Four loop Green’s functions involving the $$n=2$$ moment of the Wilson operator

Abstract We evaluate the Green’s function for the insertion of the second moment of the twist-2 flavour nonsinglet Wilson operator in a quark 2-point function in all three different single scale external momentum configurations at four loops in the $$\overline{\text{ MS }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mspace/> <mml:mtext>MS</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> scheme and the chiral limit. One configuration is where the operator is inserted at zero momentum while the other two are where a non-zero momentum flows out through the operator itself with one external quark momentum nullified. In the latter two configurations mixing of the operator with a total derivative twist-2 operator is included for renormalization group consistency. In addition we compute the correlation functions of both gauge invariant operators to four loops in the same scheme.