Four loop renormalization of 3-quark operators in QCD

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.

💡 Research Summary

This paper presents a comprehensive four‑loop renormalization of generic three‑quark operators in Quantum Chromodynamics (QCD) within the modified minimal subtraction ( MS ¯ ) scheme, using the method originally devised by Kränkl and Manashov. The authors focus on operators of the form
( O_{ijk}^{\alpha\beta\gamma}= \varepsilon_{IJK},\psi_{iI}^{\alpha}\psi_{jJ}^{\beta}\psi_{kK}^{\gamma} )
which are colour‑singlet, gauge‑invariant composites relevant for describing baryon states such as the proton, neutron, Σ and Λ hyperons. By evaluating the Green’s function with one external momentum flowing through two of the quark legs and the third leg at zero momentum, they extract the renormalization constant (Z_O) and consequently the anomalous dimensions of the operators.

The calculation is performed with the state‑of‑the‑art Forcer algorithm, implemented in the symbolic manipulation language FORM. Forcer handles the massive number of four‑loop Feynman diagrams (several thousand) and the associated tensor integrals of rank up to eight. The authors reduce the tensor structures by projecting onto transverse and longitudinal projectors, (P_{\mu\nu}(p)=\eta_{\mu\nu}-p_\mu p_\nu/p^2) and (L_{\mu\nu}(p)=p_\mu p_\nu/p^2), which renders the large 764 × 764 matrix of tensor bases block‑diagonal and sparse, dramatically improving computational efficiency.

A crucial technical ingredient is the treatment of the Dirac algebra in (d=4-2\epsilon) dimensions. Because dimensional regularisation lifts the four‑dimensional closure of the γ‑matrix algebra, the authors employ an infinite set of totally antisymmetric generalized γ‑matrices, denoted (\Gamma^{(n)}{\mu_1\ldots\mu_n}= \gamma{