Why the Quark Gluon Plasma is a Banana

When I was in 5th grade, our biology teacher dropped out for a longer period. For whatever reasons, she was temporarily replaced by our maths teacher. He was about to be retired soon, and had not taught biology for more than 20 years. Now he saw himself faced with the task of giving "sexual education" to a bunch of 11 year olds.

After having extensively discussed various forms of STDs, we all were completely sure we would never ever have sex in our life. Our teacher then proceeded to safe sex, and eventually produced a banana to explain the proper use of a condom. He was clearly struggling with words he hardly ever used in his life, and the banana seemed to us a very theoretical example for a very involved practical situation.

I thought about this banana, when I was in very interesting seminar at the KITP last week by Pavel Kovtun, titled

How Can AdS/CFT Be Useful for Heavy Ion Physics?

There, in this seminar room, all the stringy guys were talking about jet suppression, shear viscosity and central collisions! It was equally weird as hearing your maths teacher talk about sex.

In the talk, it was explained how the AdS/CFT correspondence can be used to model certain properties of heavy ion collisions. I find this an interesting topic, with a clear connection to phenomenology, which I often miss in talks at the KITP.

It also reminded me of a colloquium by Joe Polchinski that I heard about 3 years ago at LBNL. The only thing I can recall was that he said something like the quark gluon plasma is a black hole. He most likely said more, but I was too nervous to listen, since own seminar (the first seminar I ever gave outside my home institution) was scheduled for the following lunch meeting. Consequently, the only thing that was in my head while I was giving my seminar (which btw had nothing to do neither with heavy ions nor with black holes or qcd) was what the fu*k a black hole has to do with the quark gluon plasma, and why I ever agreed to give this seminar.

Had someone told me at this point that I would end up at UCSB three years later, no way I would have believed it.

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1. AdS/CFT

So, here are the rough basics that I hope are sufficient to get into the spirit. Strongly coupled QCD is messy soup, and calculations are nasty. It turns out however, that above the critical temperature, a conformal field theory is a pretty good description of the messy soup (see slide 8 of talk or picture below). Plotted is the energy density (epsilon), normalized to that of a Stefan-Bolzmann gas (subscript SB). The colored curves are lattice calculations, the black line is a conformal field theory - the maximally supersymmetric yang mills theory, which comes with two parameters: the coupling constant g, and the number N of the gauge group SU(N).

So it seems, instead of QCD at finite temperature, it's a reasonable try to model some properties of the mess with a conformal theory, like the mentioned super yang mills theory (SYM).

That by itself does not make things less messy, but now one uses the AdS/CFT correspondence! And here, the miracle happens: the 4-dimensional SYM gets translated into 5-dimensional gravity in some space called AdS₅ x S⁵ . Which is much easier to deal with.

One can then translate quantities from the 4-dimensional field theory into the 5-dimensional gravity in this specific space. E.g. the

The stress-energy tensor in SYM translates to a graviton on AdS₅ x S⁵

A state in the field theory translates to a state in the dual geometry

An operator in field theory translates to a field in the dual geometry

Etc. if you want to specify the type of particles you are looking at (fermions, gauge fields), things get more complicated, but are doable. Most importantly

The thermal equilibrium state in SYM translates to a black hole in AdS₅ x S⁵

And so, Marcus, I am very proud to say that at least my time at UCSB enabled me to answer the question what the QGP has to do with a black hole.

However, the general relativity one has now translated the field theory into, is classical only for large N. For small N, one would need to know quantum gravity - or string theory. It is usually assumed that N is 'large enough', which might or might not include N=3.

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2. Applications

Using the above translation, one can now examine observables. A prominent example is e.g. the shear viscosity and it's lower bound. But also the photon and dilepton emission rate, or the energy loss of particles traveling through the messy soup can be investigated. Besides Pavel's talk, you might want to check some of his papers. It seems to me, he is working on an enormous amount of related stuff. Some more references about the topic

• Minkowski-space correlators in AdS/CFT correspondence: recipe and applications
  D.T.Son, A.O.Starinets


• The Hydrodynamics of M-Theory
  Christopher P. Herzog


• Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics
  P.Kovtun, D.T.Son, A.O.Starinets

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3. Problems

There are most likely a whole number of theoretical, analytical and numerical problems with the approach that I am not aware of, and therefore will not mention. Starting from the question whether N=3 is indeed a large number, or the assumption of thermal equilibrium, I have the vague impression there are more subtleties than I can (or want to) imagine.

Still, I think it's a reasonable try to use AdS/CFT for SYM to model properties of heavy ion collisions. Instead of, say, some hydrodynamical description, un-quantized numerical models, or cost- and time consuming lattice calculations.

The biggest problem I have with the scenario is the formation and time evolution of the fireball. Thermal equilibrium might correspond to a black hole in the dual theory, but what describes the collision, or the following hadronization?

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The intention to use often very abstract mathematical theorems for direct applications is definitely something I welcome, esp. since the front research in theoretical physics is falling more and more apart into sub-fields.

It requires quite some courage to go to a heavy ion conference as a string-theorist, something Pavel did last month, which really surprised me. I am not sure though whether the average heavy ion physicist would already consider the approach to be more than a theoretical example for a practically very involved situation.

To come back to the banana issue: I am very relieved to tell you that I, and most of my classmates, eventually recovered from my maths teacher's lectures. We all figured out how to properly use a banana.

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Acknowledgements: I would like to thank Pavel for his patience with my questions, and I apologize for the banana. I could not resist the temptation.