\left. \alpha^{(0)}_{\mMOMs} \right|^{SU(3)} &=&
-~ 3 + 4\sigma \Nf - 66\sigma \nonumber \\
\left. \alpha^{(1)}_{\mMOMs} \right|^{SU(3)} &=&
725274 {\sigma}^{3} + 41283 {\sigma}^{2}
+ 3881196 {\sigma}^{4} - \frac{51}{2} - 96 {\sigma}^{3}\Nf^{3}
+ 5832 {\sigma}^{3}\Nf^{2} \nonumber \\
&& -~ 114048 {\sigma}^{3}\Nf
+ 132 {\sigma}^{2}\Nf^{2} - 4680 {\sigma}^{2}\Nf
- 864 {\sigma}^{4}\Nf^{3} + 42768 {\sigma}^{4}\Nf^{2}
\nonumber \\
&& -~ 705672 {\sigma}^{4}\Nf + 31 \sigma \Nf
+ \Nf + \frac {261}{2} \sigma \nonumber\\
\left. \alpha^{(2)}_{\mMOMs} \right|^{SU(3)} &=&
-~ 6562749600 {\sigma}^{6}\Nf^{2}
+ 59382298800 {\sigma}^{6}\Nf + 103680 {\sigma}^{6}\Nf^{5}
- 9720000 {\sigma}^{6}\Nf^{4} \nonumber \\
&& +~ 359251200 {\sigma}^ {6}\Nf^{3}
- \frac{542619}{2} \zeta_3 {\sigma}^{2}
- \frac{1485}{2} \zeta_3 \sigma - 4498659 {\zeta_3} {\sigma}^{3}
\nonumber \\
&& -~ 25150150080 {\sigma}^{7}\Nf^{2}
+ 207488738160 {\sigma}^{7}\Nf + 559872 {\sigma}^{7}\Nf^{5}
- 46189440 {\sigma}^{7}\Nf^{4}
\nonumber \\
&& +~ 1524251520 {\sigma}^{7}\Nf^{3}
- 17465382 \zeta_3 {\sigma}^{4} - 684712835928 {\sigma}^{7}
- 27604036251 {\sigma}^{5}
\nonumber \\
&& -~ 213252314220 {\sigma}^{6}
+ 6841137204 {\sigma}^{5}\Nf - 12384 {\sigma}^{4}\Nf^{4}
- 662256 {\sigma}^{5}\Nf^{4}
\nonumber \\
&& +~ 30458592 {\sigma}^{5}\Nf^{3}
- 658776888 {\sigma}^{5}\Nf^{2} + 5184 {\sigma}^{5}\Nf^{5}
- \frac{78139809}{2} {\sigma}^{3}
\nonumber \\
&& +~ \frac{2146905}{4} {\sigma}^{2}
- \frac{3436534431}{2} {\sigma}^{4}
+ 3408 {\sigma}^{3}\Nf^{3} - 233370 {\sigma}^{3}\Nf^{2}
+ 5290650 {\sigma}^{3}\Nf
\nonumber \\
&& +~ 7242 {\sigma}^{2}\Nf^{2}
- \frac{271143}{2} {\sigma}^{2}\Nf
- 72 {\sigma}^{2}\Nf^{3} - \frac{782}{9} \sigma \Nf^{2}
+ 954396 {\sigma}^{4}\Nf^{3}
\nonumber \\
&& -~ 27691794 {\sigma}^{4}\Nf^{2}
+ 356848173 {\sigma}^{4}\Nf
- 6 {\zeta_3} \Nf - \frac{10}{3} \Nf^{2} - \frac{9405}{8}
- 702 \zeta_3 {\sigma}^{2}\Nf^{2}
\nonumber \\
&& +~ 28026 \zeta_3 {\sigma}^{2}\Nf
+ \frac{20}{3} \zeta_3 \sigma \Nf^{2}
- 65 \zeta_3 \sigma \Nf + 432 \zeta_3 {\sigma}^{3}\Nf^{3}
- 30780 \zeta_3 {\sigma}^{3}\Nf^{2}
\nonumber \\
&& +~ 662904 \zeta_3 {\sigma}^{3}\Nf
+ 3888 \zeta_3 {\sigma}^{4}\Nf^{3}
- 192456 \zeta_3 {\sigma}^{4}\Nf^{2}
+ 3175524 \zeta_3 {\sigma}^{4}\Nf
\nonumber \\
&& +~ \frac{6517}{4} \sigma \Nf
+ \frac{551}{4} \Nf
+ 153 {\zeta_3} + \frac{909}{8} \sigma
\end{eqnarray}

