Four loop anomalous dimensions of gradient operators in phi^4 theory

TL;DR

This work computes the four-loop anomalous dimensions $\gamma_n(g)$ of gradient twist-2 operators in $\phi^4$ theory within the $\overline{\text{MS}}$ scheme as a function of the operator moment $n$, serving as a QCD-like testbed for DGLAP-type analyses. It delivers the complete $n$-dependent $\gamma_n(g)$ up to ${\cal O}(g^4)$, including harmonic sums $S_1(n)$, $S_2(n)$ and the finite sum $K_2(n)$, and verifies consistency with conservation at $n=2$. The study also connects the four-loop anomalous dimensions to splitting functions $P(g,x)$ via the Mellin transform, providing explicit expressions for $P_2(x)$, $P_3(x)$, and $P_4(x)$ and evaluating how well moment-based fits can reproduce the full $x$-dependence using carefully chosen basis functions. Overall, the results demonstrate the utility of a scalar toy model for examining the relationship between operator dimensions, Mellin moments, and parton-like splitting functions, with implications for improving QCD perturbative inputs and understanding OPE structure in related theories.

Abstract

We compute the anomalous dimensions of a set of composite operators which involve derivatives at four loops in MSbar in phi^4 theory as a function of the operator moment n. These operators are similar to the twist-2 operators which arise in QCD in the operator product expansion in deep inelastic scattering. By regarding their inverse Mellin transform as being equivalent to the DGLAP splitting functions we explore to what extent taking a restricted set of operator moments can give a good approximation to the exact four loop result.