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Energy-energy correlators in planar N=4 super Yang-Mills are now pinned down at strong coupling through order λ^{-2} (dimension-two sources) and λ^{-3/2} (any dimension), with two independent methods agreeing order by order.

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2026-08-01 08:20 UTC pith:BC7NT65X

load-bearing objection Strong, transparent computation of the strong-coupling EEC, but the 4π² discrepancy with [24] on the O(g^-3) multipole leaves the central claim conditional. the 1 major comments →

arxiv 2607.21139 v1 pith:BC7NT65X submitted 2026-07-23 hep-th

Energy-Energy Correlators at Strong Coupling

Max Jackson , Lecheng Ren , Bo Wang , Congkao Wen This is my paper
classification hep-th PACS 11.25.Tq
keywords energy-energy correlatorsN=4 super Yang-Millsstrong couplingAdS/CFT correspondenceVirasoro-Shapiro amplitudemultiple zeta valuesMellin amplitudesconformal collider bootstrap
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper pins down how much can be said about the energy-energy correlator (EEC) — the angular distribution of energy deposited on a sphere at infinity by a state created by a half-BPS operator — in planar N=4 super Yang-Mills theory at strong 't Hooft coupling. It fixes the fixed-angle expansion through order λ^{-3/2} for sources of any protected dimension p, and through order λ^{-2} for dimension-two sources, with every coefficient an explicit rational combination of Riemann zeta values (the numbers ζ(2)=π²/6, ζ(3), ζ(4), ζ(5), ...). The coefficients come from the AdS Virasoro-Shapiro amplitude, the holographic four-point scattering amplitude of gravitons, by two independent routes: integrals over the string worldsheet, and a newer short-cut that reads each coefficient off the amplitude's low-energy Wilson-coefficient expansion by analytic continuation. The two routes agree term by term, and the new λ^{-2} contribution visibly improves agreement with non-perturbative numerical bounds at intermediate coupling. Because the Wilson-coefficient short-cut bypasses worldsheet integrals, the next orders in 1/√λ become substantially cheaper to compute.

Core claim

On its own terms, the paper establishes the fixed-angle strong-coupling expansion of the EEC in planar N=4 SYM. For any source dimension p it fixes the expansion through order λ^{-3/2} with explicit zeta-value coefficients; the leading λ^{-1} stringy correction is p-independent. For p=2 the second curvature correction sets c_{2,0}=855ζ(4), c_{2,1}=−120ζ(2)+240ζ(3)−3870ζ(4), c_{2,2}=304ζ(2)−476ζ(3)+24ζ(2)ζ(3)+4692ζ(4)−40ζ(5), completing the EEC through order λ^{-2}. The paper claims the same numbers follow directly from the Wilson-coefficient expansion of the Mellin amplitude — via Borel resummation or localisation of the summation index to a half-integer — and reports term-by-term agreement

What carries the argument

Three pieces carry the argument. (1) The AdS Virasoro-Shapiro amplitude — the holographic four-point scattering amplitude of gravitons and Kaluza-Klein states — whose large-curvature expansion is expressed as worldsheet integrals of single-valued multiple polylogarithmic integrands (analytic functions built from polylogarithms of z and its conjugate). (2) The Mellin-space detector kernel K_p(t;ξ), which converts the amplitude into the EEC; integrating over t yields the closed-form angular polynomials Q_{n,p}(ξ) carrying all dependence on the detector angle. (3) The localisation formula (4.17): after a Borel transform, the s-contour integral collapses the infinite Wilson-coefficient sum to on

Load-bearing premise

The load-bearing premise is the Appendix-D analytic-continuation prescription — divergent multiple zeta values whose leading argument is ≤1 are assigned finite values by a specific ε-shift, stuffle and pole-cancellation scheme — and the paper itself flags (§4.1.2) that this choice shifts the O(λ^{-3/2}) coefficient by 4π² relative to an earlier published result, with no computation fully independent of the convention yet settling which value is physical.