in the $\mMOM$ scheme. Up to the truncation order this gives the solution to
$\bar{\gamma}_a(a)$ $=$ $\bar{\gamma}_\alpha(a)$ allowing us to find a fixed
point $a_\infty$ with $\bar{\gamma}_a(a_\infty)$ $=$ $0$. When $\Nf$ $=$ $16$
and $S$ $=$ $1$ we have

\begin{eqnarray}
\left. \alpha^{(0)}_{\mMOMs} \right|_{\Nf=16} &=& -~ 3 \nonumber \\
\left. \alpha^{(1)}_{\mMOMs} \right|_{\Nf=16} &=& -~ 9.5 \nonumber \\
\left. \alpha^{(2)}_{\mMOMs} \right|_{\Nf=16} &=& 243.558910
\end{eqnarray}

which gives

\begin{eqnarray}
\gamma_a(a,\bar{\alpha}) &=& -~ 0.333333 a ~+~ 103.666667 a^2 ~+~
152.000000 a^3 ~+~ 1060.437500 a^4 \nonumber \\
&& +~ 1929.093750 a^5 ~+~ O(a^6)
\end{eqnarray}

to the two loop level. By this we mean the two loop solution of the equations
$\bar{\gamma}_a(a)$ $=$ $0$ and $\bar{\gamma}_\alpha(a)$ $=$ $0$. These have a
positive fixed point at $a_\infty$ $=$ $0.0032000819$ implying
$\alpha_\infty$ $=$ $-$ $3.030400777$ at this approximation order. The
respective two loop $\mMOM$ fixed point values from [42] are
$0.0032001941$ and $-$ $3.0301823312$ which shows our $\Nf$ $=$ $16$ solution
is in reasonable correspondence. We note that by solving this way we have
introduced an additional truncation in the series for $\alpha$. So it is not
unexpected that the fixed point found in this way does not result in fully the
same value. However by including higher order corrections we would expect that
there would be a degree of convergence. Repeating the exercise at the three
loop level we find two real fixed point solutions on the $(a,\alpha)$ plane
which are $(0.0031380752,-3.0274132638)$ and $(0.1624143964,1.8817663994)$. The
first of these has a clear counterpart in the fixed point analysis of [42]
which is $(0.0031380724,-3.0274210489)$ and is an infrared stable fixed point.
The second solution is not as straightforward to place with a known fixed point
in [42] but may map to the stable fixed point

2|c||$\bar{\alpha}$   2|c||[42]   2|c|Difference

$\Nf$     Loop     $a_{\infty}$     $\alpha_{\infty}$    $a_{\infty}$     $\alpha_{\infty}$    $\delta_a$   $\delta_\alpha$

8   2    $0.049337$     $-3.863406$    $0.052219$   $-3.795565$   $-0.002882$   $-0.067841$

4   $0.064323$      $-3.359243$      $0.060959$   $-3.485534$   $~0.003364$   $~0.126291$

5    $0.046971$    $-2.951933$     $0.047937$   $-3.192836$   $-0.000966$   $~0.240903$

9   2   $0.046323$     $-3.764323$    $0.048504$   $-3.705259$   $-0.002181$   $ -0.059064$

4    $0.056983$     $-2.993633$      $0.055612$   $-3.212828$   $~0.001371$   $~0.219195 $

