Five-loop Konishi in N=4 SYM

Burkhard Eden; Emery Sokatchev; Gregory P. Korchemsky; Paul Heslop; Vladimir A. Smirnov

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Five-loop Konishi in N=4 SYM

Burkhard Eden, Paul Heslop, Gregory P. Korchemsky, Vladimir A. Smirnov, Emery Sokatchev

High Energy Physics - Theory · Submitted 26 Feb 2012

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The authors' main claim We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.

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arXiv ID1202.5733 (version 3)

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CitationBurkhard Eden, Emery Sokatchev, Gregory P. Korchemsky, Paul Heslop, Vladimir A. Smirnov. "Five-loop Konishi in N=4 SYM." arXiv:1202.5733 (2012). https://pith.science/paper/VMFTR2AC

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We present a new method for computing the Konishi anomalous dimension in N=4 SYM at weak coupling. It does not rely on the conventional Feynman diagram technique and is not restricted to the planar limit. It is based on the OPE analysis of the four-point correlation function of stress-tensor multiplets, which has been recently constructed up to six loops. The Konishi operator gives the leading contribution to the singlet SU(4) channel of this OPE. Its anomalous dimension is the coefficient of the leading single logarithmic singularity of the logarithm of the correlation function in the double short-distance limit, in which the operator positions coincide pairwise. We regularize the logarithm of the correlation function in this singular limit by a version of dimensional regularization. At any loop level, the resulting singularity is a simple pole whose residue is determined by a finite two-point integral with one loop less. This drastically simplifies the five-loop calculation of the Konishi anomalous dimension by reducing it to a set of known four-loop two-point integrals and two unknown integrals which we evaluate analytically. We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a new method for computing the Konishi anomalous dimension in N=4 SYM at weak coupling.	contribution 1.00	0.95	0.21	asserted 1.00
s1	It does not rely on the conventional Feynman diagram technique and is not restricted to the planar limit.	method 0.66	0.65	0.05	asserted 1.00
s2	It is based on the OPE analysis of the four-point correlation function of stress-tensor multiplets, which has been recently constructed up to six loops.	method 0.89	0.12	0.04	asserted 1.00
s3	The Konishi operator gives the leading contribution to the singlet SU(4) channel of this OPE.	background 0.66	0.25	0.04	asserted 1.00
s4	Its anomalous dimension is the coefficient of the leading single logarithmic singularity of the logarithm of the correlation function in the double short-distance limit, in which the operator positions coincide pairwise.	method 0.76	0.28	0.07	asserted 1.00
s5	We regularize the logarithm of the correlation function in this singular limit by a version of dimensional regularization.	method 1.00	0.93	0.03	asserted 1.00
s6	At any loop level, the resulting singularity is a simple pole whose residue is determined by a finite two-point integral with one loop less.	method 0.88	0.79	0.07	asserted 1.00
s7	This drastically simplifies the five-loop calculation of the Konishi anomalous dimension by reducing it to a set of known four-loop two-point integrals and two unknown integrals which we evaluate analytically.	method 0.62	0.95	0.29	asserted 1.00
s8	We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.	result 1.00	0.97	0.65	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.72 · evidence from proof or derivation 0.96.

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Original abstract

We present a new method for computing the Konishi anomalous dimension in N=4 SYM at weak coupling. It does not rely on the conventional Feynman diagram technique and is not restricted to the planar limit. It is based on the OPE analysis of the four-point correlation function of stress-tensor multiplets, which has been recently constructed up to six loops. The Konishi operator gives the leading contribution to the singlet SU(4) channel of this OPE. Its anomalous dimension is the coefficient of the leading single logarithmic singularity of the logarithm of the correlation function in the double short-distance limit, in which the operator positions coincide pairwise. We regularize the logarithm of the correlation function in this singular limit by a version of dimensional regularization. At any loop level, the resulting singularity is a simple pole whose residue is determined by a finite two-point integral with one loop less. This drastically simplifies the five-loop calculation of the Konishi anomalous dimension by reducing it to a set of known four-loop two-point integrals and two unknown integrals which we evaluate analytically. We obtain an analytic result at five loops in the planar limit and observe perfect agreement with the prediction based on integrability in AdS/CFT.

