Deep Inelastic Scattering in Conformal QCD

TL;DR

This work formulates a conformal Regge framework for four-point functions with vector and scalar operators, deriving an AdS3 impact-parameter representation and a Regge pole structure for the amplitude valid at any coupling. It specializes to weak coupling, where the BFKL pomeron governs high-energy behavior, and decomposes the vector current impact factor into transverse spin-0 and spin-2 components, providing explicit formulas for QCD and N=4 SYM. A key result is that the R-current (and more generally half-BPS operators) exhibits vanishing transverse spin at leading order, leading to a spin-0–dominated Regge trajectory in these cases, with strong-coupling consistency via AdS gravity. The paper further connects Regge physics to DIS structure functions and unitarization, offering concrete prescriptions for extracting spin-content residues and facilitating extrapolations to strong coupling and AdS/CFT contexts.

Abstract

We consider the Regge limit of a CFT correlation function of two vector and two scalar operators, as appropriate to study small-x deep inelastic scattering in N=4 SYM or in QCD assuming approximate conformal symmetry. After clarifying the nature of the Regge limit for a CFT correlator, we use its conformal partial wave expansion to obtain an impact parameter representation encoding the exchange of a spin j Reggeon for any value of the coupling constant. The CFT impact parameter space is the three-dimensional hyperbolic space H3, which is the impact parameter space for high energy scattering in the dual AdS space. We determine the small-x structure functions associated to the exchange of a Reggeon. We discuss unitarization from the point of view of scattering in AdS and comment on the validity of the eikonal approximation. We then focus on the weak coupling limit of the theory where the amplitude is dominated by the exchange of the BFKL pomeron. Conformal invariance fixes the form of the vector impact factor and its decomposition in transverse spin 0 and spin 2 components. Our formalism reproduces exactly the general results predict by the Regge theory, both for a scalar target and for gamma*-gamma* scattering. We compute current impact factors for the specific examples of N=4 SYM and QCD, obtaining very simple results. In the case of the R-current of N=4 SYM, we show that the transverse spin 2 component vanishes. We conjecture that the impact factors of all chiral primary operators of N=4 SYM only have components with 0 transverse spin.

Figures (7)

• Figure 1: Conformal compactification of the $(y^+,y^-)$ Minkowski plane. The Regge kinematics corresponds to having all points close to null infinity.
• Figure 2: (a) Conformal compactification of the $(y^+,y^-)$ Minkowski plane and its relation to the Poincaré patches $\mathcal{P}_i$. In the Regge limit the insertion points $x_i$ approach the origin of the Poincaré patches $\mathcal{P}_i$. The vertical dashed lines are identified in the boundary of AdS. (b) The $y_i$ Minkowski plane is shown as the central Poincaré patch on the boundary of global AdS. The other patches are also represented.
• Figure 3: The conformal partial waves used in the decomposition of the BFKL propagator are obtained from the integral of the product of two 3-point functions. One 3-point function has scalars of dimension zero at $z_1$ and $z_3$ and a spin $n$ operator of dimension $1+i\nu$ at $z_7$, while the other has scalars of dimension zero at $z_2$ and $z_4$ and a spin $n$ operator of dimension $1-i\nu$ at $z_7$.
• Figure 4: The functions $\chi_{\mu}(u)$ and $\psi_{\mu}^{mn}(u)$, respectively used as a complete basis for the spin $n=0$ and $n=2$ components of the impact factor $V^{mn}(u)$. These functions are given by the integral over $z_5$ of the spin $n$ bulk to boundary propagator from $x$ to $z_5$ with dimension $1+i\mu$, multiplied by the 3-point function of scalars with zero dimension at $z_1$ and $z_3$ and a spin $n$ operator of dimension $1-i\mu$ at $z_5$.
• Figure 5: Full BFKL amplitude, written as a product of the conformal basis for the left and right impact factors and for the BFKL propagator. After integrating over all $z$'s except $z_7$ one obtains the integral representation of the $\Omega_{i\nu}$ function.