NEQNET: Non-equilibrium Phenomena

Cosmology, turbulence, markets, non-equilibrium QFT and much more. No nonsense, just science

321. Holographic hydrodynamics

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.

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:

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 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 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 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 , 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 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

where stand for a set of non-perturbative corrections in , which I will not discuss here. Plugging in (corresponding to ) and , the ratio increases from to . For comparison, lattice results for in pure Yang Mills indicate that this ratio is somewhere between and at , so we’re still in the right ball park. Of course one should be skeptic of these results, since these were derived for 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 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 ,

with 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 . 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

where are the coefficients of respectively, and is proportional to the charge of the background.

The coefficient goes like , 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 ? Once again, in these superconformal gauge theories we have , leading to

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 . 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

where 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 . 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.

320. Video of the day: STS-119

Here I’ve collected some videos by NASA related to the recent somewhat problematic Space Shuttle mission STS-119. The mission’s goal is to the fourth set of solar arrays and batteries to the station, you will find complete walkthrough of the mission below.

319. Turbulence. Dynamical approach

When we study a turbulent flow, whether turbulence is realized in fluid, plasma, etc., one of the most interesting and complicated questions is the one about transition to turbulence: how exactly the smooth motion of the field becomes turbulent, chaotic, independent of external noise?

It seems to be impossible to answer to this question using statistical approach to turbulence, although I hope to discuss some recent ideas involving diagrammatic methods some day.

The first attempts to develop another approach were made by L. Landau in 1944 and E. Hopf in 1948. In their models, behaviour of a fluid is getting more chaotic (i.e., turbulent) since instabilities (hierarchy of instabilities, actually) with incommensurable time scales develop for the flow. The velocity field becomes more disordered if more and more excitations with incommensurable time scales become present in the flow. In the latter case, velocity autocorrelation function rapidly falls off with time, although it is possible to observe some regularity pattern in the flow if the observation process takes longer than the Poincare recurrence time

,

where is a constant of the order 1, while is the number of excitations with incommensurable time scales. Although this model seems to be reasonable, Ruelle and Takens have shown later that the corresponding attractor is actually unstable, so this very complicated quasi-periodic motion can be really realized in Nature and has definitely nothing to do with the generic situation.

We have got better understanding of the transition phenomenon after discovery of dynamical chaos – randomess of dynamics in deterministic systems. One famous result in chaos theory is existence of strange attractors. Strange attractor is an attracting set of trajectories in the phase space of the system such that (almost) all trajectories in the set correspond to saddle points and are therefore unstable. As it turns out, many rather simple (as well as complex) systems get attracted to this set after a small number of bifurcations.

It is currently believed that a strange attractor should exist in the phase space of the Navier-Stokes equation. Once the system gets attracted to it, the turbulence pattern is fully developed (and regime of turbulence remains stationary in the statistical sense).

Dynamical approach to turbulence and chaos theory is especially useful if we consider transition from the laminar regime to the regime chaotic in time. The simplest scenarios (for the Taylor-Quette flow for example) of such transition include intermittency, infinite sequence of period doubling and breakdown of quasi-periodic motion. There are many much more complicated scenarios observed in Nature, but those three are a kind of canonical ones. The same scenarios are observed in lattice simulations of Navier-Stokes hydrodynamics on large lattices at .

It is also possible to explain spatial chaos using these considerations. For that, we have to consider Lagrangian formulation of hydrodynamics. Introducing Lagrangian markers, we rewrite the Navier-Stokes equation on the form

,

where is velocity of the flow satisfying to the Navier-Stokes equation. The point is that even if the dynamics of is regular, dynamics of can be chaotic. Such regime is called the regime of Largangian turbulence. It seems that Lagrangian turbulence can be developed into usual turbulence of the velocity field, but I don’t think I can say more about that at this point

318. Glueballs and gluelumps as bound states of transverse constituent gluons

Fabien Buisseret is a postdoc at the Nuclear Physics department of the University of Mons-Hainaut. His interests include various approaches to QCD. Dmitry.

