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

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 MSbar 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.