5    $0.042375$     $-2.887326$     $0.043123$   $-3.003521$   $-0.000748$   $~0.116195 $

10   2    $0.042400$     $-3.657192$     $0.043898$   $-3.608171$   $-0.001498$   $-0.049021 $

4    $0.046825$     $-3.006516$     $0.046629$   $-3.124397$   $~0.000196$   $~ 0.117881$

5   $0.038413$     $-2.930885$   $0.039060$   $-2.975110$   $-0.000647$   $~0.044225 $

11   2    $0.037453$     $-3.543073$      $0.038351$   $-3.505155$   $-0.000898$   $-0.037918 $

3     $0.043328$     $-3.551534$     $0.042908$   $-3.513882$   $~0.000420$   $-0.037652 $

4    $0.038483$     $-3.055520$     $0.038443$   $-3.114044$   $~0.000040$   $~0.058524 $

5    $0.034366$     $-3.013171$     $0.034890$   $-3.025124$   $-0.000524$   $~0.011953 $

12   2    $0.031475$     $-3.424915$     $0.031919$   $-3.398484$   $-0.000444$   $-0.026431 $

3    $0.031275$     $-3.329545$    $0.031269$   $-3.323533$   $~0.000006$   $-0.006012 $

4    $0.030877$     $-3.101381$    $0.030859$   $-3.126726$   $~0.000018$   $~0.025345 $

5    $0.029525$     $-3.092376$     $0.029873$   $-3.093637$   $-0.000348$   $~0.001261 $

13   2    $0.024644$     $-3.308050$     $0.024812$   $-3.292241$   $-0.000168$   $-0.015809 $

3    $0.023188$     $-3.213527$     $0.023188$   $-3.213214$   $~0.000000$   $-0.000313 $

4    $0.023552$     $-3.122593$     $0.023543$   $-3.131342$   $~0.000009$   $~0.008749 $

5    $0.023430$     $-3.131575$     $0.023575$   $-3.130751$   $-0.000145$   $-0.000824 $

14   2    $0.017346$     $-3.199477$     $0.017389$   $-3.191979$   $-0.000043$   $-0.007498 $

3    $0.016146$     $-3.138109$     $0.016146$   $-3.138534$   $~0.000000$   $~0.000425 $

4    $0.016447$     $-3.112174$     $0.016445$   $-3.114255$   $~0.000002$   $~0.002081 $

5    $0.016525$     $-3.119407$     $0.016555$   $-3.118834$   $-0.000030$   $-0.000573 $

15   2    $0.010064$     $-3.105672$     $0.010070$   $-3.103310$   $-0.000006$   $-0.002362 $

3    $0.009537$     $-3.080460$     $0.009537$   $-3.080626$   $~0.000000$   $~0.000166 $

4    $0.009636$     $-3.076696$     $0.009636$   $-3.076936$   $~0.000000$   $~0.000240 $

5    $0.009658$     $-3.078272$     $0.009660$   $-3.078166$   $-0.000002$   $-0.000106 $

16   2    $0.003200$     $-3.030401$      $0.003200$   $-3.030182$   $~0.000000$   $-0.000219 $

3    $0.003138$     $-3.027413$     $0.003138$   $-3.027421$   $~0.000000$   $ ~0.000008$

4    $0.003143$     $-3.027352$     $0.003143$   $-3.027354$   $~0.000000$   $~0.000002 $

5    $0.003143$     $-3.027378$     $0.003143$   $-3.027377$   $~0.000000$   $-0.000001$

Fixed points solutions for $a_\infty$ and $\alpha_\infty$ to the
curve $\gamma_a(a,\bar{\alpha})$ $=$ $-$ $\gamma_\alpha(a,\bar{\alpha})$, in
the columns headed with $\bar{\alpha}$ and the corresponding solutions for the
fixed points in the $(a,\alpha)$ plane given in [42] together with the
difference $\delta$ in values in the final pair of columns up to the five loop
level in the $\mMOM$ scheme.

