\left( d_3^{abc} \varsigma^b \varsigma^c \right)^2 \right\}
\end{equation}
are several formal quartic vertex structures. Such a Lagrangian with distinct
couplings would be non-trivial and would therefore require a large computation
to determine its renormalization group functions even at low loop order in
order to explore this double emergence conjecture further.

\sect{General action.}

While we considered a generalization of the Wess-Zumino model to include
interactions with group valued tensor couplings which were real in
(\ref{lagwzt}) that was not the most general cubic supersymmetric chiral
theory. Instead the most general action involves tensors that themselves
undergo renormalization which we will determine to five loops in this section
extending thereby the four loop work of [91]. In other words the bare
action has the form
\begin{equation}
S ~=~ \int d^4 x \! \left[ \int d^2 \theta d^2 \bar{\theta} \,
\bar{\Phi}_0^i (x,\bar{\theta}) e^{-2 \theta {\partialline} \bar{\theta}}
\Phi_0^i (x,\theta) \,+\,
\frac{d_0^{ijk}}{3!} \int d^2 \theta \, \Phi_0^i \Phi_0^j \Phi_0^k \,+\,
\frac{\bar{d}_0^{ijk}}{3!} \int d^2 \bar{\theta} \,
\bar{\Phi}_0^i \bar{\Phi}_0^j \bar{\Phi}_0^k
\right]
\label{lagwzgend}
\end{equation}
where the tensor couplings are bare in contrast to (\ref{lagwzt}). The
corresponding renormalized quantities are defined by
\begin{equation}
\Phi_0^i ~=~ Z^{ij} \Phi^j ~~~,~~~
\bar{\Phi}_0^i ~=~ Z_\Phi^{ij} \bar{\Phi}^j
\end{equation}
for the superfields and
\begin{equation}
d_0^{ijk} ~=~ Z_d^{ijk|pqr} d^{pqr} ~~~,~~~
\bar{d}_0^{ijk} ~=~ Z_{\bar d}^{ijk|pqr} \bar{d}^{pqr}
\end{equation}
for the tensor couplings. However, the tensor renormalization constants are not
independent due to the supersymmetry Ward identity which implies that
$Z_d^{ijk|pqr}$ and its conjugate are constrained to satisfy
\begin{equation}
Z_\Phi^{il} Z_\Phi^{jm} Z_\Phi^{kn} Z_d^{lmn|pqr} d^{pqr} ~=~ d^{ijk} ~.
\label{susywigen}
\end{equation}
We have determined the conditions these place on the vertex counterterms to
five loops and implemented them within our automatic {\sc Form} programme to
renormalize (\ref{lagwzgend}). Once $Z_\Phi^{ij}$ has been calculated to this
order in either the $\MSbar$ or MOM schemes then the renormalization group
functions are deduced from
\begin{equation}
\gamma_\Phi^{ik} Z_\Phi^{kj} ~=~
\beta^{pqr} \frac{\partial ~}{\partial d^{pqr}} Z_\Phi^{ij} ~+~
\bar{\beta}^{pqr} \frac{\partial ~}{\partial \bar{d}^{pqr}} Z_\Phi^{ij}
\end{equation}
where the $\beta$-functions are defined by
\begin{equation}
\beta^{ijk} ~=~ \mu \frac{d~}{d\mu} d^{pqr} ~~~,~~~
\bar{\beta}^{ijk} ~=~ \mu \frac{d~}{d\mu} \bar{d}^{pqr} ~.
\end{equation}
The explicit form of the tensor $\beta$-function is found via the supersymmetry
Ward identity (\ref{susywigen}) which implies, [91],
\begin{equation}
\beta^{ijk} ~=~ -~ \epsilon d^{ijk} ~+~ d^{ijp} \gamma_\Phi^{kp} ~+~
d^{ipk} \gamma_\Phi^{jp} ~+~ d^{pjk} \gamma_\Phi^{ip} ~.
\end{equation}
We have followed this prescription and as a check have reproduced the four loop
$\MSbar$ result of [91] for $\gamma_\Phi^{ij}$. That result was expressed
as a sum of tensors which have a close correspondence with the individual four
loop graphs of the superfield $2$-point function. In other words it contained
$19$ tensors which were presented in a relatively compact way. At five loops
there are $63$ five loop graphs as indicated in Table \ref{feynnum} and we take
a similar approach here. First if we formally define the field anomalous
dimension tensor by
\begin{equation}
\gamma^{ij}_\Phi ~=~ \sum_{L=1}^5 \sum_{r=1}^{k_L} c^{\cal S}_{Lr} T^{ij}_{Lr}
\end{equation}
where ${\cal S}$ denotes the renormalization scheme, $c^{\cal S}_{Lr}$ are the
numerical coefficients of the tensors $T^{ij}_{Lr}$, $L$ labels the loop order
and $r$ identifies the specific tensor. The explicit expression for each tensor
is provided in Appendix A which also records the connection to the underlying
five loop graphs of the $2$-point function.

Having set this notation we have determined the values for each of the
coefficients. For the $\MSbar$ scheme to four loops we have
\begin{eqnarray}
c^{\MSbars}_{11} &=& \frac{1}{2} ~~,~~
c^{\MSbars}_{21} ~=~ - \frac{1}{2} ~~,~~
c^{\MSbars}_{31} ~=~ \frac{3}{2} \zeta_3 ~~,~~
c^{\MSbars}_{32} ~=~ - \frac{1}{8} ~~,~~
c^{\MSbars}_{33} ~=~ - \frac{1}{4} ~~,~~
c^{\MSbars}_{34} ~=~ 1 \nonumber \\
c^{\MSbars}_{41} &=& - 10 \zeta_5 ~~,~~
c^{\MSbars}_{42} ~=~ \frac{3}{4} \zeta_4 - \frac{3}{2} \zeta_3 ~~,~~
c^{\MSbars}_{43} ~=~ \frac{3}{4} \zeta_4 - \frac{3}{2} \zeta_3 ~~,~~
c^{\MSbars}_{44} ~=~ \frac{3}{2} \zeta_4 - 3 \zeta_3 \nonumber \\
c^{\MSbars}_{45} &=& - \frac{1}{8} + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{46} ~=~ \frac{1}{3} ~~,~~
c^{\MSbars}_{47} ~=~ - \frac{3}{4} \zeta_4 - \frac{3}{2} \zeta_3 ~~,~~
c^{\MSbars}_{48} ~=~ \frac{5}{24} ~~,~~
c^{\MSbars}_{49} ~=~ \frac{1}{3} \nonumber \\
c^{\MSbars}_{410} &=& \frac{1}{3} ~~,~~
c^{\MSbars}_{411} ~=~ - \frac{1}{8} + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{412} ~=~ \frac{5}{12} - \frac{1}{2} \zeta_3 ~~,~~
c^{\MSbars}_{413} ~=~ - \frac{5}{2}
\end{eqnarray}
which are in agreement with [47, 91]. At five loops we find
\begin{eqnarray}
c^{\MSbars}_{51} &=& \frac{9}{2} \zeta_3^2 ~~,~~
c^{\MSbars}_{52} ~=~ - \frac{143}{16} \zeta_5 - \frac{9}{32} \zeta_4 + \frac{1}{16} \zeta_3 ~~,~~
c^{\MSbars}_{53} ~=~ - \frac{143}{16} \zeta_5 - \frac{9}{32} \zeta_4 + \frac{1}{16} \zeta_3 \nonumber \\
c^{\MSbars}_{54} &=& - \frac{1}{32} + \frac{3}{32} \zeta_4 - \frac{1}{16} \zeta_3 ~~,~~
c^{\MSbars}_{55} ~=~ \frac{67}{4} \zeta_5 - \frac{9}{8} \zeta_4 + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{56} ~=~ \frac{3}{16} + \frac{3}{32} \zeta_4 - \frac{5}{16} \zeta_3
\nonumber \\
c^{\MSbars}_{57} &=& 18 \zeta_5 - \frac{9}{8} \zeta_4 + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{58} ~=~ 18 \zeta_5 - \frac{9}{8} \zeta_4 + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{59} ~=~ - \frac{25}{4} \zeta_6 + \frac{25}{2} \zeta_5 + \frac{1}{2} \zeta_3^2 \nonumber \\
c^{\MSbars}_{510} &=& - \frac{25}{4} \zeta_6 + \frac{25}{2} \zeta_5 + \frac{1}{2} \zeta_3^2 ~~,~~
c^{\MSbars}_{511} ~=~ \frac{441}{8} \zeta_7 ~~,~~
c^{\MSbars}_{512} ~=~ - \frac{25}{4} \zeta_6 + \frac{25}{2} \zeta_5 + \frac{1}{2} \zeta_3^2 \nonumber \\
c^{\MSbars}_{513} &=& - \frac{79}{4} \zeta_5 - \frac{9}{16} \zeta_4 + \frac{1}{8} \zeta_3 ~~,~~
c^{\MSbars}_{514} ~=~ \frac{441}{16} \zeta_7 ~~,~~
c^{\MSbars}_{515} ~=~ \frac{1}{6} \nonumber \\
c^{\MSbars}_{516} &=& - \frac{25}{4} \zeta_6 + \frac{25}{2} \zeta_5 - \frac{5}{2} \zeta_3^2 ~~,~~
c^{\MSbars}_{517} ~=~ - \frac{25}{4} \zeta_6 + \frac{25}{2} \zeta_5 - \frac{5}{2} \zeta_3^2 \nonumber \\
c^{\MSbars}_{518} &=& - \frac{25}{8} \zeta_6 + \frac{25}{4} \zeta_5 - \frac{11}{4} \zeta_3^2 ~~,~~
c^{\MSbars}_{519} ~=~ 9 \zeta_3^2 ~~,~~
c^{\MSbars}_{520} ~=~ 9 \zeta_3^2 \nonumber \\
c^{\MSbars}_{521} &=& - \frac{153}{8} \zeta_5 - \frac{9}{16} \zeta_4 + \frac{1}{8} \zeta_3 ~~,~~
c^{\MSbars}_{522} ~=~ - \frac{153}{8} \zeta_5 - \frac{9}{16} \zeta_4 + \frac{1}{8} \zeta_3 \nonumber \\
c^{\MSbars}_{523} &=& - \frac{143}{8} \zeta_5 - \frac{21}{16} \zeta_4 + \frac{31}{8} \zeta_3 ~~,~~
c^{\MSbars}_{524} ~=~ - \frac{143}{8} \zeta_5 - \frac{21}{16} \zeta_4 + \frac{31}{8} \zeta_3 \nonumber \\
c^{\MSbars}_{525} &=& 18 \zeta_5 - \frac{9}{8} \zeta_4 + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{526} ~=~ 18 \zeta_5 - \frac{9}{8} \zeta_4 + \frac{1}{4} \zeta_3 \nonumber \\
c^{\MSbars}_{527} &=& - \frac{143}{8} \zeta_5 - \frac{9}{16} \zeta_4 + \frac{1}{8} \zeta_3 ~~,~~
c^{\MSbars}_{528} ~=~ - \frac{143}{8} \zeta_5 - \frac{9}{16} \zeta_4 + \frac{1}{8} \zeta_3 \nonumber \\
c^{\MSbars}_{529} &=& \frac{41}{2} \zeta_5 - \frac{9}{8} \zeta_4 + \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{530} ~=~ \frac{41}{2} \zeta_5 - \frac{21}{8} \zeta_4 + \frac{31}{4} \zeta_3 \nonumber \\
c^{\MSbars}_{531} &=& \frac{1}{2} \zeta_5 + \frac{3}{32} \zeta_4 - \frac{13}{16} \zeta_3 ~~,~~
c^{\MSbars}_{532} ~=~ 0 ~~,~~
c^{\MSbars}_{533} ~=~ \frac{3}{16} + \frac{3}{32} \zeta_4 - \frac{5}{16} \zeta_3 \nonumber \\
c^{\MSbars}_{534} &=& \frac{3}{16} + \frac{3}{32} \zeta_4 - \frac{5}{16} \zeta_3 ~~,~~
c^{\MSbars}_{535} ~=~ - \frac{1}{16} + \frac{3}{16} \zeta_4 - \frac{1}{8} \zeta_3 ~~,~~
c^{\MSbars}_{536} ~=~ - \frac{3}{32} \zeta_4 - \frac{1}{16} \zeta_3 \nonumber \\
c^{\MSbars}_{537} &=& - \frac{7}{6} ~~,~~
c^{\MSbars}_{538} ~=~ \frac{1}{3} ~~,~~
c^{\MSbars}_{539} ~=~ \frac{25}{4} \zeta_6 + \frac{15}{2} \zeta_5 - \frac{1}{2} \zeta_3^2 ~~,~~
c^{\MSbars}_{540} ~=~ \frac{1}{8} \zeta_5 + 2 \zeta_3 \nonumber \\
c^{\MSbars}_{541} &=& \frac{1}{8} \zeta_5 + 2 \zeta_3 ~~,~~
c^{\MSbars}_{542} ~=~ - \frac{9}{4} \zeta_5 + 4 \zeta_3
c^{\MSbars}_{543} ~=~ \frac{3}{16} - \frac{3}{16} \zeta_4 - \frac{1}{4} \zeta_3 ~~,~~
c^{\MSbars}_{544} ~=~ - \frac{5}{6} \nonumber \\
c^{\MSbars}_{545} &=& \frac{1}{2} \zeta_5 + \frac{3}{32} \zeta_4 - \frac{13}{16} \zeta_3 ~~,~~
c^{\MSbars}_{546} ~=~ \frac{1}{2} \zeta_5 + \frac{3}{32} \zeta_4 - \frac{13}{16} \zeta_3 ~~,~~
c^{\MSbars}_{547} ~=~ 0 ~~,~~
c^{\MSbars}_{548} ~=~ 0 \nonumber \\
c^{\MSbars}_{549} &=& \frac{3}{16} + \frac{3}{32} \zeta_4 - \frac{5}{16} \zeta_3 ~~,~~
c^{\MSbars}_{550} ~=~ \frac{3}{16} + \frac{3}{32} \zeta_4 - \frac{5}{16} \zeta_3 ~~,~~
c^{\MSbars}_{551} ~=~ - \frac{3}{32} \zeta_4 - \frac{1}{16} \zeta_3 \nonumber \\
c^{\MSbars}_{552} &=& - \frac{3}{32} \zeta_4 - \frac{1}{16} \zeta_3 ~~,~~
c^{\MSbars}_{553} ~=~ - \frac{7}{6} ~~,~~
c^{\MSbars}_{554} ~=~ - \frac{7}{6} ~~,~~
c^{\MSbars}_{555} ~=~ - \frac{1}{16} + \frac{3}{16} \zeta_4 - \frac{1}{8} \zeta_3 \nonumber \\
c^{\MSbars}_{556} &=& \frac{3}{16} + \frac{3}{32} \zeta_4 - \frac{5}{16} \zeta_3 ~~,~~
c^{\MSbars}_{557} ~=~ - \zeta_5 + \frac{21}{16} \zeta_4 + \frac{25}{8} \zeta_3 ~~,~~
c^{\MSbars}_{558} ~=~ - \frac{1}{8} \nonumber \\
c^{\MSbars}_{559} &=& - \frac{5}{6} - \frac{3}{16} \zeta_4 + \frac{7}{8} \zeta_3 ~~,~~
c^{\MSbars}_{560} ~=~ - \frac{5}{6} - \frac{3}{16} \zeta_4 + \frac{7}{8} \zeta_3 ~~,~~
c^{\MSbars}_{561} ~=~ \frac{3}{16} - \frac{3}{8} \zeta_4 + \frac{1}{8} \zeta_3 \nonumber \\
c^{\MSbars}_{562} &=& - \frac{1}{4} + \frac{3}{16} \zeta_4 + \frac{9}{8} \zeta_3 ~~,~~
c^{\MSbars}_{563} ~=~ 7 ~.
\end{eqnarray}

