  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.
