Mesons and baryons are considered in soft-wall holographic approach based on the correspondence of string theory in AdS space and conformal field theory in physical space-time. The model generates Regge trajectories linear in and for the hadronic mass spectrum. Results obtained for heavy-light meson masses and decay constants are consistent with predictions of HQET. In the baryon sector applications to the nucleon electromagnetic form factors and generalized parton distributions are discussed.

Mesons and baryons in the holographic soft-wall model

Valery E. Lyubovitskij111On leave of absence from Department of Physics, Tomsk State University, 634050 Tomsk, Russia , Thomas Gutsche, Ivan Schmidt, Alfredo Vega

and Particle Physics, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

Based on the gauge/gravity duality [1] a class of AdS/QCD approaches was recently successfully developed for describing the phenomenology of hadronic properties. In order to break conformal invariance and incorporate confinement in the infrared region two alternative AdS/QCD backgrounds have been suggested in the literature: the “hard-wall” approach [2], based on the introduction of an IR brane cutoff in the fifth dimension, and the “soft-wall” approach [3, 4, 5, 8, 9, 6, 7, 10, 11, 12], based on using a soft cutoff. In series of papers [9, 10, 11, 12] we developed the soft-wall approaches, which have been successfully applied for the study of meson and baryon properties. Here we present a summary of recent results: meson mass spectrum and decay constants of light and heavy mesons, nucleon electromagnetic form factors and generalized parton distributions. Our starting point are the effective dimensional actions formulated in AdS space in terms of boson or fermion bulk fields, which serve as holographic images of mesons and baryons. For illustration we consider the simplest actions — for scalar fields ([10]

and fermions [7, 11]:

where and are the scalar and fermion bulk fields, is the covariant derivative acting on the fermion field, are the Dirac matrices, is the dilaton field, is the AdS radius, is the dilaton potential. and are the masses of scalar and fermion bulk fields defined as and . Here and are the dimensions of scalar and fermion fields, which due to the QCD/gravity correspondence are related to the scaling dimensions (twists ) of the corresponding interpolating operators, where  [4]. These actions give information about the propagation of bulk fields inside AdS space (bulk-to-bulk propagators), from inside to the boundary of the AdS space (bulk-to-boundary propagators) and bound state solutions - profiles of the Kaluza-Klein (KK) modes in extra-dimension, which correspond to the hadronic wave functions in impact space. We suppose a free propagation of the bulk field along the Poincaré coordinates with four-momentum , and a constrained propagation along the -th coordinate (due to confinement imposed by the dilaton field). In particular, it was shown [4] that the extra-dimensional coordinate corresponds to the light-front impact variable. It was also shown [8] that in case of the scattering problem the sign of the dilaton profile is important to fulfill certain model-independent constraints. But we recently showed [12], that in case of the bound state problem the sign of the dilaton profile is irrelevant, if the action is properly set up. Moreover, in solving the bound-state problem, it is more convenient to move the dilaton field from the exponential prefactor to the effective potential [4, 12]. Then we use a KK expansion for the bulk fields factorizing the dependence on Poincaré coordinates and the holographic variable . E.g. in case of scalar field it is given by , where is the radial quantum number, is the tower of the KK modes dual to scalar mesons and are their extra-dimensional profiles (wave-functions) satisfying the Schrödinger-type equation with the potential depending on dilaton field. Then using the obtained wave functions we calculate matrix elements describing hadronic processes. Finally, we present the results of our calculations for mesonic decay constants (Table 1) and spectrum (Tables 2 and 3), nucleon helicity-independent generalized parton distributions (GPDs) in Fig.1. Note, by construction we reproduce the power scaling of nucleon electromagnetic (EM) form factors at large and our predictions for the EM radii are compare well with data:

Table 1. Decay constants (MeV) of pseudoscalar mesons.

Meson Data Our

Table 2. Masses of light mesons

Meson Mass [MeV]
0 0,1,2,3 0
0,1,2,3 0 0
0 0,1,2,3
0,1,2,3 1 1
0,1,2,3 1 1
0,1,2,3 1 1
0,1,2,3 0 1
0,1,2,3 0 1
0,1,2,3 1 1

Table 3. Masses of heavy-light mesons

Meson Mass [MeV]
0 0,1,2,3 0 1857 2435 2696 2905
0 0,1,2,3 1 2015 2547 2797 3000
0 0,1,2,3 0 1963 2621 2883 3085
0 0,1,2,3 1 2113 2725 2977 3173
0 0,1,2,3 0 5279 5791 5964 6089
0 0,1,2,3 1 5336 5843 6015 6139
0 0,1,2,3 0 5360 5941 6124 6250
0 0,1,2,3 1 5416 5992 6173 6298
Figure 1: Nucleon helicity-independent GPDs in momentum space

The authors thank Stan Brodsky and Guy de Téramond for useful discussions and remarks. This work was supported by Federal Targeted Program “Scientific and scientific-pedagogical personnel of innovative Russia” Contract No. 02.740.11.0238, by FONDECYT (Chile) under Grant No. 1100287. A.V. acknowledges the financial support from FONDECYT (Chile) Grant No. 3100028. V.E.L. would like to thank Departamento de Física y Centro Científico Tecnológico de Valparaíso (CCTVal), Universidad Técnica Federico Santa María, Valparaíso, Chile for warm hospitality.


  • [1] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998).
  • [2] J. Polchinski, M. J. Strassler, Phys. Rev. Lett. 88, 031601 (2002).
  • [3] A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006).
  • [4] S. J. Brodsky, G. F. de Teramond, Phys. Rev. Lett. 96, 201601 (2006); Phys. Rev. D 77, 056007 (2008); Nucl. Phys. Proc. Suppl. 199, 89 (2010).
  • [5] O. Andreev, Phys. Rev. D 73, 107901 (2006); H. Forkel, M. Beyer and T. Frederico, JHEP 0707, 077 (2007).
  • [6] G. F. de Teramond and S. J. Brodsky, AIP Conf. Proc. 1257, 59 (2010).
  • [7] Z. Abidin and C. E. Carlson, Phys. Rev. D 79, 115003 (2009).
  • [8] A. Karch, E. Katz, D. T. Son, M. A. Stephanov, JHEP 1104, 066 (2011).
  • [9] A. Vega and I. Schmidt, Phys. Rev. D 78, 017703 (2008); A. Vega and I. Schmidt, Phys. Rev. D 79, 055003 (2009); A. Vega, I. Schmidt, T. Branz, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 80, 055014 (2010); A. Vega, I. Schmidt, Phys. Rev. D82, 115023 (2010).
  • [10] T. Branz, T. Gutsche, V. E. Lyubovitskij, I. Schmidt, A. Vega, Phys. Rev. D 82, 074022 (2010).
  • [11] A. Vega, I. Schmidt, T. Gutsche, V. E. Lyubovitskij, Phys. Rev. D 83, 036001 (2011).
  • [12] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, arXiv:1108.0346 [hep-ph].
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