Energy correlators at leading order — the elliptic sector in closed form

The energy–energy correlator — the two-point energy correlation a collider actually measures — is computed at leading order in the maximally supersymmetric theory from the four-point correlator. A 2025 classification mapped that computation completely: 141 master integrals across 44 sectors, including elliptic sectors for which no values at controlled precision had been tabulated — the published generic-angle values are numerical plots (arXiv:2506.02061). This page solves the complete elliptic sector on its angular slice: all eight of its bounded-region masters as exact combinations over the sector's elliptic periods, both wall constants in closed form at weight two, the differential system in terminal ε-form, everything validated against independent evaluations to at least 30 digits. It is the first sector of a larger goal — the full leading-order closure — and this page will grow with that campaign.

What BootLoops did:

• Closed the elliptic sector of the leading-order four-point correlator in N=4 super Yang–Mills on a one-parameter angular slice, the sector a 2025 classification identified and left unevaluated; the initial data of the transport remain written integrals evaluated numerically, their reduction to closed form still open
• All eight of the sector's masters close exactly over the curve's periods, with both wall constants in exact dilogarithms, validated to at least 30 digits
• A first step toward the full leading-order closure of the energy correlator, sector by sector

The question

The benchmark theory for energy correlators is N=4 super Yang–Mills, and at leading order the observable is determined by the four-point correlator. Ma et al. classified the full computation into 141 master integrals in 44 sectors (arXiv:2506.02061), identifying which sectors are polylogarithmic and which live on curved geometries — two elliptic curves and one genus-2 hyperelliptic curve — and evaluated none of the curved ones. The elliptic sector solved here is the first of those to be evaluated. The NNLO story, where the same machinery solved the residual elliptic integral of the 2019 NNLO two-point correlator in closed form, has its own page.

The result

The geometry. For the leading-order four-point correlator (141 master integrals across 44 sectors, classified by Ma et al.), the maximal cutthe integral with every propagator put on shell — it isolates the sector's intrinsic geometry of one elliptic sector is evaluated on the one-parameter angular slice $u=\zeta_{34}\in(0,1)$: one detector-pair angle left floating, the other pairwise angles fixed at the rational point $(\zeta_{12},\zeta_{13},\zeta_{14},\zeta_{23},\zeta_{24})=(\tfrac17,\tfrac15,\tfrac13,\tfrac25,\tfrac37)$. Solving the cut conditions eliminates two variables linearly and clears the last constraint to a quadratic $\mathcal Q(x_4;x_3,u)$ whose $x_4$-discriminant is $(2x_3-5)^2\,P(x_3;u)$ with $P$ a genus-1 quartic — a non-isotrivial rational elliptic surface with seven singular fibres, annihilated by an exact order-2 Picard–Fuchs operator derived by Griffiths–Dwork. Each of the eight maximal-cut masters — their contour definitions and the period vector $I_0,\dots,I_3$ are written out in the mathematics section — is an exact $\mathbb Q(u)$-linear combination of that period vector. None of the four-point integrals on these curved geometries had been evaluated before; the source classification evaluated no master integral at generic angle.

The full sector. Beyond the maximal-cut periods, the complete elliptic sector of the leading-order four-point correlator is solved on the same slice: all eight of its real bounded-region masters close as exact combinations $M = c\cdot J + K\cdot y$ over the curve's period functions, with the inhomogeneous layer supplied by a flux-corrected variation of parameters whose wall term was derived analytically rather than fit. The sector's two wall constants come out in closed form at weight two. The $u$-independent one, $K_1$, is a pure dilogarithm identity — fifteen dilogarithms at rational arguments in all, every coefficient exact, written out in the mathematics section; the $u$-dependent companion $K_0(u)$, on $\tfrac15<u<1$ with $\mathrm{Li}_2$ read as the real part of its principal branch where an argument exceeds one, is six dilogarithms at cross-ratio arguments over the prefactor $\tfrac{5}{2(1-u)}$, plus logarithm bilinears — symbolic identities, not integer-relation fits. The sector's differential system also reaches its terminal $\varepsilon$-form: $\varepsilon\,\tilde A + S$ with the rigid piece $S$ carrying exactly two Eichler-type slots with exact coefficients, and the whole structure governed by three exact quadratic curves — their equations are in the mathematics section — with the letter alphabet closing over them with zero new letters. A standalone evaluator, s00-evaluate.py, reproduces the full eight-master closure at any rational angle on the three anchored stretches of the slice, $0.300\le u\le 0.406$, $0.406\le u\le 0.557$, and $0.626\le u\le 0.678$, at any requested precision (mpmath only, with the closed-form $K_0,K_1$ wired in; every printed digit is one the run itself certified).

