Computation of quark mass anomalous dimension at O(1/N_f^2) in quantum chromodynamics

TL;DR

This work develops and applies a large $N_f$ expansion framework to compute $d$-dimensional critical exponents in QCD, exploiting an equivalence with a non-abelian Thirring model to access the quark wave function and mass renormalization exponents at $O(1/N_f^2)$. By introducing auxiliary couplings to restore multiplicative renormalizability and employing both renormalization-group and Schwinger–Dyson formalisms, the authors obtain explicit expressions for the exponents in $d$-dimensions, and verify agreement with known four-loop MSbar results while providing new six-loop coefficients. The study also addresses gauge issues, demonstrates gauge-invariant consistency up to $O(1/N_f^2)$, and presents cross-checks between RG and SD approaches, including in three dimensions. Overall, it offers a robust nonperturbative method to access high-order QCD RG functions and reinforces the QCD–NATM critical equivalence, with potential applications to twist‑2 operators and lattice tests.

Abstract

We present the formalism to calculate d-dimensional critical exponents in QCD in the large N_f expansion where N_f is the number of quark flavours. It relies in part on demonstrating that at the d-dimensional fixed point of QCD the critical theory is equivalent to a non-abelian version of the Thirring model. We describe the techniques used to compute critical two and three loop Feynman diagrams and as an application determine the quark wave function, eta, and mass renormalization critical exponents at O(1/N_f^2) in d-dimensions. Their values when expressed in relation to four dimensional perturbation theory are in exact agreement with the known four loop MSbar results. Moreover, new coefficients in these renormalization group functions are determined to six loops and O(1/N_f^2). The computation of the exponents in the Schwinger Dyson approach is also provided and an expression for eta in arbitrary covariant gauge is given.

Figures (12)

• Figure 1: The effective gluon propagator for $u,v\neq 1$.
• Figure 2: Diagrams contributing to the computation of $\eta_2$. The first graph represents the gluon self energy diagrams of figure \ref{['fig2']}.
• Figure 3: New external momenta routing for the diagram Fig. \ref{['fig2']}, (d).
• Figure 4: The diagrams contributing to the gluon self-energy at $O(1/N_{\!f}^2)$.
• Figure 5: Three loop gluon self energy diagram with $\epsilon$ and $\Delta$ regularizations.