321. Holographic hydrodynamics

Posted by Miguel Paulos

March 25, 2009

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Miguel Paulos is a PhD student at DAMTP, U. of Cambridge working on non-equilibrium AdS/CFT. Dmitry.

In this post I will describe recent work done by myself, Robert Myers, and Aninda Sinha to understand strongly coupled plasmas with a finite chemical potential. For more details and full references see 0903.2834.

Let us start with some motivation, and review some of the work leading up to our paper. In recent years, we have seen a shift from trying to test the AdS/CFT correspondence to actually using it as a tool to understand the strongly coupled dynamics of gauge theories. In this context, it is interesting to study the real-time dynamics of strongly coupled conformal plasmas for several reasons:

The quark-gluon plasma (QGP) created at RHIC shows conformal behaviour just above the deconfinement temperature.
The QGP seems to behave like a strongly coupled liquid, and as such can be modelled by hydrodynamics.
Real-time dynamics are not easily obtainable on the lattice.

Holographic principle

At present the best analytical tool one has available for the study of strong coupling hydrodynamics is precisely the AdS/CFT correspondence. The problem of course is that a gravity dual for QCD is not available. What one would really want is a quantitative description of the strongly coupled QGP, but we can nevertheless gain valuable insight by looking for generic qualitative features of gauge theory plasmas, or universal results. This is not a hopeless task, since it is well known that in statistical systems close to conformal points there is in fact universality. An example of this in the context of gauge-gravity duality, is given by the shear viscosity to entropy ratio which, as long as the gravity dual is Einstein gravity, has the simple form:

$\frac{\eta}{s}=\frac{1}{4\pi}.$

The universality of this ratio has been shown in a variety of cases, including theories with different gauge groups and/or matter content, with or without chemical potentials and so on. The universality of this result is directly related to generic properties of black hole horizons (see Son, Starinets 0704.0240). Remarkably, the experimental value for this ratio in the QGP seems to be very close to this theoretical prediction. Of course, this result is valid only in the strict $N,\lambda \to \infty$ limit. To go beyond this limit, one must also go beyond Einstein gravity.

As is well known, the AdS/CFT correspondence in its strongest form maps the full string theory in AdS space to some gauge theory on the boundary. On the gravity side, the effective action for the massless modes is supergravity, but there is also an infinite series of higher derivative corrections. According to the AdS/CFT map, these corrections to the supergravity action map to $1/\sqrt{\lambda}$ corrections on the field theory side. Generically these higher derivative terms also encode information about finite dynamics of the gauge theory. As a step towards a more realistic description of the QGP it is interesting to compute the coupling constant dependence of ratios such as $\eta/s$ and so that naturally leads us to consider working with higher derivative actions.

Typically one can only treat these higher derivative terms perturbatively, except in special cases such as the Gauss-Bonnet terms, which still lead to second order equations of motion. For instance, in string theory higher derivative terms come multiplied by powers of $\alpha'/L^2$ , so that as long as the radius of curvature of the background is large compared to the string length one is justified in performing the perturbative expansion.

Higher derivative corrections to $\eta/s$ were first considered by Buchel, Liu and Starinets, which worked with a particular set of higher derivative corrections in type IIB supergravity. These are the first corrections coming from the type IIB superstring, and take the form of a complicated contraction of four Weyl tensors. Later, in 0806.2156 we showed why one is justified in taking into account only these terms, and not others involving for instance the Ramond-Ramond five-form. The final result, written in field theory variables is given by

$\frac \eta s=\frac 1{4\pi} \left (1+ \frac{15 \zeta(3)}{\lambda^{3/2}}+\frac 5{16} \frac{\lambda^{1/2}}{N^2}+\mbox{n.p.}\right)$

where n.p. stand for a set of non-perturbative corrections in $1/\lambda$ , which I will not discuss here. Plugging in $\lambda=6\pi$ (corresponding to $\alpha_s=0.5$ ) and N=3 , the ratio increases from $1/4\pi \simeq 0.08$ to $\simeq 0.11$ . For comparison, lattice results for $\eta/s$ in pure SU(3) Yang Mills indicate that this ratio is somewhere between 0.10 and 0.17 at T=1.65 T_c , so we’re still in the right ball park. Of course one should be skeptic of these results, since these were derived for N=4 SYM. However, in a related paper0808.1837 we show that for a large class of supersymmetric CFT’s this correction is universal, and so one may hope that the conformal phase of QCD might fall in the same universality class.

