Holographic Baryons

Last week Deog Ki Hong was explaining how baryons can be realized in holographic QCD. Holographic QCD is a new sport discipline that consists in modelling the symmetries and dynamics of strongly coupled QCD using weakly coupled theories in more-than-four dimensions. This approach is inspired by the AdS/CFT conjecture that links N=4 superconformal gauge theories with large number of colours and large t'Hooft coupling to higher dimensional supergravity. QCD, however, is neither supersymmetric nor conformal and it is unclear whether a holographic dual exists. In fact, one can argue that it does not. Nevertheless, some bottom-up, phenomenological constructions turned out to be quite successful, against all odds.

There are two roads that lead to holographic QCD. That of Sakai and Sugimoto, rooted in string theory, uses the language of D8-branes embedded in a D4-brane background. A more pedestrian approach takes its origin from the paper of Erlich et al. , who skip the stringy preamble and exploit 5D gauge theories in curved backgrounds. The global chiral symmetry of QCD - $U(2)_L x U(2)_R$ (or U(3)xU(3) if we wish to accommodate strangeness) - is promoted to a local symmetry group in 5D. Besides, the 5D set-up includes a bifundamental scalar field with a vacuum expectation value. The Higgs mechanism breaks the local symmetry group to the diagonal $U(2)_V$, which mimics chiral symmetry breaking by quark condensates in QCD.

So far, most of the studies were focused on the meson sector. Spin 1 mesons (like the rho meson) are identified with Kaluza-Klein modes of the 5D gauge fields. The spin 0 pions are provided by the fifth components of the 5D gauge fields (mixed with pseudoscalars from the Higgs field). Employing usual methods of higher dimensional theories, one can integrate out all heavy Kaluza-Klein modes to obtain a low-energy effective theory for pions. The result can be compared with the so-called chiral lagrangian - the effective theory of low-energy QCD that is used to describe pions and their interactions. Coefficients of the lowest-order operators in the chiral lagrangian have been measured in experiment. Holographic QCD predicts values of (some combinations of) these coefficients, and the results agree with observations. Furthermore, holographic QCD predicts various form factors of the vector mesons that also have been measured in experiment. Again, there is a reasonable agreement with observations. The accuracy is comparable to that achieved in certain 4D approximate models based on large N QCD. All in all, a rather simplistic model provides quite an accurate description of low-lying mesons in low energy QCD.

Baryons are more tricky. In the string picture, they are represented by D5 branes wrapping S5, which sounds scaring. In the 5D field theory picture, they are identified with instanton solitons - still somewhat frightening. But it turns out that these instantons can be effectively described by a pair of 5D spinor fields. Now, study of fermions in a 5D curved background is a piece of cake and has been done ever so often in different contexts. The original instanton picture together with the AdS/CFT dictionary puts some constraints on the fermionic lagrangian (the 5D spinor mass and the Pauli term).

With this simple model at hand, one can repeat the same game that was played with mesons: look at the low-energy effective theory, compare it with the chiral lagrangian predictions and cry out of joy. There are two points from Deog Ki's talk that seem particularly interesting. One is the anomalous magnetic moment of baryons. Holographic QCD predicts that those of proton and neutron should sum up to zero. In reality, $\mu_p = 1.79 \mu_N$, $\mu_n = - 1.91 \mu_N$ where $\mu_N = e/2 m_N$ is the nuclear magneton. The other interesting point concerns electric dipole moments. In the holographic model the electric dipole moment of the neutron can be simply connected to the CP-violating theta angle in QCD (something that seems messy and unintuitive in other approaches) . There is another sum rule that the electric dipole moments of protons and neutrons should sum up to zero.

In summary, simple 5D models yield surprisingly realistic results. Of course, more conventional approaches to QCD may achieve a similar or better level of precision. The drawback is also that the holographic approach has no rigorous connection to QCD, so that it's not clear what is the applicability range and when should we expect the model to fail. Nevertheless, the 5D approach provides a simple and intuitive picture of low-energy QCD phenomena. The experience that is gained could also be useful in case we stumble upon some new strong interactions in the LHC.


Although technological consciousness at CERN TH is clearly improving, some convenors have not yet discovered the blessings of modern means of communication. Translating to English: slides from this talk are not available. Here you can find partly overlapping slides from some conference talk. If you long for more details, check out these papers.