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Improved Holographic QCD

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From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence

Part of the book series: Lecture Notes in Physics ((LNP,volume 828))

Abstract

We provide a review to holographic models based on Einstein-dilaton gravity with a potential in five dimensions. Such theories, for a judicious choice of potential are very close to the physics of large-N YM theory both at zero and finite temperature. The zero temperature glueball spectra as well as their finite temperature thermodynamic functions compare well with lattice data. The model can be used to calculate transport coefficients, like bulk viscosity, the drag force and jet quenching parameters, relevant for the physics of the Quark–Gluon Plasma.

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Notes

  1. 1.

    There are very few observables also that are not in agreement with YM. They are discussed in detail in [25].

  2. 2.

    These are the leading transport coefficients in the derivative expansion. There are subleading coefficients that have been calculated recently for \({\mathcal{N}}=4\) SYM [37, 38]. However, at the present level of accuracy, they cannot affect substantially the comparison to experimental data [5].

  3. 3.

    Most are reviewed in [51].

  4. 4.

    Similar models of Einstein-dilaton gravity were proposed independently in [26] to describe the finite temperature physics of large \(N_c\) YM. They differ in the UV as the dilaton corresponds to a relevant operator instead of the marginal case we study here. The gauge coupling \(e^{\Upphi}\) also asymptotes to a constant instead of zero in such models.

  5. 5.

    This relation is well motivated in the UV, although it may be modified at strong coupling (see Sect. 4.3. The quantities we will calculate do not depend on the explicit relation between \(\lambda\) and \(\lambda_t\).

  6. 6.

    With a slight abuse of notation we will denote \(V(\lambda)\) the function \(V(\Upphi)\) expressed as a function of \(\lambda\equiv e^\Upphi\).

  7. 7.

    We call “bad singularities” those that do not have a well defined spectral problem for the fluctuations without imposing extra boundary conditions.

  8. 8.

    The periods and three-space volumes of the thermal gas solution are related to the black-hole solution values by requiring that the geometry of the two solutions are the same on the (regulated) boundary. See [29] for details.

  9. 9.

    As the dilaton is now not constant there is a non-trivial question: in which frame is the metric AdS. In [25] it was argued that this should be the case in the string frame. The difference of course between the string and Einstein frame is subleading in the UV as the coupling constant vanishes logarithmically. But this may not be the case in the IR where we have very few criteria to check. In the model we are using we impose that the space is asymptotically AdS in the Einstein frame as this is the only choice consistent with the whole framework.

  10. 10.

    We would like to thank K. Kajantie for asking the question, suggesting to compare with lattice data, and providing the appropriate references.

  11. 11.

    The string scale factor is not a monotonic function on the whole manifold [23, 24], and this is the reason that it was not taken as a global energy scale. In particular in the UV, \(e^{A_s}\) decreases until it reaches a minimum. The existence of the minimum is crucial for confinement. After this minimum \(e^{A_s}\) increases and diverges at the IR singularity.

  12. 12.

    Further studies of IHQCD with different potentials can be found in [69].

  13. 13.

    Moreover, higher order \(\beta\)-function coefficients are known to be scheme-dependent.

  14. 14.

    Note that this is conceptually different from the \({\mathcal{N}}=4\) case. There, near the boundary, the theory is strongly coupled and this number must be calculated in string theory. It is different by a factor of 3/4 from the free sYM answer. Here near the boundary the theory is free. Therefore the number of degrees of freedom can be directly inferred.

  15. 15.

    The remaining two are the value \(f(0)\) which should be set to one for the solution (4.9) to obey the right UV asymptotics, and an unphysical degree of freedom in the reparametrization of the radial coordinate.

  16. 16.

    See e.g. [71–73], in which results for \(N_c=8\) do not different significantly from those for \(N_c=3\) as well as the recent high-precision data by Panero [74, 75].

  17. 17.

    Spin 1 excitations of the metric can be shown to be non-normalizable.

  18. 18.

    A further reason is that, unlike the scalar and (to some extent) the pseudoscalar sector that we are considering, the action governing the higher Regge slopes is less and less universal as one goes to higher masses. Only a precise knowledge of the underline string theory is expected to provide detailed information for such states.

  19. 19.

    This action was justified in [23–25]. The dilaton dependent coefficient \(Z(\lambda)\) is encoding both the dilaton dependence as well as the UV curvature dependence of the axion kinetic terms in the associated string theory. We cannot determine it directly from the string theory, but we pin it down by a combination of first principles and lattice input, as we explain further below.

  20. 20.

    Difference in the various numerical factors in this equation w.r.t [92] is due to our different normalization of the dilaton kinetic term.

  21. 21.

    As far as the thermodynamics of the gluon plasma is concerned, the temperatures below \(T_c\) (on the large BH branch) has little importance, because for \(T<T_c\) the plasma is in the confined phase.

  22. 22.

