{\cal O}^{W_3}_{\mu\nu\sigma} &=& \bar{\psi} \gamma_\mu D_\nu D_\sigma \psi \end{aligned}$$ and $D_\mu$ is the usual covariant derivative. These are the same definitions as used in earlier computations, [@28; @29; @30]. The total derivative operators in the same respective levels satisfy these same template definitions. (We have suppressed the free flavour indices but note that these are flavour non-singlet operators.)

Next we recall the key points concerning the mixing of the operators in levels $W_2$ and $W_3$. First, as we are working with massless quarks there are no lower dimensional operators to be included and there is no mixing between levels. Next the particular choice of operators, (\[opdef\]), means that whilst there is mixing the mixing matrix of renormalization constants is upper triangular and given by, [@36], $$Z^{W_2}_{ij} ~=~ \left(
\begin{array}{cc}
Z^{W_2}_{11} & Z^{W_2}_{12} \\
0 & Z^{W_2}_{22} \\
\end{array}
\right)
\label{mat2}$$ and $$Z^{W_3}_{ij} ~=~ \left(
\begin{array}{ccc}
Z^{W_3}_{11} & Z^{W_3}_{12} & Z^{W_3}_{13} \\
0 & Z^{W_3}_{22} & Z^{W_3}_{23} \\
0 & 0 & Z^{W_3}_{33} \\
\end{array}
\right) ~.
\label{mat3}$$ Again we avoid a clumsy index on the matrix elements by using a numerical map to the respective sets $\{W_2,\partial W_2\}$ and $\{W_3,\partial W_3,\partial 
\partial W_3\}$ respectively. Here the superscript indicates the level. 