Finite ’t Hooft coupling corrections to multiple physical observables in strongly
coupled
Article funded by SCOAP3
k^{2n} \, \frac{d\Gamma_\gamma}{dk} \,. \label{eq:Gamma_n} \end{equation} \looseness=-1 For the entropy density (or equivalently, the pressure $p = s \, T/4$), the first finite-coupling correction is modest; the $O(\gamma^1)$ term does not exceed the leading $O(\gamma^0)$ term as long as $\lambda > 1.72$. For the shear viscosity or viscosity ratio $\eta/s$, the corresponding crossover points where the first corrections equal the leading term occur at $\lambda \approx 7.4$ or 6.9, respectively. For the electric conductivity, this crossover lies at $\lambda \approx 39.7$, while the crossovers for the photoemission moments $\Gamma_0$ and $\Gamma_1$ are at 16.3 and 17.2, respectively. All these values are below, or at least within, our 10--40 range of benchmark values for $\lambda$. The situation, however, is rather different for the quasinormal mode frequencies shown in table~\ref{table:1}. The first order corrections exceed the leading order term when $\lambda < 71.2$ (for $\omega_2^{\rm shear})$ or $\lambda < 382.2$ (for $\omega_2^{\rm EM})$, suggesting that their $\lambda = \infty$ limits are likely to give poor predictions for the values of these quantities in the phenomenologically interesting range of 't Hooft couplings. A priori, it is not clear whether the above behavior of the QNM frequencies is due to an abnormally large first term in an otherwise well-behaved expansion, or whether the quantities in question are particularly sensitive to finite coupling corrections, so that their expansions in $\gamma$ have an abnormally small range of utility. Deciding between these alternatives would, in principle, require an all orders determination of the strong coupling expansion, which is far beyond the reach of present day technology. In this paper, we adopt a far more modest goal. We will investigate a simple resummation applicable to quasinormal mode frequencies and related observables which takes into account a subset of higher order terms in the expansion in powers of $\gamma$, and see if this improves the behavior of the resulting series. At the very least, this investigation should be helpful in inferring the range of utility of the above first order results. ]]>
\sqrt{-G} \left[ R_{10} -\frac 12 {(\partial \phi)^2} -\frac 1{4\cdot 5!} {(F_5)^2} \right], \\ S_{\rm IIB}^{(1)} &=& \frac{L^6}{2\kappa_{10}}\int d^{10}x \> \sqrt{-G} \, e^{-\frac 32\phi} \left( \mathcal{C}+\mathcal{T} \right)^4, \label{eq:SI} \end{eqnarray} where $\kappa_{10}$ is the 10D Newton constant, $R_{10}$ the 10D Ricci-scalar, $\phi$ the dilaton field, and $F_5$ the five-form field strength, while $\mathcal{C}$ stands for the Weyl tensor, and $\mathcal{T}$ for a tensor built from the gradient of $F_5$ plus terms quadratic in $F_5$~\cite{Gubser-ml-1998nz, Paulos-ml-2008tn, Myers-ml-2008yi}. After a reduction to 5D and the extraction of terms at most quadratic in the emergent 5D bulk gauge field dual to the EM conserved current, one eventually finds a $\gamma$-corrected Maxwell equation which, once again, may be put into the Schrodinger-like form~(\ref{psieom})~\cite{Hassanain-ml-2011ce,Vuorinen-ml-2013}. The needed field redefinition becomes $ \Psi(u)=\Sigma(u) E_{\bot}(u)\,, $ with \begin{equation} \Sigma(u) \equiv \frac{\sqrt{f(u)}}{1+\gamma \, p(u)} \,,\quad p(u)=\frac{u^2}{288} \left[ 11700 -343897 \, u^2 -37760 \, u^3 \, \hat{q}^2 +87539 \, u^4 \right] , \end{equation} while the resulting $\gamma$-corrected effective potential reads \begin{eqnarray} V(u) &=& -\frac{u+\hat{\omega}^2-\hat{q}^2f(u)}{u f(u)^2} \nonumber \\ &&{} +\frac{\gamma}{144 f(u)} \Bigl[ -11700 +2098482 \, u^2 -4752055 \, u^4 +1838319 \, u^6 \nonumber \\ &&\qquad\qquad\quad {} +\hat{q}^2 \left( 4770 \, u^{-1} +11700 \, u -953781 \, u^3+ 1011173 \, u^5 \right) \nonumber \\ &&\qquad\qquad\quad {} -\hat{\omega}^2 \left( 4770 \, u^{-1} +28170 \, u -1199223 \, u^3 \right) \Bigr] \,. \label{eq:Vcorrected} \end{eqnarray} Given the above potential, solutions to the equation $\Psi'' = V \, \Psi$ may be expanded in a power series in $\gamma$. The black brane horizon at $u = 1$ is a regular singular point of the equation, where infalling solutions behave locally as $\Psi(u) \sim (1{-}u)^r$ with characteristic exponent $r = \frac 12 (1-i \omegahat)$. Hence, one may expand the solutions of interest as \begin{equation} \Psi(u)=(1{-}u)^r\left[\Phi^{(0)}(u) +\gamma \, \Phi^{(1)}(u) +\cdots \right] \,, \end{equation} where $\Phi^{(0)}$ and $\Phi^{(1)}$ have the near-horizon Frobenius expansions, \begin{equation} \Phi^{(0)}(u) \sim \sum_{n=0}^{\infty} \> a_n \, (1{-}u)^n, \hspace{1cm} \Phi^{(1)}(u) \sim \sum_{n=0}^{\infty} \> b_n \, (1{-}u)^n \,. \label{eq:Frob} \end{equation} One may further determine the coefficients $\{ a_n \}$ and $\{ b_n \}$ recursively by inserting these expansions into eq.~(\ref{psieom}) and collecting like powers of $\gamma$ and $(1{-}u)$. Without loss of generality, one may set $a_0 = 1$ and $b_0 = 0$. \begin{figure} \centering \includegraphics[width=0.49\textwidth]{pdf-pic1.pdf} \includegraphics[width=0.49\textwidth]{pdf-pic2.pdf} ]]>
a_n(\hat{\omega}^{(0)})=0 \,, \hspace{1cm} \sum_{n=0}^{N} \> b_n(\hat{\omega}^{(0)},\hat{\omega}^{(1)})=0 \,, \end{equation} whose roots yield (approximations to) $\omegahat^{(0)}$ and $\omegahat^{(1)}$. Carrying out this procedure for values of $N$ sufficiently large that the results are stable turns out to be a viable computational strategy~\cite{Vuorinen-ml-2013}. Our results for the first few QNMs are displayed in figure~\ref{fig1} and reported in table~\ref{table:EM} below, and fully agree with the findings of ref.~\cite{Vuorinen-ml-2013}. We have also computed the photoemission spectrum~(\ref{eq:photoemission}) by solving the $\gamma$-corrected equation for the transverse electric field, as in ref.~\cite{Hassanain-ml-2011ce}, and then evaluated the first few moments~(\ref{eq:Gamma_n}) of the spectrum, obtaining the results shown in table~\ref{table:1}. As an alternative to the above approach, in which the QNM frequencies and mode functions are explicitly expanded in powers of $\gamma$, one may directly solve the Schrodinger equation~(\ref{psieom}) with the potential~(\ref{eq:Vcorrected}) evaluated at some chosen value of $\gamma$. Spectral methods provide an efficient numerical approach~\cite{Boyd-ml-2001}. We write \begin{equation} \Psi(u)=(1{-}u)^r \, \Phi(u) \,, \label{psiphi} \end{equation} so that the function $\Phi(u)$ is regular at both the horizon and boundary. In terms of $\Phi$, the explicit $\gamma$-corrected QNM equation takes the form \begin{align} -u (1 {+} u) &(1 {-} u^2) \, \Phi''(u) + u (1 {+} u)^2 (1 - i\hat{\omega}) \, \Phi'(u) + K(u) \, \Phi(u) = 0 \,, \label{eqphi2} \\ \noalign{\noindent with} K(u) &\equiv (1 - \hat{\omega}^2) + \frac 14 u (1 - 3 \hat{\omega}^2) - \frac 14 u^2 (1 + \hat{\omega}^2) \nonumber \\ &\quad {} + \frac \gamma{144} \, (1 {+} u) \Bigl[ 11700 \, u + 2098482 \, u^3 - 4752055 \, u^5 + 1838319 \, u^7 \nonumber \\ &\quad\qquad\qquad\qquad {} + \qhat^2 \left(4770 + 11700 \, u^2 - 953781 \, u^4 + 1011173 \, u^6 \right) \nonumber \\ &\quad\qquad\qquad\qquad {} + \omegahat^2 \left( -4770 - 28170 \, u^2 + 1199223 \, u^4 \right) \Bigr] \,. \label{eq:phieqn} \end{align} Next, we expand $\Phi$ in a truncated series of Chebyshev polynomials, \begin{equation} \Phi(u) = \sum_{n=0}^N \> c_n \, T_n(2u{-}1) \,, \label{eq:Cheb} \end{equation} \looseness=-1 with $T_n(z) \equiv \cos(n \cos^{-1}z)$.