Magnon dispersion to four loops in the ABJM and ABJ models

C. Sieg; J.A. Minahan; O. Ohlsson Sax

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Magnon dispersion to four loops in the ABJM and ABJ models

J.A. Minahan, O. Ohlsson Sax, C. Sieg

High Energy Physics - Theory · Submitted 17 Aug 2009

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira.

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The ABJM model is a superconformal Chern-Simons theory with N=6 supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension. We find that all coefficients have maximal transcendentality. We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The ABJM model is a superconformal Chern-Simons theory with N=6 supersymmetry which is believed to be integrable in the planar limit.	background 1.00	0.04	0.02	hedged possibility 0.76
s1	However, there is a coupling dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling.	gap 0.99	0.05	0.03	asserted 0.98
s2	We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension.	contribution 0.85	0.98	0.19	asserted 1.00
s3	We find that all coefficients have maximal transcendentality.	result 1.00	0.96	0.19	asserted 1.00
s4	We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira.	result 0.98	0.97	0.22	asserted 1.00
s5	We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.	contribution 0.43	0.73	0.08	speculative 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from simulation or computation 0.54.

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Original abstract

The ABJM model is a superconformal Chern-Simons theory with N=6 supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension. We find that all coefficients have maximal transcendentality. We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.

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Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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The one-loop scalar-sector spin chain of planar N=2 superconformal QCD contains two-site 'dimer' impurities and passes the Yang-Baxter test in the limit where the interpolating quiver coupling is switched off.
Exact Slope and Interpolating Functions in ABJM Theory
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The exact slope function of planar ABJM theory is derived via the Quantum Spectral Curve as a ratio of determinants in h(λ), and a closed-form conjecture for h(λ) itself is proposed.
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A two-loop computation of generalized latitude Wilson loops in ABJM theory yields an all-loop conjecture for the Bremsstrahlung function that matches the known two-loop cusp result.

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Scattering Amplitudes/Wilson Loop Duality In ABJM Theory
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The two-loop four-point amplitude in ABJM theory equals the two-loop light-like four-polygon Wilson loop up to a scheme-dependent constant, extending the amplitude/Wilson-loop duality to three dimensions.
Y-system, TBA and Quasi-Classical Strings in AdS4 x CP3
hep-th 2009-12 conditional novelty 6.0 of 10

The authors derive TBA equations for planar ABJM from conjectured mirror Bethe equations, show the published Y-system must be modified for the momentum-carrying nodes, and match their strong-coupling solution to the q...
Anomalous dimensions at four loops in N=6 superconformal Chern-Simons theories
hep-th 2009-12 conditional novelty 6.0 of 10

The four-loop coefficient of the ABJM magnon dispersion function is h4(σ) = -(4+σ²)ζ(2), and the four-loop wrapping correction for the length-four 20 operator is (32-16ζ(2))λ̄⁴, matching the Y-system prediction.
Y-system and Quasi-Classical Strings
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Solving the conjectured Y-system at strong coupling reproduces the one-loop quantum string spectrum in AdS3 x S1, including all exponential finite-size corrections, matching the algebraic-curve quantization.
Roadmap on Wilson loops in 3d Chern-Simons-matter theories
hep-th 2019-10 accept novelty 5.0 of 10

The review organizes the state of the art on ABJ(M) Wilson loops and introduces a manifestly reparametrization-invariant superconnection formulation of the 1/6 BPS fermionic loop.
Wilson lines as superconformal defects in ABJM theory: a formula for the emitted radiation
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The Bremsstrahlung function of the 1/2 BPS cusp in ABJM theory equals one over four pi squared times the derivative of the deformed circular Wilson loop with respect to the deformation parameter, derived from defect t...
Review of AdS/CFT Integrability, Chapter IV.3: N=6 Chern-Simons and Strings on AdS4xCP3
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The chapter consolidates how planar ABJM theory and IIA strings on AdS4xCP3 both reduce to an alternating OSp(6|4) spin chain, and catalogues where the Bethe equations, S-matrix and algebraic curve differ from the AdS5 case.

How the literature uses it

45 arXiv papers cite this in 84 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 57 sentences without a confident label

Uses its data or values 6 papers

The four-loop coefficient $e_4,4=-16$ was computed in [ref] , while $e_4,2= 4 h_2$ was argued in [ref] .

Integrability of N=6 Chern-Simons Theory at Six Loops and Beyond · Hamiltonian and Spectrum From Integrability · 0.96
Since at weak coupling $h(λ)$ is known up to the forth order [ref] , we easily find f_CS(λ)= 4 λ^2 - 24 ζ_2 λ^4 + O(λ^6) and the ansatz ([ref]) provides a prediction for the four–loop expression of the finite remainder $F_4^(4)$ (in the notation of [ref] ) for the ABJM four–point scattering amplitude [ref]

Scattering in ABJ theories · BDS–like ansatz · 0.85
And (on the string side) $Δ$ is the energy, $J$ and $Q$ are $CP^3$ angular momenta, and $S$ is an $AdS_4$ angular momentum. ] [equation] Four-loop gauge theory calculations [ref] show that $h_4=-4ζ(2)≈-6.58$.