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Cited by 10 Pith papers

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

Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons
hep-th 2025-12 conditional novelty 7.0 of 10

A general two-loop formula is obtained for all n-point 10d null polygon correlators in planar N=4 SYM, expressed in a conformal-integral basis and verified numerically at 11 points.
Energy correlations in the end-point region
hep-th 2019-05 conditional novelty 7.0 of 10

The energy-energy correlation in both end-point regions is expressed as a compact function of twist-two conformal data, reproducing known NNLO results and predicting the next order.
Quantum spectral curve as a tool for a perturbative quantum field theory
hep-th 2014-11 accept novelty 7.0 of 10

A recursive algorithm solves the quantum spectral curve perturbatively for any sl(2) operator in planar N=4 SYM, yielding ten-loop conformal dimensions and a proof that they are multiple zeta-values.
Leading singularities and off-shell conformal integrals
hep-th 2013-03 accept novelty 7.0 of 10

The paper analytically evaluates the two unknown three-loop conformal integrals in the N=4 SYM stress-tensor four-point function, completing the three-loop correlator, and evaluates one four-loop integral as a demonstration.
Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM
hep-th 2012-01 accept novelty 7.0 of 10

Permutation symmetry plus light-cone and short-distance constraints fix the planar four-point correlator integrand in N=4 SYM through six loops, matching the known six-loop four-particle amplitude.

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Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
hep-th 2016-08 conditional novelty 6.0 of 10

Two new graphical rules, combined with the known rung rule, uniquely fix the planar N=4 SYM four-point correlator and amplitude through ten loops.
Cutting through form factors and cross sections of non-protected operators in N=4 SYM
hep-th 2014-10 accept novelty 6.0 of 10

A correction rule lets on-shell unitarity compute Konishi form factors in D=4-2epsilon, yielding two-loop two-point and one-loop three-point results and an infrared-finite decay rate whose logarithm gives the known tw...
Correlation functions of local composite operators from generalized unitarity
hep-th 2012-09 accept novelty 6.0 of 10

Generalized unitarity, previously used for scattering amplitudes, is extended to compute momentum-space correlation functions of local operators, yielding new n-point correlators of BPS and twist-2 operators in N=4 su...
Integrability of the AdS_5 x S^5 superstring and its deformations
hep-th 2013-10 unverdicted novelty 2.0 of 10

The review organizes how integrability and the thermodynamic Bethe ansatz give exact finite-volume spectra for the AdS5 x S5 superstring and several integrable deformations.
DGLAP and BFKL equations in $\mathcal{N}=4$ SYM: from weak to strong coupling
hep-th 2019-08 unverdicted

A review of known results on DGLAP and BFKL equations in N=4 SYM, from weak-coupling perturbation theory to strong coupling via AdS/CFT and integrability, with no new derivations.

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34 arXiv papers cite this in 46 sentences

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Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We show numerically that the corresponding result perfectly agrees with the one recently obtained via the generalized Luscher formulae.

Five-loop Konishi from the Mirror TBA · 0.85

Papers citing it · 30 sentences without a confident label

Uses its method or tool 2 papers

The method to compute the first two integrals in ([ref]) was presented in the appendix of [ref] .

Extracting OPE coefficient of Konishi at four loops · Master integrals · 0.98
[ref] the IBP reduction was performed by the c++ version of the code FIRE [ref] .

Leading singularities and off-shell conformal integrals · Asymptotic expansions · 0.98

Uses its data or values 3 papers

These have been computed in [ref] [equation] The constants $b_i$ and $c_i$ are obtained using [equation] Plugging these values in ([ref]) we obtain again ([ref]) and also the values for $j_22,1$ and $j_22,1$ [equation]

Extracting OPE coefficient of Konishi at four loops · Master integrals · 0.87
We can extract the divergence as follows: [equation] Therefore, we obtain the OPE limit constraint on $F_n$: [equation] where we have used $γ_𝒦|_a^1=3$, which could be obtained either from eq.(1.1) of [ref] , or by plugging in the $n=5$ result eq.[ref].

The Cusp Limit of Correlators and A New Graphical Bootstrap for Correlators/Amplitudes to Eleven Loops · Essentially one-loop argument of the triangle rule · 0.88
Similarly, the OPE coefficient with $T_2$ gives [ref] C_T_2 T_2 𝒪_𝒦 = 1 - 3 2 N g__YM^2/4 π^2 + …= 1 - 6N λ/m+ O(1/m^2) .

