4 distribution function, to the exponent ω . For small x the cross section depends on a scaling variable τ defined by exchange of a Reggeon. The value of the exponent ω may be fixed experimentally using geometric scaling [32]. The use of a single Reggeon to describe general features of the crossover from a dilute system of partons within the hadron to a very dense one is well known [2, 33]. In the context of AdS/CFT, similar arguments for external scalar states were given in [8], so we shall be brief. To estimate the AdS black disk size, consider the linear term in the expansion of e iχ in (4), and approximate χ with the contribution of a single Reggeon exchange [18] where Q$_{s}$ = Q$_{s}$ ( x ) is the saturation scale. The exponent λ is determined experimentally to be λ = 0 . 321 ± 0 . 056 [37]. Geometric scaling occurs over a large kinematical window that includes the region Q ∼ Q$_{s}$ . In particular, in this near saturation region, and provided Q ∼ > 1 GeV so that α$_{s}$ is small and runs slowly, one can relate the exponents ω and λ by associating the growth of the cross section to a single Reggeon exchange. At zero momentum transfer the amplitude (2) becomes τ = ( Q Q$_{s}$ ) 2 , Q 2 s = M 2 ( 1 x ) λ , (12) T ab ≈ 2 is ∫ dr r 3 d ¯ r ¯ r 3 Ψ ab τ µ ( r ) Φ(¯ r ) t µ $_{τ}$( s, r, ¯ r ) χ µ τ ≈ 1 N 2 ∫ dν S j ( ν ) − 1 (11) × [ β ( ν ) δ µ τ + ˜ β ( ν ) ($_{∇}$ $^{µ}$∇ τ − 1 3 δ µ τ )] Ω$_{iν}$ ( L ) , t µ $_{τ}$( s, r, ¯ r ) = ∫ dl$_{⊥}$ iχ µ $_{τ}$( S, L ) . T ij ≈ 2 is ∫ dr r 3 d ¯ r ¯ r 3 Ψ ij µ µ ( r ) Φ(¯ r ) t$_{T}$ ( s, r, ¯ r ) (13) t$_{T}$ = i s ∫ dν 3 β ( ν ) + (3 iν − 4) ˜ β ( ν ) 6 πN 2 ( sr ¯ r ) j ( ν ) ( ¯ r r ) − iν . with Using the Regge representation (11), we can do the in- tegration in l$_{⊥}$ of the harmonic function Ω$_{iν}$ ( L ) and its derivatives [18]. In particular, the components T ij are where, in the linear approximation, where ∇ µ is the Levi-Civita connection on H$_{3}$ and Ω$_{iν}$( L ) = ν sin νL/ (4 π $^{2}$sinh L ) is a basis of harmonic functions on H$_{3}$ . The Regge spin j = j ( ν ) and the residues β ( ν ) and ˜ β ( ν ) depend on the ’t Hooft coupling ¯ α$_{s}$ = α$_{s}$N/π and are even functions of ν . This is exact in a conformal gauge theory such as N = 4 SYM theory and approximate in QCD at weak coupling. The explicit form of the Regge residues β ( ν ) and ˜ β ( ν ) was computed to leading order in perturbation theory in [18]. These residues are purely imaginary and satisfy Im χ µ τ > 0. For fixed S , the phase shift χ ( S, L ) vanishes at large impact parameter L → ∞ . As the impact pa- rameter decreases the phase will grow and reach order unity at L$_{s}$ ( S ). For impact parameters L < L$_{s}$ the am- plitude will then be that of an absorptive black disk and the details in the form of the phase shift for L < L$_{s}$ are not important to the computation of the hadronic tensor in (4). To estimate the size L$_{s}$ of the AdS black disk one can do a saddle point approximation to the ν integral in (11), exactly as in [8]. At weak coupling the Reggeon is the perturbative BFKL hard pomeron [34] and one ob- tains ω ≈ 2 . 44¯ α$_{s}$ . This value of ω is actually to large to match experiments, and next to leading order correction to the BFKL spin are important. A similar estimation of the saturation region can be done using the dipole for- malism [33], however, we remark that here L$_{s}$ is the black disk size in the AdS impact parameter space H$_{3}$ . Saturation effects at strong coupling in DIS have been analysed using the AdS/CFT duality in [6]. We can also show that at strong coupling and for a conformal gauge theory, the size of the AdS black disk still satisfies (3). In this case for large impact parameter the phase shift is predominantly real and is given by the AdS gravi- Reggeon trajectory. In particular, the residue β ( ν ) has poles at ν = ± 2 i , as required for graviton exchange [35], while ˜ β ( ν ) vanishes [36]. From conservation of energy- momentum j ( ± 2 i, ¯ α$_{s}$ ) = 2 for any value of the coupling, so the ν integral in (11) can be done by contour defor- mation. One obtains χ ∼ e 3( L$_{s}$ − L $^{)}$, with L$_{s}$ given by (3) with exponent ω = 1 / 3. Thus, for L < L$_{s}$ the phase shift becomes a rapid varying phase for the L integration in the hadronic tensor (4) and can therefore be dropped. We finish by relating the phenomenon of geometric scaling [32], associated to the growth of the hadron gluon This work was partially supported by grants CERN/FP/109306/2009, SFRH/BPD/34052/2006, NSF PHY05-51164 and by the FCT POCI programme. Acknowledgments where now the scaling variable τ = τ ( r, ¯ r ). The same scaling holds for the component T ++ of the amplitude. Integrating over r and ¯ r , we conclude that the structure functions F$_{T}$ and F$_{L}$ will depend on τ given in (12) with exponent λ = 2 ω/ (1 − ω ). From the experimental value of λ we may therefore estimate ω = 0 . 138 ± 0 . 021. Using this value, we predict F$_{L}$/F$_{T}$ ≈ 0 . 36 deep inside saturation. t$_{T}$ ∼ 1 s τ − (1+ iν$_{s}$ ) 1 − ω 2 , τ = ¯ r 2 r 2 ( sr $^{2}$) − 2 ω 1 − ω , Since the minimum value of the AdS impact parameter L is given by ln(¯ r/r ), the region near saturation corre- sponds to setting ln(¯ r/r ) ∼ L$_{s}$ ( sr ¯ r ), which will be the dominant region in the radial integration in (13) for a suitable choice of the external kinematics s and Q . In this case the ν integral for the amplitude t$_{T}$ is dominated by the saturation saddle point ν$_{s}$ , and it scales as