Four loop renormalization of 3-quark operators in QCD

TL;DR

This work performs a four-loop renormalization of a general SU(3) 3-quark operator in the $\overline{\mathrm{MS}}$ scheme using the Forcer framework and a gamma-matrix basis to obtain the anomalous dimensions for four core baryon operators related to nucleon structure. It establishes a detailed mapping of Dirac structures through evanescent operators, yielding explicit perturbative expressions for the operator anomalous dimensions as functions of the coupling and number of flavors, and then analyzes these dimensions at the Banks–Zaks fixed point within the conformal window. By expanding around $\Delta_{\mathrm{BZ}}=11-\tfrac{2}{3}N_f$ and employing high-order terms plus Padé approximants, the study assesses the convergence and reliability of perturbation theory for the baryon operators, finding reasonable convergence down to $N_f\approx 12$ for some cases. The results provide benchmark values for operator exponents and offer potential cross-checks with lattice QCD and guidance for extending the baryon-operator program in conformal and Beyond-Standard-Model contexts.

Abstract

We renormalize generalized 3-quark operators that relate to baryon states using the method devised by Kränkl and Manashov at four loops in the MSbar scheme. The anomalous dimensions of the four core operators used to compute nucleon matrix elements are determined and their associated critical exponents are studied in the conformal window using the Banks-Zaks expansion.

Figures (4)

• Figure 1: Graphical structure of the Green's function $G_{(\alpha^\prime \beta^\prime \gamma^\prime | \alpha \beta \gamma)}^{(i^\prime j^\prime k^\prime | ijk)}(p)$ defined in (\ref{['gfdef']}).
• Figure 2: Four loop graph contribution to $G_{(\alpha^\prime \beta^\prime \gamma^\prime | \alpha \beta \gamma)}^{(i^\prime j^\prime k^\prime | ijk)}(p)$ containing the $\gamma$ matrix structure $\Gamma_{484}$.
• Figure 3: Plots of the leading order (LO), next-to-leading order (NLO), next-to-next-to-leading order (NNLO), next-to-next-to-next-to-leading order (NNNLO), anomalous dimension of the four baryon operators at the Banks-Zaks fixed point for $8$$\leq$$N_{\!f}$$\leq$$16$.
• Figure 4: Plots of the Padé estimates for the anomalous dimension of the two spin-$\hbox{\small{$\frac{1}{2}$}}$ baryon operators at the Banks-Zaks fixed point for $8$$\leq$$N_{\!f}$$\leq$$16$.