One may notice that the correction is always positive. This goes in line with the KSS conjecture, which states that $\eta/s=1/4\pi$ is an absolute lower bound for this ratio. However, this has been disproven in several cases; for instance, taking the Gauss-Bonnet term as the higher derivative corrections leads to a negative sign correction to $\eta/s$ ,

$\frac \eta s=\frac 1{4\pi} (1-8\lambda).$

with $\lambda{}$ the coefficient of the Gauss-Bonnet term in the action. It was also shown in 0812.2521 that including fundamental matter generically one always violates the bound once higher derivative corrections are included.

One may wonder what’s the interplay between finite chemical potential and higher derivative corrections on $\eta/s$ . This was precisely what was studied in 0903.2834 (see also 0903.3244). The idea there is to start with Einstein-Maxwell gravity and then consider the most general set of higher derivative corrections involving at most four derivatives. Even though in principle there is a great number of these terms, we show there that using field redefinitions one can narrow these to a manageable five.

We take the planar AdS-Reissner-Nordstrom black hole and find the leading order corrections coming from these higher derivative terms. From this one can compute corrections to the thermodynamics of the dual field theory. We go on to find the corrections to the shear viscosity, and find that the ratio is modified to

$\frac \eta s=\frac 1{4\pi}\left(1-8 c_1+4(c_1+ 6 c_2)Q^2\right)$

where c_1,c_2 are the coefficients of $R_{a b c d}^2, R_{abcd}F^{ab}F^{cd}$ respectively, and is proportional to the charge of the background.

The coefficient c_1 goes like c-a , the two central charges of the four dimensional CFT. For generic superconformal gauge theories, this number is positive, and so one sees that accordingly the bound is violated when the charge is zero. This is just the previously cited result for Gauss-Bonnet corrections. But what about c_2 ? Once again, in these superconformal gauge theories we have c_2=-c_1/2 , leading to

$\frac \eta s=\frac 1{4\pi}\left(1-8 c_1(1+Q^2)\right)$

We see that turning on a chemical potential only makes the bound violation worse. As for the QCD applications, the chemical potential for the QGP is not expected to be large, but nevertheless as precision increases one might hope that its effects become measurable.

In the same paper, we also look at the DC conductivity $\sigma$ . Recently there have been some proposals for other ratios which seem to be universal involving the conductivity (see 0806.0110). At zero chemical potential we compute these and find

$\frac{\sigma T^2}{\eta e^2}=1-\frac{10}3 c_1+16 c_2,$
$\frac{\sigma T}{e^2\Xi}=\frac 1{2\pi}\left(1-2c_1+16 c_2\right),$

where $\Xi$ is the charge susceptibility. It has been suggested that the first ratio should always be less or equal to one, while the following should be larger or equal than $1/2\pi$ . Reasoning as above, we see that our results agree with the first proposal but seem to contradict the second.

To summarize, holographic hydrodynamics is a fantastic tool to compute the properties of strongly coupled plasmas. We are only now beginning to understand what are the good quantities to look for, and which might show universal behaviour. On a more speculative note, one has the exciting possibility that through methods such as these we might perhaps provide a first definitive experimental prediction of gauge-gravity duality as applied to strongly coupled plasmas.

AdS/CFT, Hydrodynamics, Journal club, Master Postdoc, Quantum field theory, String theory

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