    Note that \(c_b\) is also a function of \(\lambda_h\). As both the fluctuation (4.59) and the boundary conditions are smooth at \(\lambda_h=\lambda_{\min}\), one concludes that \(c_b\) also is smooth at this point.

  23. 23.

    See the discussion at appendix B of [93].

  24. 24.

    Various coefficients in these equations differ from [26] due to our different normalization of the dilaton kinetic term.

  25. 25.

    This argument may break down for two (dependent) reasons: First of all the adiabatic approximation becomes lees good near \(\Upphi_c\). This is because, in this region \(V'/V\) varies relatively more rapidly as a function of \(\Upphi\). Secondly, precisely because of this, even though \(\Upphi_c\) is not far away from \(\Upphi_*\) the difference can result in a considerable change in the value of \(\zeta/s\) through (4.76).

  26. 26.

    Since this theory contains two derivatives only, \(\frac{\eta}{s}\) has the universal value \(1/4\pi\).

  27. 27.

    This representation ignores the fact that the \(kinetic\) mass of a moving quark may be different from the static mass [17].

  28. 28.

    It would stop at the confinement radius if the latter were closer to the boundary than the horizon, i.e. if \(r_*(T) < r_h(T)\). However, in the model we are considering, in the large BH branch we find that the relation \(r_h < r_* \) is always satisfied.

  29. 29.

    For a possible field theoretical explanation of this phenomenon, see [103].

  30. 30.

    This is itself an approximation, since as we know both from experiment and in our holographic model, the plasma is strongly coupled up to temperatures of a few \(T_c.\)

  31. 31.

    Even if we choose \(\epsilon=r_{\min}\), the new singularity at \(r=r_{\min}\) is also integrable as suggested from (4.131).

  32. 32.

    In practise, the previous discussion including regularizing the UV is academic. The numerical calculation is done with a finite cutoff where the boundary conditions for the couplings are imposed.

  33. 33.

    In this case, the value of \(\ell_s\) appearing in (4.143) is fixed as explained in Sect. 4.4.

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Acknowledgements

We would like to thank the numerous colleagues that have shared their insights with us via discussions and correspondence since 2007 that this line of research has started: O. Aharony, L. Alvarez-Gaumé, B. Bringoltz, R. Brower, M. Cacciari, J. Casalderrey-Solana, R. Emparan , F. Ferrari, B. Fiol, P. de Forcrand, R. Granier de Cassagnac, L. Giusti, S. Gubser, K. Hashimoto, T. Hertog, U. Heinz, D. K. Hong, G. Horowitz, E. Iancu, K. Intriligator, K. Kajantie, F. Karsch, D. Kharzeev, D. Kutasov, H. Liu, B. Lucini, M. Luscher, J. Mas, D. Mateos, H. B. Meyer, C. Morningstar, V. Niarchos, C. Nunez, Y. Oz, H. Panagopoulos, S. Pal, M. Panero, I. Papadimitriou, A. Paredes, A. Parnachev, G. Policastro, S. Pufu, K. Rajagopal, F.Rocha, P. Romatchke, C. Salgado, F. Sannino, M. Shifman, E. Shuryak, S. J. Sin, C. Skenderis, D. T. Son, J. Sonnenschein, S. Sugimoto, M. Taylor, M. Teper, J. Troost, A. Tseytlin, A. Vainshtein, G. Veneziano, A. Vladikas, L. Yaffe, A. Yarom and U. Wiedemann.

This work was partially supported by a European Union grant FP7-REGPOT-2008-1-CreteHEPCosmo-228644, and ANR grant NT05-1-41861. Work of LB has been partly funded by INFN, Ecole Polytechnique (UMR du CNRS 7644), MEC and FEDER under grant FPA2008-01838, by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) and by Xunta de Galicia (Consellería de Educación and grant PGIDIT06PXIB206185PR).

E. K. thanks the organizers and especially E. Papantonopoulos for organizing a very interesting and stimulating school.

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Authors and Affiliations

  1. Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE, Utrecht, The Netherlands

    U. Gursoy

  2. Department of Physics, Crete Center for Theoretical Physics, University of Crete, 71003, Heraklion, Greece

    E. Kiritsis

  3. Department of Particle Physics and IGFAE, University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Liuba Mazzanti

  4. Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN, 47907-2036, USA

    G. Michalogiorgakis

  5. APC, Université Paris 7, (UMR du CNRS 7164), Bâtiment Condorcet, 75205, Paris Cedex 13, France

    Francesco Nitti

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  1. U. Gursoy
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  1. Dept.Physics, National Technical University, Zografou Campus, Athens, 157 80, Greece

    Eleftherios Papantonopoulos

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Gursoy, U., Kiritsis, E., Mazzanti, L., Michalogiorgakis, G., Nitti, F. (2011). Improved Holographic QCD. In: Papantonopoulos, E. (eds) From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence. Lecture Notes in Physics, vol 828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04864-7_4

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