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Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match

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Jorge Escobedo, Nikolay Gromov, Amit Sever, Pedro Vieira

High Energy Physics - Theory · Submitted 28 Apr 2011 · 40 references

Pith's verdict: A real weak/strong match for three-point functions, with honest caveats about the sketch and the SU(3) generality.

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CitationAmit Sever, Jorge Escobedo, Nikolay Gromov, Pedro Vieira. "Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match." arXiv:1104.5501 (2011). https://pith.science/paper/GU3XJ55W
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The paper

The paper claims

In a classical long-string limit, weak and strong coupling give the same three-point function of single-trace operators in planar $N=4$ super Yang–Mills, provided the two heavy operators differ only by roots added to existing classical cuts

The reading

The paper asks whether a three-point correlation function of single-trace operators in planar $N=4$ super Yang–Mills takes the same value when computed at weak coupling (tree-level Wick contractions dressed by integrability) and at strong coupling (a hanging string in $AdS_5$). The authors work in a corner where two operators are long — length $J$ taken to infinity with the density of excitations held fixed — and the third is short, with $j$ fields and $j$ much smaller than $J$. On the weak side they replace the exact, highly entangled Bethe eigenstates of the two long operators by unentangled product states built from Landau–Lifshitz coherent states, which collapses a factorial number of Wick contractions into a single integral over where the short operator is inserted. On the strong side they take the holographic formula for two heavy strings plus one light chiral primary and evaluate it in the Frolov–Tseytlin limit, where the worldsheet integral localizes on one time slice and the same single integral survives. The two answers coincide, and the paper also pins down which pairs of long operators actually qualify, since the result turns out to depend sharply on how the shortfall in R-charge is made up.

What carries the argument

Three ingredients carry the argument. First, Landau–Lifshitz coherent states: the exact Bethe eigenstates of the long operators are replaced by unentangled tensor products $|u(1)⟩⊗...⊗|u(J)⟩$ with $u(σ)$ slowly varying, which reproduces classical averages such as the energy $⟨H⟩ = (λ/2J)∫(dσ/2π) ∂_σ u ∂_σ u$ up to $1/J$ corrections, and turns the three-point contraction into $C^{123}_{••◦} = [(j_1+j_2)!/(j_1! j_2!)] √(j!/(j_1! j_2! j_3!)) ∫(J dσ/2π) u_1^{j_1} u_2^{j_2} u_3^{j_3}$. Second, the Frolov–Tseytlin limit of the holographic three-point function, in which the factor $1/...$ localizes the Euclidean time integral at $τ≈1/κ$ while the profile $u$ varies slowly, so the two-dimensional worldsheet integral collapses to the same one-dimensional $σ$ integral. Third, the back-reaction criterion: comparing the overlap $I = exp[(J/2π)∫dσ log(1+...)]$ for the three ways of adding the missing roots shows that only deformations scaling as $δu ∼ l/J$ (roots added to existing cuts) leave $I=1$, while $δu ∼ √l/J$ (roots outside the cuts) or roots at infinity give $|I| ∼ e^{-cl}$ or power suppression.

Extended reading notes

Core claim

The central claim is an exact match, in the Frolov–Tseytlin classical limit, between the weak-coupling and strong-coupling values of the ratio $r=|C^{123}_{••◦}/C^{123}_{◦◦◦}|$, where the denominator is the known three-chiral-primary structure constant with the same charges. Both sides reduce to $r = |(v_1^{j_1} v_2^{j_2} v_3^{j_3})^{-1} (2J)^{-1} ... |$ — more precisely, to the normalized integral of $u_1^{j_1} u_2^{j_2} u_3^{j_3}$ over the spin-chain coordinate $σ$ evaluated on the classical profile of the heavy state, with $v_a = J_a/J$ the point-like-string direction. The weak-coupling derivation builds this from coherent states and a conformal-like gauge choice $(j_1+j_2-j_3) u_1 u_2 u_3 ...$ that kills the overlap between the two heavy operators, which is exactly the Virasoro constraint of the strong-coupling side. The strong-coupling derivation gets it from the localization of the boundary-to-bulk propagator at $τ=0$ in the Frolov–Tseytlin limit $κ→∞$ with $κ ∂_τ u$ fixed. The paper then shows, numerically against the exact tree-level formulae and analytically in a solvable case, that the match holds only when the extra Bethe roots that reconcile the R-charges of the two heavy operators are added to cuts that were already open; roots sent to infinity suppress the answer by powers $J^{-n/2}$, and roots placed at finite values outside the cuts change it by an order-one amount.

Load-bearing premise

The weak-coupling side rests on replacing the exact, highly entangled Bethe eigenstates of the two long operators by unentangled product coherent states and assuming this replacement captures the leading large-$J$ three-point contraction; the authors describe their derivation as a sketch and leave the detailed filling-in to the reader.