Large charge meets semiclassics in $\mathcal{N}=4$ super Yang-Mills · HHLL Four-point functions: OPE analysis · 0.93

Compares with it 4 papers

In a separate publication [ref] we will show how to obtain its five-loop value, in full agreement with the prediction of Ref. [ref] , without making use of the conventional Feynman diagram technique.

Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM · Introduction and summary of the results · 0.87
The five-loop result was predicted from integrability in [ref] , and confirmed in [ref] from an OPE analysis of the four-point correlation function of stress-tensor multiplets.

Cutting through form factors and cross sections of non-protected operators in N=4 SYM · Introduction · 0.92
In comparison, the recent quantum field theory based computation reached the same order [ref] .

Quantum spectral curve as a tool for a perturbative quantum field theory · Introduction and historical overview · 0.98

Cites it as background 7 papers

The interaction between four-point correlation functions and amplitudes has been the focus of much work in this direction [ref] .

Local integrands for the five-point amplitude in planar N=4 SYM up to five loops · Introduction · 0.96
The OPE expansion of the correlation function G-Mellin1 has been studied in Refs. [ref] .

Energy correlations in the end-point region · Conformal partial wave expansion · 0.96
The OPE has long been recognized as deeply intertwined with the anomalous dimensions of operators (see e.g. [ref] ), inspiring the idea of extracting these fundamental quantities from the Wilson coefficients of the expansion [ref] .

Four-loop Anomalous Dimensions of Scalar-QED Theory from Operator Product Expansion · Introduction · 0.83

What it cites · 38 sentences without a confident label

Uses the method or tool of 6 papers

In this paper, we apply the results of Refs. [ref] to perform the OPE analysis of the four-point correlation function of the stress-tensor multiplets.

Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM · Introduction · 0.92
In this paper, we apply the results of Refs. [ref] to perform the OPE analysis of the four-point correlation function of the stress-tensor multiplets.

Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM · Introduction · 0.92
To solve the IBP relations, i.e. to represent every integral on the right-hand side of C4 as a linear combination of master integrals, we apply the C++ version of the code FIRE [ref] .

Algorithm FIRE -- Feynman Integral REduction · Konishi anomalous dimension at five loops · 1.00

Uses data or values from 2 papers

Finally, we combine together the relations w – w12, make use of the results of Ref. [ref] for the master integrals shown in Fig. [ref] and obtain from C4-master the following result for $C_4$ or equivalently, five-loop Konishi anomalous dimension [equation] This relation is the main result of the paper.

Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals · Konishi anomalous dimension at five loops · 0.98
The explicit expression for $F^(3)$ in terms of the basis scalar integrals is [equation] Here the integrals $g,h$ were defined in eq:g+h while the remaining integrals $L, E$ and $H$ can be found in [ref] .

Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM · Twist-two anomalous dimensions · 0.95

Compares with 9 papers

In this manner, we obtain an analytic result for the Konishi anomalous dimension at five loops in planar $𝒩=4$ SYM theory and observe perfect agreement with the prediction based on integrability in AdS/CFT [ref] .

Five loop Konishi from AdS/CFT · Introduction · 0.97
In this manner, we obtain an analytic result for the Konishi anomalous dimension at five loops in planar $𝒩=4$ SYM theory and observe perfect agreement with the prediction based on integrability in AdS/CFT [ref] .

5-loop Konishi from linearized TBA and the XXX magnet · Introduction · 0.97
In this manner, we obtain an analytic result for the Konishi anomalous dimension at five loops in planar $𝒩=4$ SYM theory and observe perfect agreement with the prediction based on integrability in AdS/CFT [ref] .

Four-loop perturbative Konishi from strings and finite size effects for multiparticle states · Introduction · 0.97

Cites as background 8 papers

Moreover, if considered in the planar limit and restricted to the light cone, it is dual to the four-particle scattering amplitudes [ref] .

From correlation functions to scattering amplitudes · Introduction · 0.87
Moreover, if considered in the planar limit and restricted to the light cone, it is dual to the four-particle scattering amplitudes [ref] .

More on the duality correlators/amplitudes · Introduction · 0.87
It has been realized recently that the four-point correlation function of the so-called stress-tensor multiplets in $𝒩=4$ super-Yang-Mills theory (SYM) has a new symmetry [ref] .

Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM · Introduction · 0.93

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