The Uses of Conformal Symmetry in QCD

TL;DR

The paper surveys how conformal symmetry informs QCD calculations, particularly for light-cone dominated processes. It presents a systematic conformal framework (SL(2,R)) to classify operators, build conformal towers, and perform conformal partial wave expansions that diagonalize leading evolution kernels (ER-BL) and reveal integrable structures in multi-parton sectors. It then extends to higher-twist analyses, renormalon considerations, and Wandzura-Wilczek contributions, and discusses how conformal Ward identities and a conformal subtraction scheme enable calculations beyond leading order, including two-photon processes and the Regge limit via BFKL. The review highlights exact and approximate solvability (e.g., integrable spin chains for three-quark systems) and outlines how conformal symmetry constrains perturbative expansions, with broad implications for both phenomenology and formal QCD. Practical impact includes RG-improved predictions for distribution amplitudes and boosted understanding of off-forward processes through COPE and conformal schemes.

Abstract

The Lagrangian of Quantum Chromodynamics is invariant under conformal transformations. Although this symmetry is broken by quantum corrections, it has important consequences for strong interactions at short distances and provides one with powerful tools in practical calculations. In this review we give a short exposition of the relevant ideas, techniques and applications of conformal symmetry to various problems of interest.

Figures (10)

• Figure 1: The twist-four two-particle pion distribution amplitude $g_1(u)$ in the renormalon model (\ref{['ren-5']}) (solid line) compared with contributions of the first two orders in the conformal partial wave expansion (\ref{['ren-2']}) (dashed line) with the coefficients estimated using QCD sum rules BF90. The asymptotic distribution amplitude corresponding to the first term in Eq. (\ref{['ren-2']}) is shown by the dotted curve. The normalization is adjusted so that $\int_0^1 du\, g_1(u) =1$.
• Figure 2: Examples of a 'vertex' correction (a), 'exchange' diagram (b) and self-energy insertion (c) contributing to the renormalization of quark-antiquark operators in Feynman gauge. Path-ordered gauge factors are shown by the dashed lines. The set of all diagrams includes possible permutations.
• Figure 3: Examples of a 'vertex' correction (a), 'exchange' diagram (b) and self-energy insertion (c) contributing to the renormalization of three-quark operators in Feynman gauge. Path-ordered gauge factors are shown by the dashed lines. The set of all diagrams includes possible permutations.
• Figure 4: The spectrum of eigenvalues for the helicity-3/2 (left) and helicity-1/2 (right) Hamiltonians ${\cal H}_{3/2}(N)$ and ${\cal H}_{1/2}(N)$, respectively. The lines of the largest and the smallest ${E}_{3/2}$ are indicated on the plot for ${E}_{1/2}$ by dots for comparison. The solid curves in the left panel show three different trajectories for the anomalous dimensions (see text and Fig. \ref{['figure3']}).
• Figure 5: The spectrum of eigenvalues for the conserved charge ${\mathbb Q}$.