While this gives a flavour of the situation at the first few orders for
$\Nf$ $=$ $16$ we have analysed the remaining values of $\Nf$ in the conformal
window and the results are recorded in Table <ref>. In that table
columns $3$ and $4$ record the solutions for all values of $\Nf$ in the
conformal window for the $SU(3)$ group. The next pair of columns records the
values of the critical point solutions of (<ref>) from [42] for
comparison. The final two columns record the difference $\delta_a$ and
$\delta_\alpha$ of the respective values of $a_\infty$ and $\alpha_\infty$. At
the upper end of the conformal window, where perturbation theory is more
reliable, there is a clear indication that the solution for $\bar{\alpha}$ is
in good agreement with the fixed point of [42]. This continues to be the
case for lower values of $\Nf$ but the agreement becomes less pronounced except
at high loop order. Below around $\Nf$ $=$ $10$ one strays into territory where
the critical coupling value ceases to be small and perturbation theory may be
less reliable for drawing a precise conclusion. However in the region of
validity it does support the premise that the Crewther relation reflects
conformal properties. Indeed to complete this $\mMOM$ analysis we have
calculated the value of $C_{{\mbox{\footnotesize{Bjr}}}}^{\mMOMs}(a,\alpha)
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ for $\Nf$ $=$ $16$ at
successive loop orders and present the outcome in Table <ref>.
Again we observe that in effect the product evaluates to $3$ similar to the
earlier cases.

4|c|$\mMOM$ 2 loop

$a_{\infty}$     $\alpha_{\infty}$   $O(a^3)$   $O(a^4)$
$0.0032000819$     $-3.0304007777$     $2.9999982454$     $3.0000012447$
4|c|$\mMOM$ 3 loop

$a_{\infty}$     $\alpha_{\infty}$   $O(a^3)$   $O(a^4)$
$0.0031380752$     $-3.0274132638$     $2.9999973440$     $3.0000001217$
$0.1624143964$     $1.8817663994$     $9.7789795843$     $20.2267682549$
4|c|$\mMOM$ 4 loop

$a_{\infty}$     $\alpha_{\infty}$   $O(a^3)$   $O(a^4)$
$0.0031430140$     $-3.0273515306$
$2.9999974127$     $3.0000002081$
$0.0922743518$     $0.7554173647$     $4.1917263936$     $6.5855502329$
4|c|$\mMOM$ 5 loop

$a_{\infty}$     $\alpha_{\infty}$   $O(a^3)$   $O(a^4)$
$0.0031434057$    $-3.0273781422$     $2.9999974182$     $3.0000002149$
$0.0498699873$     $-3.9546945229$      $3.1876456375$    $3.2609125084$

Values of $C_{{\mbox{\footnotesize{Bjr}}}}^{\mMOMs}(a,\alpha)
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ from the solution for
$\bar{\alpha}$ at successive loop orders for $\Nf$ $=$ $16$.

Loop    $a_\infty$     $\bar{\alpha}(a_\infty)$   $S=1$    [42]
2   0.003311    $-2.952784 - 0.000061 S - 0.000245 S^2$   $-2.953090$   $-2.952985$
3   0.003162    $-2.945674 - 0.000400 S + 0.000218 S^2$   $-2.945856$   $-2.945839$

$ - 6.171139\times10^{-8} S^3$
4   0.003170    $-2.945920 - 0.000033 S + 0.000016 S^2 $   $-2.945937$   $-2.945935$

$+1.111287 \times10^{-7} S^3 - 2.848273\times10^{-10} S^4$
0.091700    $-3.819444 + 8.65019 S - 4.55718 S^2 $   $0.2776970$   $-4.407403$

$+ 0.00413990 S^3 - 6.895175\times 10^{-6} S^4$
5   0.003171    $-2.945966 + 1.06321\times10^{-6} S - 8.15566\times 10^{-7} S^2  $   $-2.945966$   $-2.945965$

$- 3.72127\times 10^{-8} S^3+ 8.257949\times 10^{-10} S^4$

$- 3.041220\times 10^{-14} S^5$
0.046999    $-4.691390 + 2.754610 S - 1.377941 S^2  $   $-3.321283$    $-3.682521$

$- 0.006615 S^3+0.000053 S^4 - 1.467961\times10^{-9} S^5$

Values of $\bar{\alpha}(a_\infty)$ for $\Nf$ $=$ $16$ in the $\MSbar$
scheme as a function of $S$.

