163. What is AdS/QCD?
HEP-TH/PH — By Josh Erlich on January 3, 2009 at 12:00 pm
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This is a a guest post by Josh Erlich from the College of William and Mary, Williamsburg. Dmitry.
This is a summary of AdS/QCD based on a recent review talk I gave at the Confinement 8 conference in Mainz, Germany. For more details and a list of references, the talk is posted on the arxiv as arxiv:0812.4976.
AdS/QCD is a catch-all term which describes extra-dimensional models of QCD. The starting point is the suggestion that the towers of radial excitions of QCD resonances with given quantum numbers might be identified with Kaluza-Klein modes of fields in more than four dimensions. The original motivation for these models was the AdS/CFT correspondence, but it is well understood that QCD with three colors does not have a supergravity dual. Hence, AdS/QCD is really just a class of models of hadronic resonances with many features of previous models built in: chiral symmetry breaking, sum rules and hidden local symmetry, to name a few. These models generally make more accurate predictions at low energies than should have been expected. Brodsky and de Teramond have noticed an intriguing relation between AdS/QCD models and hadronic wavefunctions in the light-front formalism. The AdS/CFT correspondence has also motivated models of electroweak symmetry breaking and condensed matter systems, the latter of which have been discussed on this blog.
AdS/QCD models come in two types: top-down models engineered from brane constructions in string theory, and bottom-up models that are more phenomenological.
The top-down model most closely related to QCD is the D4-D8 system of Sakai and Sugimoto. At low energies the model is nonsupersymmetric and confines with non-Abelian chiral symmetry breaking, just like QCD. The D8-D8 strings are identified with mesons, and it is straightforward to calculate low-energy observables like meson masses and decay constants, as well as some finite-temperature observables. The model also contains solitonic excitations identified as baryons, similar in spirit to the old Skyrme model. On the down side, like all known top-down AdS/QCD models, the D4-D8 system also contains non-QCD-like states with hadronic-scale masses.
Removing the shackles of string theory allows us to consider a more general class of models. The prototypical bottom-up models involve a gauge theory in a slice of five-dimensional Anti-de Sitter space. Kaluza-Klein modes of the gauge fields are identified with the vector and axial-vector mesons of QCD, via their spins, charges and parities. The 5D gauge invariance is manifested as the approximate 4D global chiral symmetry in the effective 4D theory. The gauge invariance is broken either by boundary conditions or by the Higgs mechanism to a diagonal isospin subgroup.
It is possible to build in some features of QCD at higher energies, like agreement with UV correlators of isospin currents, and the Regge spectrum. However, AdS/QCD models generally disagree with QCD at high energies. In my opinion, our best hope to understand why AdS/QCD fares so well at low energies is to understand which predictions are independent of the details of these models. This is the notion of AdS/QCD universality. A famous example of universality is the AdS/CFT prediction of the ratio of equilibrium shear viscosity to entropy density, 
. As shown by Kovtun, Son and Starinets, this is
a universal prediction that relies only on the properties of black hole horizons. I am hopeful that we will better understand the
success of AdS/QCD by identifying classes of low-energy hadronic observables that are relatively model-independent.
10 Comments
Dear Josh
Thank you very much for the post! Could you explain in a few words for undergrads, what is Son-Stephanov hidden local symmetry model?
Cheers,
Dmitry.
Hi Dmitry,
The idea of hidden local symmetry rests in the old observation by Sakurai and others in the 1960′s that the rho mesons act a lot like massive vector fields. Bando and others formalized this idea and called it hidden local symmetry. Son and Stephanov in hep-ph/0304182 formulated an extension of the hidden local symmetry models via a deconstructed 5D SU(2) gauge theory, i.e. a 4D
gauge theory with large k. Thinking of the SU(2)’s as living at lattice sites along a line, there are scalar fields that transform as bifundamentals of neighboring SU(2) gauge groups. When those scalar fields get vevs, the 
gauge group is broken, and the massive gauge fields are interpreted as rho and a1 mesons.
Below a scale set by the vevs, the action can be written as a 5D gauge theory with some background spacetime geometry with two boundaries, latticized in one dimension. In this sense, it’s just like the Sakai-Sugimoto model.
What Son and Stephanov did that’s so interesting is that they explained in this context why AdS/CFT works. Let’s say the SU(2)’s at each of the two ends of the latticized line are originally just global symmetries, but then we weakly gauge them. The gauge fields couple to conserved currents, so we think of these gauge fields as sources for the
currents associated with the global symmetries of the theory. To calculate the correlator of a product of currents you vary the generating functional, which in the classical limit is 
where S is the action, with respect to the gauge fields. This is precisely what AdS/CFT tells you to do, if you think of the two SU(2)’s as living at the UV boundaries of the 5D spacetime geometry.
Okay, I guess I didn’t really address this to undergrads, but Son and Stephanov explain it better in their paper.
