Hadron structure functions at small $x$ from string theory

Deep inelastic scattering of leptons from hadrons at small values of the Bjorken parameter $x$ is studied from superstring theory. In particular, we focus on single-flavored scalar and vector mesons in the large $N$ limit. This is studied in terms of different holographic dual models with flavor Dp-branes in type IIA and type IIB superstring theories, in the strong coupling limit of the corresponding dual gauge theories. We derive the hadronic tensor and the structure functions for scalar and polarized vector mesons. In particular, for polarized vector mesons we obtain the eight structure functions at small values of the Bjorken parameter. The main result is that we obtain new relations of the Callan-Gross type for several structure functions. These relations have similarities for all different Dp-brane models that we consider. This would suggest their universal character, and therefore, it is possible that they hold for strongly coupled QCD in the large $N$ limit.


DIS basics
: The typical interaction in the DIS inclusive process between a lepton and a Hadron via a virtual photon. The state X is not measured.

Some Definitions
• q = k − k is the momentum transfer and y ≡ P·q P·k is the fractional energy loss of the lepton.
DIS is the study of the lepton-hadron scattering when q 2 → ∞, with x fixed.

D3D7 brane model/N = 2 SYM
The perturbation induced by the electro-magnetic currents J µ from the boundary is a metric fluctuation of the form where A m is a U(1) gauge field and v i is a killing vector on the sphere 2 .
These fluctuations will interact with scalar or vector brane fields (transversal or longitudinal brane fluctuations)

Structure Functions
Finally, we obtain the following results for the structure functions (at leading order) g 1 = −2g 2 = 1 4x 2 (I 1 + I 0 ) , where and since I 0,2∆+3 I 1,2∆+3 = 2∆+3 ∆+2 one recovers relations of the Callan-Gross type: • DIS scattering of leptons from spin-0 and spin-1 mesons at small x at strong coupling and in the large N limit has been investigated in terms of superstring theory (in the large N limit).
• For polarized vector mesons the 8 structure functions were obtained, along with the Callan-Gross type relations 3 • This results have similarities for all the different Dp-brane models. This could be a signal of a universal behaviour for confining gauge theories with a dual description in terms of probe Dp-branes.
• Future work: calculating the differential cross section for DIS and apply this techniques to other process, using the OPE of vertex operators and other techniques in order to describe the string theory scattering.
Thank you for listening! Any questions?
Some important steps • Using the optical theorem to relate DIS and forward Compton scattering, in order to relate our calculations and the hadronic tensor.
• Calculate the leading amplitude and finding an effective action S eff (first in flat space-time).
• Obtaining the field solutions on the curved background (AdS 5 × S 3 ) with boundary conditions.
• Inserting these solutions in the S eff and folding the amplitude in AdS (since the interaction can be considered local) 4 .
• Comparing the resulting W µν with the most general one and extract the structure functions F , b and g .
Gauge Theory and Brane models N = 2 SYM gauge theory: • It is derived from N = 4 SYM with 8 broken susy.
• It has matter in the fundamental rep. and mesons, like QCD. However recall that QCD is logarithmically running in the UV.
• At high energies it becomes conformal, so we can use the gauge-string duality. • In the interaction region the induced metric can be approximated by AdS 5 × S 3 .
• We will use N f = 1 and some other models 6 : D4D8D8 and D4D6D6.
Formally, both for scalar and vector mesons one should compute the two open vs two closed strings amplitude from a vertex operator worldsheet integral on the disk of the form 7 o (x, 2µ , k 2 ) : : This gives a sum of terms with an α -independent kinematic term and a pre-factor that carries the α dependence of the form For us, |t| << 1 << s . Only the one that has a 1 t pole in the pre-factor and a kinetic term that can be obtained from supergravity is important in this regime. We begin with the action where G ab = ∂aB b − ∂ b Ba and choose the static gauge H ab →ĝ ab ≡ g ab + 2g I (a ∂ b) X I + g IJ ∂aX I ∂ b X J , g ab = η ab + 2κh ab , g IJ = δ IJ + 2κh IJ , g aI = 2κh aI .
By expanding (1) we get the usual propagator for Ba and the interaction lagrangians This parameter region characterized by x exp − √ gN is more complicated because it goes deeper into string theory issues: • The locality approximation breaks down. This is because the (α s) α t that we had set to 1 cannot be neglected because we must take into account the transverse momentum transfer. Thus one has to include m α t/2 ∼ (α s) α t/2 ∼ x −α t/2 ∼ x −α ∇ 2 /2 .
• The differential operator acts on the solutions as a diffusion operator in the r direction, coming from the growth of the strings.
• The odd q 2 -dependence both in the structure functions obtained in the previous regimes (x 1/ √ gN and 1/ √ gN x exp − √ gN) is fixed, and so is the divergent moment issue.
In the last ten years a lot of progress has been made in this direction, specially in the spin zero (glueball) case. 10