We have repeated this exercise for the MOM scheme and found to four loops
\begin{eqnarray}
c^{\MOMs}_{11} &=& \frac{1}{2} ~~,~~
c^{\MOMs}_{12} ~=~ - \frac{1}{2} ~~,~~
c^{\MOMs}_{31} ~=~ \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{32} ~=~ \frac{1}{4} ~~,~~
c^{\MOMs}_{33} ~=~ \frac{1}{2} \nonumber \\
c^{\MOMs}_{34} &=& 1 \nonumber \\
c^{\MOMs}_{41} &=& - 10 \zeta_5 ~~,~~
c^{\MOMs}_{42} ~=~ - \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{43} ~=~ - \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{44} ~=~ - 3 \zeta_3 \nonumber \\
c^{\MOMs}_{45} &=& - \frac{3}{4} + \frac{1}{2} \zeta_3 ~~,~~
c^{\MOMs}_{46} ~=~ - \frac{5}{4} + \frac{1}{2} \zeta_3 ~~,~~
c^{\MOMs}_{47} ~=~ - \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{48} ~=~ - \frac{3}{4} \nonumber \\
c^{\MOMs}_{49} &=& - \frac{5}{4} ~~,~~
c^{\MOMs}_{410} ~=~ - \frac{5}{4} ~~,~~
c^{\MOMs}_{411} ~=~ - \frac{3}{4} ~~,~~
c^{\MOMs}_{412} ~=~ - \frac{3}{2} \nonumber \\
c^{\MOMs}_{413} &=& - \frac{5}{2} ~.
\end{eqnarray}
To two loops the respective coefficients are the same as those of the $\MSbar$
scheme consistent with earlier expectations. At three and four loops a few of
the coefficients also match between schemes aside from the primitive graphs. At
next order the coefficients are
\begin{eqnarray}
c^{\MOMs}_{51} &=& \frac{9}{2} \zeta_3^2 ~~,~~;
c^{\MOMs}_{52} ~=~ - \frac{75}{8} \zeta_5 + \frac{3}{4} \zeta_3 ~~,~~
c^{\MOMs}_{53} ~=~ - \frac{75}{8} \zeta_5 + \frac{3}{4} \zeta_3 \nonumber \\
c^{\MOMs}_{54} &=& \frac{3}{4} - \frac{1}{2} \zeta_3 ~~,~~
c^{\MOMs}_{55} ~=~ 15 \zeta_5 + 3 \zeta_3 ~~,~~
c^{\MOMs}_{56} ~=~ \frac{9}{4} - \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{57} ~=~ 15 \zeta_5 + 3 \zeta_3 \nonumber \\
c^{\MOMs}_{58} &=& 15 \zeta_5 + 3 \zeta_3 ~~,~~
c^{\MOMs}_{59} ~=~ 10 \zeta_5 ~~,~~
c^{\MOMs}_{510} ~=~ 10 \zeta_5 ~~,~~
c^{\MOMs}_{511} ~=~ \frac{441}{8} \zeta_7 \nonumber \\
c^{\MOMs}_{512} &=& 10 \zeta_5 ~~,~~
c^{\MOMs}_{513} ~=~ - \frac{175}{8} \zeta_5 + \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{514} ~=~ \frac{441}{16} \zeta_7 ~~,~~
c^{\MOMs}_{515} ~=~ \frac{7}{4} - \zeta_3 \nonumber \\
c^{\MOMs}_{516} &=& 10 \zeta_5 - 3 \zeta_3^2 ~~,~~
c^{\MOMs}_{517} ~=~ 10 \zeta_5 - 3 \zeta_3^2 ~~,~~
c^{\MOMs}_{518} ~=~ 5 \zeta_5 - 3 \zeta_3^2 ~~,~~
c^{\MOMs}_{519} ~=~ 9 \zeta_3^2 \nonumber \\
c^{\MOMs}_{520} &=& 9 \zeta_3^2 ~~,~~
c^{\MOMs}_{521} ~=~ - \frac{85}{4} \zeta_5 + \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{522} ~=~ - \frac{85}{4} \zeta_5 + \frac{3}{2} \zeta_3 \nonumber \\
c^{\MOMs}_{523} &=& - \frac{75}{4} \zeta_5 + 3 \zeta_3 ~~,~~
c^{\MOMs}_{524} ~=~ - \frac{75}{4} \zeta_5 + 3 \zeta_3 ~~,~~
c^{\MOMs}_{525} ~=~ 15 \zeta_5 + 3 \zeta_3 \nonumber \\
c^{\MOMs}_{527} &=& - \frac{75}{4} \zeta_5 + \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{528} ~=~ - \frac{75}{4} \zeta_5 + \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{529} ~=~ \frac{65}{4} \zeta_5 + 3 \zeta_3 \nonumber \\
c^{\MOMs}_{530} &=& \frac{65}{4} \zeta_5 + 6 \zeta_3 ~~,~~
c^{\MOMs}_{531} ~=~ \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{532} ~=~ \frac{9}{8} - \frac{1}{2} \zeta_3 ~~,~~
c^{\MOMs}_{533} ~=~ \frac{9}{4} - \frac{5}{4} \zeta_3 \nonumber \\
c^{\MOMs}_{534} &=& \frac{9}{4} - \frac{5}{4} \zeta_3 ~~,~~
c^{\MOMs}_{535} ~=~ \frac{3}{2} - \frac{3}{4} \zeta_3 ~~,~~
c^{\MOMs}_{536} ~=~ \frac{9}{4} - \zeta_3 ~~,~~
c^{\MOMs}_{537} ~=~ \frac{7}{2} - \frac{3}{2} \zeta_3 \nonumber \\
c^{\MOMs}_{538} &=& \frac{7}{2} ~~,~~
c^{\MOMs}_{539} ~=~ 10 \zeta_5 ~~,~~
c^{\MOMs}_{540} ~=~ 3 \zeta_3 ~~,~~
c^{\MOMs}_{541} ~=~ 3 \zeta_3 ~~,~~
c^{\MOMs}_{542} ~=~ 6 \zeta_3 \nonumber \\
c^{\MOMs}_{543} &=& 3 - \frac{1}{2} \zeta_3 ~~,~~
c^{\MOMs}_{544} ~=~ \frac{9}{2} - \frac{1}{2} \zeta_3 ~~,~~
c^{\MOMs}_{545} ~=~ \frac{3}{2} \zeta_3 ~~,~~
c^{\MOMs}_{546} ~=~ \frac{3}{2} \zeta_3 \nonumber \\
c^{\MOMs}_{547} &=& \frac{9}{8} ~~,~~
c^{\MOMs}_{548} ~=~ \frac{9}{8} ~~,~~
c^{\MOMs}_{549} ~=~ \frac{9}{4} ~~,~~
c^{\MOMs}_{550} ~=~ \frac{9}{4} ~~,~~
c^{\MOMs}_{551} ~=~ \frac{9}{4} \nonumber \\
c^{\MOMs}_{552} &=& \frac{9}{4} ~~,~~
c^{\MOMs}_{553} ~=~ \frac{7}{2} ~~,~~
c^{\MOMs}_{554} ~=~ \frac{7}{2} ~~,~~
c^{\MOMs}_{555} ~=~ \frac{3}{2} ~~,~~
c^{\MOMs}_{556} ~=~ \frac{9}{4} \nonumber \\
c^{\MOMs}_{557} &=& 3 \zeta_3 ~~,~~
c^{\MOMs}_{558} ~=~ \frac{9}{4} ~~,~~
c^{\MOMs}_{559} ~=~ \frac{9}{2} ~~,~~
c^{\MOMs}_{560} ~=~ \frac{9}{2} ~~,~~
c^{\MOMs}_{561} ~=~ 3 \nonumber \\
c^{\MOMs}_{562} &=& \frac{9}{2} ~~,~~
c^{\MOMs}_{563} ~=~ 7 ~.
\end{eqnarray}
To assist with the derivation of both sets of coefficients from the value of
$Z_\Phi^{ij}$ in each scheme we have recorded the explicit expression in
Appendix B. Indeed by providing them for each specific tensor means the
divergence structure of all the individual diagrams are provided to five loops.
More tensors appear in $Z_\Phi^{ij}$ than $\gamma_\Phi^{ij}$. The extra ones
arise in terms with poles in $\epsilon$ higher than the simple one. They
correspond to connected one-particle reducible Feynman graphs of the $\Phi$
$2$-point function. Such topologies and hence tensors clearly cannot appear in
the final expression for $\gamma_\Phi^{ij}$ in either scheme which is a
non-trivial check on the overall expression. This is because it is the
generalization of the observation that in a conventional coupling constant
renormalization the coefficients of the non-simple poles in $\epsilon$ are
determined by the lower order renormalization constants.

\sect{XYZ model.}

As an application of the general tensor renormalization we consider a
particular theory that is connected to the Wess-Zumino model which was examined
in [40, 103]. It was investigated in [40] due to its connection with a
one dimensional conformal manifold. In particular several theories are of
interest for the case when the Wess-Zumino model has three chiral superfields
as they lie on the manifold. These are the XYZ model and a version of the model
itself with three copies. First we recall the relevant properties of the more
general model in order to extend the four loop analysis of [91] to five
loops here. As indicated in [40] the model involves three chiral
superfields and their anti-chiral counterparts with superpotential
\begin{equation}
W(\Phi_i) ~=~ g_1 \Phi_1 \Phi_2 \Phi_3 ~+~
\frac{g_2}{6} \left( \Phi_1^3 + \Phi_2^3 + \Phi_3^3 \right)
\label{pot3d}
\end{equation}
and its conjugate where $g_1$ and $g_2$ are complex coupling constants.
Therefore the non-zero tensor coupling entries are
\begin{equation}
d^{123} ~=~ g_1 ~~,~~
d^{111} ~=~ d^{222} ~=~ d^{333} ~=~ g_2 ~~,~~
\bar{d}^{123} ~=~ \bar{g}_1 ~~,~~
\bar{d}^{111} ~=~ \bar{d}^{222} ~=~ \bar{d}^{333} ~=~ \bar{g}_2 ~.
\label{cc3d}
\end{equation}
These variables were mapped to others which are similar to polar coordinates in
geometry through, [40, 104],
\begin{equation}
r^2 ~=~ 2 g_1 \bar{g}_1 ~+~ g_2 \bar{g}_2 ~~~,~~~
\tau ~=~ \frac{g_2}{g_1} ~~~,~~~
\bar{\tau} ~=~ \frac{\bar{g}_2}{\bar{g}_1}
\label{parm3d}
\end{equation}
where the parameter $\tau$ takes values in $\Cc \Pp(1)$, [104]. Using
these combinations certain values of $\tau$ and $\bar{\tau}$ allow one to
define various different theories with the justification recorded in [40].
We have provided these in Table \ref{modeldef} where the first three were given
in [40] and cWZ${}^3$ is used as shorthand to denote the three copy
Wess-Zumino model. This is also equivalent to the parameter choice of the final
row of Table \ref{modeldef} which was not noted in [40] and will be
another useful limit for checking results. For the $\Zz_2$~$\times$~$\Zz_2$
symmetric model the complex number $\omega$ and its conjugate appear are
\begin{equation}
\omega ~=~ -~ \frac{1}{2} ~+~ \frac{\sqrt{3}}{2} i ~~~,~~~
\bar{\omega} ~=~ -~ \frac{1}{2} ~-~ \frac{\sqrt{3}}{2} i ~.
\end{equation}

\begin{center}
\begin{tabular}{|c|c||c|}
\hline
$\tau$ & $\bar{\tau}$ & Theory \\
\hline
$0$ & $0$ & XYZ model \\
$1$ & $1$ & cWZ${}^3$ \\
$(1-\sqrt{3})\omega^2$ & $(1-\sqrt{3})\bar{\omega}^2$ &
$\Zz_2$~$\times$~$\Zz_2$ symmetric \\
$\infty$ & $\infty$ & Wess-Zumino model (\ref{lagwz}) \\
\hline
\end{tabular}
\end{center}
\begin{center}
\caption{Definition of various models from the values of $\tau$ and
\label{modeldef}
\end{center}
\end{table}}

With (\ref{parm3d}) the anomalous dimension is formally written as
\begin{equation}
\gamma_\Phi(r,\tau,\bar{\tau}) ~=~ \sum_{i=1}^\infty f_i(\tau,\bar{\tau}) r^{2i}
\label{gam3d}
\end{equation}
where the coefficients are given by
\begin{eqnarray}
f_1(\tau,\bar{\tau}) &=& \frac{1}{2} ~~,~~
f_2(\tau,\bar{\tau}) ~=~ -~ \frac{1}{2} ~~,~~
f_3(\tau,\bar{\tau}) ~=~ \frac{5}{8} ~+~
\frac{3}{2} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \zeta_3 \nonumber \\
f_4(\tau,\bar{\tau}) &=& -~ \frac{9}{8} ~+~
\left[ \frac{9}{4} \zeta_4 - \frac{15}{2} \zeta_3 \right]
\frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \nonumber \\
&& -~ 10 \frac{[ (2+\tau\bar{\tau})^4 - 8 (1-\tau^3)(1-\bar{\tau}^3)]}
{[2+\tau\bar{\tau}]^4} \zeta_5 \nonumber \\
f_5(\tau,\bar{\tau}) &=& \frac{79}{32} ~+~
\left[ \frac{3}{8}
+ \frac{423}{16} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \right] \zeta_3
-~ \frac{441}{32}
\frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \zeta_4 \nonumber \\
&& +~ \left[
\frac{305}{4} \frac{[ (2+\tau\bar{\tau})^4 - 8 (1-\tau^3)(1-\bar{\tau}^3)]}
- \frac{153}{8} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \right] \zeta_5 \nonumber \\
&& -~ \frac{225}{8}
\frac{[ (2+\tau\bar{\tau})^4 - 8 (1-\tau^3)(1-\bar{\tau}^3)]}
{[2+\tau\bar{\tau}]^4} \zeta_6 \nonumber \\
&& -~ \left[ \frac{45}{4}
- \frac{9}{2} \frac{[ (2+\tau\bar{\tau})^4 - 8 (1-\tau^3)(1-\bar{\tau}^3)]}
- \frac{45}{2} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \right] \zeta_3^2 \nonumber \\
&& +~ \frac{1323}{16}
\frac{[ (2+\tau\bar{\tau})^4 - 10 (1-\tau^3)(1-\bar{\tau}^3)]}
{[2+\tau\bar{\tau}]^4} \zeta_7
\label{gam3dc}
\end{eqnarray}
with $f_1$ to $f_4$ in accord with [40]. It is straightforward to check
that $f_i(1,1)$~$=$~$f_i(\infty,\infty)$ for $i$~$=$~$1$ to $5$. Moreover  the
$f_i(1,1)$ correspond to the respective coefficients of (\ref{gam5}). While we
have checked the values $f_i(\tau,\bar{\tau})$ to four loops and found
$f_5(\tau,\bar{\tau})$ using (\ref{cc3d}) and (\ref{parm3d}) they could also
have been derived from (\ref{gam5T}) from the simple identifications
\begin{eqnarray}
T_2 &=& 1 ~~,~~
T_5 ~=~ \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]}
{[2+\tau\bar{\tau}]^3} \nonumber \\
T_{71} &=& \frac{[ (2+\tau\bar{\tau})^4 - 8 (1-\tau^3)(1-\bar{\tau}^3)]}
{[2+\tau\bar{\tau}]^4} ~~,~~
T_{94} ~=~ \frac{[ (2+\tau\bar{\tau})^4 - 10 (1-\tau^3)(1-\bar{\tau}^3)]}
{[2+\tau\bar{\tau}]^4} ~~~~
\end{eqnarray}
thereby making the connection with the primitive graphs for the conformal
manifold case. It is worth remarking that given this relation between the $T_i$
invariants one could in principle repeat the analysis of [40] and that
which follows here for non-supersymmetric scalar $\phi^3$ theory. While that
theory is renormalizable in six dimensions the four loop renormalization group
functions have been expressed in terms of the four $T_i$ that appear here for
chiral $\phi^3$ theory.