1. Generalities

Among other exotic hadrons like hybrid mesons and tetraquarks, QCD allows the existence of purely gluonic bound states, called glueballs, whose structure and properties deserve a lot of interest theoretically. A recent review on glueballs can be found in arXiv:0810.4453. Much effort is also devoted to the experimental detection of a clear glueball signal, but no unambiguous candidate has been found yet (arXiv:0812.0600). An important achievement in the field has been the computation of the glueball spectrum in quenched lattice QCD (hep-lat/9901004, hep-lat/0510074), that is the mass spectrum of pure gauge QCD. Gluelump masses have also been obtained on the lattice (hep-lat/9811010, hep-ph/0310130). These are another hypothetical type of gluonic hadrons, where a spinless, static, color-octet source is added to the gauge field. Gluelumps have actually been a first attempt to model gluino-gluon bound states. A plot of some lattice data concerning glueballs and gluelumps is given in Figure 1, where the masses are expressed in units of the lattice energy scale in order to avoid larger error bars due to the determination of .

Figure 1: Some low-lying glueball and gluelump masses obtained in quenched lattice QCD, plotted in lattice units. The boxes separate the spectrum into states with a given number of constituent gluons.

Apart from lattice QCD, the glueball spectrum has also been computed by using effective approaches like Coulomb gauge QCD (hep-ph/0308268) and potential models – see for example the pioneering work of Barnes [Z. Phys C 10, 275 (1981)]. Potential models have been very succesful in the meson and baryon sectors, where those hadrons are described as bound states of valence quarks and antiquarks. The idea that low-energy QCD allows for an effective description of hadrons as bound states of constituent particles is actually not new: The classification of baryons and mesons with the quark hypothesis is a first historical example of the viability of such a picture. Still in potential models, glueballs and gluelumps are assumed to be bound states of constituent gluons. In the present post I give arguments, taken from previous works (arXiv:0802.0088, arXiv:0806.3174, arXiv:0808.2399, arXiv: 0902.1028, arXiv:0902.4836), showing the relevance of such a picture. Notice that the properties of constituent gluons are not unanimously accepted: Should they be massive or massless? With a spin- or helicity-degree of freedom? As shown in the aforementioned works, it is crucial for the constituent gluons to be transverse, that is with helicity-1, in order to understand the lattice data. This will appear more clearly in the following. Concerning the mass, I assume that a constituent gluon has a zero bare mass, as a “usual” gluon does. But, since it is confined inside a glueball, it can gain a dynamical, or constituent, mass generated by the confining interaction. Recall that confinement is due to low-energy interactions between color-charged particles, either quarks or gluons.

2. The constituent gluon picture

Here is our basic assumption: A description of gluelumps and glueballs in terms of states with a given number of transverse constituent gluons is a satisfactory approximation of what these gluonic hadrons exactly are. Large-Nc QCD provides a hint of the relevance of that hypothesis (see the reviews hep-ph/9802419, hep-ph/0701061). In the limit where the number of colours becomes infinite, baryons are pure -quark states and model-independent mass formulae can be obtained within this framework, leading to a very accurate description of light and heavy baryon resonances. The Large- limit thus appears to capture the essential features of QCD, in the baryonic sector in particular. But the number of valence gluons is a good quantum number for glueballs in this limit (arXiv:0710.4185). By analogy, it suggests that the Fock-space expansion of a given glueball at may be dominated by a particular component, characterised by its number of constituent gluons. Let us now identify that component for the different states of Figure 1.

Bound states of two gluons are the most studied purely gluonic systems in the literature. The color wave function of a two-gluon system is the symmetric configuration , leading to a positive charge conjugation and to a symmetric spin-space wave function in virtue of the Pauli principle. Jacob and Wick’s helicity formalism [Ann. Phys.7, 404 (1959)] can be applied to build the spin-space wave function of a two-transverse gluon system. It appears that no state exists, in agreement with Yang’s theorem stating that a vector meson cannot decay into two photons. The average value of the squared orbital angular momentum can give an idea of the mass hierarchy of the different states. One obtains the following ordering for the lightest two-gluon glueballs: , , , . Two is the minimal number of constituent gluons to build a color-singlet glueball. Looking at Figure 1, it is rather clear that the lightest glueballs are located in the positive-C sector. Moreover, the mass hierarchy corresponds to the one expected for a two-gluon bound state, and the absence of low-lying glueballs is confirmed.

