Eikonal Methods in AdS/CFT: BFKL Pomeron at Weak Coupling

TL;DR

The paper develops a position-space formulation of high-energy Regge physics in N=4 SYM, unifying weak- and strong-coupling regimes via AdS/CFT. It shows that at weak coupling the planar four-point function is governed by the BFKL pomeron with a coupling- and ν-dependent trajectory, and it constructs a conformally invariant kernel and a basis of impact factors in position space. A key result is the explicit expression for the BFKL amplitude as a ν-integral with impact factors V(ν) and ȼV(ν), and the demonstration that only the n=0 transverse-spin component contributes for scalar external operators. The framework naturally leads to an AdS eikonal interpretation of multi-pomeron exchanges and raises questions about unitarity and saturation in AdS, hinting at rich connections between string dynamics in AdS and high-energy QCD phenomena.

Abstract

We consider correlators of N=4 super Yang Mills of the form A ~ < O_1 O_2 O*_1 O*_2 >, where the operators O_1 and O_2 are scalar primaries. In particular, we analyze this correlator in the planar limit and in a Lorentzian regime corresponding to high energy interactions in AdS. The planar amplitude is dominated by a Regge pole whose nature varies as a function of the 't Hooft coupling g^2. At large g, the pole corresponds to graviton exchange in AdS, whereas at weak g, the pole is that of the hard perturbative BFKL pomeron. We concentrate on the weak coupling regime and analyze pomeron exchange directly in position space. The analysis relies heavily on the conformal symmetry of the transverse space E^2 and of its holographic dual hyperbolic space H_3, describing with an unified language, both the weak and strong 't Hooft coupling regimes. In particular, the form of the impact factors is highly constrained in position space by conformal invariance. Finally, the analysis suggests a possible AdS eikonal resummation of multi-pomeron exchanges implementing AdS unitarity, which differs from the usual 4-dimensional eikonal exponentiation. Relations to violations of 4-dimensional unitarity at high energy and to the onset of nonlinear effects and gluon saturation become immediate questions for future research.

Figures (13)

• Figure 1: CFT points $\mathbf{x}_i$ on the boundary of global AdS. Shown is the relevant Lorentzian kinematics, with $\mathbf{x}_{4}$ in the future of $\mathbf{x}_{1}$, $\mathbf{x}_{3}$ in the future of $\mathbf{x}_{2}$, and with the pairs $\mathbf{x}_{1}$, $\mathbf{x}_{2}$ and $\mathbf{x}_{3}$, $\mathbf{x}_{4}$ spacelike related. This choice corresponds to a $2$ to $2$ interaction in the bulk of AdS.
• Figure 2: (a) The kinematics (\ref{['in100']}) and (\ref{['in100BIS']}) for the Lorentzian amplitude $\hat{\mathcal{A}}$. (b) For this kinematics we show the relevant analytic continuation in $z,\bar{z}$ for $\hat{\mathcal{A}}$, starting from the Euclidean amplitude $\mathcal{A}$ with $\bar{z}=z^{\star}$.
• Figure 3: Exchange of a BFKL pomeron. At leading order in the coupling constant, the kernel $F$ is given by the exchange of a pair of transverse gluons in a color singlet state.
• Figure 4: Hyperbolic $3$--space H$_3$ seen as the unit mass--shell in $\mathbb{M}^4$, given by points $\mathbf{w}\in\mathrm{M}$ with $\mathbf{w}^{2}=-1$. The boundary $\partial\mathrm{H}_3$ is then naturally identified with lines in the light--cone, given by points $\mathbf{z}\in\partial\mathrm{M}$ with $\mathbf{z}^{2}=0$, defined up to rescalings $\mathbf{z}\sim\alpha\mathbf{z}$.
• Figure 5: Integral representation of the $n=0$ component of the BFKL kernel.