The main topic of study in [40] was the evaluation of the critical
exponents of the dimension two bilinear operators denoted by $\Delta_i$ where
$i$~$\in$~$\{ \mathbf{1}, \mathbf{2}, \mathbf{2^\prime},
\mathbf{2^{\prime\prime}}, \mathbf{2^{\prime\prime\prime}} \}$ correspond to
the different representations of the
$\mathbf{3}$~$\otimes$~$\bar{\mathbf{3}}$ decomposition of the $9$ operators.
These operator dimensions were determined in three dimensions using conformal
bootstrap methods as well as resumming four dimensional perturbation theory.
For the latter the matrix of operator anomalous dimensions was computed to four
loops prior to being evaluated at the Wilson-Fisher fixed point. The critical
point eigenvalues of this matrix then corresponded to the critical exponents
$\Delta_i$, [40]. We are now in a position to extend the four loop
analysis of [40] to five loops in order to compare with the bootstrap
exponent estimates. First, the location of the Wilson-Fisher fixed point has to
be found. Since the $\beta$-function is synonymous with
$\gamma_\Phi(r,\tau,\bar{\tau})$ in this model then the $\epsilon$
expansion of the critical value of $r$, denoted by $r_\ast$, is given by
solving $\gamma_\Phi(r_\ast,\tau,\bar{\tau})$~$=$~$\third \epsilon$. From
(\ref{gam3d}) and defining
\begin{equation}
r_\ast^2 ~=~ \sum_{i=1}^\infty h_i \epsilon^i
\end{equation}
the various coefficients of the critical coupling are
\begin{eqnarray}
h_1 &=& \frac{1}{3f_1} ~~,~~
h_2 ~=~ -~ \frac{f_2}{9f_1^3} ~~,~~
h_3 ~=~ \frac{2f_2^2}{27f_1^5} ~-~ \frac{f_3}{27f_1^4} ~~,~~
h_4 ~=~ \frac{5f_2f_3}{81f_1^6} ~-~ \frac{5f_2^3}{81f_1^7} ~-~
\frac{f_4}{81f_1^5} \nonumber \\
h_5 &=& \frac{14f_2^4}{243f_1^9} ~-~ \frac{7f_2^2f_3}{81f_1^8} ~+~
\frac{f_3^2}{81f_1^7} ~+~ \frac{2f_2f_4}{81f_1^7} ~-~ \frac{f_5}{243f_1^6} ~.
\end{eqnarray}

The $3$~$\times$~$3$ matrix of mass anomalous dimensions that was constructed
in [40] is defined by
\begin{equation}
\gamma_M^{ij} ~=~ \mu \frac{d M^{ij}}{d \mu}
\end{equation}
where the matrix $M^{ij}$ corresponds to the mass dimension $2$ matrix
$(m^2)^{ij}$ of [40] which is computed from $\gamma_\phi^{ij}$ using
\begin{eqnarray}
\gamma_M^{ij} &=& -~ 2 M^{ij} ~+~
\left[ M^{ps} d^{sqr} + M^{qs} d^{psr} + M^{rs} d^{pqs} \right]
\frac{\partial \gamma_\Phi^{ij}}{d^{pqr}} \nonumber \\
&& +~ \left[ M^{ps} \bar{d}^{sqr} + M^{qs} \bar{d}^{psr} +
M^{rs} \bar{d}^{pqs} \right] \frac{\partial \gamma_\Phi^{ij}}{\bar{d}^{pqr}} ~.
\end{eqnarray}
The next stage is to construct the $9$~$\times$~$9$ matrix, $\Delta^{ijkl}$,
the eigenvalues of which produce the scaling dimensions of the bilinear
operators. It has $81$ elements since the matrix is labelled by the pairs of
indices $(ij)$ and $(kl)$ and defined by
\begin{equation}
\Delta^{ijkl} ~=~ d \delta^{ik} \delta^{jl} ~+~
\frac{\partial \gamma_M^{ij}}{\partial M^{kl}} ~.
\end{equation}
Following the prescription given in [40] we have extended the four loop
expressions for the five critical exponents $\Delta_i$ to the next order. In
particular we found
\begin{eqnarray}
\Delta_{\mathbf 1} &=& 2 ~-~ \frac{4}{3} \epsilon^2 ~+~
\left[ \frac{4}{9}
+ \frac{16}{3} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3} \right] \zeta_3 \epsilon^3
\nonumber \\
-~ \left[ \frac{28}{27}
+ \frac{112}{9} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3} \zeta_3
- 8 \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3} \zeta_4
\right. \nonumber \\
&& \left. ~~~~
+~ \frac{320}{9} \frac{[(\tau \bar{\tau}+2)^4 - 8 (1-\tau^3) (1-\bar{\tau}^3)]}
{[2+\tau \bar{\tau}]^4} \zeta_5 \right] \epsilon^4
\nonumber \\
+~ \left[ \frac{76}{81}
+ \left[ \frac{496}{27}
\frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3}
+ \frac{32}{27} \right] \zeta_3
- \frac{56}{3} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3} \zeta_4
\right. \nonumber \\
&& \left. ~~~~
+~ \left[ \frac{3520}{27}
\frac{[(\tau \bar{\tau}+2)^4 - 8 (1-\tau^3) (1-\bar{\tau}^3)]}
{[2+\tau \bar{\tau}]^4}
- \frac{544}{9} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3} \right] \zeta_5
\right. \nonumber \\
&& \left. ~~~~
-~ \frac{800}{9}
\frac{[(\tau \bar{\tau}+2)^4 - 8 (1-\tau^3) (1-\bar{\tau}^3)]}
{[2+\tau \bar{\tau}]^4} \zeta_6
\right. \nonumber \\
&& \left. ~~~~
+ \left[ \frac{256}{81}
\left[ \frac{9}{2} \frac{[(2+\tau \bar{\tau})^4 - 8(1-\tau^3)(1-\bar{\tau}^3)]}
{[2+\tau \bar{\tau}]^4}
- \frac{45}{4}
+ \frac{45}{2} \frac{[(\tau^3+2)(\bar{\tau}^3+2) + 18 \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3} \right]
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~
- \frac{64}{3} \frac{[(\tau^3+2) (\bar{\tau}^3+2) + 18 \tau \bar{\tau} ]^2}
{[2+\tau \bar{\tau}]^6} \right] \zeta_3^2
\right. \nonumber \\
&& \left. ~~~~
+~ \frac{784}{3} \frac{[(2+\tau \bar{\tau})^4 - 10 (1-\tau^3) (1-\bar{\tau}^3)]}
{[2+\tau \bar{\tau}]^4} \zeta_7 \right] \epsilon^5 ~+~ O(\epsilon^6)
\label{op31}
\end{eqnarray}
for the singlet operator as well as
\begin{eqnarray}
\Delta_{\mathbf 2} &=& 2 ~-~ \frac{4}{[2+\tau \bar{\tau}]} \epsilon
+ \frac{4}{3} \tau \bar{\tau}
\frac{[1 - \tau \bar{\tau}]}{[2+\tau \bar{\tau}]^2} \epsilon^2
\nonumber \\
+~ \tau \bar{\tau}
\left[ \frac{4}{9} \frac{[1 - \tau \bar{\tau}][10 - \tau \bar{\tau}]}
{[2+\tau \bar{\tau}]^3}
+ \frac{16}{3} \frac{[3 (1 - \tau \bar{\tau})^2
+ (1 - \tau^3)(1 - \bar{\tau}^3)]}{[2+\tau \bar{\tau}]^4} \zeta_3 \right]
\epsilon^3
\nonumber \\
+~ \tau \bar{\tau}
\left[ \frac{4}{27} \frac{[ 7 \tau^2 \bar{\tau}^2 - 26 \tau \bar{\tau} + 100 ]
[1 - \tau \bar{\tau}]}{[2+\tau \bar{\tau}]^4}
\right. \nonumber \\
&& \left. ~~~~~~~~
-~ \frac{16}{9} \frac{[ 2 (1-\tau \bar{\tau}) (2+\tau \bar{\tau})^2
+ [3 (1-\tau \bar{\tau})^2 + (1-\tau^3) (1-\bar{\tau}^3) ]
(2+7 \tau \bar{\tau}) ]}{[2+\tau \bar{\tau}]^5} \zeta_3
\right. \nonumber \\
&& \left. ~~~~~~~~
+~ 8 \frac{[ 3 (1-\tau \bar{\tau})^2 + (1-\tau^3) (1-\bar{\tau}^3) ]}
{[2+\tau \bar{\tau}]^4} \zeta_4
\right. \nonumber \\
&& \left. ~~~~~~~~
-~ \frac{320}{27}
\frac{[3 \tau \bar{\tau} (2+\tau \bar{\tau}) (1-\tau \bar{\tau})^2
+ 8 (1-\tau^3) (1-\bar{\tau}^3) ]}
{[2+\tau \bar{\tau}]^5} \zeta_5 \right] \epsilon^4
\nonumber \\
+~ \tau \bar{\tau}
\left[ \frac{4}{81} \frac{[19 \tau^2 \bar{\tau}^2 - 38 \tau \bar{\tau} + 100 ]
[1-\tau \bar{\tau}] [10-\tau \bar{\tau}]}{[2+\tau \bar{\tau}]^5}
\right. \nonumber \\
&& \left. ~~~~~~~~
+~ \frac{16}{27}
\left[ \frac{}{} 31 \tau^2 \bar{\tau}^2 (1-\tau^3) (1-\bar{\tau}^3)
- 32 \tau \bar{\tau} (1-\tau^3) (1-\bar{\tau}^3)
+ 28 (1-\tau^3) (1-\bar{\tau}^3)
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~~~
-~ (2 \tau^4 \bar{\tau}^4+85 \tau^3 \bar{\tau}^3-237 \tau^2 \bar{\tau}^2
+ 148 \tau \bar{\tau}-52) (1-\tau \bar{\tau}) \right]
\frac{\zeta_3}{[2+\tau \bar{\tau}]^6}
\right. \nonumber \\
&& \left. ~~~~~~~~
-~ \frac{8}{3} \left[ \frac{}{}
7 (1-\tau^3) (1-\bar{\tau}^3) \tau \bar{\tau}
+ 21 \tau \bar{\tau} (1-\tau \bar{\tau})^2 + 12 (1-\tau \bar{\tau})
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~
+ 2 (1-\tau^3) + 2 (1-\bar{\tau}^3) \right]
\frac{\zeta_4}{[2+\tau \bar{\tau}]^5}
\right. \nonumber \\
&& \left. ~~~~~~~~
-~ \frac{16}{81} \left[ \frac{}{}
3 (70 \tau^3 \bar{\tau}^3-386 \tau^2 \bar{\tau}^2
- 566 \tau \bar{\tau}+207) (2+\tau \bar{\tau}) (1-\tau \bar{\tau})
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~~~
- 144 \tau^2 \bar{\tau}^2 (1-\tau^3) (1-\bar{\tau}^3)
+ 1916 \tau \bar{\tau} (1-\tau^3) (1-\bar{\tau}^3)
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~~~
+ 1336 (1-\tau^3) (1-\bar{\tau}^3) \right]
\frac{\zeta_5}{[2+\tau \bar{\tau}]^6}
\right. \nonumber \\
&& \left. ~~~~~~~~
-~ \frac{800}{27}
\frac{[ 8 (1-\tau) (1-\bar{\tau}) (1+\tau+\tau^2) (1+\bar{\tau}+\bar{\tau}^2)
+ 3 \tau \bar{\tau} (2+\tau \bar{\tau}) (1-\tau \bar{\tau})^2 ]}
{[2+\tau \bar{\tau}]^5} \zeta_6
\right. \nonumber \\
&& \left. ~~~~~~~~
+ \frac{128}{27} \left[ \frac{}{}
(18 \tau^4 \bar{\tau}^4-72 \tau^3 \bar{\tau}^3+137 \tau^2 \bar{\tau}^2
- 205 \tau \bar{\tau}-34) (1-\tau \bar{\tau})
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~~~
+ 6 \tau^3 \bar{\tau}^3 (1-\tau^3) (1-\bar{\tau}^3)
+ 46 \tau^2 \bar{\tau}^2 (1-\tau^3) (1-\bar{\tau}^3)
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~~~
- 77 \tau \bar{\tau} (1-\tau^3) (1-\bar{\tau}^3)
+ 9 (1-\tau^3)^2 + 9 (1-\bar{\tau}^3)^2
\right. \right. \nonumber \\
&& \left. \left. ~~~~~~~~~~~~~~~~~~
+ 52 (1-\tau^3) + 52 (1-\bar{\tau}^3)
\right]
\frac{\zeta_3^2}{[2+\tau \bar{\tau}]^7}
\right. \nonumber \\
&& \left. ~~~~~~~~
+~ \frac{392}{3}
\frac{[ \tau \bar{\tau} (1+2 \tau \bar{\tau}) (1-\tau \bar{\tau})^2
+ 4 (1-\tau^3) (1-\bar{\tau}^3)]}{[2+\tau \bar{\tau}]^5} \zeta_7 \right]
\epsilon^5
~+~ O(\epsilon^6) ~.
\label{op32}
\end{eqnarray}
Electronic expressions for these are included in the attached data file. While
we have also calculated expressions for $\Delta_\mathbf{2^\prime}$,
$\Delta_\mathbf{2^{\prime\prime}}$ and $\Delta_\mathbf{2^{\prime\prime\prime}}$
explicitly they can also be deduced from the following mappings given in
\begin{eqnarray}
&& \Delta_{\mathbf 2} ~\to~ \Delta_{\mathbf 2^\prime}: ~~~~~~~
\tau ~\to~ \frac{[\tau+2]}{[\tau-1]} ~~~,~~~
\bar{\tau} ~\to~ \frac{[\bar{\tau}+2]}{[\bar{\tau}-1]}
\nonumber \\
&& \Delta_{\mathbf 2} ~\to~ \Delta_{\mathbf 2^{\prime\prime}}: ~~~~~~~
\tau ~\to~ \frac{[\omega\tau+2]}{[\omega\tau-1]} ~~~,~~~
\bar{\tau} ~\to~
\frac{[\bar{\omega}\bar{\tau}+2]}{[\bar{\omega}\bar{\tau}-1]}
\nonumber \\
&& \Delta_{\mathbf 2} ~\to~ \Delta_{\mathbf 2^{\prime\prime\prime}}: ~~~~~~~
\tau ~\to~ \frac{[\omega^2\tau+2]}{[\omega^2\tau-1]} ~~~,~~~
\bar{\tau} ~\to~
\frac{[\bar{\omega}^2\bar{\tau}+2]}{[\bar{\omega}^2\bar{\tau}-1]} ~.
\end{eqnarray}
We note that each expression resulting from applying the mappings to
$\Delta_{\mathbf 2}$ is consistent with the direct five loop evaluation which
provides a useful check on the critical exponents. Another consistency check
is that setting both $\tau$ and $\bar{\tau}$ to be equal to $1$ or $\infty$ in
$\Delta_{\mathbf 1}$ reproduces the coefficients of $\epsilon$ in
(\ref{omeps}). The discrepancy in the $O(\epsilon)$ term is due to the
canonical part of $\Delta^{ijkl}$.