Quantum Strings and the AdS4/CFT3 Interpolating Function · Introduction · 0.98

Compares with it 8 papers

It reads [equation] Now, by subtracting from [ref] the range five contribution and inserting $h_4(σ)=-(4+σ^2)ζ(2)$, the subtracted dilatation operator becomes [equation] The dilatation operator for length four states then reads [equation] and it coincides with the results obtained in terms of component fields [ref] .

Superspace calculation of the four-loop spectrum in N=6 supersymmetric Chern-Simons theories · Wrapping interactions · 0.99
Expansion at weak/strong coupling gives h(λ) = λ-π^2 λ^3/3+5 π^4 λ^5/12-893 π^6 λ^7/1260+O(λ^9), h(λ) = √(1/2(λ-1/24) )-log2/2 π+O( e^- π√(8λ)), which reproduces all known coefficients at weak and at strong coupling i.e. in total $4$ nontrivial coefficients [ref] .

Exact Slope and Interpolating Functions in ABJM Theory · Comparison with Localization · 0.85
This expansion can be compared against the direct 4-loop perturbative calculation of [ref] (later also confirmed in [ref] ), which gives h_4-loop^2(λ_1,λ_2)=λ_1λ_2+h_4$λ_1λ_2$^2+h_4,σλ_1λ_2(λ_1-λ_2)^2 where h_4=-2π^2/3, h_4,σ=-π^2/6 , perfectly matching our result!

On the Exact Interpolating Function in ABJ Theory · Weak coupling limit · 1.00

Cites it as support 1 paper

Indeed, it was explicitly shown in [ref] and [ref] by direct computations, that this is true at four-loop order: the contributions to the dilatation operator of the diagrams that could lead to interactions between the two types of magnons cancel.

The Leading Order Dressing Phase in ABJM Theory · Introduction · 0.98

Cites it as background 7 papers

The two-loop dilatation operator of $N=6$ superconformal Chern-Simons gauge theory has been studied in [ref] - [ref] .

On the absence of reflection in AdS4/CFT3 · Local charges and the reflectionless property · 0.91
Preliminary results can be found in literature, concerning integrability [ref] and related Yangian symmetry [ref] .

Scattering Amplitudes/Wilson Loop Duality In ABJM Theory · Introduction · 0.94
Analogous checks have been done for AdS$_4$/CFT$_3$ – the four-loop wrapping corrections obtained in [ref] were reproduced in [ref] , [ref] , while in [ref] the proposed Y-system and TBA were shown to be remarkably consistent with the algebraic curve quantization [ref] .

Numerical results for the exact spectrum of planar AdS4/CFT3 · Introduction · 0.86

What it cites · 39 sentences without a confident label

Uses the method or tool of 1 paper

Following the strategy of [ref] , we first subtract from the asymptotic dilatation operator [ref] the range five interactions (given by $S_2$ and the reflected diagram).

Anomalous dimension with wrapping at four loops in N=4 SYM · Four-loop wrapping interactions · 0.98

Uses data or values from 1 paper

Adding $D_4,odd^sub$ to $D_4,odd^w$, we find that the anomalous dimension of the 20 is given by [equation] where we have included the lower orders [ref] .

The Bethe ansatz for superconformal Chern-Simons · Four-loop wrapping interactions · 0.90

Compares with 7 papers

Hence the four-loop wrapping contribution to the anomalous dimension is [equation] which agrees with the GKV $Y$-system prediction [ref] .

Integrability for the Full Spectrum of Planar AdS/CFT · Four-loop wrapping interactions · 1.00
The result for $h_4$ in ([ref]) is strikingly similar to the leading wrapping correction for length four operators, which has been obtained by Gromov, Kazakov and Vieira [ref] by means of the proposed $Y$-system [ref] for the ABJM model.[This result is not altered by a correction [ref] of the originally proposed $Y$-system.] In this paper we explicitly compute the wrapping correction with our methods and find agreement with their result.

Y-system, TBA and Quasi-Classical Strings in AdS4 x CP3 · Introduction and summary of results · 0.99
The result for $h_4$ in ([ref]) is strikingly similar to the leading wrapping correction for length four operators, which has been obtained by Gromov, Kazakov and Vieira [ref] by means of the proposed $Y$-system [ref] for the ABJM model.[This result is not altered by a correction [ref] of the originally proposed $Y$-system.] In this paper we explicitly compute the wrapping correction with our methods and find agreement with their result.

Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal · Introduction and summary of results · 0.99

Cites as background 7 papers

In the ABJM case, the fundamental fields are in bifundamental representations of $U(N)×U(N)$ and the corresponding spin chain is of alternating type [ref] .

Integrable Spin Chain in Superconformal Chern-Simons Theory · Introduction and summary of results · 0.80
In the ABJM case, the fundamental fields are in bifundamental representations of $U(N)×U(N)$ and the corresponding spin chain is of alternating type [ref] .

The Bethe ansatz for superconformal Chern-Simons · Introduction and summary of results · 0.80
In the ABJM case, the fundamental fields are in bifundamental representations of $U(N)×U(N)$ and the corresponding spin chain is of alternating type [ref] .

Spin Chains in N=6 Superconformal Chern-Simons-Matter Theory · Introduction and summary of results · 0.80

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