Loop    $a_\infty$     $\bar{\alpha}(a_\infty)$   $S=1$    [42]
2   0.003311    $-3.032990 - 0.005566S - 0.000559S^2$   $-3.039142$   $-3.040393$

$ -0.000027 S^3$
3   0.003162    $-3.028600 -0.001502 S +0.000424 S^2$   $-3.029633$   $-3.029389$

$ +0.000045 S^3 -4.220754\times10^{-7} S^4 -1.523738\times 10^{-8} S^5$
4   0.003170    $-3.027400  -0.000072 S + 0.000095 S^2$   $-3.027390$   $-3.027317$

$ -0.000014 S^3 + 1.097536\times 10^{-6} S^4 +5.156952\times 10^{-8} S^5$

$-4.450271\times 10^{-10} S^6 -1.186739\times 10^{-11} S^7$
0.091700    $29.170852 + 37.509021 S  -7.270245 S^2$   58.084230   $-7.001486$

$  -1.363470 S^3 + 0.036484 S^4 +0.001606 S^5$

$ -0.000011 S^6 -2.872889\times 10^{-7} S^7$
5   0.003171    $-3.027362 + 7.648957\times 10^{-6} S + 0.000011 S^2$   $-3.027348$   $-3.027350$

$  -3.795632\times 10^{-6}S^3  -7.923119\times 10^{-7} S^4 $

$ -5.731136\times 10^{-8} S^5 +1.945960\times 10^{-9} S^6 $

$ -5.788853\times 10^{-11} S^7-5.007868\times 10^{-13} S^8$

$  -1.052508\times 10^{-14} S^9$
0.046999    $3.423566 + 9.178962 S -4.964152 S^2$   7.849534   $-6.454812$

$ + 0.302490 S^3  -0.086406 S^4 -0.005043 S^5$

$ +0.000114 S^6 +3.328853\times 10^{-6} S^7-2.417239\times 10^{-8} S^8  $

$- 5.080334\times 10^{-10} S^9$

Values of $\bar{\alpha}(a_\infty)$ for $\Nf$ $=$ $16$ in the $\RI$
scheme as a function of $S$.

One question that arises is whether the choice of $S$ affects the analysis and
if so to what extent. While we could consider the numerical fixed points for an
array of values of $S$, instead we will consider the fixed points in the
$\MSbar$ and $\RI$ schemes since the running of the coupling constant is
$\alpha$ independent in both these schemes. This will allow us to calculate the
fixed points as a function of $S$. The procedure is the same as for the
$\mMOM$ scheme except that $\gamma_a(a_\infty)$ $=$ $0$ immediately determines
$a_\infty$. So solving
$\gamma_\alpha(a,\alpha)$ $+$ $S\gamma_a(a,\alpha)$ $=$ $0$ means the $S$
dependence appears solely in $\bar{\alpha}(a_\infty)$ and allows us to study
the relation to the loop order. Repeating the earlier example for
$\Nf$ $=$ $16$ we find

\begin{eqnarray}
\alpha_{\MSbars}^{(0)}&=& -~ 2.777778 ~-~ 0.222222 S \nonumber \\
\alpha_{\MSbars}^{(1)}&=& -~ 52.851852 ~+~ 67.092593 S ~-~
0.074074S^2 \nonumber\\
\alpha_{\MSbars}^{(2)}&=& -~ 78.713043 ~+~ 968.858403 S ~+~ 45.225583 S^2 ~-~
0.006173 S^3 \nonumber\\
\alpha_{\MSbars}^{(3)}&=& 5792.714293  ~-~ 7038.053127 S ~-~
6394.380328 S^2 ~+~ 5.436159 S^3 ~-~ 0.008942 S^4 \nonumber\\
\alpha_{\MSbars}^{(4)}&=& 29279.884317 ~-~ 325028.480677 S ~-~
166111.075407 S^2 ~-~ 1468.595029 S^3 \nonumber \\
&& +~ 10.989272 S^4 ~-~ 0.000301 S^5
\end{eqnarray}

