We develop a method for integrating out the heavy Kaluza-Klein modes of scalar type as well as those of vector and axial-vector types, in a class of hard-wall bottom-up approaches of holographic QCD models, including the Dirac-Born-Infeld and Chern-Simons parts. By keeping only the lowest-lying vector mesons, we first obtain an effective chiral Lagrangian of the vector mesons based on the hidden local symmetry, and all the low-energy constants in the HLS Lagrangian are expressed in terms of holographic integrals and, consequently, are fully determined by the holographic geometry and a few constants of mesons. We find that the Gell-Mann–Oakes–Renner relation is manifestly reproduced at the lowest order of derivative expansion. We also explicitly show that a naive inclusion of the Chern-Simons term cannot reproduce the desired chiral anomaly in QCD, and hence, some counterterms should be provided: This implies that the holographic QCD models of hard-wall type cannot give definite predictions for the intrinsic parity-odd vertices involving vector and axial-vector mesons. After integrating out the vector mesons from the HLS Lagrangian, we further obtain the Lagrangian of chiral perturbation theory for pseudoscalar mesons with all the low-energy constants fully determined.

• Received 22 April 2014

DOI:https://doi.org/10.1103/PhysRevD.89.115012