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We develop a method for integrating out the heavy Kaluza-Klein modes ofscalar type as well as those of vector and axial-vector types, in a class ofhard-wall bottom-up approaches of holographic QCD models, including theDirac-Born-Infeld and Chern-Simons parts. By keeping only the lowest-lyingvector mesons, we first obtain an effective chiral Lagrangian of the vectormesons based on the hidden local symmetry, and all the low-energy constants inthe HLS Lagrangian are expressed in terms of holographic integrals and,consequently, are fully determined by the holographic geometry and a fewconstants of mesons. We find that the Gell-Mann--Oakes--Renner relation ismanifestly reproduced at the lowest order of derivative expansion. We alsoexplicitly show that a naive inclusion of the Chern-Simons term cannotreproduce the desired chiral anomaly in QCD, and hence, some countertermsshould be provided: This implies that the holographic QCD models of hard-walltype cannot give definite predictions for the intrinsic parity-odd verticesinvolving vector and axial-vector mesons. After integrating out the vectormesons from the HLS Lagrangian, we further obtain the Lagrangian of chiralperturbation theory for pseudoscalar mesons with all the low-energy constantsfully determined.

Chiral effective theories from holographic QCD with scalars