AdS black disk model for small-x DIS Lorenzo Cornalba, Miguel S. Costa 1 and Jo˜ ao Penedones 2 $^{1}$Centro de F´ısica do Porto e Departamento de F´ısica e Astronomia da Faculdade de Ciˆ encias da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal $^{2}$Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA Using the approximate conformal invariance of QCD at high energies we consider a simple AdS black disk model to describe saturation in DIS. Deep inside saturation the structure functions have the same power law scaling, F T ∼ F L ∼ x − $^{ω}$, where ω is related to the expansion rate of the black disk with energy. Furthermore, the ratio F L /F T is given by the universal value 1+ ω 3+ $_{ω}$, independently of the target. For γ ∗ − γ ∗ scattering at high energies we obtain explicit expressions and ratios for the total cross sections of transverse and longitudinal photons in terms of the single parameter ω . Deep inelastic scattering (DIS) has been of primary importance for understanding the partonic structure of hadrons. A classical example is the Callan-Gross relation The direct measurement of the longitudinal structure function F$_{L}$ is an important test to QCD. For low x such measurements were done at HERA, still outside the sat- uration region [20]. Other data from γ ∗ − γ ∗ scattering at LEP2 [21] are also outside this regime. Future ex- periments, such as LHeC [22] or ILC [23], will enter the saturated domain, and will be able to test the model here proposed. To define the scattering amplitude of an off-shell pho- ton by a scalar target in a conformal field theory, we consider the momentum space correlation function, involving a conserved current j a and a scalar primary operator O of dimension Δ. We shall be interested in the limit of large s = − ( k$_{1}$ + k$_{2}$ ) 2 with fixed momentum transfer t = − ( k$_{1}$ + k$_{3}$ ) 2 = − q 2 ⊥ and virtualities k 2 $_{i}$. As shown in [8, 18, 24], this limit is conveniently described by the impact parameter representation where the phase shift χ µ τ is a tensor that encodes all the dynamical information and depends on The scalar function Φ is associated with the operator O and the tensor function Ψ ab τ µ with the current operator j $^{a}$. Their explicit form was given in [18]. It is important to note that this representation is valid for any value of the coupling, since it relies only on conformal invariance. The above conformal representation is quite natural from the view point of the dual AdS scattering process, with transverse space given by the three-dimensional hy- where the constant ω , as explained below, is related to the expansion rate of the AdS black disk with energy, independently of the details of the target hadron, We propose that this is a good description deep inside satu- ration, for 1 GeV ∼ < Q ≪ Q$_{s}$ where Q$_{s}$ is the saturation scale. We also consider γ ∗ − γ ∗ scattering, obtaining uni- versal ratios for the polarized cross sections. A new approach to DIS using the AdS/CFT duality [3] was put forward in [4], and further explored in [5-19]. We shall consider the AdS/CFT approach to saturation in DIS [6, 8], and model the saturated hadron with an AdS black disk (a black disk in conformal QCD), extending the results of [8] by including the polarization of the hard probe [18]. We will show that deep inside saturation the AdS black disk model predicts the scaling law F$_{L}$ ∼ F$_{T}$ ∼ x − $^{ω}$, with the universal ratio In this Letter we shall focus on the kinematical limit of fixed Q and x ≪ 1, which corresponds to the Regge limit of large center of mass energy, s ≈ Q $^{2}$/x , with other kinematical invariants fixed. In this kinematical window one observes a power like growth in 1 /x of the hadron structure functions [2]. The hadron becomes a dense gluon medium so that the picture of the hadron made of weakly interacting partons is no longer valid. Although the coupling α$_{s}$ may still be small, say for hard probes with Q ∼ > 1 GeV, to understand low x structure functions one needs to include diagrams such as those of order [ α$_{s}$ ln(1 /x )] n due to the kinematical enhancement. Thus, for hard probes, low x DIS is the ideal ground to explore the approximate conformal symmetry of QCD in a situation where the gluon density inside the hadron is so high that nonlinear effects are nevertheless important. which holds in the limit of large photon virtuality Q at fixed Bjorken x , and shows that quarks have spin 1/2 [1]. F$_{2}$ = 2 xF$_{1}$ , (2 π ) $^{4}$δ ( ∑ k$_{j}$ ) i T $^{ab}$( k$_{j}$ ) = 〈 j $^{a}$( k$_{1}$ ) O ( k$_{2}$ ) j $^{b}$( k$_{3}$ ) O ( k$_{4}$ ) 〉 , T $^{ab}$( k$_{j}$ ) ≈ 2 is ∫ dl$_{⊥}$ e iq$_{⊥}$ · l$_{⊥}$ (2) × ∫ dr r 3 d ¯ r ¯ r 3 Ψ ab τ µ ( r ) Φ(¯ r ) [ 1 − e iχ ( S,L ) ] µ τ , S = r ¯ rs , cosh L = r 2 + ¯ r 2 + l 2 ⊥ 2 r ¯ r . F$_{L}$ F$_{T}$ ≈ F$_{2}$ − 2 xF$_{1}$ 2 xF$_{1}$ ≈ 1 + ω 3 + ω , (1) II. ADS BLACK DISK I. INTRODUCTION arXiv:1001.1157v2 [hep-ph] 8 Mar 2011