51. Planck 2008: day 4 - Soft wall AdS/QCD
DmitryThe last talk of the 4th day was “Soft wall AdS/QCD” by Tony Gherghetta. The issue is of course desire to find an adequite description of QCD in the regime of strong coupling. One idea that captured everyone’s attention for several years is of course AdS/CFT correspondence.
According to AdS/CFT duality, supergravity on 
describes physics of 
SYM at large ‘t Hooft coupling 
. This is not quite what we need though because:
a) SYM is conformal field theory, while QCD is asymptotically free (coupling becomes stronger in the IR).
b) that QCD we are interested in from the practical point of view actually corresponds to 
.
On the other hand, we can try to take AdS/CFT as zero approximation for description of strong coupling QCD, since the latter is approximately conformal at large energies (I have an impression that the word “approximately” here is more about what one has to believe in rather than something one can consistently check, but I may be wrong). So, we start with 
geometry describing 4d physics of YM CFT at high energies and then somehow introduce a cutoff in 5th dimension in order to induce confinement for the 4d Yang-Mills theory. This cutoff can be either introduced by hands (this is what is called “hard wall AdS/QCD limit”) or dynamically (such as by introducing non-trivial dilaton background - situation of “soft wall AdS/QCD” under present discussion).
As it turns out, in order to reproduce linear Regge trajectories, one has to introduce a dilaton background which is quadratic w.r.t. conformal coordinate 
of AdS:
(1).
The question however is whether it is possible to continue the reverse engeneering process a step further compared to Stephanov et al. and find a selfconsistent solution of supergravity equations such that the gravitational background is and the dilaton background is given by (1).
Tony has introduced a model naturally predicting such a background. The price to pay is that he had to introduce an additional scalar field 
(he wants to identify it with closed string tachyon of non-critical string theory - note that here happens a trick: we don’t want to find consistent 10 dim background of the 
form, but limit the discussion to 5 dim instead) and a very nontrivial potential 
for the dilaton and the field . The effective action for the 5-dim theory is (I am not sure I have to present it here, but I will)
where
.
Looks scary, doesn’t it? ;-)
Since the potential is reverse-ingeneered, it was impossible for me to understand the physical meaning of the potential form (or how it would follow from a stringy model).
Let me finish with a couple of notes:
1) It would be interesting to understand how this model works for the 2D QCD which is exactly solvable - it would correspond to gravity in 
, which is presumably topological (it is a theory of scattering of conical defects produced by the quanta of 
and ).
2) It will be hard to explain the nature of the field and the corresponding terms in the potential from the stringy point of view (we should stick to 10 dim physics in this case). As Tony shows, also has to have a runaway behavior (linear growth with ). I wonder - can be one of the moduli which don’t get stabilized?
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