Having determined the five loop corrections to $\Delta_i$ we can now extract
estimates for them in three dimensions. First we record the explicit
expressions for the $\epsilon$ expansion of the various exponents for each of
the three theories. We have
\begin{eqnarray}
\Delta_\mathbf{1}^{\mbox{\footnotesize{XYZ}}} &=& 2 ~-~ \frac{4}{3} \epsilon^2
~+~ 4 (6 \zeta_3 + 1 ) \frac{\epsilon^3}{9}
~+~ 4 ( 27 \zeta_4 - 42 \zeta_3 - 120 \zeta_5 - 7 ) \frac{\epsilon^4}{27}
\nonumber \\
&& +~ 2 (72 \zeta_3^2 + 420 \zeta_3 - 378 \zeta_4 + 1416 \zeta_5 - 1800 \zeta_6
+ 3969 \zeta_7 + 38 ) \frac{\epsilon^5}{81} ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2}^{\mbox{\footnotesize{XYZ}}} &=& 2 ~-~ 2 \epsilon
\nonumber \\
\Delta_\mathbf{2^\prime}^{\mbox{\footnotesize{XYZ}}} &=&
\Delta_\mathbf{2^{\prime\prime}}^{\mbox{\footnotesize{XYZ}}} ~=~
\Delta_\mathbf{2^{\prime\prime\prime}}^{\mbox{\footnotesize{XYZ}}} \nonumber \\
&=& 2 ~-~ \frac{2}{3} \epsilon ~-~ \frac{4}{9} \epsilon^2
~+~ 4 (12 \zeta_3 - 1) \frac{\epsilon^3}{27}
~+~ 4 ( 54 \zeta_4 - 56 \zeta_3 - 160 \zeta_5 - 3 ) \frac{\epsilon^4}{81}
\nonumber \\
&& +~ 2 (528 \zeta_3^2 + 248 \zeta_3 - 504 \zeta_4 + 1467 \zeta_5
- 2400 \zeta_6 + 5292 \zeta_7 - 14 ) \frac{\epsilon^5}{243} \nonumber \\
&& +~ O(\epsilon^6)
\label{expxyz}
\end{eqnarray}
\begin{eqnarray}
\Delta_\mathbf{1}^{\mbox{\footnotesize{cWZ${}^3$}}} &=&
\Delta_\mathbf{2^\prime}^{\mbox{\footnotesize{cWZ${}^3$}}} \nonumber \\
&=& 2 ~-~ \frac{4}{3} \epsilon^2 ~+~  4 (12 \zeta_3 + 1 ) \frac{\epsilon^3}{9}
~+~ 4 ( 54 \zeta_4 - 84 \zeta_3 - 240 \zeta_5 - 7 ) \frac{\epsilon^4}{27}
\nonumber \\
&& +~ 4 (576 \zeta_3^2 + 396 \zeta_3 - 378 \zeta_4 + 1416 \zeta_5
- 1800 \zeta_6 + 5292 \zeta_7 + 19 ) \frac{\epsilon^5}{81} ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2}^{\mbox{\footnotesize{cWZ${}^3$}}} &=&
\Delta_\mathbf{2^{\prime\prime}}^{\mbox{\footnotesize{cWZ${}^3$}}} ~=~
\Delta_\mathbf{2^{\prime\prime\prime}}^{\mbox{\footnotesize{cWZ${}^3$}}} ~=~
2 ~-~ \frac{4}{3} \epsilon
\label{expcwz3}
\end{eqnarray}
\begin{eqnarray}
\Delta_\mathbf{1}^{\Zz_2\times\Zz_2} &=& 2 ~-~ \frac{4}{3} \epsilon^2
~+~ 4 (9 \zeta_3 + 1 ) \frac{\epsilon^3}{9}
~+~ 2 ( 81 \zeta_4 - 126 \zeta_3 - 300 \zeta_5 - 14 ) \frac{\epsilon^4}{27}
\nonumber \\
&& +~ (2376 \zeta_3^2 + 2424 \zeta_3 - 2268 \zeta_4 + 5856 \zeta_5
- 9000 \zeta_6 + 22491 \zeta_7 + 152) \frac{\epsilon^5}{162} ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2}^{\Zz_2\times\Zz_2} &=&
\Delta_\mathbf{2^{\prime\prime}}^{\Zz_2\times\Zz_2} \nonumber \\
&=& 2 ~+~ 2 \frac{( 26 - 15 \sqrt{3} )}{(71 \sqrt{3} - 123)} \epsilon
~+~ \frac{( 265 - 153 \sqrt{3} )}{3 (71 \sqrt{3} - 123)} \epsilon^2
\nonumber \\
&& +~ (1590 \sqrt{3} \zeta_3 - 41 \sqrt{3} - 2754 \zeta_3 + 71 )
\frac{\epsilon^3}{9 (71 \sqrt{3} - 123) }
\nonumber \\
&& +~ ( 14310 \sqrt{3} \zeta_4 - 17452 \sqrt{3} \zeta_3
- 53000 \sqrt{3} \zeta_5 - 1011 \sqrt{3} + 30228 \zeta_3 - 24786 \zeta_4
\nonumber \\
&& ~~~~
+ 91800 \zeta_5 + 1751) \frac{\epsilon^4}{54 (71 \sqrt{3} - 123)}
\nonumber \\
&& +~ (177624 \sqrt{3} \zeta_3^2 + 107664 \sqrt{3} \zeta_3
- 157068 \sqrt{3} \zeta_4 + 451070 \sqrt{3} \zeta_5 - 795000 \sqrt{3} \zeta_6
\nonumber \\
&& ~~~~
+ 1912617 \sqrt{3} \zeta_7 - 1602 \sqrt{3} - 307656 \zeta_3^2 - 186480 \zeta_3
+ 272052 \zeta_4 - 781293 \zeta_5
\nonumber \\
&& ~~~~
+ 1377000 \zeta_6 - 3312792 \zeta_7 + 2774)
\frac{\epsilon^5}{324 (71 \sqrt{3} - 123)} ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2^\prime}^{\Zz_2\times\Zz_2} &=&
\Delta_\mathbf{2^{\prime\prime\prime}}^{\Zz_2\times\Zz_2} \nonumber \\
&=& 2 ~+~ 2 \frac{( 97 - 56 \sqrt{3} )}{(71 \sqrt{3} - 123)} \epsilon
~+~ \frac{(11 \sqrt{3} - 19)}{3 (71 \sqrt{3} - 123)} \epsilon^2
\nonumber \\
&& +~ (114 \sqrt{3} \zeta_3 + 41 \sqrt{3} - 198 \zeta_3 - 71)
\frac{\epsilon^3}{9 (71 \sqrt{3} - 123)}
\nonumber \\
&& +~ ( 1026 \sqrt{3} \zeta_4 - 724 \sqrt{3} \zeta_3 - 3800 \sqrt{3} \zeta_5
+ 301 \sqrt{3} + 1260 \zeta_3 - 1782 \zeta_4
\nonumber \\
&& ~~~~
+ 6600 \zeta_5 - 521) \frac{\epsilon^4}{54 (71 \sqrt{3} - 123)}
\nonumber \\
&& +~ (6408 \sqrt{3} \zeta_3^2 + 1392 \sqrt{3} \zeta_3 - 6516 \sqrt{3} \zeta_4
+ 48983 \sqrt{3} \zeta_5 - 57000 \sqrt{3} \zeta_6
\nonumber \\
&& ~~~~
+ 122598 \sqrt{3} \zeta_7 + 2170 \sqrt{3} - 11160 \zeta_3^2 - 2448 \zeta_3
+ 11340 \zeta_4 - 84996 \zeta_5
\nonumber \\
&& ~~~~
+ 99000 \zeta_6 - 213003 \zeta_7 - 3758)
\frac{\epsilon^5}{324 (71 \sqrt{3} - 123)} ~+~ O(\epsilon^6) ~.
\label{expz2sq}
\end{eqnarray}
We note that both $\Delta_\mathbf{1}^{\mbox{\footnotesize{cWZ${}^3$}}}$ and
$\Delta_\mathbf{2}^{\mbox{\footnotesize{cWZ${}^3$}}}$ are indeed consistent
with (\ref{omeps}) as expected after allowance is made for the canonical
dimension contribution of $2$~$-$~$2\epsilon$. For several exponents the series
truncates at $O(\epsilon)$ and no order symbol is included. This is because
these are exact to all orders in $\epsilon$ and their three dimensional values
tally precisely with those of [40]. In deriving (\ref{expxyz}),
(\ref{expcwz3}) and (\ref{expz2sq}) we have encoded (\ref{op31}) and
(\ref{op32}) together with the $\tau$ and $\bar{\tau}$ dependent expressions
for $\Delta_{\mathbf 2^\prime}$, $\Delta_{\mathbf 2^{\prime\prime}}$ and
$\Delta_{\mathbf 2^{\prime\prime\prime}}$ in one programme and then evaluated
each explicitly. For the XYZ and the cWZ${}^3$ cases we find that several
non-exact exponents are equal and this agrees with [40]. However in the
$\Zz_2$~$\times$~$\Zz_2$ case we disagree with the equivalences recorded in
Table 2 of [40] for the $\mathbf{2^{\prime\prime}}$ and
$\mathbf{2^{\prime\prime\prime}}$ dimensions. Instead we found
$\Delta^{\Zz_2\times\Zz_2}_{\mathbf{2}}$~$=$~$\Delta^{\Zz_2\times\Zz_2}_{\mathbf{2^{\prime\prime}}}$ and
To see the alternating sign pattern and the magnitude of the coefficients the
numerical values of the non-exact exponents are
\begin{eqnarray}
\Delta_\mathbf{1}^{\mbox{\footnotesize{XYZ}}} &=&
2 ~-~ 1.333333 \epsilon^2 ~+~ 3.649929 \epsilon^3 ~-~ 22.621480 \epsilon^4
~+~ 95.728196 \epsilon^5 ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2^\prime}^{\mbox{\footnotesize{XYZ}}} &=&
\Delta_\mathbf{2^{\prime\prime}}^{\mbox{\footnotesize{XYZ}}} ~=~
\Delta_\mathbf{2^{\prime\prime\prime}}^{\mbox{\footnotesize{XYZ}}} \nonumber \\
&=& 2 ~-~ 0.666667 \epsilon ~-~ 0.444444 \epsilon^2 ~+~ 1.988842 \epsilon^3
~-~ 8.779169 \epsilon^4 ~+~ 40.471457 \epsilon^5 \nonumber \\
&& +~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{1}^{\mbox{\footnotesize{cWZ${}^3$}}} &=&
\Delta_\mathbf{2^\prime}^{\mbox{\footnotesize{cWZ${}^3$}}} \nonumber \\
&=& 2 ~-~ 1.333333 \epsilon^2 ~+~ 6.855415 \epsilon^3 ~-~ 44.205924 \epsilon^4
~+~ 290.935250 \epsilon^5 ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{1}^{\Zz_2\times\Zz_2} &=&
2 ~-~ 1.333333 \epsilon^2 ~+~ 5.252672 \epsilon^3 ~-~ 28.805134 \epsilon^4
~+~ 145.920995 \epsilon^5 ~+~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2}^{\Zz_2\times\Zz_2} &=&
\Delta_\mathbf{2^{\prime\prime}}^{\Zz_2\times\Zz_2} \nonumber \\
&=& 2 ~-~ 1.577350 \epsilon ~+~ 0.051567 \epsilon^2 ~+~ 0.278877 \epsilon^3
~-~ 0.888082 \epsilon^4 ~+~ 5.331310 \epsilon^5 \nonumber \\
&& +~ O(\epsilon^6)
\nonumber \\
\Delta_\mathbf{2^\prime}^{\Zz_2\times\Zz_2} &=&
\Delta_\mathbf{2^{\prime\prime\prime}}^{\Zz_2\times\Zz_2} \nonumber \\
&=& 2 ~-~ 0.422650 \epsilon ~-~ 0.718233 \epsilon^2 ~+~ 2.926608 \epsilon^3
~-~ 16.028343 \epsilon^4 ~+~ 78.326933 \epsilon^5
\nonumber \\
&& +~ O(\epsilon^6) ~.
\end{eqnarray}
For the exponents which have an $O(\epsilon)$ term the series are alternating
when the canonical value of $(2$~$-$~$2\epsilon)$ is allowed for.

\begin{center}
\begin{tabular}{|c||r|r|}
\hline
Pad\'{e} & $\Delta_{\mathbf 1}~~~$ & $\Delta_{\mathbf 2^\prime}~~~$ \\
\hline
$[2,1]$ & 1.859277 & 1.632346 \\
$[1,2]$ & 1.868528 & 1.660704 \\
\hline
$[3,1]$ & 1.777975 & 1.633073 \\
$[2,2]$ & ----------- & 1.633070 \\
$[1,3]$ & 1.797562 & 1.639170 \\
\hline
$[4,1]$ & 1.669152 & 1.638139 \\
$[3,2]$ & ----------- & 1.632229 \\
$[2,3]$ & ----------- & 1.637434 \\
$[1,4]$ & 1.705650 & 1.637537 \\
\hline
\end{tabular}
\end{center}
\begin{center}
\caption{Pad\'{e} approximants at three, four and five loops for non-exact
operator dimensions in the XYZ model.}
\label{tabxyz}
\end{center}
\end{table}}

\begin{center}
\begin{tabular}{|c||r|r|}
\hline
Pad\'{e} & $\Delta_{\mathbf 1}~~~$ \\
\hline
$[2,1]$ & 1.906650 \\
$[1,2]$ & 1.910813 \\
\hline
$[3,1]$ & 1.869530 \\
$[2,2]$ & ----------- \\
$[1,3]$ & 1.874821 \\
\hline
$[4,1]$ & 1.879670 \\
$[3,2]$ & 1.877593 \\
$[2,3]$ & 1.879319 \\
$[1,4]$ & 1.879929 \\
\hline
\end{tabular}
\end{center}
\begin{center}
\caption{Pad\'{e} approximants at three, four and five loops for the non-exact
operator dimensions in cWZ${}^3$ model.}
\label{tabcwz3}
\end{center}
\end{table}}
In [44] the perturbative expansion was used to estimate the exponents in
three dimensions in order to compare them with the conformal bootstrap
calculation. Therefore we have extended that study here using the same method.
This was to construct the Pad\'{e} approximants for the five loop non-exact
exponents. The results for each of the three theories are given in Tables
\ref{tabxyz}, \ref{tabcwz3} and \ref{tabz2sq} where the Pad\'{e} approximants
for three and four loops are also given. The $[L,0]$ and $[0,L]$ approximants
at each loop order $L$ are excluded as they either do not converge or are
singular in $2$~$<$~$d$~$<$~$4$. There are no entries in each table for some
operator dimensions. This is because for those cases the Pad\'{e} approximant
is also singular above three dimensions. So because there is no continuous
connection down from four dimensions to three in these cases any evaluation at
the latter dimension is unreliable. What is generally evident for each of the
theories is that the five loop Pad\'{e} approximants are similar especially in
the cases where there are no singularities. Table \ref{tabsum} summarizes the
situation at three, four and five loops for each of the three theories and also
records the conformal bootstrap results of [40]. Each loop estimate is the
average of the Pad\'{e} approximants in the individual table of each theory.
In [40] the three loop Pad\'{e} approximants were used to compare with the
bootstrap. By providing the same data for the next two loop orders gives an
overall indication of the trend of including higher order loops. For the XYZ
model the $\Delta_\mathbf{1}$ estimates are decreasing towards the bootstrap
value and is a significant improvement on the three loop estimate. The
estimates for the other exponents are slowly decreasing away from the value
given in [40]. It might be tempting to surmise that the operator
dimensions in the XYZ model have been interchanged since swapping them would
give agreement to a few percent. However this is not the case from analysing
(\ref{expxyz}). A similar feature occurs for the non-exact exponent of the
cWZ${}^3$ theory although the five loop value is within $2\%$ of the bootstrap
value. The situation for the three non-exact dimensions for the
$\Zz_2$~$\times$~$\Zz_2$ case is somewhat mixed. Clearly the estimate for
$\Delta_{\mathbf 2}^{\Zz_2\times\Zz_2}$ is within less than a percentage of the
value of [40] and is stable at each loop order. For the other operators
the tolerance is around $5\%$ but the trend with loop order is not as settled.

\begin{center}
\begin{tabular}{|c||r|r|r|}
\hline
Pad\'{e} & $\Delta_{\mathbf 1}~~~$ & $\Delta_{\mathbf 2}~~~$ &
$\Delta_{\mathbf 2^\prime}~~~$ \\
\hline
$[2,1]$ & 1.887757 & ----------- & 1.729559 \\
$[1,2]$ & 1.893722 & 1.253242 & 1.747789 \\
\hline
$[3,1]$ & 1.842132 & 1.237664 & 1.706973 \\
$[2,2]$ & ----------- & 1.237098 & 1.702425 \\
$[1,3]$ & 1.850355 & ----------- & 1.716789 \\
\hline
$[4,1]$ & 1.813663 & 1.245205 & 1.684017 \\
$[3,2]$ & ----------- & 1.255392 & ----------- \\
$[2,3]$ & ----------- & 1.243920 & ----------- \\
$[1,4]$ & 1.821597 & 1.253878 & 1.692667 \\
\hline
\end{tabular}
\end{center}
\begin{center}
\caption{Pad\'{e} approximants at three, four and five loops for the non-exact
operator dimensions in $\Zz_2$~$\times$~$\Zz_2$ model.}
\label{tabz2sq}
\end{center}
\end{table}}

\begin{center}
\begin{tabular}{|c|c||r|r|r||r|}
\hline
Model & Dimension & $3$ loop & $4$ loop & $5$ loop & [40] \\
\hline
XYZ & $\Delta_{\mathbf 1}$ & 1.863902 & 1.787768 & 1.687401 & 1.639 \\
& $\Delta_{\mathbf 2^\prime}$ & 1.646525 & 1.635104 & 1.636335 & 1.681 \\
\hline
\rule{0pt}{12pt}
cWZ${}^3$ & $\Delta_{\mathbf 1}$ & 1.908732 & 1.872175 & 1.879128 & 1.910 \\
\hline
$\Zz_2$~$\times$~$\Zz_2$ & $\Delta_{\mathbf 1}$ & 1.890740 & 1.846243 & 1.817630 & 1.898 \\
& $\Delta_{\mathbf 2}$ & 1.253242 & 1.237381 & 1.249599 & 1.259 \\
& $\Delta_{\mathbf 2^\prime}$ & 1.738674 & 1.708729 & 1.688342 & 1.727 \\
\hline
\end{tabular}
\end{center}
\begin{center}
\caption{Averages of three, four and five loop Pad\'{e} approximants for
non-exact operator dimensions compared with conformal bootstrap results.}
\label{tabsum}
\end{center}
\end{table}}

\begin{center}
\includegraphics[width=15.0cm,height=8.7cm]{wzfivefigtw61prcr.pdf}
\end{center}
\caption{Six loop product primitive graphs.}
\label{figtw61pr}
\end{figure}}