which lead to expressions for $\bar{\alpha}(a_\infty)$ given in Table
<ref> at successive loop order as a function of $S$. Included in the
table are the evaluation of $\bar{\alpha}(a_\infty)$ at $S$ $=$ $1$ and the
critical point value from [42] to compare with. Clearly in the first
instance the $S$ $=$ $0$ values are an accurate estimate of the eventual
$S$ $=$ $1$ value. Equally as the loop order increases the dependence on $S$
washes out quickly if one uses the coefficients of $S$ itself as a guide. At
four and five loops additional fixed points emerged. In each case the
coefficients of $S$ in the polynomial are relatively large for low powers which
is due to the large critical coupling constant value.

6|c|Curci-Ferrari Gauge

2|c||$\bar{\alpha}$   2|c|[42]

$N_f$     Loop     $a_{\infty}$     $\alpha_{\infty}$    $a_{\infty}$     $\alpha_{\infty}$

8   3    $0.116505$     $5.425313$    $0.116505$   $-0.525277$

9   2    $0.416667$     $7.333333$    $0.416667$   $-2.387888$

3    $0.081803$     $1.386711$    $0.081803$   $-1.239118$

10   2    $0.175676$     $0.207207$    $0.175676$   $-2.981627$

3    $0.060824$     $-0.762250$    $0.060824$   $-1.988405$

11   2    $0.098214$     $-2.202381$    $0.098214$   $-3.548192$

3    $0.046039$     $-2.180305$    $0.046039$   $-2.744498$

12   2    $0.060000$     $-3.480000$    $0.060000$   $-4.078489$

3    $0.034607$     $-3.235417$    $0.034607$   $-3.479125$

13   2    $0.037234$     $-4.312057$   $0.037234$   $-4.569040$

3    $0.025191$     $-4.075904$    $0.025191$   $-4.169158$

14   2    $0.022124$     $-4.923304$    $0.022124$   $-5.020489$

3    $0.017070$     $-4.768441$    $0.017070$   $-4.796907$

15   2    $0.011364$     $-5.409091$    $0.011364$   $-5.435844$

3    $0.009818$     $-5.342638$    $0.009818$   $-5.347954$

16   2    $0.003311$     $-5.816777$    $0.003311$   $-5.819105$

3    $0.003162$     $-5.808281$    $0.003162$   $-5.808458$

8   3    $0.116505$     $-14.454928$    $0.116505$   $-14.732061$

9   2    $0.416667$     $-5.748610$    $0.416667$   $-1.775685$

3    $0.081803$     $-9.342567$    $0.081803$   $-6.679483$

10   2    $0.175676$     $-5.644755$    $0.175676$   $-2.041475$

3    $0.060824$     $-7.302268$    $0.060824$   $-9.374058$

11   2    $0.098214$     $-5.611372$    $0.098214$   $-2.385628$

3    $0.046039$     $-6.350328$    $0.046039$   $-1.943195$

12   2    $0.060000$     $-5.594904$     $0.060000$   $-2.822900$

3    $0.034607$     $-5.885931$    $0.034607$   $-2.691103$

13   2    $0.037234$     $-5.585093$    $0.037234$   $-3.356225$

3    $0.025191$     $-5.668820$    $0.025191$   $-3.414438$

14   2    $0.022124$     $-5.578581$    $0.022124$   $-3.968278$

3    $0.017070$     $-5.583155$    $0.017070$   $-4.074876$

15   2    $0.011364$     $-5.573944$    $0.011364$   $-4.620179$

3    $0.009818$     $-5.563763$    $0.009818$   $-4.684705$

16   2    $0.003311$     $-5.570474$    $0.003311$   $-5.263519$

3    $0.003162$     $-5.568155$    $0.003162$   $-5.273175$

Fixed points solutions for $a_\infty$ and $\alpha_\infty$ to the curve
$\gamma_a(a,\bar{\alpha})$ $=$ $-$ $\gamma_\alpha(a,\bar{\alpha})$ in the
Curci-Ferrari gauge and MAG in the $\MSbar$ scheme.

