Residues of Correlators in the Strongly Coupled N=4 Plasma

TL;DR

The paper investigates how strongly coupled ${\mathcal N}=4$ SYM plasma responds to small perturbations by analyzing the residues of quasinormal-mode poles of R-charge current correlators in the AdS/CFT framework. Using an exact zero-momentum solution and numerical pole-residue calculations at finite momentum, it shows that residues are generally complex, with the hydrodynamic diffusion mode possessing a purely imaginary residue, while other modes have residues that grow with momentum. The work reveals a damped-oscillatory hydrodynamic residue and zeros at special momenta, leading to a hydrodynamic regime that decouples at high $q$, and introduces a hydrodynamic onset time $\tau_H$ governing the validity of hydrodynamics, with implications for modeling the quark-gluon plasma. By providing a pole-based representation of retarded correlators and examining spectral features, the study clarifies how strong coupling shapes time-dependent responses and connects hydrodynamic behavior to finite-$q$ dynamics in the plasma.

Abstract

Quasinormal modes of asymptotically AdS black holes can be interpreted as poles of retarded correlators in the dual gauge theory. To determine the response of the system to small external perturbations it is not enough to know the location of the poles: one also needs to know the residues. We compute them for R-charge currents and find that they are complex except for the hydrodynamic mode, whose residue is purely imaginary. For different quasinormal modes the residue grows with momentum q, whereas for the hydrodynamic mode it behaves as a damped oscillation with distinct zeroes at finite q. Similar to collective excitations at weak coupling the hydrodynamic mode decouples at short wavelengths. Knowledge of the residues allows as well to define the time scale t_H from when on the system enters the hydrodynamic regime, restricting the validity of hydrodynamic simulations to times t>t_H.

Figures (2)

• Figure 1: (Top) Real and imaginary parts of the residues for the first four quasinormal modes in the transverse component $E_T$ (bottom) Idem for longitudinal component $E_L$. The $n^2$ scaling is necessary to recover the asymptotic behaviour of the spectral function at large frequencies. Close to the crossing with the diffusion mode ${\mathfrak{q}}\sim 1$, the residues of the longitudinal component present peaks. The residues grow with momentum, this is also reflected in the growth of the spectral function.
• Figure 2: In the lower figure, we show the real and imaginary parts of the quasinormal frequencies in the longitudinal channel and the value of the frequency for the diffusion mode. The quasinormal frequencies remain fairly stable as momentum increases, until there is a 'crossing' with the diffusion mode (it reaches a special value ${\mathfrak{w}}=-in$). Then, there is a qualitative change in the behaviour of the quasinormal frequency that starts approaching the real axis. This behaviour can also be oberved in previous computations of quasinormal frequencies Nunez:2003eq. The residue of the diffusion mode, shown in the upper figure, behaves according to hydrodynamics $\sim q^2$ for small momentum. For larger momentum, it shows an oscillatory and decaying behaviour. The zeroes of the residue coincide with the 'crossing' values. Quasinormal frequencies in the transverse channel approach smoothly the real axis as momentum increases.