\sect{Beyond five loops.}

While our focus to this point has been on the five loop renormalization group
functions, the next stage in studying (\ref{lagwz}) would be to extend this to
six loops. Given what we have established here it is worth giving guidance on
what would be required for that as several common features emerged. First, at
six loops there are $324$ Feynman graphs contributing to the $\Phi$ $2$-point
function. The content of $\gamma_\Phi(a)$ at that order will involve rationals
as well as what we term irrationals. The majority of these will be $\zeta_n$
for $n$~$=$~$3$ to $9$. In addition their products such as $\zeta_3 \zeta_5$
and $\zeta_3^3$, which are both present in the six loop $\phi^4$
$\beta$-function [32, 34], should appear if the structure of the
renormalization group functions of this non-supersymmetric paradigm theory is
valid. That would therefore imply the potential additional presence of the
multiple zeta $\zeta_{3,5}$. As noted earlier the $O(1/N^3)$ expression for the
exponent $\eta$, [49], may indicate that such an irrational is actually
absent. However if it were present it would have to arise in a primitive graph
whose $O(N)$ group theory factor is beyond $O(1/N^3)$. Alternatively candidate
primitive graphs from $\phi^4$ theory may be excluded because of the
restriction the chiral symmetry places on the graph topologies.

\begin{center}
\includegraphics[width=9.95cm,height=4.0cm]{wzfivefigv3cr.pdf}
\end{center}
\caption{Three loop planar vertex correction.}
\label{figv3}
\end{figure}}

Of the $324$ graphs it turns out that $17$ of these are primitive. One feature
to emerge from the five loop evaluation of the Feynman graphs was the
appearance of what was termed the product primitives. These are $2$-point
graphs with vertex subgraphs. As the vertex function is finite, we noted that
the simple pole can be deduced from the finite value of the vertex itself. At
six loops we have illustrated the $8$ graphs of the total primitives that are
product primitives in Figure \ref{figtw61pr} where the vertex $V_3$ is defined
in Figure \ref{figv3}. The residue of the simple pole in $\epsilon$ of each of
the graphs will be proportional to $\zeta_3 \zeta_5$ and have a group factor of
$T_2 T_5 T_{71}$ for (\ref{lagwzt}). The explicit coefficient of this residue
requires the implementation of the $D$-algebra. This is also an issue for the
remaining non-product primitives especially as the power of the irreducible
scalar products increases with loop order. The remaining graphs intermediate to
those with rational contributions and the primitives correspond to the
decoration of the lower loop primitives with an extra one loop bubble. A subset
of these should be calculable with the use of subtractions and {\sc Forcer}.
The remainder of this type, similar to the non-product primitives, could only
be reliably evaluated with a five loop version of {\sc Forcer}.

\begin{center}
\includegraphics[width=15.0cm,height=8.70cm]{wzfivefighoprcr.pdf}
\end{center}
\caption{Higher order product primitive graphs.}
\label{fighopr}
\end{figure}}

Next we note that the concept of product primitives naturally continues at
higher loop order. We have provided several examples in Figure \ref{fighopr} to
illustrate the point. A new vertex function $V_4$ has been defined in Figure
\ref{figv4} where the actual $3$-point function is isolated by amputating the
right external vertex. In Figure \ref{fighopr} the graphs are $8$, $10$, $9$
and $13$ loops respectively from top left to bottom right. The simple pole
residue of each would be $\zeta_3^2 \zeta_5$, $\zeta_5^3$, $\zeta_3 \zeta_5^2$
and $\zeta_3 \zeta_5^2 \zeta_7$ in the same respective order with the equally
associated group factors of $T_2 T_5^2 T_{71}$, $T_2 T_{71}^3$,
$T_2 T_5 T_{71}^2$ and $T_2 T_5 T_{71}^2 T_{94}$. So there is a clear
association of each group factor with a specific $\zeta_n$.

\begin{center}
\includegraphics[width=14.0cm,height=4.0cm]{wzfivefigv4cr.pdf}
\end{center}
\caption{Definition of four loop primitive vertex.}
\label{figv4}
\end{figure}}

Finally we return to the rational part of $\gamma_\Phi(a)$ and note that it is
possible to deduce the contribution in the $\MSbar$ scheme purely from the five
loop computation. This is because the rational part of the six loop MOM scheme
is known from the Hopf algebra solution of the Dyson-Schwinger equation given
in [81]. As we showed earlier the five loop MOM expression for
$\gamma_\Phi(a)$ could be deduced from the $\MSbar$ expression by using the
coupling constant map (\ref{ccmommap}) and the formalism of (\ref{betamomms})
and (\ref{gammamomms}). To extract the rational part at six loops requires one
ingredient which is the finite part of the $\Phi$ $2$-point function at five
loops. This is because the coupling constant mapping at $L$ loops determines
the $(L+1)$ loop renormalization group functions from (\ref{betamomms}) and
(\ref{gammamomms}) once they are available at $L$ loops in one specific scheme.
Previously the MOM five loop $\beta$-function was deduced from the $\MSbar$
one. Here we reverse the process given the result of [81]. So all that is
required is the rational part of the $\Phi$ $2$-point function at five loops.
As these are the bubble graphs which are simple to evaluate to the finite part
we have applied the formalism to find the rational piece of the six loop
$\MSbar$ $\beta$-function which is
\begin{eqnarray}
\beta(a) &=& \frac{3}{2} a^2 ~-~ \frac{3}{2} a^3 ~+~
\left[ 36 \zeta_3 + 15 \right] \frac{a^4}{8} ~+~
\left[ 54 \zeta_4 - 180 \zeta_3 - 240 \zeta_5 - 27 \right] \frac{a^5}{8}
\nonumber \\
&& +~ \left[ 1512 \zeta_3^2 + 2574 \zeta_3 - 1323 \zeta_4 + 5484 \zeta_5
- 2700 \zeta_6 + 7938 \zeta_7 + 237 \right] \frac{a^6}{32} \nonumber \\
&& +~ \left[ -~ \frac{369}{20} ~+~ \mbox{non-rational contribution}
\right] a^7 ~+~ O(a^8)
\label{bet6rat}
\end{eqnarray}
and we note that the alternating sign pattern of the rationals is maintained.
To determine the non-rational contribution of (\ref{bet6rat}) is of course a
more strenuous exercise.

\sect{Discussion.}

We have completed a comprehensive study of the Wess-Zumino model at five loops.
This has proceeded in two phases with the initial one outlining the algorithm
for carrying out the computation of the five loop Feynman graphs that are
required for the $\beta$-function of the original model of [1]. Once
established the second part addressed applications to various extensions of the
core theory by allowing the fields to lie in various symmetry groups or take
the couplings to be general tensors. One consequence was to extend the
precision of the $\epsilon$ expansion of critical exponents to a new order.
This is important in the context of other methods such as the conformal
bootstrap and the functional renormalization group techniques. These have been
applied to several problems like the emergent supersymmetric fixed point that
is present in Gross-Neveu-Yukawa systems which relate to materials in Nature
and could be the first manifestation of supersymmetry in reality. As a
corollary the five loop Wess-Zumino renormalization could be a useful
independent check on any future higher order renormalization of that system.
However, to effect such a calculation in the Gross-Neveu-Yukawa model in four
dimensions at five loops would be a massive undertaking especially given the
number of graphs that would need to be evaluated. At four loops either $7384$
or $188531$ Feynman graphs were determined in [15] where the two totals
depend on whether real or complex scalars were used together with their
respective Dirac or left and right handed Weyl fermions. These are
substantially larger numbers than the four loop ones given in Table
\ref{feynnum}. This is primarily due to the fact that unlike the component
Wess-Zumino model each interaction of the Gross-Neveu-Yukawa system has an
independent coupling constant. Consequently all the $3$- and $4$-point vertices
have to be renormalized separately in the absence of any Ward identities. One
interesting aspect of the $\epsilon$ expansion analysis was the close agreement
of the five loop estimates with other methods for the Gross-Neveu-Yukawa system
as is evident from Table \ref{expd3sum}. While the five loop results appear
competitive with the latest bootstrap estimates there is still not precise
agreement. Whether this is an indication of some discrepancy or not, such as
non-perturbative contributions outside the scope of perturbation theory, is
worth pursuing. If so it should not violate the underlying supersymmetry in the
extension from four to three dimensions in an $\epsilon$ expansion approach.
The other case where we produced exponent estimates to compare with bootstrap
methods, which concerned the one dimensional conformal manifold connected to
the XYZ model, we found values that in some instances were close to the values
quoted in [40]. This suggests that perhaps higher orders in $\epsilon$
would be necessary to produce a more accurate comparison. While we have
sketched out some basic ideas as to how a six loop computation could proceed
again such a task is not trivial. Perhaps the graphical function methods of
[31, 32, 33] offers the best direction to follow especially if the method
could be adapted to superspace in the first instance rather than have to use a
component Lagrangian. Such a six loop renormalization would give insight into
whether there are multiple zetas in the $\beta$-function of the Wess-Zumino
model. This is the order where $\zeta_{3,5}$ first appears in its
non-supersymmetric cousin $\phi^4$ theory which also has no chiral symmetry. If
it was present at this order in (\ref{lagwz}) then there would be no more

\vspace{1cm}
\noindent
{\bf Acknowledgements.} We thank I. Jack for useful discussions at various
stages of this work, H. Osborn for pointing out reference [40], D. Poland
for discussions on the conformal bootstrap formalism as well as G. Dunne for
posing several questions. It is also a pleasure to thank D.J. Broadhurst for
enlightening discussions over many years concerning multiple zetas in primitive
Feynman integrals at very high loop order. This work was supported by a DFG
Mercator Fellowship and in part through the STFC Consolidated Grants
ST/J000493/1 and ST/T000988/1. The graphs were drawn with the {\sc Axodraw}
package [105].

\appendix

\sect{Tensor definitions.}

In this appendix we define the tensors $T_{Lr}^{ij}$ that appear in the
anomalous dimension of the general action (\ref{lagwzgend}). Each of these
tensors depends on the tensor couplings $d^{ijk}$ and $\bar{d}^{ijk}$. The
subscript of each dummy index $j_n$ in each of the definitions is in direct
correspondence to the label used in the {\sc Qgraf} electronic output that
defines the underlying graph. In particular the bridge between $T_{Lr}^{ij}$
and the settings of the {\tt qgraf.dat} file in partnership with the
{\tt form.sty} style file is to use the {\tt notadp} and {\tt onepi} options.
To three loops the first set of tensors is
\begin{eqnarray}
&& d^{i j_1 j_2} \bar{d}^{j j_1 j_2} ~=~ T_{11}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_5} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} ~=~ T_{21}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_7} d^{j_4 j_6 j_8} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} ~=~ T_{31}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_7 j_8} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} ~=~ T_{32}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_8 j_6 j_7} ~=~ T_{33}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_8} ~=~ T_{34}^{i j} ~.
\end{eqnarray}
For orientation $T_{11}^{ij}$ and $T_{21}^{ij}$ correspond to the graphs of
Figures \ref{figtw1} and \ref{figtw2} respectively while $T_{31}^{ij}$ is the
non-planar graph of Figure \ref{figtw3}. We note that in [91] a factor of
$\half$ was included in the definition of the tensor corresponding to
$T_{11}^{ij}$. At four loops the $13$ tensors are
\begin{eqnarray}
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_7 j_{10}} d^{j_6 j_8 j_{11}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{11}} ~=~ T_{41}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_6 j_{10}} d^{j_{11} j_7 j_8} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{11}} ~=~ T_{42}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{10} j_{11}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_6 j_{10}} \bar{d}^{j_{11} j_7 j_8} ~=~ T_{43}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_7} d^{j_4 j_6 j_9} d^{j_8 j_{10} j_{11}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{11}} ~=~ T_{44}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_9 j_{10}} d^{j_7 j_8 j_{11}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_{11} j_9 j_{10}} ~=~ T_{45}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_7 j_9} d^{j_8 j_{10} j_{11}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{11}} ~=~ T_{46}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_{10}} \bar{d}^{j_7 j_9 j_{11}} ~=~ T_{47}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_9} \bar{d}^{j_7 j_{10} j_{11}} ~=~ T_{48}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{10} j_{11}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_7 j_{10} j_{11}} \bar{d}^{j_8 j_6 j_9} ~=~ T_{49}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_7 j_{10} j_{11}} d^{j_8 j_6 j_9} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_5 j_8} \bar{d}^{j_9 j_{10} j_{11}} ~=~ T_{410}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{10} j_{11}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_9} \bar{d}^{j_8 j_{10} j_{11}} ~=~ T_{411}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_{11} j_9 j_{10}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_8 j_{11}} ~=~ T_{412}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_9 j_{10} j_{11}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_9} \bar{d}^{j_8 j_{10} j_{11}} ~=~ T_{413}^{i j}
\end{eqnarray}
where $T_{44}^{ij}$ and $T_{41}^{ij}$ respectively correspond to the graphs in
the bottom row of Figure \ref{figtw4}.