BR
Josh
Thanks! Why QCD behavior can be considered conformal at mass scales below the mass of rho meson (say, between pion mass and rho meson mass)? What is actually the precision of the physical statement that QCD is conformal (scale invariant) at these energy scales (I mean, how much is it scale invariant when compared to experiment)?
QCD is not conformal around the rho mass. Brodsky and de Teramond have argued based on Jefferson Lab data that there is a window of energies a few times the rho mass over which it seems the QCD coupling doesn’t run, so it looks conformal. At very high energies, where QCD is asynmoptotically free, it is also approximately conformal.
In AdS/QCD we choose the geometry to be
in the UV region because we ultimately do our best to match to the conformal UV behavior of QCD. Precisely where that begins is not important to us, since it just affects the UV cutoff in the model and most observables we looked at were insensitive to the cutoff. Away from the UV we do not want the model to be conformal, but the choice of a slice of 
is the simplest geometry that is consistent with the UV behavior, so that’s what we start with. It’s just a toy model that really shouldn’t work as well as it does. It seems that the precise choice of geometry away from the UV region does not have a huge impact on the fit to data.
BR
Josh
Thanks again! And another question – I hope there are not too boring for you
Why, do you think, naive AdS/CFT is so unnaturally good (it minimizes RMS error, as you say) although it corresponds to N=4 SUSY at field theory side and
? Can it be that if I introduce more parameters (say, some SUSY breaking terms in the Lagrangian), your statement about minimizing RMS error will go away?
Cheers,
Dmitry.
For some things N=4 YM indeed does quite well, but it’s certainly not QCD-like. At finite temperature, certain features of models with supergravity duals are relatively universal, like I mentioned in my post, which is probably why even N=4 YM does well for those things. However, N=4 YM certainly won’t get correct anything that crucially depends on running and confinement, like the spectrum.
SUSY and conformal invariance are not important for AdS/CFT. There are many examples of non-conformal theories that confine, with and without chiral symmetry breaking, and have supergravity duals. The Sakai-Sugimoto model is non-conformal and non-supersymmetric, for example. The large-N limit causes some problems, like infinitely narrow resonances, which is definitely not like real QCD.
I’m sure I didn’t do a great job addressing your questions, but let me know if there’s anything else I can try to help with.
Best regards,
Josh.
Hi Josh
No, I think, many things became clearer to me. Thank you very much for your answers!
Dmitry.
Stan Brodsky noticed this blog and corrected me on my answer to Dmitry’s second question. I said that according to Brodsky and de Teramond there is a window of energies above the rho mass over which the theory seems conformal. In fact, Dmitry understood correctly that the claim is that QCD is conformal in the infrared, down to Q^2=0 (as well as in the UV where asymptotic freedom makes QCD conformal). Physically, the explanation offered by Stan is that the quarks and gluons can’t have too large a wavelength because of confinement, which sets an effective IR cutoff in loop integrals contributing to the QCD beta function.
The data that supports Stan’s claim comes from a number of places. Most telling, perhaps, is the JLab data in Figs. 1 and 2 of:
Determination of the effective strong coupling constant alpha(s,g(1))(Q**2) from CLAS spin structure function data.
A. Deur, V. Burkert, J.P. Chen (Jefferson Lab) , W. Korsch (Kentucky U.) . JLAB-PHY-08-808, Mar 2008. 8pp.
Published in Phys.Lett.B665:349-351,2008.
e-Print: arXiv:0803.4119 [hep-ph]
http://www.sciencedirect.com/s.....262372e3b1
You can also look at Stan’s papers, for example:
Maximum Wavelength of Confined Quarks and Gluons and Properties of Quantum Chromodynamics.
Stanley J. Brodsky (SLAC & YITP, Stony Brook & Durham U.) , Robert Shrock (YITP, Stony Brook) . SLAC-PUB-13246, IPPP-08-37, DCPT-08-74, YITP-SB-09-11, Jun 2008. 7pp.
Published in Phys.Lett.B666:95-99,2008.
e-Print: arXiv:0806.1535 [hep-th]
Light-Front Holography and Hadronization at the Amplitude Level.
Stanley J. Brodsky, Guy de Teramond, Robert Shrock . SLAC-PUB-13306, 51DCPT-08-102, YITP-SB-08-33, Jul 2008. Temporary entry
Invited talk at 6th International Conference on Perspectives in Hadronic Physics ( Hadron 08 ), Trieste, Italy, 12-16 May 2008.
Published in AIP Conf.Proc.1056:3-14,2008.
e-Print: arXiv:0807.2484 [hep-ph]
Brodsky and de Teramond’s perspective on AdS/QCD based on the light-front formalism is quite intriguing. It relates the radial coordinate of AdS to an invariant light-front variable related to parton momentum inside the hadron. That story deserves a post of its own…
Cheers,
Josh
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