r a p h i c s > -48 (4 ϵ+1) (2 ϵ-1)ϵ(5 ϵ+1) -2.5mm < g r a p h i c s > -48 (2 ϵ-
Transcendentality of ABJM two-point functions · Two-point functions in terms of uniformly transcendental integrals · 0.94
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We obtain for the one-loop correction to the two-point function [equation] At two loops a more substantial simplification with respect to the expression in [ref] , using a different integral basis, already appears [equation] since the full contribution is expressed solely in terms of integrals with rational coefficients.
Protected and uniformly transcendental · Two-point functions in terms of uniformly transcendental integrals · 0.87
Cites as support 1 paper
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For the two more complicated master integrals, the planar topology already appears to be uniformly transcendental in three dimensions, as observed in [ref] , [equation]
Transcendentality of ABJM two-point functions · Three dimensions · 0.97
Cites as background 32 papers
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Similarly, the three-dimensional ABJM model [ref] shares many of these hidden properties [ref] .
Fractional M2-branes · Introduction · 0.88
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Similarly, the three-dimensional ABJM model [ref] shares many of these hidden properties [ref] .
Review of AdS/CFT Integrability, Chapter IV.3: N=6 Chern-Simons and Strings on AdS4xCP3 · Introduction · 0.88
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Similarly, the three-dimensional ABJM model [ref] shares many of these hidden properties [ref] .
N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals · Introduction · 0.88
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