Editorial extensions

If this is right

  • The weak/strong match for structure constants is as general as the corresponding match for the spectrum: it holds for any classical string solution with nontrivial motion on $S^5$ that admits the Frolov–Tseytlin limit, not just the folded string checked numerically.
  • The classical three-point function of two heavy and one light operator is determined entirely by the overlap of the two heavy coherent states, so the answer is a single $σ$ integral rather than the determinant-and-partition sums of the exact tree-level formula.
  • The result extends to non-protected short operators: the only change is a combinatorial prefactor $D$ read off from the coefficients of the traces in $O_3$, giving $C^{123}_{•••} = D∫(J dσ/2π) u_1^{j_1} u_2^{j_2} u_3^{j_3}$, exact and independent of the classical limit.
  • The same coherent-state method applies to any large-$N$ gauge theory with a semiclassical spin-chain limit, since integrability is needed only for the strong-coupling comparison, not for the weak-coupling computation.
  • A concrete sharpening of the operator-identification problem: 'the two heavy operators are roughly conjugate' is not enough to fix the answer, and the paper supplies the rule (add roots to existing cuts) that makes the statement precise.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the tree-level match is the first term of a systematic $1/κ^2$ expansion on both sides, then the natural next test is the order-$1/κ^2$ correction: the paper notes that expanding the holographic formula to next order should be compared with the one-loop Hamiltonian insertion plus two-loop wavefunction correction at weak coupling, and an optimistic reading is agreement up to two loops.
  • The back-reaction analysis suggests a general principle beyond this model: any semiclassical three-point function of two heavy and one light state is dominated by wavefunction overlap, so the same 'which deformation is soft' criterion should govern which heavy-heavy-light correlators are computable classically in other holographic pairs.
  • The result gives a target for an all-loop conjecture: since the spectrum match in this limit was a stepping stone to discretized Bethe equations, the single-integral structure constant formula is a candidate classical limit that a conjectured finite-$J$ formula should reproduce.
  • The paper's own remark that the exact tree-level expression is far more complicated than the coherent-state answer suggests the simplification should be derivable directly from the exact formula; doing so would indicate how to handle the case where all three operators are classical.
Original abstract

We compute three-point functions of single trace operators in planar N=4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,512 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.77 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.60 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We compute three-point functions of single trace operators in planar N=4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We compute three-point functions of single trace operators in planar N=4 SYM. contribution 0.98 0.95 0.05 asserted 1.00
s1 We consider the limit where one of the operators is much smaller than the other two. method 0.58 0.60 0.03 asserted 0.89
s2 We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. result 1.00 0.94 0.85 asserted 1.00
s3 To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation. method 0.66 0.94 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. (a) Three-point function of SU(2) operators at tree level considered in [4]. All contractions are such that R-charge is preserved. This is the simplest non-trivial configuration which is not extremal. The tree-level problem is then trivial to enunciate (although very hard to solve in full generality): we need to pick three operators like (3) with different kinds of scalars, as repre￾sented in figure 1. Then we need to sum over all possible Wick contractions of these three operators as represented in the same figure. This problem is complex because the number of terms in the Bethe…
Figure 2. The setup considered in this paper for three-point function of
Figure 3. The Landau-Lifshitz field theory arises naturally both at weak and strong coupling. How
Figure 4. Two Bethe roots configurations for a folded string with
Figure 5. Three different types of deformations of a classical state: (A) A global symmetry trans
Figure 6. Fits of the numerical data (black curves) compared to the analytical prediction (dashed
Figure 7. Numerical data for the folded string example for the case (A) against the analyt
Figure 8. Numerical data for the three-point function with operator
Figure 9. At one loop, the three-point function gets two types of corrections. One type is the
Figure 10. Fits of the numerical data (black curves) compared to the analytical prediction (dashed

Figures reproduced from arXiv:1104.5501 by the authors.

References
How later papers used it

45 later papers cite it

Evidence: substantial. 40 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 115 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 5 papers with this label
How the literature uses it

45 arXiv papers cite this in 115 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 · 70 sentences without a confident label

Contests it 1 paper
  • This means that the match between the tree-level gauge theory answer and the leading Frolov-Tseytlin limit on the string side [ref] found in [ref] does not extend beyond one-loop.

    Holographic 3-point function at one loop · Introduction, summary and conclusion · contest 0.79

Builds on it 3 papers
  • The work of [ref] can thus be seen as a natural extension of the work of [ref] to 3-point correlation functions, using the prescription of [ref] for two semi-classical operators and a light chiral primary operator in the $SU(3)$ sector of $=4$ SYM theory.

    Holographic 3-point function at one loop · Introduction, summary and conclusion · 0.97

  • The determinant representations allowed to generalise the results of [ref] to the case of 3 non-BPS fields in the EGSV configuration [ref] , to the case when one of the fields is $su(3)$ type [ref] , and extend the result to the one-loop order [ref] .

