Baryon Distribution Amplitudes in QCD

TL;DR

This paper introduces an integrable framework for leading-twist three-quark baryon distribution amplitudes, revealing a hidden quantum number that governs scale dependence in the λ=3/2 sector and reducing the evolution to a simple three-term recurrence. The authors construct an exact solution for the lowest anomalous dimension and develop a systematic WKB expansion for large moments, while also analyzing the λ=1/2 sector and revealing a finite mass gap interpreted as scalar diquark formation. They cast the problem in terms of conformal SL(2,R) symmetry, connect it to the XXX spin chain with noncompact spin s=-1 via Baxter/Bethe approaches, and establish dual conformal bases that streamline the diagonalization and spectral analysis. The results yield analytic and semi-analytic characterizations of the spectra and eigenfunctions, with implications for end-point behavior and the parton interpretation of three-quark distributions, and point to extensions to other multi-parton amplitudes. Overall, the work provides a powerful, integrable toolkit to understand the scale evolution of baryon distribution amplitudes in QCD and highlights deep connections to integrable models in high-energy physics.

Abstract

We develop a new theoretical framework for the description of leading twist light-cone baryon distribution amplitudes which is based on integrability of the helicity $λ=3/2$ evolution equation to leading logarithmic accuracy. A physical interpretation is that one can identify a new `hidden' quantum number which distinguishes components in the $λ=3/2$ distribution amplitudes with different scale dependence. The solution of the corresponding evolution equation is reduced to a simple three-term recurrence relation. The exact analytic solution is found for the component with the lowest anomalous dimension for all moments $N$, and the WKB-type expansion is constructed for other levels, which becomes asymptotically exact at large $N$. Evolution equations for the $λ=1/2$ distribution amplitudes (e.g. for the nucleon) are studied as well. We find that the two lowest anomalous dimensions for the $λ=1/2$ operators (one for each parity) are separated from the rest of the spectrum by a finite `mass gap'. These special states can be interpreted as scalar diquarks.

Figures (13)

• Figure 1: 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 2: The spectrum of eigenvalues for the conserved charge $Q$ (a) and for the helicity-3/2 Hamiltonian $H_{3/2}$ (b), see text.
• Figure 3: The trajectories for $q$ as given by Eq. (\ref{['q-low']}). The crosses give the values of the function $f(q,N)$ calculated using the exact eigenvalues $q$.
• Figure 4: The dependence of the energy ${\cal E}$ on the charge $q$ for $N=30$. The solid curve is calculated using Eq. (\ref{['Energy']}) and the exact values of energy for quantized $q$ are shown by crosses.
• Figure 5: Two different analytic continuations of eigenvalues for the integrable Hamiltonian ${\cal H}_{3/2}$, see text.