At five loops the $63$ different tensors are
\begin{eqnarray}
&& d^{i j_1 j_2} d^{j_3 j_9 j_{10}} d^{j_4 j_{11} j_{12}} d^{j_5 j_7 j_{13}} d^{j_6 j_8 j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_{13} j_9 j_{11}} \bar{d}^{j_{14} j_{10} j_{12}} ~=~ T_{51}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_9 j_{10}} d^{j_4 j_{11} j_{12}} d^{j_5 j_6 j_{13}} d^{j_7 j_8 j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_{13} j_9 j_{11}} \bar{d}^{j_{14} j_{10} j_{12}} ~=~ T_{52}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_{13} j_9 j_{11}} d^{j_{14} j_{10} j_{12}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_9 j_{10}} \bar{d}^{j_4 j_{11} j_{12}} \bar{d}^{j_5 j_6 j_{13}} \bar{d}^{j_7 j_8 j_{14}} ~=~ T_{53}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_9 j_{10}} d^{j_4 j_{11} j_{12}} d^{j_5 j_6 j_{13}} d^{j_7 j_8 j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_{13} j_9 j_{10}} \bar{d}^{j_{14} j_{11} j_{12}} ~=~ T_{54}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_{10} j_{11}} d^{j_6 j_{12} j_{13}} d^{j_7 j_8 j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{14} j_{12}} \bar{d}^{j_{13} j_{10} j_{11}} ~=~ T_{55}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_{10} j_{11}} d^{j_6 j_{12} j_{13}} d^{j_7 j_8 j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{12} j_{13}} \bar{d}^{j_{14} j_{10} j_{11}} ~=~ T_{56}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_{10} j_{11}} d^{j_6 j_7 j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{13} j_{14}} \bar{d}^{j_{12} j_{10} j_{11}} ~=~ T_{57}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{13} j_{14}} d^{j_{12} j_{10} j_{11}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_{10} j_{11}} \bar{d}^{j_6 j_7 j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{58}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_{10} j_{11}} d^{j_6 j_7 j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{12} j_{13}} \bar{d}^{j_{14} j_{10} j_{11}} ~=~ T_{59}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{12} j_{13}} d^{j_{14} j_{10} j_{11}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_{10} j_{11}} \bar{d}^{j_6 j_7 j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{510}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_{10} j_{11}} d^{j_6 j_7 j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{13}} \bar{d}^{j_{12} j_{11} j_{14}} ~=~ T_{511}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_{10} j_{11}} d^{j_6 j_7 j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{12} j_{10}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{512}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_7 j_{10}} d^{j_6 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{13} j_{14}} \bar{d}^{j_{10} j_{11} j_{12}} ~=~ T_{513}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_7 j_{10}} d^{j_6 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{13}} \bar{d}^{j_{10} j_{12} j_{14}} ~=~ T_{514}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_7 j_{10}} d^{j_6 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{12}} \bar{d}^{j_{10} j_{13} j_{14}} ~=~ T_{515}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_7 j_{10}} d^{j_6 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{12}} \bar{d}^{j_{10} j_{13} j_{14}} ~=~ T_{516}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{11} j_{12}} d^{j_{10} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_7 j_{10}} \bar{d}^{j_6 j_8 j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{517}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_7 j_{10}} d^{j_6 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{518}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_6 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{13}} \bar{d}^{j_{10} j_{12} j_{14}} ~=~ T_{519}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{11} j_{13}} d^{j_{10} j_{12} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_6 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{520}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_6 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{12}} \bar{d}^{j_{10} j_{13} j_{14}} ~=~ T_{521}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{11} j_{12}} d^{j_{10} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_6 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{522}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_6 j_{10}} d^{j_8 j_{13} j_{14}} d^{j_{11} j_7 j_{12}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{523}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_6 j_{10}} \bar{d}^{j_8 j_{13} j_{14}} \bar{d}^{j_{11} j_7 j_{12}} ~=~ T_{524}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_6 j_{10}} d^{j_7 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{12}} \bar{d}^{j_{10} j_{13} j_{14}} ~=~ T_{525}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{11} j_{12}} d^{j_{10} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_6 j_{10}} \bar{d}^{j_7 j_8 j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{526}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_9} d^{j_4 j_6 j_{10}} d^{j_7 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{527}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_9 j_{10} j_{12}} d^{j_{11} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_9} \bar{d}^{j_4 j_6 j_{10}} \bar{d}^{j_7 j_8 j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{528}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_7} d^{j_4 j_6 j_9} d^{j_8 j_{10} j_{11}} d^{j_{14} j_{12} j_{13}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{12} j_{13}} \bar{d}^{j_{10} j_{11} j_{14}} ~=~ T_{529}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_7} d^{j_4 j_6 j_9} d^{j_8 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{530}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_9 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{13}} \bar{d}^{j_{10} j_{12} j_{14}} ~=~ T_{531}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_9 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{11} j_{12}} \bar{d}^{j_{10} j_{13} j_{14}} ~=~ T_{532}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_9 j_{10}} d^{j_7 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_{10} j_{13} j_{14}} \bar{d}^{j_{11} j_9 j_{12}} ~=~ T_{533}^{i j} \nonumber \\
&& d^{i j_3 j_4} d^{j_1 j_5 j_6} d^{j_2 j_7 j_8} d^{j_{10} j_{13} j_{14}} d^{j_{11} j_9 j_{12}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_5 j_6} \bar{d}^{j_4 j_9 j_{10}} \bar{d}^{j_7 j_8 j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{534}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_9 j_{10}} d^{j_7 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{535}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_7 j_9} d^{j_8 j_{10} j_{11}} d^{j_{14} j_{12} j_{13}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{12} j_{13}} \bar{d}^{j_{10} j_{11} j_{14}} ~=~ T_{536}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_5 j_6} d^{j_4 j_7 j_9} d^{j_8 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_3 j_4} \bar{d}^{j_1 j_5 j_6} \bar{d}^{j_2 j_7 j_8} \bar{d}^{j_9 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{537}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_{12}} \bar{d}^{j_7 j_{13} j_{14}} \bar{d}^{j_9 j_{10} j_{11}} ~=~ T_{538}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_{12}} \bar{d}^{j_7 j_{10} j_{13}} \bar{d}^{j_9 j_{11} j_{14}} ~=~ T_{539}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_{12}} \bar{d}^{j_7 j_9 j_{13}} \bar{d}^{j_{14} j_{10} j_{11}} ~=~ T_{540}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_6 j_8 j_{12}} d^{j_7 j_9 j_{13}} d^{j_{14} j_{10} j_{11}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_8 j_9} \bar{d}^{j_5 j_{10} j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{541}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_{10}} \bar{d}^{j_7 j_9 j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{542}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} d^{j_{14} j_{12} j_{13}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_9} \bar{d}^{j_7 j_{12} j_{13}} \bar{d}^{j_{10} j_{11} j_{14}} ~=~ T_{543}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_8 j_9} d^{j_5 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_8 j_9} \bar{d}^{j_7 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{544}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{11} j_{13}} d^{j_{10} j_{12} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{545}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_6 j_9 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_5 j_8} \bar{d}^{j_9 j_{11} j_{13}} \bar{d}^{j_{10} j_{12} j_{14}} ~=~ T_{546}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{10} j_{13}} d^{j_{14} j_{11} j_{12}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{547}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_6 j_9 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_5 j_8} \bar{d}^{j_9 j_{10} j_{13}} \bar{d}^{j_{14} j_{11} j_{12}} ~=~ T_{548}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{10} j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{549}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_6 j_9 j_{10}} d^{j_7 j_{11} j_{12}} d^{j_8 j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_5 j_8} \bar{d}^{j_9 j_{10} j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{550}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{12} j_{13}} d^{j_{10} j_{11} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_7 j_{10} j_{11}} \bar{d}^{j_8 j_6 j_9} \bar{d}^{j_{14} j_{12} j_{13}} ~=~ T_{551}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_7 j_{10} j_{11}} d^{j_8 j_6 j_9} d^{j_{14} j_{12} j_{13}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_5 j_8} \bar{d}^{j_9 j_{12} j_{13}} \bar{d}^{j_{10} j_{11} j_{14}} ~=~ T_{552}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{10} j_{12}} d^{j_{11} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_7 j_{10} j_{11}} \bar{d}^{j_8 j_6 j_9} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{553}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_7 j_{10} j_{11}} d^{j_8 j_6 j_9} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_6 j_7} \bar{d}^{j_4 j_5 j_8} \bar{d}^{j_9 j_{10} j_{12}} \bar{d}^{j_{11} j_{13} j_{14}} ~=~ T_{554}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{12} j_{13}} d^{j_{10} j_{11} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_9} \bar{d}^{j_8 j_{10} j_{11}} \bar{d}^{j_{14} j_{12} j_{13}} ~=~ T_{555}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_6 j_7} d^{j_4 j_5 j_8} d^{j_9 j_{10} j_{12}} d^{j_{11} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_9} \bar{d}^{j_8 j_{10} j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{556}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_9 j_{11} j_{13}} d^{j_{10} j_{12} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{557}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_9 j_{11} j_{12}} d^{j_{10} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_{11} j_{12}} \bar{d}^{j_8 j_{13} j_{14}} ~=~ T_{558}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_{10} j_{13} j_{14}} d^{j_{11} j_9 j_{12}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_8 j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{559}^{i j} \nonumber \\
&& d^{i j_1 j_3} d^{j_2 j_4 j_5} d^{j_6 j_9 j_{10}} d^{j_7 j_8 j_{11}} d^{j_{12} j_{13} j_{14}} \bar{d}^{j j_1 j_2} \bar{d}^{j_3 j_4 j_6} \bar{d}^{j_5 j_7 j_8} \bar{d}^{j_{10} j_{13} j_{14}} \bar{d}^{j_{11} j_9 j_{12}} ~=~ T_{560}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_9 j_{10} j_{12}} d^{j_{11} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_9 j_{10}} \bar{d}^{j_7 j_8 j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{561}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_9 j_{12} j_{13}} d^{j_{10} j_{11} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_9} \bar{d}^{j_8 j_{10} j_{11}} \bar{d}^{j_{14} j_{12} j_{13}} ~=~ T_{562}^{i j} \nonumber \\
&& d^{i j_1 j_2} d^{j_3 j_4 j_6} d^{j_5 j_7 j_8} d^{j_9 j_{10} j_{12}} d^{j_{11} j_{13} j_{14}} \bar{d}^{j j_1 j_3} \bar{d}^{j_2 j_4 j_5} \bar{d}^{j_6 j_7 j_9} \bar{d}^{j_8 j_{10} j_{11}} \bar{d}^{j_{12} j_{13} j_{14}} ~=~ T_{563}^{i j} ~.
\end{eqnarray}
Again to assist with orientation the graphs in the top row of Figure
\ref{figpr5} are respectively $T_{511}^{ij}$ and $T_{514}^{ij}$. Those of the
lower row correspond to the tensors $T_{51}^{ij}$ and $T_{519}^{ij}$.

\sect{Renormalization constants.}

In this appendix we record the explicit form of the wave function
renormalization constant for the action with the general tensor couplings
(\ref{lagwzgend}). This is primarily to illustrate the structure of such a
tensor renormalization constant as well as to provide the numerical value of
each pole in $\epsilon$ for each tensor. To record the result in a compact way
we decompose the renormalization constant $Z^{ij}_\Phi$ into a basis of tensors
as well as the residues of the respective poles giving
\begin{equation}
Z^{ij}_\Phi ~=~ \delta^{ij} ~+~
\sum_{L=1}^5 \sum_{q=0}^L \sum_{r=1}^{k_L} a^{\cal S}_{Lq|Lr} T^{ij}_{Lr}
\frac{1}{\epsilon^q} ~+~
\sum_{L=2}^5 \sum_{q=0}^L \sum_{r=1}^{d_L} b^{\cal S}_{Lq|Lr} D^{ij}_{Lr}
\frac{1}{\epsilon^q}
\end{equation}
where $k_L$ is defined in the last column of Table \ref{feynnum}. The
coefficients $a^{\cal S}_{Lq|Lr}$ and $b^{\cal S}_{Lq|Lr}$ have pairs of
labels. The first pair identifies the loop order and the power of the
$\epsilon$ pole while the second pair relates to the relevant tensor. The label
${\cal S}$ denotes either the $\MSbar$ or MOM scheme. Clearly
$a^{\MSbars}_{L0|Lr}$~$=$~$0$ and $b^{\MSbars}_{L0|Lr}$~$=$~$0$ as $q$~$=$~$0$
would indicate the finite part of the renormalization constant.

In addition to the tensors $T_{Lr}^{ij}$ that ultimately appear in the related
renormalization group functions, other ones arise for poles in $\epsilon$ of
order higher than the simple one. These are denoted by $D_{Lr}^{ij}$ and those
that arise to five loops are
\begin{eqnarray}
D_{21}^{ij} &=& \left( T_{11}^2 \right)^{ij} ~~,~~
D_{31}^{ij} ~=~ \left( T_{11}^3 \right)^{ij} ~~,~~
D_{32}^{ij} ~=~ \left( T_{21} T_{11} \right)^{ij} \nonumber \\
D_{41}^{ij} &=& \left( T_{11}^4 \right)^{ij} ~~,~~
D_{42}^{ij} ~=~ \left( T_{21}^2 \right)^{ij} ~~,~~
D_{43}^{ij} ~=~ \left( T_{21} T_{11}^2 \right)^{ij} ~~,~~
D_{44}^{ij} ~=~ \left( T_{31} T_{11} \right)^{ij} \nonumber \\
D_{45}^{ij} &=& \left( T_{32} T_{11} \right)^{ij} ~~,~~
D_{46}^{ij} ~=~ \left( T_{33} T_{11} \right)^{ij} ~~,~~
D_{47}^{ij} ~=~ \left( T_{34} T_{11} \right)^{ij} \nonumber \\
D_{51}^{ij} &=& \left( T_{11}^5 \right)^{ij} ~~,~~
D_{52}^{ij} ~=~ \left( T_{21}^2 \right)^{ij} ~~,~~
D_{53}^{ij} ~=~ \left( T_{21} T_{11}^3 \right)^{ij} ~~,~~
D_{54}^{ij} ~=~ \left( T_{21}^2 T_{11} \right)^{ij} \nonumber \\
D_{55}^{ij} &=& \left( T_{31} T_{21} \right)^{ij} ~~,~~
D_{56}^{ij} ~=~ \left( T_{32} T_{21} \right)^{ij} ~~,~~
D_{57}^{ij} ~=~ \left( T_{33} T_{21} \right)^{ij} ~~,~~
D_{58}^{ij} ~=~ \left( T_{34} T_{21} \right)^{ij} \nonumber \\
D_{59}^{ij} &=& \left( T_{31} T_{11}^2 \right)^{ij} ~~,~~
D_{510}^{ij} ~=~ \left( T_{32} T_{11}^2 \right)^{ij} ~~,~~
D_{511}^{ij} ~=~ \left( T_{33} T_{11}^2 \right)^{ij} ~~,~~
D_{512}^{ij} ~=~ \left( T_{34} T_{11}^2 \right)^{ij} \nonumber \\
D_{513}^{ij} &=& \left( T_{41} T_{11} \right)^{ij} ~~,~~
D_{514}^{ij} ~=~ \left( T_{42} T_{11} \right)^{ij} ~~,~~
D_{515}^{ij} ~=~ \left( T_{43} T_{11} \right)^{ij} ~~,~~
D_{516}^{ij} ~=~ \left( T_{44} T_{11} \right)^{ij} \nonumber \\
D_{517}^{ij} &=& \left( T_{45} T_{11} \right)^{ij} ~~,~~
D_{518}^{ij} ~=~ \left( T_{46} T_{11} \right)^{ij} ~~,~~
D_{519}^{ij} ~=~ \left( T_{47} T_{11} \right)^{ij} ~~,~~
D_{520}^{ij} ~=~ \left( T_{48} T_{11} \right)^{ij} \nonumber \\
D_{521}^{ij} &=& \left( T_{49} T_{11} \right)^{ij} ~~,~~
D_{522}^{ij} ~=~ \left( T_{410} T_{11} \right)^{ij} ~~,~~
D_{523}^{ij} ~=~ \left( T_{411} T_{11} \right)^{ij} ~~,~~
D_{524}^{ij} ~=~ \left( T_{412} T_{11} \right)^{ij} \nonumber \\
D_{525}^{ij} &=& \left( T_{413} T_{11} \right)^{ij} ~.
\end{eqnarray}
Graphically these correspond to the product of one-particle irreducible graphs.
Their coefficients in the Laurent expansion in $\epsilon$ are determined by
lower loop orders consistent with the renormalization group function.

