Jet Physics from Static Charges in AdS

TL;DR

This work reframes soft jet interactions via Wilson lines in radial coordinates, mapping Minkowski space to $\mathbb{R}\times\mathrm{AdS}$ so that the anomalous dimension becomes a static AdS energy dependent on cusp angles $\beta_{ij}$. A central tool is the conformal gauge, which eliminates $A_\tau$–$A_i$ mixing and simplifies multi-loop calculations, enabling a clean, largely pairwise structure of the cusp anomalous dimension at two loops. In the lightlike limit, the energy develops an imaginary, linearly growing piece tied to Sudakov physics, while an IR regulator reveals the no-branching Sudakov factor; the analysis also uncovers formal connections to Witten diagrams in AdS. Together, these results provide a geometric, calculationally efficient framework for jet-soft dynamics and point toward leveraging AdS/CFT techniques for higher-loop jet physics.

Abstract

Soft interactions with high-energy jets are explored in radial coordinates which exploit the approximately conformal behavior of perturbative gauge theories. In these coordinates, the jets, approximated by Wilson lines, become static charges in Euclidean AdS. The anomalous dimension of the corresponding Wilson line operator is then determined by the potential energy of the charges. To study these Wilson lines we introduce a "conformal gauge" which does not have kinetic mixing between radial and angular directions, and show that a number of properties of Wilson lines are reproduced through relatively simple calculations. For example, certain non-planar graphs involving multiple Wilson lines automatically vanish. We also discuss the linear growth of the charges' imaginary potential energy with separation, and a relationship between Wilson line diagrams and Witten diagrams.

Figures (6)

• Figure 2: A coordinate change maps Minkowski space to $\mathbb{R}\times\mathrm{AdS}$. In this figure the outgoing Wilson lines become static charges in AdS, and their tree level energy in AdS is equal to the original one-loop anomalous dimension for the lines.
• Figure 3: In radial quantization, final state lines map to a copy of $\mathrm{AdS}_3$ at positive Minkowski times, while initial state lines map to a second copy of $\mathrm{AdS}_3$ at negative Minkowski times. Points that are spacelike separated from the origin map to $\mathrm{dS}_3$.
• Figure 4: The naive solution to Laplace's equation on the Euclidean cylinder, Eq. (\ref{['eq:euclideanwrong']}), represents the potential in the presence of additional phantom charges at diametrically opposite points on the sphere, Figure \ref{['fig:ghostsphere']}. After analytic continuation back to Minkowski signature, the phantom charges map to another copy of AdS, Figure \ref{['fig:ghosthyp']}, corresponding to phantom initial state particles.
• Figure 5: On the left is the electric field lines for two charges in flat space. The middle shows the imaginary part of the electric field for two charges in AdS, after projecting to rectangular coordinates with $x=\beta \sin\theta$ and $y=\beta \cos\theta$. The right (from Gallicchio:2010sw), shows the distribution of radiation from a color singlet scalar decaying to two jets at the LHC. The axes in this case are psuedorapidity and azimuthal angle, and the contours correspond to factors of two in the accumulated energy distribution. The rightmost plot is included to remind the reader that a color dipole radiates between the color charges, which roughly corresponds to the region where the energy density has support in the AdS picture. The sharp drop-off of the radiation pattern in the effect of color coherence. In a qualitative sense only, this corresponds to the exponential decay of the radiation away from the dipole axis in the AdS picture.
• Figure 6: 2-loop graphs contributing to the coefficient $F(\gamma_{ij},\gamma_{jk},\gamma_{ki})$ of the antisymmetric color structure in $\Gamma_\mathrm{cusp}(v_i)$ (Eq. \ref{['eq:2loopcusp']}).