    String Bits and the Spin Vertex · Introduction · 0.91

  • The strong coupling results presented in section [ref] can be generalized to the $SU(3)$ sector [ref] .

    On four-point functions and integrability in N=4 SYM: from weak to strong coupling · Conclusions and discussion · 0.83

Uses its method or tool 5 papers
  • The prescription of [ref] gives in our case [Note that the unconventional powers of $κ$ are due to a rescaling of the time coordinate (see App. [ref]).]

    On Three-point Functions in the AdS_4/CFT_3 Correspondence · The holographic approach · 0.94

  • Let us now prove analytically the match between weak and strong coupling for the four-point functions of interest.[We will follow closely the logic of section 3 of [ref] , to where we refer the reader for more details.] We will use an alternative approach to the one presented in section [ref] and used in the previous subsection.

    On four-point functions and integrability in N=4 SYM: from weak to strong coupling · Analytics: four-point functions from coherent states · 0.83

  • Following [ref] we use coherent states to approximate the two heavy operators.

    Holographic 3-point function at one loop · Gauge theory side: Non-extremal 3-point functions · 1.00

Compares with it 4 papers
  • Comparing this result with the one for the dual gauge theory, Eq. [ref], it is evident that the leading terms in both sides perfectly match as already pointed out in [ref] .

    Holographic 3-point function at one loop · Comparison with gauge theory at one loop order · 0.96

  • The tree level weak coupling result matches the leading order expansion in $g/L_1$ of the string theory result, denoted as $C_123^tree$ [ref] (see also [ref] ).

    Quantum Integrability for Three-Point Functions · Comparison with string theory · 0.98

  • In [ref] the authors evaluated the three point function of a chiral primary and two heavy (non-chiral) operators using integrability in a certain large charge limit and showed that this agreed with that obtained at strong coupling in [ref] .

    Structure constants of $\beta$ deformed super Yang-Mills · Introduction · 0.81

Cites it as background 22 papers
  • Recall from the discussion in [ref] that the other two choices for the positions of the extra roots of $Ø_1$ are i) to put them at finite positions outside the existing cuts of $Ø_2$ and ii) to put them at infinity.

    On four-point functions and integrability in N=4 SYM: from weak to strong coupling · Numerics: $SU(2)$ folded string · 0.86

  • In [ref] , a class of tree-level 3-point functions of states that live in scalar $su(2)$ sectors[More than one $su(2)$ sector is involved in these 3-point functions.] were formulated in XXX spin-$1/2$ chain terms, and Izergin-Korepin sum expressions were obtained for their structure constants.

    Variations on Slavnov's scalar product · Tree-level SYM$_4$ structure constants that can be expressed in determinant form · 0.92

  • In [ref] , Escobedo, Gromov, Sever and Vieira studied 3-point functions of three gauge-invariant single-trace length-$N_i$ operators, $_i(x_i)$, $i ∈{1, 2, 3}$, that are composed of elementary scalars in $SU(2)$ subsectors of $𝒩 = 4$ supersymmetric Yang-Mills theory [ In [ref] , ${_i}$, $i ∈{1, 2, 3}$ are chosen such that their lengths $L_i$ satisfy non-extremal length conditions, $L_i < L_j + L_k$ for any distinct ${i, j, k}$, and further, they are characterized by rapidity variables ${x_i}$, such that they are non-BPS (${x_i}$ has finitely many elements that are finite, rather than infinite), and have well-defined conformal dimensions (the elements of ${x_i}$ satisfy Bethe equations).].

    Partial domain wall partition functions · A brief history of partial domain wall configurations · 0.94

What it cites · 69 sentences without a confident label

Builds on 2 papers
Uses the method or tool of 1 paper
  • After some slight modifications described in the main text of [ref] , the codes given in the appendices of [ref] can (and were) used to compute the cases when some roots are at infinity.] For example, if we wanted to reproduce the first point for $α=1/4$ in figure [ref], we would simply run to obtain $r=0.976868082298$.

    Tailoring Three-Point Functions and Integrability · Tailoring numerics · 0.94

Compares with 5 papers
  • We will first check this claim by comparing [ref] with the exact tree-level results of [ref] .

    Tailoring Three-Point Functions and Integrability · $SU(2)$ Folded string warm-up · 0.93

  • Our result is analogous to the match observed in the spectrum problem [ref] and we hope it can be equally inspiring in guessing an all-loop interpolation for structure constants as done for the spectrum in [ref] .

    Bethe Ansatz for Quantum Strings · Conclusion and open problems · 0.94

  • Our result is analogous to the match observed in the spectrum problem [ref] and we hope it can be equally inspiring in guessing an all-loop interpolation for structure constants as done for the spectrum in [ref] .

    Long-Range PSU(2,2|4) Bethe Ansaetze for Gauge Theory and Strings · Conclusion and open problems · 0.94

Cites as background 22 papers

Discussion

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