The lightest glueballs are heavier than those with positive-C. That point can be intuitively understood by remarking that at least three gluons are needed to make a negative-C state, the color wave function being in that case the totally symmetric one . Another color singlet exists; it is totally antisymmetric and leads to glueballs. Three-gluon helicity states can also be built in principle with the helicity formalism, and it can be deduced that the lightest ones have , followed by . If gluons were not transverse, a light state would be allowed, in disagreement with lattice QCD. The three-gluon sector corresponds to the green box in Figure 1. It is now worth looking at the glueball. It cannot be made of three gluons since is not allowed in this case. But a four gluon state can generate those quantum numbers: The glueball could be located in a four-gluon sector, the purple box in Figure 1.

Finally, let us make some comments about gluelumps. They are lighter that the lightest glueballs. Such a low mass can be understood as follows: The presence of the static color-octet source makes possible for a single constituent gluon to be bound in a color singlet. The charge conjugation of such a state is always negative, and the helicity formalism demands that , in agreement with the lattice data.

Roughly speaking, the mass of a gluonic state should grow like , with the number of gluons and the constituent gluon mass. With , one obtains the squares in Figure 1. They are all located near the average mass of a given sector, as expected. Since typically, is around .

3. Interactions between gluons

Now we go beyond the above qualitative description by showing how to build an explicit potential model for glueballs. Consider the case of two-gluon glueballs. The simplest Hamiltonian describing a system of two massless particles is a spinless Salpeter one, given by

(1)

It is well-known from lattice QCD that the potential energy between a static quark-antiquark pair is accurately fitted by a funnel form, i.e.

.

The linear term stands for the confining interaction generated by a flux tube of tension , while the Coulomb term encodes one-gluon-exchange effects. The funnel potential has been widely used in quark models to reproduce accurately the meson and baryon experimental mass spectra. Can it be used as a good gluon-gluon potential? The potential energy between two static color-octet sources has been computed in lattice QCD. It is still compatible with a funnel shape for different values of the parameters (hep-lat/0006022). In particular, the string tension has to be scaled by a factor (9/4) following the Casimir scaling hypothesis, and the color factor of the Coulomb term has also to be modified. However, massless constituent gluons are far from being static sources, and it would be interesting to find an alternative way of computing the gluon-gluon potential.

The radial wave function of the scalar glueball has been computed in lattice QCD (hep-lat/0603030), with a mass that is in agreement with previous calculations. The idea is now to find the local radial potential needed in (1) to reproduce the lattice mass and wave function. Such a task can be performed numerically and the result is shown in Figure 2. The computed gluon-gluon potential has a long-range confining part and a short-range attractive singular part. Actually, it is compatible with a funnel form, validating the use of such a potential in effective approaches.

Figure 2: Effective potential between two transverse gluons in the scalar channel (dashed line and gray area), computed from the lattice wave function R(r) (dots and dashed-dotted line) and the spinless Salpeter Hamiltonian (1).

By using parameters in agreement with the potential of Figure 2, one can build a glueball model, based on the helicity formalism and on the spinless Salpeter Hamiltonian (1), that reproduces accurately the glueball spectrum in the positive-C sector. The gluelump spectrum is also nicely reproduced, while three- and four-gluon generalizations of that approach remain to be done.

4. Gluon plasma

The constituent gluon picture eventually finds an application in finite-temperature QCD, in relation with the celebrated quark-gluon-plasma. Remember that the “confined world” of the three previous sections is a zero-temperature one. Theoretical as well as experimental results (mainly obtained at RHIC) support the idea that there exists a critical temperature in QCD, beyond which the hadronic matter is deconfined; that is the quark-gluon-plasma. For temperature slightly above the critical one Tc, lattice computations of the free and internal energies between two static color sources in various representations show that the color interactions are still nonnegligible, although no longer confining (see for example hep-lat/0309121 and arXiv:0711.2251). There exists consequently a strongly coupled phase near the critical temperature, where quarks and gluons are deconfined but not free as it could have been expected.