For the $\MSbar$ scheme the residue of the poles to three loops are
\begin{eqnarray}
a^{\MSbars}_{11|11} &=& - \frac{1}{4} ~~,~~
a^{\MSbars}_{22|21} ~=~ - \frac{1}{8} ~~,~~
a^{\MSbars}_{21|21} ~=~ \frac{1}{8} ~~,~~
a^{\MSbars}_{33|32} ~=~ - \frac{1}{48} ~~,~~
a^{\MSbars}_{33|33} ~=~ - \frac{1}{24} \nonumber \\
a^{\MSbars}_{33|34} &=& - \frac{1}{24} ~~,~~
a^{\MSbars}_{32|32} ~=~ \frac{1}{48} ~~,~~
a^{\MSbars}_{32|33} ~=~ \frac{1}{24} ~~,~~
a^{\MSbars}_{32|34} ~=~ \frac{1}{8} ~~,~~
a^{\MSbars}_{31|31} ~=~ - \frac{1}{4} \zeta_3 \nonumber \\
a^{\MSbars}_{31|32} &=& \frac{1}{48} ~~,~~
a^{\MSbars}_{31|33} ~=~ \frac{1}{24} ~~,~~
a^{\MSbars}_{31|34} ~=~ - \frac{1}{6}
\end{eqnarray}
with those at four loop being given by
\begin{eqnarray}
a^{\MSbars}_{44|45} &=& - \frac{1}{64} ~~,~~
a^{\MSbars}_{44|46} ~=~ - \frac{1}{64} ~~,~~
a^{\MSbars}_{44|48} ~=~ - \frac{1}{192} ~~,~~
a^{\MSbars}_{44|49} ~=~ - \frac{1}{64} \nonumber \\
a^{\MSbars}_{44|410} &=& - \frac{1}{64} ~~,~~
a^{\MSbars}_{44|411} ~=~ - \frac{1}{64} ~~,~~
a^{\MSbars}_{44|412} ~=~ - \frac{1}{96} ~~,~~
a^{\MSbars}_{44|413} ~=~ - \frac{1}{96} \nonumber \\
a^{\MSbars}_{43|45} &=& \frac{1}{64} ~~,~~
a^{\MSbars}_{43|46} ~=~ \frac{1}{24} ~~,~~
a^{\MSbars}_{43|48} ~=~ \frac{1}{64} ~~,~~
a^{\MSbars}_{43|49} ~=~ \frac{1}{24} ~~,~~
a^{\MSbars}_{43|410} ~=~ \frac{1}{24} \nonumber \\
a^{\MSbars}_{43|411} &=& \frac{1}{64} ~~,~~
a^{\MSbars}_{43|412} ~=~ \frac{1}{32} ~~,~~
a^{\MSbars}_{43|413} ~=~ \frac{1}{16} \nonumber \\
a^{\MSbars}_{42|42} &=& - \frac{1}{16} \zeta_3 ~~,~~
a^{\MSbars}_{42|43} ~=~ - \frac{1}{16} \zeta_3 ~~,~~
a^{\MSbars}_{42|44} ~=~ - \frac{1}{8} \zeta_3 ~~,~~
a^{\MSbars}_{42|45} ~=~ \frac{1}{64} \nonumber \\
a^{\MSbars}_{42|46} &=& - \frac{5}{192} ~~,~~
a^{\MSbars}_{42|47} ~=~ - \frac{3}{16} \zeta_3 ~~,~~
a^{\MSbars}_{42|48} ~=~ - \frac{1}{192} ~~,~~
a^{\MSbars}_{42|49} ~=~ - \frac{5}{192} \nonumber \\
a^{\MSbars}_{42|410} &=& - \frac{5}{192} ~~,~~
a^{\MSbars}_{42|411} ~=~ \frac{1}{64} ~~,~~
a^{\MSbars}_{42|412} ~=~ - \frac{1}{96} ~~,~~
a^{\MSbars}_{42|413} ~=~ - \frac{19}{96} \nonumber \\
a^{\MSbars}_{41|41} &=& \frac{5}{4} \zeta_5 ~~,~~
a^{\MSbars}_{41|42} ~=~ - \frac{3}{32} \zeta_4 + \frac{3}{16} \zeta_3 ~~,~~
a^{\MSbars}_{41|43} ~=~ - \frac{3}{32} \zeta_4 + \frac{3}{16} \zeta_3 \nonumber \\
a^{\MSbars}_{41|44} &=& - \frac{3}{16} \zeta_4 + \frac{3}{8} \zeta_3 ~~,~~
a^{\MSbars}_{41|45} ~=~ \frac{1}{64} - \frac{1}{32} \zeta_3 ~~,~~
a^{\MSbars}_{41|46} ~=~ - \frac{1}{24} \nonumber \\
a^{\MSbars}_{41|47} &=& \frac{3}{32} \zeta_4 + \frac{3}{16} \zeta_3 ~~,~~
a^{\MSbars}_{41|48} ~=~ - \frac{5}{192} ~~,~~
a^{\MSbars}_{41|49} ~=~ - \frac{1}{24} ~~,~~
a^{\MSbars}_{41|410} ~=~ - \frac{1}{24} \nonumber \\
a^{\MSbars}_{41|411} &=& \frac{1}{64} - \frac{1}{32} \zeta_3 ~~,~~
a^{\MSbars}_{41|412} ~=~ - \frac{5}{96} + \frac{1}{16} \zeta_3 ~~,~~
a^{\MSbars}_{41|413} ~=~ \frac{5}{16} ~.
\end{eqnarray}
The coefficients of the connected higher pole tensors are
\begin{eqnarray}
b^{\MSbars}_{22|21} &=& - \frac{1}{32} ~~,~~
b^{\MSbars}_{33|31} ~=~ - \frac{1}{128} ~~,~~
b^{\MSbars}_{33|32} ~=~ - \frac{1}{32} ~~,~~
b^{\MSbars}_{32|32} ~=~ \frac{1}{32} \nonumber \\
b^{\MSbars}_{44|41} &=& - \frac{5}{2048} ~~,~~
b^{\MSbars}_{44|42} ~=~ - \frac{1}{128} ~~,~~
b^{\MSbars}_{44|43} ~=~ - \frac{3}{256} ~~,~~
b^{\MSbars}_{44|45} ~=~ - \frac{1}{192} \nonumber \\
b^{\MSbars}_{44|46} &=& - \frac{1}{96} ~~,~~
b^{\MSbars}_{44|47} ~=~ - \frac{1}{96} \nonumber \\
b^{\MSbars}_{43|42} &=& \frac{1}{64} ~~,~~
b^{\MSbars}_{43|43} ~=~ \frac{3}{256} ~~,~~
b^{\MSbars}_{43|45} ~=~ \frac{1}{192} ~~,~~
b^{\MSbars}_{43|46} ~=~ \frac{1}{96} ~~,~~
b^{\MSbars}_{43|47} ~=~ \frac{1}{32} \nonumber \\
b^{\MSbars}_{42|42} &=& - \frac{1}{128} ~~,~~
b^{\MSbars}_{42|44} ~=~ - \frac{1}{16} \zeta_3 ~~,~~
b^{\MSbars}_{42|45} ~=~ \frac{1}{192} ~~,~~
b^{\MSbars}_{42|46} ~=~ \frac{1}{96} \nonumber \\
b^{\MSbars}_{42|47} &=& - \frac{1}{24}
\end{eqnarray}
where obviously there can be no one loop coefficient.

Given that there are more tensors at five loops we record the data for this
part of $Z_\Phi^{ij}$ by the order of the pole. First, the leading pole
coefficients are
\begin{eqnarray}
a^{\MSbars}_{55|54} &=& - \frac{1}{320} ~~,~~
a^{\MSbars}_{55|56} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|515} ~=~ - \frac{1}{320} ~~,~~
a^{\MSbars}_{55|532} ~=~ - \frac{1}{480} \nonumber \\
a^{\MSbars}_{55|533} &=& - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|534} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|535} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|536} ~=~ - \frac{1}{240} \nonumber \\
a^{\MSbars}_{55|537} &=& - \frac{1}{240} ~~,~~
a^{\MSbars}_{55|538} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|543} ~=~ - \frac{1}{320} ~~,~~
a^{\MSbars}_{55|544} ~=~ - \frac{1}{320} \nonumber \\
a^{\MSbars}_{55|547} &=& - \frac{1}{480} ~~,~~
a^{\MSbars}_{55|548} ~=~ - \frac{1}{480} ~~,~~
a^{\MSbars}_{55|549} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|550} ~=~ - \frac{1}{160} \nonumber \\
a^{\MSbars}_{55|551} &=& - \frac{1}{240} ~~,~~
a^{\MSbars}_{55|552} ~=~ - \frac{1}{240} ~~,~~
a^{\MSbars}_{55|553} ~=~ - \frac{1}{240} ~~,~~
a^{\MSbars}_{55|554} ~=~ - \frac{1}{240} \nonumber \\
a^{\MSbars}_{55|555} &=& - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|556} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{55|558} ~=~ - \frac{1}{960} ~~,~~
a^{\MSbars}_{55|559} ~=~ - \frac{1}{320} \nonumber \\
a^{\MSbars}_{55|560} &=& - \frac{1}{320} ~~,~~
a^{\MSbars}_{55|561} ~=~ - \frac{1}{320} ~~,~~
a^{\MSbars}_{55|562} ~=~ - \frac{1}{480} ~~,~~
a^{\MSbars}_{55|563} ~=~ - \frac{1}{480}
\end{eqnarray}
\begin{eqnarray}
a^{\MSbars}_{54|54} &=& \frac{1}{320} ~~,~~
a^{\MSbars}_{54|56} ~=~ \frac{1}{64} ~~,~~
a^{\MSbars}_{54|515} ~=~ \frac{1}{80} ~~,~~
a^{\MSbars}_{54|532} ~=~ \frac{11}{1920} \nonumber \\
a^{\MSbars}_{54|533} &=& \frac{1}{64} ~~,~~
a^{\MSbars}_{54|534} ~=~ \frac{1}{64} ~~,~~
a^{\MSbars}_{54|535} ~=~ \frac{1}{160} ~~,~~
a^{\MSbars}_{54|536} ~=~ \frac{11}{960} \nonumber \\
a^{\MSbars}_{54|537} &=& \frac{11}{480} ~~,~~
a^{\MSbars}_{54|538} ~=~ \frac{1}{40} ~~,~~
a^{\MSbars}_{54|543} ~=~ \frac{3}{320} ~~,~~
a^{\MSbars}_{54|544} ~=~ \frac{1}{60} \nonumber \\
a^{\MSbars}_{54|547} &=& \frac{11}{1920} ~~,~~
a^{\MSbars}_{54|548} ~=~ \frac{11}{1920} ~~,~~
a^{\MSbars}_{54|549} ~=~ \frac{1}{64} ~~,~~
a^{\MSbars}_{54|550} ~=~ \frac{1}{64} \nonumber \\
a^{\MSbars}_{54|551} &=& \frac{11}{960} ~~,~~
a^{\MSbars}_{54|552} ~=~ \frac{11}{960} ~~,~~
a^{\MSbars}_{54|553} ~=~ \frac{11}{480} ~~,~~
a^{\MSbars}_{54|554} ~=~ \frac{11}{480} \nonumber \\
a^{\MSbars}_{54|555} &=& \frac{1}{160} ~~,~~
a^{\MSbars}_{54|556} ~=~ \frac{1}{64} ~~,~~
a^{\MSbars}_{54|558} ~=~ \frac{1}{160} ~~,~~
a^{\MSbars}_{54|559} ~=~ \frac{1}{60} \nonumber \\
a^{\MSbars}_{54|560} &=& \frac{1}{60} ~~,~~
a^{\MSbars}_{54|561} ~=~ \frac{3}{320} ~~,~~
a^{\MSbars}_{54|562} ~=~ \frac{1}{80} ~~,~~
a^{\MSbars}_{54|563} ~=~ \frac{1}{48}
\end{eqnarray}
are the quartic pole ones. Continuing the triple pole coefficients are
\begin{eqnarray}
a^{\MSbars}_{53|563} &=& - \frac{11}{96} ~~,~~
a^{\MSbars}_{53|562} ~=~ - \frac{13}{480} ~~,~~
a^{\MSbars}_{53|561} ~=~ - \frac{1}{320} ~~,~~
a^{\MSbars}_{53|560} ~=~ - \frac{11}{320} \nonumber \\
a^{\MSbars}_{53|559} &=& - \frac{11}{320} ~~,~~
a^{\MSbars}_{53|558} ~=~ - \frac{13}{960} ~~,~~
a^{\MSbars}_{53|556} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{53|555} ~=~ \frac{1}{160} \nonumber \\
a^{\MSbars}_{53|554} &=& - \frac{7}{120} ~~,~~
a^{\MSbars}_{53|553} ~=~ - \frac{7}{120} ~~,~~
a^{\MSbars}_{53|550} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{53|549} ~=~ - \frac{1}{160} \nonumber \\
a^{\MSbars}_{53|544} &=& - \frac{11}{320} ~~,~~
a^{\MSbars}_{53|543} ~=~ - \frac{1}{320} ~~,~~
a^{\MSbars}_{53|538} ~=~ - \frac{17}{480} ~~,~~
a^{\MSbars}_{53|537} ~=~ - \frac{7}{120} \nonumber \\
a^{\MSbars}_{53|535} &=& \frac{1}{160} ~~,~~
a^{\MSbars}_{53|534} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{53|533} ~=~ - \frac{1}{160} ~~,~~
a^{\MSbars}_{53|515} ~=~ - \frac{17}{960} \nonumber \\
a^{\MSbars}_{53|56} &=& - \frac{1}{160} ~~,~~
a^{\MSbars}_{53|54} ~=~ \frac{1}{320} ~~,~~
a^{\MSbars}_{53|557} ~=~ - \frac{3}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|546} ~=~ - \frac{9}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{53|545} &=& - \frac{9}{160} \zeta_3 ~~,~~
a^{\MSbars}_{53|542} ~=~ - \frac{3}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|541} ~=~ - \frac{3}{80} \zeta_3 ~~,~~
a^{\MSbars}_{53|540} ~=~ - \frac{3}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{53|531} &=& - \frac{9}{160} \zeta_3 ~~,~~
a^{\MSbars}_{53|530} ~=~ - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|529} ~=~ - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|528} ~=~ - \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{53|527} &=& - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{53|526} ~=~ - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|525} ~=~ - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|524} ~=~ - \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{53|523} &=& - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{53|522} ~=~ - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{53|521} ~=~ - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{53|513} ~=~ - \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{53|58} &=& - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|57} ~=~ - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|55} ~=~ - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{53|53} ~=~ - \frac{1}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{53|52} &=& - \frac{1}{160} \zeta_3
\end{eqnarray}
\begin{eqnarray}
a^{\MSbars}_{52|52} &=& - \frac{3}{320} \zeta_4 + \frac{3}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|53} ~=~ - \frac{3}{320} \zeta_4 + \frac{3}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|54} ~=~ \frac{1}{320} - \frac{1}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{52|55} &=& - \frac{3}{80} \zeta_4 + \frac{3}{40} \zeta_3 ~~,~~
a^{\MSbars}_{52|56} ~=~ - \frac{3}{320} - \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|57} ~=~ - \frac{3}{80} \zeta_4 + \frac{3}{40} \zeta_3 \nonumber \\
a^{\MSbars}_{52|58} &=& - \frac{3}{80} \zeta_4 + \frac{3}{40} \zeta_3 ~~,~~
a^{\MSbars}_{52|59} ~=~ \frac{1}{4} \zeta_5 ~~,~~
a^{\MSbars}_{52|510} ~=~ \frac{1}{4} \zeta_5 ~~,~~
a^{\MSbars}_{52|512} ~=~ \frac{1}{4} \zeta_5 \nonumber \\
a^{\MSbars}_{52|513} &=& - \frac{3}{160} \zeta_4 + \frac{3}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|515} ~=~ \frac{1}{120} ~~,~~
a^{\MSbars}_{52|516} ~=~ \frac{1}{4} \zeta_5 ~~,~~
a^{\MSbars}_{52|517} ~=~ \frac{1}{4} \zeta_5 \nonumber \\
a^{\MSbars}_{52|518} &=& \frac{1}{8} \zeta_5 ~~,~~
a^{\MSbars}_{52|521} ~=~ - \frac{3}{160} \zeta_4 + \frac{3}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|522} ~=~ - \frac{3}{160} \zeta_4 + \frac{3}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{52|523} &=& - \frac{3}{160} \zeta_4 + \frac{7}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|524} ~=~ - \frac{3}{160} \zeta_4 + \frac{7}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|525} ~=~ - \frac{3}{80} \zeta_4 + \frac{3}{40} \zeta_3 \nonumber \\
a^{\MSbars}_{52|526} &=& - \frac{3}{80} \zeta_4 + \frac{3}{40} \zeta_3 ~~,~~
a^{\MSbars}_{52|527} ~=~ - \frac{3}{160} \zeta_4 + \frac{3}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|528} ~=~ - \frac{3}{160} \zeta_4 + \frac{3}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{52|529} &=& - \frac{3}{80} \zeta_4 + \frac{3}{40} \zeta_3 ~~,~~
a^{\MSbars}_{52|530} ~=~ - \frac{3}{80} \zeta_4 + \frac{7}{40} \zeta_3 ~~,~~
a^{\MSbars}_{52|531} ~=~ \frac{3}{320} \zeta_4 + \frac{9}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{52|532} &=& - \frac{19}{1920} ~~,~~
a^{\MSbars}_{52|533} ~=~ - \frac{3}{320} - \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|534} ~=~ - \frac{3}{320} - \frac{1}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{52|535} &=& \frac{1}{160} - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|536} ~=~ - \frac{19}{960} + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|537} ~=~ \frac{19}{480} ~~,~~
a^{\MSbars}_{52|538} ~=~ \frac{1}{60} \nonumber \\
a^{\MSbars}_{52|539} &=& \zeta_5 ~~,~~
a^{\MSbars}_{52|540} ~=~ - \frac{9}{160} \zeta_4 + \frac{3}{16} \zeta_3 ~~,~~
a^{\MSbars}_{52|541} ~=~ - \frac{9}{160} \zeta_4 + \frac{3}{16} \zeta_3 \nonumber \\
a^{\MSbars}_{52|542} &=& -\frac{ 9}{80} \zeta_4 + \frac{3}{8} \zeta_3 ~~,~~
a^{\MSbars}_{52|543} ~=~ - \frac{1}{320} - \frac{3}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|544} ~=~ \frac{1}{240} \nonumber \\
a^{\MSbars}_{52|545} &=& \frac{3}{320} \zeta_4 + \frac{9}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|546} ~=~ \frac{3}{320} \zeta_4 + \frac{9}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|547} ~=~ - \frac{19}{1920} \nonumber \\
a^{\MSbars}_{52|548} &=& - \frac{19}{1920} ~~,~~
a^{\MSbars}_{52|549} ~=~ - \frac{3}{320} - \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|550} ~=~ - \frac{3}{320} - \frac{1}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{52|551} &=& - \frac{19}{960} + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|552} ~=~ - \frac{19}{960} + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|553} ~=~ \frac{19}{480} \nonumber \\
a^{\MSbars}_{52|554} &=& \frac{19}{480} ~~,~~
a^{\MSbars}_{52|555} ~=~ \frac{1}{160} - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|556} ~=~ - \frac{3}{320} - \frac{1}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{52|557} &=& \frac{3}{40} \zeta_4 + \frac{9}{40} \zeta_3 ~~,~~
a^{\MSbars}_{52|558} ~=~ \frac{1}{240} ~~,~~
a^{\MSbars}_{52|559} ~=~ \frac{1}{240} + \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{52|560} &=& \frac{1}{240} + \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{52|561} ~=~ - \frac{1}{320} - \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{52|562} ~=~ \frac{1}{120} + \frac{1}{20} \zeta_3 \nonumber \\
a^{\MSbars}_{52|563} &=& \frac{19}{48}
\end{eqnarray}
as the coefficients for the double pole. Finally the simple poles that lead to
$\gamma_\Phi^{ij}$ are
\begin{eqnarray}
a^{\MSbars}_{51|51} &=& - \frac{9}{20} \zeta_3^2 ~~,~~
a^{\MSbars}_{51|52} ~=~ \frac{143}{160} \zeta_5 + \frac{9}{320} \zeta_4 - \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|53} ~=~ \frac{143}{160} \zeta_5 + \frac{9}{320} \zeta_4 - \frac{1}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{51|54} &=& \frac{1}{320} - \frac{3}{320} \zeta_4 + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|55} ~=~ - \frac{67}{40} \zeta_5 + \frac{9}{80} \zeta_4 - \frac{1}{40} \zeta_3 \nonumber \\
a^{\MSbars}_{51|56} &=& - \frac{3}{160} - \frac{3}{320} \zeta_4 + \frac{1}{32} \zeta_3 ~~,~~
a^{\MSbars}_{51|57} ~=~ - \frac{9}{5} \zeta_5 + \frac{9}{80} \zeta_4 - \frac{1}{40} \zeta_3 \nonumber \\
a^{\MSbars}_{51|58} &=& - \frac{9}{5} \zeta_5 + \frac{9}{80} \zeta_4 - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{51|59} ~=~ \frac{5}{8} \zeta_6 - \frac{5}{4} \zeta_5 - \frac{1}{20} \zeta_3^2 \nonumber \\
a^{\MSbars}_{51|510} &=& \frac{5}{8} \zeta_6 - \frac{5}{4} \zeta_5 - \frac{1}{20} \zeta_3^2 ~~,~~
a^{\MSbars}_{51|511} ~=~ - \frac{441}{80} \zeta_7 ~~,~~
a^{\MSbars}_{51|512} ~=~ \frac{5}{8} \zeta_6 - \frac{5}{4} \zeta_5 - \frac{1}{20} \zeta_3^2 \nonumber \\
a^{\MSbars}_{51|513} &=& \frac{79}{40} \zeta_5 + \frac{9}{160} \zeta_4 - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|514} ~=~ - \frac{441}{160} \zeta_7 ~~,~~
a^{\MSbars}_{51|515} ~=~ - \frac{1}{60} \nonumber \\
a^{\MSbars}_{51|516} &=& \frac{5}{8} \zeta_6 - \frac{5}{4} \zeta_5 + \frac{1}{4} \zeta_3^2 ~~,~~
a^{\MSbars}_{51|517} ~=~ \frac{5}{8} \zeta_6 - \frac{5}{4} \zeta_5 + \frac{1}{4} \zeta_3^2 \nonumber \\
a^{\MSbars}_{51|518} &=& \frac{5}{16} \zeta_6 - \frac{5}{8} \zeta_5 + \frac{11}{40} \zeta_3^2 ~~,~~
a^{\MSbars}_{51|519} ~=~ - \frac{9}{10} \zeta_3^2 ~~,~~
a^{\MSbars}_{51|520} ~=~ - \frac{9}{10} \zeta_3^2 \nonumber \\
a^{\MSbars}_{51|521} &=& \frac{153}{80} \zeta_5 + \frac{9}{160} \zeta_4 - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|522} ~=~ \frac{153}{80} \zeta_5 + \frac{9}{160} \zeta_4 - \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{51|523} &=& \frac{143}{80} \zeta_5 + \frac{21}{160} \zeta_4 - \frac{31}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|524} ~=~ \frac{143}{80} \zeta_5 + \frac{21}{160} \zeta_4 - \frac{31}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{51|525} &=& - \frac{9}{5} \zeta_5 + \frac{9}{80} \zeta_4 - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{51|526} ~=~ - \frac{9}{5} \zeta_5 + \frac{9}{80} \zeta_4 - \frac{1}{40} \zeta_3 \nonumber \\
a^{\MSbars}_{51|527} &=& \frac{143}{80} \zeta_5 + \frac{9}{160} \zeta_4 - \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|528} ~=~ \frac{143}{80} \zeta_5 + \frac{9}{160} \zeta_4 - \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{51|529} &=& - \frac{41}{20} \zeta_5 + \frac{9}{80} \zeta_4 - \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{51|530} ~=~ - \frac{41}{20} \zeta_5 + \frac{21}{80} \zeta_4 - \frac{31}{40} \zeta_3 \nonumber \\
a^{\MSbars}_{51|531} &=& - \frac{1}{20} \zeta_5 - \frac{3}{320} \zeta_4 + \frac{13}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|533} ~=~ - \frac{3}{160} - \frac{3}{320} \zeta_4 + \frac{1}{32} \zeta_3 \nonumber \\
a^{\MSbars}_{51|534} &=& - \frac{3}{160} - \frac{3}{320} \zeta_4 + \frac{1}{32} \zeta_3 ~~,~~
a^{\MSbars}_{51|535} ~=~ \frac{1}{160} - \frac{3}{160} \zeta_4 + \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{51|536} &=& \frac{3}{320} \zeta_4 + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|537} ~=~ \frac{7}{60} ~~,~~
a^{\MSbars}_{51|538} ~=~ - \frac{1}{30} \nonumber \\
a^{\MSbars}_{51|539} &=& - \frac{5}{8} \zeta_6 - \frac{3}{4} \zeta_5 + \frac{1}{20} \zeta_3^2 ~~,~~
a^{\MSbars}_{51|540} ~=~ - \frac{1}{80} \zeta_5 - \frac{1}{5} \zeta_3 ~~,~~
a^{\MSbars}_{51|541} ~=~ - \frac{1}{80} \zeta_5 - \frac{1}{5} \zeta_3 \nonumber \\
a^{\MSbars}_{51|542} &=& \frac{9}{40} \zeta_5 - \frac{2}{5} \zeta_3 ~~,~~
a^{\MSbars}_{51|543} ~=~ - \frac{3}{160} + \frac{3}{160} \zeta_4 + \frac{1}{40} \zeta_3 ~~,~~
a^{\MSbars}_{51|544} ~=~ \frac{1}{12} \nonumber \\
a^{\MSbars}_{51|545} &=& - \frac{1}{20} \zeta_5 - \frac{3}{320} \zeta_4 + \frac{13}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|546} ~=~ - \frac{1}{20} \zeta_5 - \frac{3}{320} \zeta_4 + \frac{13}{160} \zeta_3 \nonumber \\
a^{\MSbars}_{51|549} &=& - \frac{3}{160} - \frac{3}{320} \zeta_4 + \frac{1}{32} \zeta_3 ~~,~~
a^{\MSbars}_{51|550} ~=~ - \frac{3}{160} - \frac{3}{320} \zeta_4 + \frac{1}{32} \zeta_3 \nonumber \\
a^{\MSbars}_{51|551} &=& \frac{3}{320} \zeta_4 + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|552} ~=~ \frac{3}{320} \zeta_4 + \frac{1}{160} \zeta_3 ~~,~~
a^{\MSbars}_{51|553} ~=~ \frac{7}{60} ~~,~~
a^{\MSbars}_{51|554} ~=~ \frac{7}{60} \nonumber \\
a^{\MSbars}_{51|555} &=& \frac{1}{160} - \frac{3}{160} \zeta_4 + \frac{1}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|556} ~=~ - \frac{3}{160} - \frac{3}{320} \zeta_4 + \frac{1}{32} \zeta_3 \nonumber \\
a^{\MSbars}_{51|557} &=& \frac{1}{10} \zeta_5 - \frac{21}{160} \zeta_4 - \frac{5}{16} \zeta_3 ~~,~~
a^{\MSbars}_{51|558} ~=~ \frac{1}{80} ~~,~~
a^{\MSbars}_{51|559} ~=~ \frac{1}{12} + \frac{3}{160} \zeta_4 - \frac{7}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{51|560} &=& \frac{1}{12} + \frac{3}{160} \zeta_4 - \frac{7}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|561} ~=~ - \frac{3}{160} + \frac{3}{80} \zeta_4 - \frac{1}{80} \zeta_3 \nonumber \\
a^{\MSbars}_{51|562} &=& \frac{1}{40} - \frac{3}{160} \zeta_4 - \frac{9}{80} \zeta_3 ~~,~~
a^{\MSbars}_{51|563} ~=~ - \frac{7}{10}
\end{eqnarray}
from which it is straightforward to see the connection with $c^{\MSbars}_{5r}$.

