Various holographic approaches to QCD in five dimensions are explored using input both from the putative five-dimensional non-critical string theory as well as QCD. It is argued that a gravity theory in five dimensions coupled to a dilaton and an axion may capture the important qualitative features of pure YM theory. A part of the effects of higher α ′ -corrections is resummed into a dilaton potential. The potential is shown to be in one-to-one correspondence with the exact β-function of QCD, and its knowledge determines the full structure of the vacuum solution. The geometry near the UV boundary is that of AdS 5 with logarithmic corrections reflecting the asymptotic freedom of QCD. We find that all relevant confining backgrounds have an IR singularity of the "good" kind that allows unambiguous spectrum computations. Near the singularity the 't Hooft coupling is driven to infinity. Asymptotically linear glueball masses can also be achieved. The classification of all confining asymptotics, the associated glueball spectra and meson dynamics are addressed in a companion paper ArXiv:0707.1349

This paper is a continuation of ArXiv:0707.1324 where improved holographic theories for QCD were set up and explored. Here, the IR confining geometries are classified and analyzed. They all end in a "good" (repulsive) singularity in the IR. The glueball spectra are gapped and discrete, and they favorably compare to the lattice data. Quite generally, confinement and discrete spectra imply each other. Asymptotically linear glueball masses can also be achieved. Asymptotic mass ratios of various glueballs with different spin also turn out to be universal. Meson dynamics is implemented via space filling D 4 −D 4 brane pairs. The associated tachyon dynamics is analyzed and chiral symmetry breaking is shown. The dynamics of the RR axion is analyzed, and the non-perturbative running of the QCD θ-angle is obtained. It is shown to always vanish in the IR. Appendices 66 A. Characterization of confining backgrounds 66 A.1 Unbounded conformal coordinate 67 A.1.1 Logarithmic divergence 67 A.1.2 Power-law divergence 68 A.2 Finite range of the conformal coordinate 69 A.2.1 Finite A(r 0 ) 69 A.2.2 Power-law divergence 70 A.2.3 Logarithmic divergence 71 B. Magnetic charge screening: the finite range 72 B.1 A(r 0 ) finite 72 B.2 A(r 0 ) → −∞ 72 C. Fundamental string world-sheet embeddings in the presence of a non-trivial dilaton 73 D. Singularities of the tachyon 75 E. The superpotential versus the potential 76 F. Standard AdS/QCD Glueball spectrum 78 G. A simple analytic solution with AdS and confining asymptotics 79 References 80 -2 -6 The singularity is at a finite value u IR of the u coordinate. See appendix A. 7Since we are assuming that the singularity is at r → ∞, and Φ is monotonically increasing from Φ = −∞ at r = 0, A S cannot diverge to −∞ at some finite r. Therefore, if there is a minimum for A S , the string tension is certainly finite.