What would settle it

Numerically evaluate the first-curvature worldsheet integral I^{(1)}(T) of eq. (3.27) at small finite T (the disk and circle integrals of Appendix B.3 are finite for T>0) and extrapolate the O(T^0) term: this fixes c_{1,1}(2) without ever invoking the Appendix-D regularisation. If the paper is right the extrapolated value is 24ζ(2)−240ζ(3); if the earlier published value is right it is 8π²−240ζ(3), the two differing by exactly 4π². A sharper micro-test: evaluate the limit in eq. (4.21), −(2a+1)ζ(2a+2,2) at a=−1/2+ε/2, in exact arithmetic and check whether εζ(1+ε,2) tends to +ζ(2) or −ζ(2).

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For p=2 the EEC is now an explicit analytic function of the detection angle through order λ^{-2}; the three new coefficients give the planar numerical bootstrap a sharper target at intermediate coupling, where the paper reports the two now overlap more closely.
  • For any operator dimension p the first two stringy corrections are known and the leading one is universal (p-independent), so any future all-order result for these observables must reproduce that structure.
  • The Wilson-coefficient route reduces each new order to bootstrapping the next curvature coefficient and one localised evaluation — the O(λ^{-5/2}) correction is in reach without new worldsheet integrals.
  • The flagged 4π² non-commutation between the light-ray transform and the spectral summation implies that any strong-coupling extraction of this type must specify the order of operations; the paper's convention is spectral summation first.
  • If the modularity conjecture holds, the universal λ^{-1} coefficient should be promoted to a non-holomorphic Eisenstein series in the very-strong-coupling regime — a concrete finite-coupling prediction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • My inference: the localisation-then-regularise recipe should transfer to any observable built from the same Mellin amplitude — higher-point energy correlators, heavy-state EECs, charge-flow correlators — because the hard part (the s-integral) is identical; the main new work is bootstrapping the relevant Wilson coefficients.
  • My inference: the 4π² ambiguity is settleable by a route the paper does not run — numerically evaluating the first-curvature worldsheet integrals of Appendix B.3 at small finite T and extrapolating the O(T^0) piece — since that computation never invokes the Appendix-D regularisation, unlike both methods used here, which share the same analytic-continuation convention.
  • My inference: the fixed-angle expansion's reach stops at the endpoints (collinear ξ→0 and back-to-back ξ→1), as the paper stresses; the new coefficients make it feasible to match the light-ray OPE and Sudakov data order by order in 1/√λ, which would complete the analytic picture.
  • My inference: if the S-duality completion is correct, the p-dependence of the λ^{-3/2} coefficients should encode how the Eisenstein-series structure generalises away from the stress-tensor channel — a sharper test than the p=2 coefficient alone.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper computes the strong-coupling expansion of energy-energy correlators (EEC) in planar N=4 super Yang-Mills theory for states created by half-BPS operators O_p of arbitrary dimension p. The main results are: for generic p, the EEC is determined through O(λ^{-3/2}) with coefficients c_{0,0}(p), c_{1,0}(p), c_{1,1}(p) given in (1.2)/(3.31); for p=2, the second curvature correction is computed, completing the EEC through O(λ^{-2}) with coefficients c_{2,0}, c_{2,1}, c_{2,2} given in (1.3)/(3.37). The authors use two routes: a worldsheet-integral representation of the AdS Virasoro-Shapiro amplitude (§3 and Appendix B), and a complementary method based on the low-energy Wilson coefficients of the Mellin amplitude (§4 and Appendices C, D), involving Borel resummation and an analytic-continuation/localisation procedure. They report perfect agreement between the two routes and compare the p=2 result with planar conformal bootstrap bounds, claiming the new λ^{-2} term improves agreement at intermediate coupling. The paper is technically detailed and includes closed-form expressions for the angular functions Q_{n,p}(ξ) in Appendix A, but the central coefficients are conditional on a regularization prescription that is not independently established.

