High Energy Physics - Theory · arXiv:hep-th/0509121 v2 · submitted 15 Sep 2005
Integrability of two-loop dilatation operator in gauge theories
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“Our analysis suggests that integrability is only tied to the planar limit and it is sensitive neither to conformal symmetry nor supersymmetry.”
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Abstract
We study the two-loop dilatation operator in the noncompact SL(2) sector of QCD and supersymmetric Yang-Mills theories with N=1,2,4 supercharges. The analysis is performed for Wilson operators built from three quark/gaugino fields of the same helicity belonging to the fundamental/adjoint representation of the SU(3)/SU(N_c) gauge group and involving an arbitrary number of covariant derivatives projected onto the light-cone. To one-loop order, the dilatation operator inherits the conformal symmetry of the classical theory and is given in the multi-color limit by a local Hamiltonian of the Heisenberg magnet with the spin operators being generators of the collinear subgroup of full (super)conformal group. Starting from two loops, the dilatation operator depends on the representation of the gauge group and, in addition, receives corrections stemming from the violation of the conformal symmetry. We compute its eigenspectrum and demonstrate that to two-loop order integrability survives the conformal symmetry breaking in the aforementioned gauge theories, but it is violated in QCD by the contribution of nonplanar diagrams. In SYM theories with extended supersymmetry, the N-dependence of the two-loop dilatation operator can be factorized (modulo an additive normalization constant) into a multiplicative c-number. This property makes the eigenspectrum of the two-loop dilatation operator alike in all gauge theories including the maximally supersymmetric theory. Our analysis suggests that integrability is only tied to the planar limit and it is sensitive neither to conformal symmetry nor supersymmetry.
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abstract compiled · 2026-09-23 · no language-model review
How the main claim is stated: hedged possibility 1.00
What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.74 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.42 · mentions code or data availability 0.01
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| We study the two-loop dilatation operator in the noncompact SL(2) sector of QCD and supersymmetric Yang-Mills theories with N=1,2,4 supercharges. | contribution 0.98 | 0.90 | 0.03 | asserted 0.82 |
| The analysis is performed for Wilson operators built from three quark/gaugino fields of the same helicity belonging to the fundamental/adjoint representation of the SU(3)/SU(N_c) gauge group and involving an arbitrary number of covariant derivatives projected onto the light-cone. | method 0.51 | 0.63 | 0.02 | asserted 0.96 |
| To one-loop order, the dilatation operator inherits the conformal symmetry of the classical theory and is given in the multi-color limit by a local Hamiltonian of the Heisenberg magnet with the spin operators being generators of the collinear subgroup of full (super)conformal group. | background 0.98 | 0.24 | 0.03 | asserted 1.00 |
| Starting from two loops, the dilatation operator depends on the representation of the gauge group and, in addition, receives corrections stemming from the violation of the conformal symmetry. | background 0.92 | 0.27 | 0.06 | asserted 1.00 |
| We compute its eigenspectrum and demonstrate that to two-loop order integrability survives the conformal symmetry breaking in the aforementioned gauge theories, but it is violated in QCD by the contribution of nonplanar diagrams. | result 0.99 | 0.95 | 0.74 | asserted 1.00 |
| In SYM theories with extended supersymmetry, the N-dependence of the two-loop dilatation operator can be factorized (modulo an additive normalization constant) into a multiplicative c-number. | result 1.00 | 0.83 | 0.07 | asserted 0.96 |
| This property makes the eigenspectrum of the two-loop dilatation operator alike in all gauge theories including the maximally supersymmetric theory. | result 0.68 | 0.74 | 0.12 | asserted 0.99 |
| Our analysis suggests that integrability is only tied to the planar limit and it is sensitive neither to conformal symmetry nor supersymmetry. | interpretation 0.99 | 0.71 | 0.80 | hedged possibility 1.00 |
What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.92
What was examined, and what was not
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- arXiv listing abstract v2
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- C citations
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How the labels were made
Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,599 input tokens, $0.0003 reported.
Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON
Cited by 7 Pith papers
Reviewed papers on Pith that cite this one. The full citation record
- From Scattering Amplitudes to the Dilatation Generator in N=4 SYM hep-th · Nov 2011
- Five-Loop Anomalous Dimension of Twist-Two Operators hep-th · Dec 2009
- BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin hep-th · Feb 2018
- Large spin systematics in CFT hep-th · Feb 2015
- Integrability in N=2 superconformal gauge theories hep-th · Oct 2013
- Six-Loop Anomalous Dimension of Twist-Three Operators in N=4 SYM hep-th · Mar 2010
- Anatomy of a gauge theory hep-th · Sep 2005
How the literature uses it
15 arXiv papers cite this in 21 sentences
Papers citing it · 12 sentences without a confident label
Compares with it 1 paper
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At two loops, there are 79 nonvanishing diagrams, and the result in $=2$ SYM can be written as [equation] with nested harmonic sums, recursively defined as [ref] [equation] in agreement with Ref. [ref] .
Anomalous dimensions of twist-two operators in extended N=2 and N=4 super Yang-Mills theories · Three-loop anomalous dimension in $=2$ SYM · 0.90
Cites it as support 1 paper
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Three gaugino operators have been studied at two loops in $N=1, 2, 4$ SYM by direct computation of the dilatation operator in [ref] .
Universality of three gaugino anomalous dimensions in N=4 SYM · Additional evidence for universality at two-loops · 0.98
Cites it as background 6 papers
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These operators have been studied at two loops in $N=1, 2, 4$ SYM by direct computation of the dilatation operator in [ref] .
Universality of three gaugino anomalous dimensions in N=4 SYM · Introduction · 0.84
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The first part comes from the Bethe Ansatz for the integrable spin chain, which was found first [ref] [Earlier, the similar integrability was opened in quantum chromodynamics in the Regge limit [ref] and for some of operators [ref] .] in the leading order for the simplest BMN-operators [ref] .
Six-Loop Anomalous Dimension of Twist-Three Operators in N=4 SYM · Introduction · 0.89
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At one loop the complete dilatation operator is known [ref] , while direct perturbative calculations at higher loops – without the assumption of integrability – have been performed only up to two [ref] , three [ref] and four loops [ref] in particular sectors.
Integrability and MHV diagrams in N=4 supersymmetric Yang-Mills theory · Conclusions · 0.90
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