The mathematics

The four-point definitions. Each of the eight maximal-cut masters is

$$M_N(u)=\oint_\gamma\Bigl[\frac{N(x_1,x_2,x_3,x_4)\,\bigl(1-\tfrac25 x_3\bigr)}{\partial\mathcal Q/\partial x_4}\Bigr]_{\rm odd}dx_3,\qquad N\in\{1,\,x_1,\,x_1^2,\,x_1^3,\,x_2,\,x_1x_2,\,x_2^2,\,x_3\},$$

with $x_1,\dots,x_4$ the on-shell variables of the cut and the sheet-odd half carrying the elliptic period, and each is an exact $\mathbb Q(u)$-linear combination of the period vector

$$I_k=\oint\frac{x_3^k\,dx_3}{\sqrt{P(x_3;u)}}\;(k=0,1,2),\qquad I_3=\oint\frac{dx_3}{(2x_3-5)\sqrt{P(x_3;u)}}.$$

The $u$-independent wall constant is the pure dilogarithm identity

$$K_1 \;=\; \tfrac{35}{8}\Big[\mathrm{Li}_2\big(-\tfrac{57}{7}\big)+\mathrm{Li}_2(-8)-\mathrm{Li}_2\big(-\tfrac{19}{5}\big)-\mathrm{Li}_2\big(-\tfrac{3}{2}\big)-2\,\mathrm{Li}_2\big(-\tfrac{9}{7}\big)+\mathrm{Li}_2(-1)+\cdots+\mathrm{Li}_2\big(\tfrac{27}{7}\big)\Big]\;+\;\tfrac{35}{16}\pi^2\;+\;\text{(explicit log-bilinears)},$$

fifteen dilogarithms at rational arguments in all, every coefficient exact. The three exact quadratic curves governing the terminal $\varepsilon$-form are

$$q_1 = 225u^2-240u+4,\qquad q_2 = 1225u^2-1190u+1,\qquad q_S = 45u^2-138u+5,$$

the first of which is both the quadratic apparent factor in the Gauss–Manin denominators and, up to the constant 196, the leading coefficient of the maximal-cut quartic $P(x_3;u)$. The letter alphabet closes with no new letters: the simple-pole letters of the transformed system are $u$, the degree-7 discriminant $\Delta(u)$ (with nilpotent residues), these three quadratics with their square roots, and one further polynomial letter of degree 12, of apparent type. Eichler-type kernels also enter one coupling row at positive orders in $\varepsilon$, and the columns sourced by the subsector layer carry algebraic forms that add no transcendental letter; in these two respects the terminal form stops short of a pure logarithmic one. The homogeneous solutions of the subsector layer are algebraic.

Scope

The four-point result covers the one elliptic sector on the one-parameter angular slice — its maximal-cut periods and its full eight-master closure with the wall constants in closed form — not the full 141-master four-point correlator. The remaining sectors of the leading-order classification — including the second elliptic curve and the genus-2 sector — are open; this page is the first closed sector, and the full leading-order closure is the campaign it belongs to. The slice choice is stated up front: one detector-pair angle floats, the others sit at a fixed rational point.

Paper and code

Full paper: New mathematics and new physics in the four-point energy correlator (PDF) — the elliptic sector's eight-master closure, the wall constants, and the slice evaluation; the NNLO two-point companion: The hidden sunrise in the energy-energy correlator (PDF).

References