For the coefficients of $D_{Lr}^{ij}$ we have
\begin{eqnarray}
b^{\MSbars}_{55|51} &=& - \frac{7}{8192} ~~,~~
b^{\MSbars}_{55|53} ~=~ - \frac{5}{1024} ~~,~~
b^{\MSbars}_{55|54} ~=~ - \frac{3}{512} ~~,~~
b^{\MSbars}_{55|56} ~=~ - \frac{1}{384} \nonumber \\
b^{\MSbars}_{55|57} &=& - \frac{1}{192} ~~,~~
b^{\MSbars}_{55|58} ~=~ - \frac{1}{192} ~~,~~
b^{\MSbars}_{55|510} ~=~ - \frac{1}{512} ~~,~~
b^{\MSbars}_{55|511} ~=~ - \frac{1}{256} \nonumber \\
b^{\MSbars}_{55|512} &=& - \frac{1}{256} ~~,~~
b^{\MSbars}_{55|517} ~=~ - \frac{1}{256} ~~,~~
b^{\MSbars}_{55|518} ~=~ - \frac{1}{256} ~~,~~
b^{\MSbars}_{55|520} ~=~ - \frac{1}{768} \nonumber \\
b^{\MSbars}_{55|521} &=& - \frac{1}{256} ~~,~~
b^{\MSbars}_{55|522} ~=~ - \frac{1}{256} ~~,~~
b^{\MSbars}_{55|523} ~=~ - \frac{1}{256} ~~,~~
b^{\MSbars}_{55|524} ~=~ - \frac{1}{384} \nonumber \\
b^{\MSbars}_{55|525} &=& - \frac{1}{384}
\end{eqnarray}
for the $\frac{1}{\epsilon^5}$ coefficients and
\begin{eqnarray}
b^{\MSbars}_{54|53} &=& \frac{5}{1024} ~~,~~
b^{\MSbars}_{54|54} ~=~ \frac{3}{256} ~~,~~
b^{\MSbars}_{54|56} ~=~ \frac{1}{192} ~~,~~
b^{\MSbars}_{54|57} ~=~ \frac{1}{96} \nonumber \\
b^{\MSbars}_{54|58} &=& \frac{1}{48} ~~,~~
b^{\MSbars}_{54|510} ~=~ \frac{1}{512} ~~,~~
b^{\MSbars}_{54|511} ~=~ \frac{1}{256} ~~,~~
b^{\MSbars}_{54|512} ~=~ \frac{3}{256} \nonumber \\
b^{\MSbars}_{54|517} &=& \frac{1}{256} ~~,~~
b^{\MSbars}_{54|518} ~=~ \frac{1}{96} ~~,~~
b^{\MSbars}_{54|520} ~=~ \frac{1}{256} ~~,~~
b^{\MSbars}_{54|521} ~=~ \frac{1}{96} \nonumber \\
b^{\MSbars}_{54|522} &=& \frac{1}{96} ~~,~~
b^{\MSbars}_{54|523} ~=~ \frac{1}{256} ~~,~~
b^{\MSbars}_{54|524} ~=~ \frac{1}{128} ~~,~~
b^{\MSbars}_{54|525} ~=~ \frac{1}{64}
\end{eqnarray}
for the next order. The remaining two sets of coefficients are
\begin{eqnarray}
b^{\MSbars}_{53|54} &=& - \frac{3}{512} ~~,~~
b^{\MSbars}_{53|55} ~=~ - \frac{1}{32} \zeta_3 ~~,~~
b^{\MSbars}_{53|58} ~=~ - \frac{7}{192} ~~,~~
b^{\MSbars}_{53|59} ~=~ - \frac{3}{128} \zeta_3 \nonumber \\
b^{\MSbars}_{53|510} &=& \frac{1}{512} ~~,~~
b^{\MSbars}_{53|511} ~=~ \frac{1}{256} ~~,~~
b^{\MSbars}_{53|512} ~=~ - \frac{1}{64} ~~,~~
b^{\MSbars}_{53|514} ~=~ - \frac{1}{64} \zeta_3 \nonumber \\
b^{\MSbars}_{53|515} &=& - \frac{1}{64} \zeta_3 ~~,~~
b^{\MSbars}_{53|516} ~=~ - \frac{1}{32} \zeta_3 ~~,~~
b^{\MSbars}_{53|517} ~=~ \frac{1}{256} ~~,~~
b^{\MSbars}_{53|518} ~=~ - \frac{5}{768} \nonumber \\
b^{\MSbars}_{53|519} &=& - \frac{3}{64} \zeta_3 ~~,~~
b^{\MSbars}_{53|520} ~=~ - \frac{1}{768} ~~,~~
b^{\MSbars}_{53|521} ~=~ - \frac{5}{768} ~~,~~
b^{\MSbars}_{53|522} ~=~ - \frac{5}{768} \nonumber \\
b^{\MSbars}_{53|523} &=& \frac{1}{256} ~~,~~
b^{\MSbars}_{53|524} ~=~ - \frac{1}{384} ~~,~~
b^{\MSbars}_{53|525} ~=~ - \frac{19}{384}
\end{eqnarray}
\begin{eqnarray}
b^{\MSbars}_{52|55} &=& \frac{1}{32} \zeta_3 ~~,~~
b^{\MSbars}_{52|56} ~=~ - \frac{1}{384} ~~,~~
b^{\MSbars}_{52|57} ~=~ - \frac{1}{192} ~~,~~
b^{\MSbars}_{52|58} ~=~ \frac{1}{48} ~~,~~
b^{\MSbars}_{52|513} ~=~ \frac{5}{16} \zeta_5 \nonumber \\
b^{\MSbars}_{52|514} &=& - \frac{3}{128} \zeta_4 + \frac{3}{64} \zeta_3 ~~,~~
b^{\MSbars}_{52|515} ~=~ - \frac{3}{128} \zeta_4 + \frac{3}{64} \zeta_3 ~~,~~
b^{\MSbars}_{52|516} ~=~ - \frac{3}{64} \zeta_4 + \frac{3}{32} \zeta_3 \nonumber \\
b^{\MSbars}_{52|517} &=& \frac{1}{256} - \frac{1}{128} \zeta_3 ~~,~~
b^{\MSbars}_{52|518} ~=~ - \frac{1}{96} ~~,~~
b^{\MSbars}_{52|519} ~=~ \frac{3}{128} \zeta_4 + \frac{3}{64} \zeta_3  \nonumber \\
b^{\MSbars}_{52|520} &=& - \frac{5}{768} ~~,~~
b^{\MSbars}_{52|521} ~=~ - \frac{1}{96} ~~,~~
b^{\MSbars}_{52|522} ~=~ - \frac{1}{96} ~~,~~
b^{\MSbars}_{52|523} ~=~ \frac{1}{256} - \frac{1}{128} \zeta_3  \nonumber \\
b^{\MSbars}_{52|524} &=& - \frac{5}{384} + \frac{1}{64} \zeta_3 ~~,~~
b^{\MSbars}_{52|525} ~=~ \frac{5}{64} ~.
\end{eqnarray}
The analogous expressions for $a^{\MOMs}_{Lq|Lr}$ and $b^{\MOMs}_{Lq|Lr}$ are
available in the attached data file.

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\end{document}