radial coordinate r in the Schr¨ odinger equation. The resulting mesonic spectrum has the phenomenologically successful Regge form M 2 = 4 κ $^{2}$( n + L + S/ 2) , with equal slopes in the orbital angular momentum and the radial quantum number n . The pion with n = L = S = 0 is massless for zero quark mass, consistent with chiral symmetry. We will show in this section how the LF holographic mapping of effective classical gravity in AdS space, modified by a positive-sign dilaton background, can be used to identify an analytically simple color-confining nonperturbative effective coupling α AdS s ( Q $^{2}$) as a function of the spacelike momentum transfer Q 2 = − q $^{2}$. As we shall show, this coupling incorpo- rates confinement and agrees well with effective charge observables and lattice simulations. It also exhibits an infrared fixed point at small Q 2 and asymptotic freedom at large Q $^{2}$. However, the falloff of α AdS s ( Q $^{2}$) at large Q 2 is exponential: α AdS s ( Q $^{2}$) ∼ e − Q $^{2}$/κ$^{2}$, rather than the perturbative QCD (pQCD) logarithmic falloff. We shall show in later sections that a phenomenological extended coupling can be defined which implements the pQCD behavior. As will be explained in Sec. V, the β function derived from light-front holography becomes significantly negative in the nonperturbative regime Q 2 ∼ κ $^{2}$, where it reaches a minimum, signaling the transition region from the IR conformal region, characterized by hadronic degrees of freedom, to a pQCD conformal UV regime where the relevant degrees of freedom are the quark and gluon constituents. The β function is always negative; it vanishes at large Q 2 consistent with asymptotic freedom, and it vanishes at small Q 2 consistent with an infrared fixed point. [28, 29] Let us consider a five-dimensional gauge field F propagating in AdS$_{5}$ space in the presence of a dilaton background ϕ ( z ) which introduces the energy scale κ in the five-dimensional action. At quadratic order in the field strength the action is where the metric determinant of AdS$_{5}$ is √ g = ( R/z ) $^{5}$, ϕ = κ $^{2}$z$^{2}$, and the square of the coupling g$_{5}$ has dimensions of length. On general grounds we would expect that the value of the five-dimensional coupling g 2 5 in units R = 1 is determined by a geometrical factor scaled S = − 1 4 ∫ d $^{5}$x √ g e ϕ ( z ) 1 g 2 5 F $^{2}$, (5) 8 III. NONPERTURBATIVE QCD COUPLING FROM LIGHT-FRONT HOLOG- RAPHY