2 At zero momentum transfer we can use the represen- tation (2) to write the hadronic tensor as perbolic space H$_{3}$ , whose boundary is the physical trans- verse space R $^{2}$. Using Poincar´ e coordinates, W ab ≈ 4 πis ∫ dr r 2 d ¯ r ¯ r 2 Ψ ab τ µ ( r ) Φ(¯ r ) × ∫ + ∞ | ln ¯ r/r | dL sinh L [ e iχ ( S,L ) − 1 ] µ τ , (4) ds $^{2}$( H$_{3}$ ) = dr 2 + ds $^{2}$( R $^{2}$) r 2 , where we did the angular integral in the impact parame- ter l$_{⊥}$ and traded | l$_{⊥}$ | in the radial integration for the AdS impact parameter L . Note that Φ(¯ r ) = | φ (¯ r ) | $^{2}$, where now φ (¯ r ) is the radial part of the normalizable AdS wave function dual to the state | k$_{2}$ 〉 . This wave function is lo- calized in the IR around ¯ r ∼ 1 /M . Its explicit form in the IR region, where space is no longer AdS, will not be important in what follows, because we shall consider a hard probe localized near the AdS boundary. The black disk model of the previous section gives we identify L with the geodesic distance between two points that are separated by l$_{⊥}$ along R 2 and have radial coordinates r and ¯ r . The variable S measures the lo- cal energy squared of the scattering process in AdS. The Greek indices µ and τ in (2) label tangent directions to H$_{3}$ , which are the physical polarizations of the AdS gauge field dual to the conserved current j $^{a}$. The functions Ψ $^{abµτ}$( r ) = ψ in ( r ) ψ $_{out}$( r ) , Φ(¯ r ) = φ$_{in}$ (¯ r ) φ$_{out}$ (¯ r ) , W ab ≈ − 2 πis ∫ dr r 2 d ¯ r ¯ r 2 Ψ ab µ µ ( r ) Φ(¯ r ) × [ ( sr ¯ r ) ω + ( sr ¯ r ) − ω − r ¯ r − ¯ r r ] . (5) W ab ≈ − 2 πis 1+ ω ∫ dr r 2 − ω Ψ ab µ µ ( r ) ∫ d ¯ r ¯ r 2 − ω Φ(¯ r ) . (6) k$_{1}$ = ( √$_{s,}$ − Q 2 √$_{s}$ , 0 ) , k$_{2}$ = ( M 2 √$_{s}$ $^{$_{,}$√$_{s,}$$_{0}$}$) , (7) [ 1 − e iχ ( S,L ) ] µ τ = Θ ( L$_{s}$ ( S ) − L ) δ µ τ , L$_{s}$ ( S ) ≈ ω log S . (3) Ψ ab µ µ ( r ) = − π 2 6 C r 2 ∫ ∞ 0 dudv e − u − v − Q $^{2}$r2 4 u − Q $^{2}$r2 4 v ×   sr 2 4 uv v − 1 u 0 u − 1 v 4( u − 1)( v − 1) sr 2 0 0 0 I   , (8) Ψ ij µ µ ( r ) = − δ $_{ij}$π 2 6 CQ $^{2}$r$^{4}$K2 $_{1}$( Qr ) , Ψ ++ µ µ ( r ) = − π 2 6 Csr $^{4}$K2 $_{0}$( Qr ) , W $^{ab}$( k$_{j}$ ) = i ∫ d $^{4}$y e ik 1 · $^{y}$〈 k$_{2}$ | T { j $^{a}$( y ) j $^{b}$(0) } | k$_{2}$ 〉 , III. DEEP INELASTIC SCATTERING in light-cone coordinates (+ , − , ⊥ ). Then, following [18], To give the explicit form of Ψ it is convenient to write At very low x the first term dominates and we have where the radius L$_{s}$ of the disk increases with energy as are given by the product of the radial part of the incoming and outgoing dual AdS fields. These functions are non- normalizable because they are produced by a plane wave source created by the dual operator at the boundary. We shall consider a black disk model defined by a phase shift in the impact parameter representation (2) given by Note that the size of the disk is independent of the dual AdS gauge field polarization, so that this simple model is characterized by the single parameter ω > 0. To mo- tivate this model, in section V we assume Reggeon ex- change and consider the limiting cases corresponding to the BFKL Pomeron at weak coupling and N = 4 SYM at strong coupling. We then use geometric scaling observed in DIS at low x to phenomenologically fix ω . where the matrix elements are also ordered by the light- cone coordinates. In particular, we have The total DIS cross section, and corresponding hadron structure functions, are related to the hadronic tensor where j a is the electromagnetic current and | k$_{2}$ 〉 is the target hadron state of momentum k$_{2}$ . We define the vir- tuality Q 2 = k 2 $_{1}$, target mass M 2 = − k 2 and Bjorken where i, j run over the transverse space R 2 directions and K is the Bessel function of the second kind. The constant C is determined by the conformal two point function 〈 j $^{a}$( y ) j $^{b}$(0) 〉 = C y $^{2}$η ab − 2 y $^{a}$yb ( y 2 + iϵ ) 4 . ∫ d ¯ r ¯ r 2 − ω Φ(¯ r ) = h ( ω ) M 1+ ω , x = − Q 2 2 k$_{1}$ · k$_{2}$ ≈ Q 2 s . W ab = ( η ab − k a $_{1}$k b 1 k 2 1 ) Π$_{1}$+ 2 x Q 2 ( k a 2 + k a 1 2 x ) ( k b 2 + k b 1 2 x ) Π$_{2}$ . By dimensional analysis the integral over ¯ r in (6) is Lorentz invariance and conservation restricts W ab to The structure functions F$_{i}$ satisfy 2 πF$_{i}$ = Im Π$_{i}$.