content_image/423477.jpg content_image/423485.jpg content_image/423501.jpg content_image/423473.jpg content_image/423481.jpg content_image/423482.jpg content_image/423500.jpg content_image/423479.jpg content_image/423480.jpg content_image/423483.jpg content_image/423472.jpg content_image/423478.jpg content_image/423484.jpg content_image/423498.jpg content_image/423499.jpg
perbolic space H$_{3}$ , whose boundary is the physical trans- verse space R $^{2}$. Using Poincar´ e coordinates, 

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 

<img src='content_image/423477.jpg'>

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 

where the radius L$_{s}$ of the disk increases with energy as 

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 . 

## III. DEEP INELASTIC SCATTERING 

The total DIS cross section, and corresponding hadron structure functions, are related to the hadronic tensor 

<img src='content_image/423485.jpg'>

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 

Lorentz invariance and conservation restricts W ab to 

<img src='content_image/423501.jpg'>

The structure functions F$_{i}$ satisfy 2 πF$_{i}$ = Im Π$_{i}$. 

<img src='content_image/423473.jpg'>

<img src='content_image/423481.jpg'>

<img src='content_image/423482.jpg'>

<img src='content_image/423500.jpg'>

At zero momentum transfer we can use the represen- tation (2) to write the hadronic tensor as 

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 

At very low x the first term dominates and we have 

<img src='content_image/423479.jpg'>

To give the explicit form of Ψ it is convenient to write 

<img src='content_image/423480.jpg'>

in light-cone coordinates (+ , − , ⊥ ). Then, following [18], ∞ 

<img src='content_image/423483.jpg'>

where the matrix elements are also ordered by the light- cone coordinates. In particular, we have 

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 

By dimensional analysis the integral over ¯ r in (6) is 

<img src='content_image/423472.jpg'>

<img src='content_image/423478.jpg'>

<img src='content_image/423484.jpg'>

<img src='content_image/423498.jpg'>

<img src='content_image/423499.jpg'>