\footnote{The Chebyshev polynomials $\{ T_n(z) \}$ form an orthogonal basis in the Hilbert space with inner product $ (f,g) = \int_{-1}^1 dz \> f(z) \, g(z) / \sqrt{1-z^2} $. } Requiring equation~(\ref{eqphi2}) to be satisfied at the points $u_j \equiv \frac 12 \, [1-\cos(j\pi/N)]$, $j =0, 1, \cdots N$, which comprise a Chebyshev-Gauss-Lobatto grid, converts the differential equation~(\ref{eqphi2}) into a finite matrix equation of the form $A\cdot v = 0$. Here, $v$ is a vector consisting of the $N{+}1$ coefficients $\{ c_n \}$ of the Chebyshev expansion, while $A$ denotes a matrix whose entries on the $j$'th row are obtained by evaluating eq.~(\ref{eqphi2}) at the $j$'th grid point.\footnote{The row of this matrix corresponding to the $u = 0$ endpoint automatically enforces the Dirichlet boundary condition $\Phi(0) = 0$. } Nonvanishing solutions of this homogeneous set of equations only exist when \begin{equation} \det(A)=0 \,. \label{det} \end{equation} This determinant is a polynomial in $\omegahat$, whose roots $\{ \omegahat_k^{(N)} \}$ rapidly converge (for fixed $k$) as the number of grid points $N$ increases. Evaluating these roots numerically is straightforward (for relatively modest values of $N$) given specific values of the wavevector $\qhat$ and the parameter $\gamma$. \begin{figure} \centering \includegraphics[width=0.49\textwidth]{pdf-pic3.pdf} \includegraphics[width=0.49\textwidth]{pdf-pic4.pdf} ]]>
\mathrm {Im} \frac{1}{\omega} \, G_{xy,xy}^{\rm ret}(\omega,0) \,. \label{eq:eta} \end{equation} Including $O(\gamma)$ corrections, it can be shown that this correlator is given by~\cite{Buchel-ml-2005} \begin{equation} G_{xy,xy}^{\rm ret}(\omega,q) = \lim_{u\to 0} \frac{\Nc^2 \, (r^0_h)^4}{4\pi^2} \, \frac{Z'(u)}{u \, Z(u)}, \label{eq:Gxyxy} \end{equation} where $Z(u)$ is a solution to eqs.~(\ref{eq:Z1eq})--(\ref{eq:Q}) at $\qhat=0$ that satisfies infalling boundary conditions at the horizon, and $ r^0_h = \pi T L^2 / (1 + 15 \, \gamma) $ is the horizon position of the $\gamma = 0$ geometry.\footnote{This $\gamma = 0$ horizon position, used consistently in ref.~\cite{Buchel-ml-2005}, differs from the horizon position $ r_h = \pi T L^2 / (1 + \frac{265}{16} \, \gamma) $ of the $\gamma$-corrected metric derived in ref.~\cite{Pawelczyk-ml-1998pb} and used throughout refs.~\cite{Hassanain-ml-2012uj,Hassanain-ml-2011fn,Hassanain-ml-2011ce}. } Using the Frobenius method to solve for the metric perturbation to linear order in $\gamma$ now leads to \begin{equation} G_{xy,xy}^{\rm ret}(\omega,0) = -i\frac{\Nc^2 \, (r^0_h)^4 \, \omega}{8\pi^3 T} \, (1+195 \, \gamma)+\mathcal{O}(\omega^2,\gamma^2) \,, \end{equation} from which one can extract the result found in ref.~\cite{Buchel-ml-2008sh}, $ \eta = \frac{\pi}{8} \, \Nc^2 \, T^3 \, ( 1+135 \, \gamma+ \cdots ) $. \looseness=-1 One may also apply our partial resummation scheme when evaluating the zero-frequency slope of the correlator~(\ref{eq:Gxyxy}). As with the conductivity, since the shear viscosity receives a much smaller $O(\gamma)$ correction than do the QNM frequencies, the effect of the resummation is much more modest for the this transport coefficient. At, for example, $\lambda = 1000$ the relative size of the $O(\gamma)$ correction to the viscosity is about $6 \times 10^{-4}$, and our resummation increases this deviation from the $\lambda = \infty$ value by a mere $10^{-7}$. As with conductivity, the resummed approximation for the shear viscosity leads to a somewhat larger nominal breakdown value of $\lambda$ as compared to the first order approximation ($\lambda \approx 14$ instead of $\approx 7$). ]]>