Two-Loop Renormalization-Group Evolution for the Nucleon Distribution Amplitude

TL;DR

The paper resolves a long-standing gap by deriving the two-loop renormalization-group evolution kernel for the leading-twist nucleon distribution amplitude $\Phi_N$, enabling complete NLL corrections to nucleon form factors within hard-collinear factorization. It employs an effective-field-theory framework with evanescent operators, computes the necessary renormalization factors, and solves the RG equation analytically via a conformal partial-wave expansion up to $M\le 3$, while providing a two-loop EO-to-KM matching kernel $\mathbb{K}_N$ for scheme consistency. The authors show that the NLO kernel $\mathbb{H}^{(1)}$ comprises diagonal LC and multi-particle interactions (2P, 3P) with explicit color structures, and verify results through KM anomalous dimensions and scheme matching. Numerically, the two-loop evolution yields sizable corrections to conformal moments and nucleon form factors across a range of $Q^2$, underscoring the importance of including NLL evolution in predictive hard-exclusive QCD analyses and paving the way for extensions to the full baryon octet/decuplet.

Abstract

We determine for the first time the two-loop renormalization-group (RG) equation for the nucleon light-cone distribution amplitude, which constitutes the last missing ingredient for the complete next-to-leading-logarithmic corrections to the nucleon form factors in the hard-collinear factorization framework. Applying the conformal expansion for this fundamental nucleon distribution amplitude then enables us to construct an analytic solution that captures the desired scale dependence of phenomenologically interesting series coefficients. Importantly, the two-loop RG evolutions of these central hadronic quantities can bring about noticeable impacts on the corresponding leading-logarithmic results for three sample models of the nucleon distribution amplitude.

Figures (3)

• Figure 1: Sample two-loop Feynman diagrams for the QCD matrix element $\Pi_{1}$. The circled cross marks an insertion of the three finite-distance collinear Wilson lines in (\ref{['definiton: the nucleon DA']}).
• Figure 2: Theory predictions for the RG evolution of the normalized moments ${\cal R}_{M m}$ (with $M=0, \, 1, \, 2$) of the leading-twist nucleon distribution amplitude at the LL order (dotted curves) and at the NLL order (solid curves) in QCD, obtained from the sample COZ model of $\Phi_{N}(x_i, \, \mu_0)$.
• Figure 3: Theory predictions for the leading-power hard-gluon-exchange contributions to the Dirac nucleon form factors of both the proton (upper panel) and the neutron (lower panel) at the LL, NLO, ${\rm LL}^{\prime}$ and NLL accuracy by taking the parameter set ${\tt LAT25}$ for the nucleon distribution amplitude. The perturbative uncertainties from varying the renormalization and factorization scales in the preferred intervals, $\nu^2=\mu^2 = \langle x \rangle \, Q^2$ with $1/6 \leq \langle x \rangle \leq 1/2$Huang:2024ugd, are indicated by the colour bands.