Conformal collider bootstrap in ${\mathcal N}=4$ SYM

TL;DR

This work develops a comprehensive conformal collider bootstrap framework for ${\rm SU}(N_c)$ ${\mathcal N}=4$ SYM to bound the energy-energy correlator (EEC) across couplings. By coupling analytic input (weak/strong coupling expansions, light-ray OPE, stringy corrections, and localization) with numerical bootstrap constraints (crossing, ANEC, dispersive/dispersion relations, and planar integrability data), it derives tight two-sided bounds in the planar limit and nontrivial lower bounds at finite $N_c$ on EEC multipoles $c_s$ and on smeared EEC observables. The results reveal a smooth coupling-dependent transition from jet-like single-trace dominance at weak coupling to a homogeneous, double-trace-dominated regime at strong coupling, with Padé models providing accurate finite-coupling descriptions that match the bootstrap bounds. The study also develops inversion formulas and large-spin analyses that connect endpoint behavior to high-spin data, enabling reconstruction of the EEC across angles at finite coupling. Overall, the work demonstrates the power of combining conformal bootstrap with holography, integrability, and localization to illuminate Lorentzian collider observables in a highly symmetric quantum field theory.

Abstract

We use a combination of perturbation theory, holography, supersymmetric localization, integrability, and numerical conformal bootstrap methods to constrain the energy-energy correlator in $\text{SU}(N_c)$ ${\mathcal N}=4$ SYM at finite coupling. For finite $N_c$, we derive lower bounds on the second and fourth multipoles of the energy-energy correlator at different couplings, along with a smeared energy-energy correlator as a function of the angle between the two detectors. We present evidence that our lower bounds on the multipoles are nearly saturated by the ${\cal N} = 4$ SYM theory. In the planar limit, we further use dispersive functionals to obtain tight two-sided bounds on both the first three non-trivial multipoles and on the angular dependence of the energy-energy correlator. As the coupling is varied from weak to strong, the energy-energy correlator exhibits a transition from single-trace to double-trace operator dominance in the collinear limit, which we characterize quantitatively. A similar phenomenon occurs in QCD, where a parton-hadron transition is observed as detectors are brought closer together.

Figures (22)

• Figure 1: Schematic overview of the different regimes of the EEC in a conformal collider experiment in the planar limit of $\mathcal{N}=4$ SYM. At weak coupling (the bottom region of the diagram), the EEC is known to three loops Engelund:2012reBelitsky:2013xxaBelitsky:2013ofaHenn:2019gkr, while at strong coupling (the top region of the diagram) there is a homogeneous distribution with small inhomogeneities on top of it Hofman:2008ar. At small angles (the left region of the diagram), the light-ray OPE predicts a power-law behavior $\theta^{-2+\gamma}$ with a power determined by the anomalous dimension of a spin-$3$ twist-$2$ operator Hofman:2008arKologlu:2019mfz. The value of $\gamma$ is known from integrability, and in particular $\gamma>2$ for $g>g_{\text{cr}}\sim 0.81$. Above $g_{\text{cr}}$, there is a transition where the twist-$4$ double-trace operator dominates and contributes analytic terms. In the back-to-back limit (the right region of the diagram), large logarithms invalidate the perturbative expansion and require resummation that renders the back-to-back limit finite. This regime was studied in Korchemsky:2019nzmChen:2023wahChen:2023amz using the known behavior of the local correlator in the double-lightcone limit. Notice that as the coupling constant increases, the contribution of twist-4 and higher operators becomes more important rendering the analysis of the papers above inaccurate. In this work, we explore the previously inaccessible central region using the conformal bootstrap, supplemented by input from integrability and supersymmetric localization.
• Figure 2: In orange we plot the bootstrap bounds on $c_2(g)$ in the planar limit, as obtained in \ref{['sec:planarbootstrap']}. We also plot the perturbative $3$-loop result (dashed green), the leading strong coupling result (dotted blue), as well as the strong coupling result \ref{['eq:c2SubStrongF']} up to subleading order in the strong coupling expansion (dashed blue). In black we plot a Padé approximation based on the $3$-loop result at weak coupling and the leading strong coupling result. In particular, the subleading strong prediction \ref{['eq:c2SubStrongF']} (in dashed blue) improves the agreement with the bootstrap bounds significantly relative to the leading result (in dotted blue).
• Figure 3: In orange, we show upper and lower bootstrap bounds for $\text{EEC}(z)$ in the planar limit at $g=0.4$, as obtained in \ref{['sec:planarbootstrap']}. This value of the $g$ is not accessible in perturbation theory, nor using the strong coupling expansion. In black, we also plotted a Padé approximation based on the three-loop result at weak coupling and the leading $O(1/g^2)$ behavior at strong coupling. Away from the end-points, we see that the Padé approximation does a good job at reconstructing the $\text{EEC}(z)$ for any $z$, while towards the endpoints we do not expect the Padé approximation to be accurate because the perturbative expansion that it is based on breaks down.
• Figure 4: Bootstrap lower bounds for the second EEC multipole, $c_2$, as a function of coupling $g$ for $N_c = 2,3,\ldots,10$. For each value of $N_c$, we calculate bounds up to the self-dual point $g = \sqrt{N_c/4\pi}$ (indicated with the colored triangles) and then use $S$-duality to continue to larger $g$. We see that for each value of $N_c$, our lower bounds match the value of $c_2$ in the planar limit (shown as the Padé approximant in the black dashed line; see \ref{['fig:c2Intro']}) for sufficiently small $g$, and the range of such values of $g$ increases with $N_c$.
• Figure 5: Here we show the weak-coupling expansion of the energy-energy correlator evaluated at LO, NLO and NNLO order at fixed value of the coupling $g=0.15$. In the bulk the perturbative corrections are small while they become large at the endpoints due to large logarithms. To explore the $z\to0,1$ regions we thus need to resum the perturbative expansion.