Significance. If correct, the paper would provide the definitive fixed-angle strong-coupling expansion of the EEC at these orders, extend previous results to arbitrary external charge p, and introduce a new, potentially more efficient technique for extracting EEC coefficients directly from Wilson coefficients. The paper is strong in presentation: the derivations are unusually complete, the worldsheet integrals and Wilson-coefficient bootstraps are documented in appendices, the angular functions are given in closed form, and no free parameters are introduced. The comparison with bootstrap data is a useful falsifiable check. However, the central claim is load-bearing on a specific regularization and ordering of limits in §4.1.2/Appendix D, which disagrees with the published analytic result of [24] by 4π². Since the two computational routes share the same bootstrapped AdS Virasoro-Shapiro amplitude as input, their mutual agreement does not independently adjudicate this discrepancy. The significance of the paper's new predictions is therefore conditional on resolving that physical-ordering question.

major comments (1)
  1. [§6, Conclusion] The conclusion repeats the claim of 'perfect agreement' between the two approaches without mentioning the unresolved 4π² discrepancy with [24]. This overstates the strength of the check, especially since the two approaches are not logically independent as explained above. I recommend that the conclusion be revised to state precisely what is checked and what remains open.
minor comments (5)
  1. [References [44], [45]] The manuscript cites reference [44] as 'To appear' without title or arXiv number, and [45] as 'to appear' with a placeholder arXiv number. These should be updated or removed before publication.
  2. [Eq. (4.21) vs Appendix D] The regulator in (4.21) is introduced as a→-1/2+ε/2, while Appendix D uses m→m-2ε. The relation between these two ε's (a factor of 2) is not explicitly stated; a brief clarifying sentence would avoid confusion.
  3. [Fig. 1 caption] The caption says 'green hollow diamonds denote the sum through order λ^{-2}', but the legend label reads 'SUGRA + λ^{-1} + λ^{-3/2} + λ^{-2}'. The text is consistent but the reader might appreciate a matching phrasing.
  4. [Eq. (1.1) and (3.3)] The expansion variable is written λ^{-k/2-1}, but in the abstract and elsewhere the terms are described as λ^{-1}, λ^{-3/2}, λ^{-2}. The notation is fine, but the leading term '1' in (1.1) is the supergravity contribution, while the c_{0,0} term is O(λ^{-1}); this should be explicitly restated near (1.1) to avoid confusion.
  5. [Appendix A] The seed functions (A.6) are given without derivation. A short explanation of how they are obtained from the detector kernel (A.2) would improve readability, although the recurrence (A.4) is clear.

Circularity Check

0 steps flagged

No significant circularity: EEC coefficients are genuine new integrals of the bootstrapped AdS Virasoro-Shapiro amplitude; the two-method agreement is a consistency check, and the 4π² discrepancy with [24] is a regularization ambiguity, not a circular reduction.

full rationale

The derivation chain is: the EEC is expressed as an integral of the Mellin amplitude against a known detector kernel, and the strong-coupling coefficients are obtained by expanding the AdS Virasoro-Shapiro amplitude and performing the integrals. The amplitude itself is bootstrapped from OPE data in prior work; the EEC is not used as an input anywhere. No parameter is fitted to EEC data, and no prediction is defined in terms of the quantity it claims to predict. The two methods (worldsheet integrals and low-energy Wilson-coefficient extraction) are two representations of the same bootstrapped amplitude; their agreement is a nontrivial algebraic consistency check, but it does not make the derivation circular because the output is not assumed. The acknowledged 4π² discrepancy with [24] originates from the regularization of the subtle term in (4.21) and the ordering of the spectral summation versus the s-integral. The paper states its convention explicitly and notes that the discrepancy is due to this treatment. This is a physical/mathematical ambiguity and a correctness risk, not a circular reduction: the chosen convention is not fitted to the EEC, and the worldsheet route provides an independent (though not fully independent in input) evaluation. Self-citations such as [23] are prior published results with stated assumptions and do not assume the target coefficients. Therefore, under the stated rules, there is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The computation contains no parameters fitted to data: the EEC coefficients are exact outputs of the stated analytic machinery, and the ansatz coefficients in the Wilson-coefficient bootstraps are fixed by OPE constraints, not tuned. The axioms listed are the imported amplitude data (bootstrapped worldsheet representation), the MZV-ring ansatz for the new Wilson coefficients, the paper's own regularization scheme for divergent MZVs (the least externally anchored step), and the prior-work identities connecting EEC to amplitudes. No invented entities are introduced; the Eisenstein-series completion in footnote 1 is explicitly a conjecture not used in the derivation.