While this analysis is in the $\MSbar$ scheme, similar to that of the original
Crewther study [5], it is worth repeating the exercise in the $\RI$ scheme
to ascertain whether the same picture emerges. Therefore taking $\Nf$ $=$ $16$
again we find

\begin{eqnarray}
\alpha_{\RIs}^{(0)} &=& -~ 2.777778 ~-~ 0.222222 S \nonumber \\
\alpha_{\RIs}^{(1)} &=& -~ 77.074074 ~+~ 65.430041 S ~-~ 0.168724S^2 ~-~
0.008230 S^3
\nonumber\\
\alpha_{\RIs}^{(2)} &=& -~ 712.810800 ~+~ 1384.485718 S ~+~ 95.815127 S^2 ~+~
7.080171 S^3 ~-~ 0.042219 S^4 \nonumber \\
&& -~ 0.001524 S^5 \nonumber\\
\alpha_{\RIs}^{(3)} &=& 58371.788435 ~+~  26052.906414 S ~-~
10453.261109 S^2 ~-~ 1844.453717 S^3 \nonumber \\
&& +~ 47.775386 S^4 ~+~ 2.099823 S^5 ~-~ 0.013971 S^6 ~-~ 0.000373 S^7
\nonumber \\
\alpha_{\RIs}^{(4)} &=& 1094061.824587 ~+~  115396.980509 S ~-~
836720.663333 S^2 ~+~ 98112.588891 S^3 \nonumber \\
&& -~ 18705.923877 S^4 ~-~ 1077.589340 S^5~+~ 23.656656 S^6 ~+~
0.690162 S^7 ~-~ 0.004954 S^8 \nonumber \\
&& -~ 0.000104 S^9
\end{eqnarray}

with the subsequent implications for $\alpha(a_\infty)$ recorded in Table
<ref>. We recall that as the $\RI$ $\beta$-function has the same
coefficients as those of the $\MSbar$ scheme the critical couplings will be the
same and so there will also be four and five loop solutions. However comparing
the solutions for each coefficient $\alpha_{\RIs}^{(i)}$ with its $\MSbar$
partner we note that in the $\RI$ scheme the degree of the polynomial solution
in $S$ at each loop order is higher. Despite this the same outcome emerges when
$S$ is set to unity in that the numerical value is in very good agreement with
the fixed point values given in [42]. Again the convergence to a
practically $S$ independent solution for $\alpha(a_\infty)$ transpires. This is
indicative of the idea argued earlier that the condition
$\gamma_a(a,\alpha)$ $+$ $S\gamma_\alpha(a,\alpha)$ $=$ $0$ can be used to find
the stable infrared fixed point and also that the value of $S$ is unimportant.
This means that while the value of $S$ $=$ $1$ has properties of interest
specifically to studies of the Crewther relation the relation of this
particular curve to the stable infrared fixed point is no more important than
any similar curve generated from any other value of $S$.