Let us turn our attention to a pure gluon plasma, whose equation of state is known from quenched lattice QCD (hep-lat/9506025 and Figure 3). These last data can be understood thanks to a constituent gluon approach: The gluon plasma is then modeled as an ideal Bose gas of transverse gluons with an appropriate temperature-dependent mass (hep-ph/9710463). The thermal gluon mass has to be fitted on lattice QCD, but from perturbative QCD it can be expected that it grows linearly at very large temperature. Fits are in agreement with that point. Notice that a linearly rising gluon mass is a necessary condition for the thermodynamical quantities to saturate below the Stefan-Boltzmann limit, as it can be observed in Figure 3.

Figure 3: Energy density (red line), entropy density (pink line), and pressure (blue line), in reduced form, versus temperature computed in quenched lattice QCD at zero chemical potential. The horizontal black line shows the Stefan-Boltzmann limit for a gas of transverse gluons. Gray curves are the results of the glueball-gluon gas model.

In the paragraphs above, two apparently opposite statements have been made. First, color interactions between gluons are strong near . Second, the gluon plasma can be described as an ideal gas of transverse gluons in the same temperature range. The point is that the thermal gluon mass, fitted on lattice data, takes into account effective color interactions. It even becomes singular near , while the linear behavior only appears at temperature larger than 2.5 . Using a spinless Salpeter Hamiltonian such as (1) in which the gluon-gluon potential is modified according to finite-temperature lattice results, it can be computed that the color interactions are strong enough to bind two gluons in the scalar channel up to 1.6 . Then, it seems relevant to see the gluon plasma as an ideal Bose gas containing a mixture of free gluons and glueballs, with a gueball abundance depending on the temperature. Fitting that quantity on lattice QCD, one reproduces with an excellent accuracy the gluon plasma equation of state, see Figure 3.

5. Conclusions

In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states. Gluelumps should moreover be seen as one-gluon states, allowed because of the static colour octet source.

Not only the glueball masses can be computed on the lattice, but also glueball wave functions. In the scalar channel, it appears that the lattice mass and wave function correspond to those of a spinless Salpeter Hamiltonian with a funnel potential, partly justifying the use – and the success – of such potential models in the description of glueballs.

Finally, modelling the gluon plasma as an ideal mixture of gluons and glueballs allows to understand its equation of state computed in quenched lattice QCD. Although it is no standard way of concluding, let me say that in view of the results known so far and those still to be obtained, the gluon below has good reasons to smile.

A constituent gluon, as people from http://www.particlezoo.net imagine it.

Some good QCD reading

1. W. Greiner and A. Schafer, Quantum Chromodynamics, Springer, 1995 – pedestrian introduction to QCD.
2. F. J. Yndurain, The Theory of Quark and Gluon Interactions, Springer, 1998 – a higher-level course.

317. Global crisis: one interesting plot

As it seems, I did not write about global crisis for quite a long time It is not like I see any meaning to get deeper into finding ultimate reason why the crisis had to take off (I guess the only meaningful question one might ask in the present situation – “what am I, personally, going to do about this surprising turn in my life?”), old interests do not die out too quickly. Sometimes I find some material regarding crisis here and there and really want to share it with you… but then I ask myself – are you really interested? Why do you need to know that? Well, I am going to forget about those questions this time

I have the following rather interesting plot that has quite a lot to do with the history (and the future) of the global financial crisis.

So, what are you thoughts? Is it correct? (= is Soros right calling this turmoil much worse that the Great Depression?) If it is wrong, where is the flaw?

Cheers,
Dmitry.

P.S. In the mean time – have you noticed a couple of changes on NEQNET? First of all, you can now print any post from NEQNET in a nice fashion (that is, header, footer, social bookmarking icons, Facebook Connect etc. are not to appear on the paper – this will probably save you ta page or two of paper per print-out) by clicking on “Print This Post” link below the title of the post. Instead of printing, you can also save the post of your choice as a PDF file with the same bonuses as printing it out.

I have also decided to follow Lubos Motl’s path and turn Odiogo text-to-speech support on NEQNET. Now, istead of reading a post you have option to listen it (or save text-to-speech version as MP3 file and listen to it on your way home from the office).

Finally, if you have a webcam, you can now leave video comments on NEQNET. To do this, click on the title of the post you want to comment on, scroll down until you see the comment form and press “Add video comment with Seesmic”. You don’t need to have a Seesmic account to post the comment, just press “Anonymous” in the next screen and have fun

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