axioms (5)
  • domain assumption AdS Virasoro-Shapiro amplitude admits the worldsheet integral representation A^(k)(S,T) = ∫ d²z |z|^{-2S-2}|1-z|^{-2T-2} G^(k)_tot(z,z̄;S,T) with bootstrapped SVMPL kernels (eq. 3.6).
    Imported from Alday-Hansen [19,20] and Fardelli-Hansen-Silva [21]; all of §3 rests on it. The kernels are bootstrapped (conjectured and OPE-matched), not proven, within the cited literature.
  • domain assumption Wilson coefficients ω^(1)_{a,0}(p) and α^(2)_{a,0} lie in the ring of (single-valued) multiple zeta values with uniform transcendentality weight (App. C.1, C.2).
    Stated as an assumption in C.1 ('we assume ... uniform weight 4+2a+3b'); the Euler-Zagier ansatz is 'the simplest possible choice' consistent with localization results. If the true coefficients contain non-MZV pieces, method 2 misses them.
  • ad hoc to paper The localization formula (4.17) with the regularization rules (D.13), (D.15), (D.16) assigns unique finite values to divergent MZV limits, with individual ε-poles cancelling after summation.
    This is the paper's own scheme, developed in App. D. Its physical uniqueness is not proven; §4.1.2 states the 4π² discrepancy with [24] originates exactly here ('the subtle contribution in (4.21)').
  • domain assumption The strong-coupling EEC is given by the Mellin/worldsheet integral identities (2.11), (3.1), (3.7)-(3.9), including the exchange of Borel and Mellin-Barnes integrals.
    Connector formula established in [23] by three of the present authors, together with [32] for the detector kernel. Load-bearing: it maps amplitude data to the EEC.
  • domain assumption Detector kernel K_p(t;ξ) from [32] (eq. 2.12) and the Q-function recurrence (A.4) are correct for all p.
    Imported from prior work; the appendix 'derives' Q_{n,p} from this kernel, so any error there propagates into every displayed EEC.

pith-pipeline@v1.3.0-alltime-deepseek · 36599 in / 19634 out tokens · 181614 ms · 2026-08-01T08:20:41.140637+00:00 · methodology

0 comments
read the original abstract

We study energy-energy correlators (EEC) in planar $\mathcal{N}=4$ super Yang-Mills theory at strong 't Hooft coupling $\lambda$. We consider the EEC in states created by half-BPS operators of arbitrary dimension $p$, and determine the corresponding event-shape function up to order $\lambda^{-3/2}$ from the worldsheet representation of the AdS Virasoro-Shapiro amplitude with Kaluza-Klein external states. For $p=2$ we compute the second curvature correction, which completes the EEC through order $\lambda^{-2}$; the new contribution improves the agreement with recently derived non-perturbative bounds at intermediate coupling. We further develop a complementary method in which the strong-coupling expansion coefficients of the EEC are extracted directly from the Wilson coefficients of low-energy expansion of the AdS Virasoro-Shapiro amplitude, and find the two approaches in perfect agreement.

discussion (0)

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