Sensitivity of holographic ${\cal N}=4$ SYM plasma hydrodynamics to finite coupling corrections

Gauge theory/string theory holographic correspondence for ${\cal N}=4$ supersymmetric Yang-Mills theory is well under control in the planar limit, and for large (infinitely large) 't Hooft coupling, $\lambda\to \infty$. Certain aspects of the correspondence can be extended including ${\cal O}(\lambda^{-3/2})$ corrections. There are no reliable first principle computations of the ${\cal N}=4$ plasma non-equilibrium properties beyond the stated order. We show extreme sensitivity of the non-hydrodynamic spectra of holographic ${\cal N}=4$ SYM plasma to ${\cal O}(\lambda^{-3})$ corrections, challenging any conclusions reached from 'resummation' of ${\cal O}(\lambda^{-3/2})$ corrections.


Introduction and Summary
The most studied example of the holographic correspondence relating gauge theories and string theory is for the maximally supersymmetric SU(N) N = 4 Yang-Mills theory (SYM) and type IIB string theory in AdS 5 × S 5 [1]. The number of colors N of the SYM is related to the 5-form flux on the string theory side. Furthermore, the asymptotic AdS 5 (or S 5 ) radius L in units of the string length α ′ = ℓ 2 s along with the asymptotic value of the string coupling g s establishes a correspondence to the 't Hooft coupling λ on the SYM side: While there has been tremendous progress over the years in developing the correspondence e.g., see [2], understanding the full parameter space {N, λ} is elusive. How much is exactly known depends on what questions one asks. Thermal or non-equilibrium states of SYM plasma at strong coupling are under control in the planar limit, g Y M → 0 N → ∞ with λ kept fixed, and (in addition) for large 't Hooft coupling λ ≫ 1. Only first subleading corrections ∝ O(λ −3/2 ) are computationally accessible [3]. Here is a sample of SYM plasma results including first subleading corrections in the limit λ → ∞: The thermal equilibrium free energy density of the SYM plasma is [4,5] The shear viscosity to the entropy density ratio is [6][7][8] The speed of the sound waves and the bulk viscosity is [11] c 2 s = 1 3 + 0 · γ + · · · , ζ s = 0 · γ + · · · . (1.4) A sample of the second-order transport coefficients (see [9,10] for further details) is [12,13] (1.5) The plasma conductivity is [14] where σ ∞ is the plasma conductivity at infinite 't Hooft coupling.
In expressions (1.2)-(1.6) we introduced Notice that as one proceeds from the corrections to the equilibrium quantities (1.2) to the first-order (1.3), the second-order (1.5) transport, the conductivity (1.6), the relative "strength" of the corrections grow. The correction strength is even more dramatic, ∝ (10 4 − 10 5 ) · γ to the spectra of the non-hydrodynamic plasma excitations (the QNMs of the dual gravitational background) [16,17]. This observation led the authors of [18] to propose the idea of an effective resummation of γ-corrections. In a nutshell, on α ′ -corrected gravity side of the holographic correspondence one typically gets higher-derivative bulk equations of motion. One can use the smallness of γ to eliminated the higher-derivatives, reducing the equations to the second-order ones, where γ corrections affect the first order derivatives at the most -this is precisely what was done for example in computation of the shear viscosity in [6]. The next (new) step is to 'forget' that γ must be small in transformed equations, and instead treat the equations non-perturbatively in γ. There are two effects of such a resummation at finite γ: • it is possible to compute finite-γ corrections to SYM observables at infinitely large 't Hooft coupling; • one can discover new phenomena, which are absent in an infinite 't Hooft coupling limit. 1 The reference [14] corrects the earlier computation [15].
It is the latter aspect of the resummation that should be subject to additional scrutiny in drawing physical conclusions. In particular, following the resummation approach of [18], in [19] a new branch of the QNMs was found -these are (purported) SYM plasma excitations with Re(w) = 0. The physics of these new excitations was crucial to draw conclusions regarding properties of N = 4 spectral function at intermediate 't Hooft coupling [20].
To our knowledge, there is no discussion in the literature, even at a phenomenological level, how robust is the resummation approach of [18]? In this note we address this question focusing on Re(w) = 0 branch of the QNMs identified in [19]. In the absence of the reliable corrections to type IIB supergravity we proceed as follows. Recall the tree level type IIB low-energy effective action in ten dimensions taking into account the leading order string corrections [21,22] where W in a certain scheme is proportional to the fourth power of the Weyl tensor A consistent (for the purpose of QNM spectra computation) Kaluza-Klein reduction of (1.8) on S 5 results in where W is a five-dimensional equivalent of (1.9). We would like to stress that an • (c) The proposal of [18] is to treat equations in (b) as exact in γ.
Clearly, there is no physical justification of step (c) where one extends, without any modifications, EOMs valid at O(γ) only. On can easily invent infinitely many resummation schemes in the spirit of [18]. Here is one of them: where k ≥ 1 is an arbitrary integer. The new truncation and resummation procedure of γ-corrections is as good (or as bad) as the one proposed in [18]. The purpose of our paper is precisely to test the robustness of the different k resummation schemes. Specifically, we consider the simplest extension of the five-dimensional effective action (1.10): where we study a family of a constant α such that |αγ| 1. Notice that the (phenomenological) action (1.11) is assumed to be exact up to order γ 2 . At α = 0 the effective action (1.11) is just k = 2 representative of the new resummation scheme explained above. The order O(α) term is one of the potential terms that could arise from real string theory computations -we do not claim that it is a dominant one (there could be other terms at this order); neither do be know the precise value of α. The purpose of introducing this α-term is to illustrate that physical observables does not necessarily have to be monotonic in γ. Given (1.11), the corrections at order γ 2 arise from the second-order perturbation due to γW term, and directly due to the first-order term in α. In the next section we present results of the computations. In both cases, • setting α = 0 but treating (1.10) as (1.11); • fixing γ = 10 −3 and exploring |α| 100, we find a dramatic variation in the spectrum of QNMs on the branch with Re(w) = 0.
Thus, we conclude that physics extracted from (1.

Technical details
To facilitate comparison and readability, we follow notations of [19].
The two reduction procedures are not equivalent: specifically, G 1,2 differs 2 . Expression (2.5) represents the result of the latter of the two reduction schemes 3 .
As in [19], with the temperature given by (2.3), and {ω, q} begin the frequency and the momentum of the non-hydro SYM plasma excitation.
We focus on QNMs with Re(w) = 0 at q = 0. Thus, we need to solve (numerically) (2.4) for z 1 , defined as We confirm the computations of the QNM frequencies determined in [19] and presented in Fig. 5 there.