Brane Gymnastics

A fun weekend in Beijing with some stunning sunsets seen from my apartment balcony in the quickly rising temperatures of the evenings.


Also, a couple of photos of the Nanchang campus just before we left.



I mentioned a few things I wanted to talk about last time and was way too tired. Now, Sunday evening I'm still pretty knackered but can at least cover one of the topics, though it is physics and I'm going to have to talk in a little stringy detail I'm afraid.

It's this paper by Nick Evans, my former PhD supervisor amongst others. Nick told me before that they were working on this so I was eagerly awaiting the results. The topic is meson spectroscopy from AdS/CFT though from a very different perspective to the work which was first performed by Karch and Katz (and KMMW). The idea of this original work was that in order to introduce quarks (fundamental matter) into the gauge/gravity duality, you need some new object for a string to end on.

A string stretching between the D3-brane and the new object knows that its ends are different, even when the two objects are brought to the 'same position'. From the fundamental construction of the correspondence, the D3-branes label a colour charge for the string and the other object should label a new charge, a flavour charge. Therefore, something with one end on a D3-brane and one end on the new object is in the fundamental of the colour gauge group and the fundamental of the flavour group.

There are two other types of string we can talk about too. There are the original strings which were there before we introduced the new object. These are simply the adjoint strings of the N=4 hypermultiplets. Then there is another new type of string which just sits on the new object and is therefore in the adjoint of the flavour group.

In the supergravity limit we can forget about the fact that this is a string and treat it as a point particle. We can write down an action for this which we treat as an effective field theory.

It turns out that the correct new object to introduce is a D7 brane. There are several reasons for this choice, one of which is the supersymmetry preserving properties of a system of Dp-D(p-4) branes.

Another, more phenomenologically interesting reason for this choice is that because there are two perpendicular directions to a D7-brane, there is an SO(2) symmetry of the scalar fields living on it. Of course, the scalar fields are charged in the adjoint of the flavour group and are not coloured and therefore correspond to mesons. The SO(2) symmetry simply corresponds to the usual chiral symmetry which we know gets broken by non-perturbative effects in QCD, so this symmetry is of considerable interest to us.

When we introduce a D7-brane into AdS5xS5, we can study the well-behaved brane-flows in the two directions perpendicular to the brane as a function of the radial direction in the AdS space (which corresponds to an energy scale on the field theory side).

The behaviour of the flow at large energies can tell us about the mass of quarks and vev of the quark bilinear and, in the massless quark limit, we find that the chiral symmetry is preserved. This is not a suprising result because in a supersymmetric field theory we don't expect to get a chiral condensate which would break the chiral symmetry.

So, we need to go to a more complicated geometry in order to find any QCD-like properties. The geometry which was first studied in the context of chiral symmetry breaking was the Constable-Myers geometry which is a rather strange deformation with a flowing dilaton, a naked singularity and an unstable gauge theory. However, it's realistic enough for us to be able to ask relevent questions about QCD.

In particular, when we study the solution of the D7-brane in this background, we find that in the massless quark limit there is a chiral condensate and the geometrical SO(2) symmetry is broken as the brane flows into the corresponding IR region of the geometry. Not only this but we can study the fluctuations of the brane, which correspond to mesons, find the eigenvalues of these fluctuations and discover that we have a massless mode in the massless quark limit, corresponding to the pion. In fact we have all the ingredients we would expect for a theory which exhibits chiral symmetry breaking (Note also that the relationship between quark mass, meson mass and chiral condensate agree with the Gell-Mann-Oakes-Renner relation in the small quark mass limit).

So, this is all well and good but we have only introduced one flavour of quark. We can introduce two D7 branes in parallel and study the DBI action of these but we find that because certain important superpotential terms, there is never an SU(Nf) axial symmetry to break and so we can't study the interesting effects that would go with this. We get no new results for multiple branes as compared to just one.

So, step forward a couple of years to last week and we have this new paper. Their idea is to study what happens when we have a brane corresponding to a heavy quark and a brane corresponding to a massless quark. What can the string stretching between them tell us?

In fact, I'll jump to the punchline before getting back to the technicalities. It turns out that by setting two of the parameters correctly using experimental results and using the same non-supersymmetric background that was used to study chiral symmetry breaking, we can get the mass of the lightest B-meson to within around 20% of the correct answer. This is a pretty nice result in what is essentially a rather naive model which is not that much like QCD and is certainly not being used in the range of validity of the strong-coupling limit of QCD. What I mean by this is that at the energy of the B-meson mass, QCD is weakly coupled, whereas, the field theory dual to the supergravity theory being used is conformal but strongly coupled at these scales.

In fact, I'm not going to go much into the technicalities of the paper which I'm currently working through and reproducing the results, but the basic outline is this:

Construct the flows for the two D7-branes, corresponding to the heavy and light quarks. Write down the Polyakov action for a string stretching between them. Use the constaint equations for the strings and the conjugate operators of position in each of the eight directions perpendicular to the string to write down an operator equation which acts on a field living in the eight dimensions of the D7-brane.

We can study the large energy behaviour of this field which tells us about the operator ub(bar) in the lagrangian. We know that there is no mixing term like this and so the first order behaviour for the field is zero.

On top of this zero are the fluctuations which correspond to the meson itself. Using a plane-wave ansatz (because we are not interesting in interactions), we can set the wave-function equal to f(r)exp(ik.x) where r corresponds to four of the directions (three of them being on the five sphere) and x represents the directions in Minkowski space-time. With this ansatz k^2->-M^2 and we can find the eigenvalues of this equation which give us the meson mass. This all has to be performed numerically even though the metric for the Constable-Myers geometry has a nice analytical form, but when this is done, a 20% agreement to experiment is found. Nice!

Ermm, so there it is. The paper is nicely written and not too long so you would probably understand more if you actually read the paper yourself but hopefully there are a couple of random pieces of insight in there for you to chew on. I can see some obvious extensions to this but I'm not getting to tell you now because, if they work out, I might get a paper out of it :-P