Four loop anomalous dimensions of gradient operators in ${\phi^4}$ theory

Abstract. We compute the anomalous dimensions of a set of composite operators which involve derivatives at four loops in $\bar{\rm MS}$ 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.