To this point we have concentrated on the linear covariant gauge with this
approach. However we have also examined the two nonlinear gauges in depth and
for the kinematic schemes of [34, 35] although we are limited to three
loops in this case. The outcome is in essence the same as the linear gauge
case except that the leading value of $\bar{\alpha}$ is in the neighbourhood
of $\alpha_{\CFs}$ $=$ $-$ $6$ for the Curci-Ferrari gauge. Due to the nature
of $\gamma_\alpha(a,\alpha)$ for the MAG the leading value of $\alpha$ is
$\alpha_{\MAGs}$ $=$ $-$ $5.56$. For both gauges the same property is present
as the linear gauge in that over the range of the conformal window the higher
order corrections do not unduly alter these respective $\alpha$ values.
Equally for values of $\Nf$ down to around $12$ the solutions for
$\bar{\alpha}$ are in keeping with the fixed point solutions of [42]. We
would expect this to be improved for lower values of $\Nf$ when higher order
corrections become available. This viewpoint can be seen in Table
<ref>. Similar tables are also available in the data file
associated with this article but they are more comprehensive in that they
include extra fixed points which have a mapping to the $\bar{\alpha}$ solution.
In addition we include the results for all the schemes and gauges we have
discussed in the article. Examining these wider tables it is evident that for
certain schemes and gauges the connection with the fixed points of [42]
and the solutions found by the method of this section is remarkably stable even
at the lower end of the conformal window. Although for the MOM schemes of
[34, 35] only a few perturbative orders are available. We note that in
several schemes there were no fixed point solutions for various $\Nf$ values in
[42]. In that instance there are no entries but we did find solutions for
$\bar{\alpha}$. It is not clear whether this is indicating that there will be
solutions for fixed points in these cases when higher order perturbative
results become available in those schemes.

Wider perspective.

It is worth taking stock of the wider perspective that our investigations have
arrived at. We begin by putting the choice of the left hand side of
(<ref>) in a field theory context. Clearly the combined renormalization
group function

\begin{equation}
\hat{\gamma}_\alpha(a,\alpha) ~=~ \gamma_\alpha(a,\alpha) ~+~
\gamma_a(a,\alpha)
\label{modgammaalpha}
\end{equation}

plays an important role in the analysis of the Crewther relation extension as
well as being connected to the special choice of $\alpha$ $=$ $-$ $3$ in the
linear covariant gauge as illustrated in the previous section. To understand
the origin of (<ref>) we recall the purely gluonic sector of the
Lagrangian of a non-abelian gauge theory is

\begin{equation}
L^{\mbox{\footnotesize{gluonic}}} ~=~
-~ \frac{1}{4} G^a_{\mu\nu} G^{a \, \mu\nu} ~-~
\frac{1}{2\alpha} \left( \partial^\mu A^a_\mu \right)^2
\label{laggluon}
\end{equation}

for a linear covariant gauge fixing. The field strength tensor is given by

\begin{equation}
G^a_{\mu\nu} ~=~ \partial_\mu A^a_\nu ~-~ \partial_\nu A^a_\mu ~+~
g f^{abc} A^b_\mu A^c_\nu
\label{fsdef}
\end{equation}

where $g$ is the coupling and $f^{abc}$ are the gauge group structure
constants. While this is invariably the usual presentation of the gluonic
sector and the one used for perturbative computations one can always redefine
the fields and variables without altering any physical predictions. Therefore
defining the rescaling

\begin{equation}
\hat{A}^a_\mu ~=~ g A^a_\mu
\end{equation}

we have

\begin{equation}
L^{\mbox{\footnotesize{gluonic}}} ~=~
-~ \frac{1}{4a} \hat{G}^a_{\mu\nu} \hat{G}^{a \, \mu\nu} ~-~
\frac{1}{2\alpha a} \left( \partial^\mu \hat{A}^a_\mu \right)^2
\label{laggluonhat}
\end{equation}

where $a$ $=$ $g^2/(16\pi^2)$ and $\hat{G}^a_{\mu\nu}$ corresponds to
(<ref>) at the formal value of $g$ $=$ $1$. Formulating the gluonic
sector in this way identifies the renormalization of the dimension four
operator $\hat{G}^a_{\mu\nu} \hat{G}^{a \, \mu\nu}$ with its associated
coupling which is $1/a$ and whose renormalization group function is related to
the $\beta$-function. One can view the second term of (<ref>) in
the same light and regard the coupling associated with the linear gauge fixing
term as $\alpha a$. Clearly both operators are independent which can be seen by
focussing on the quadratic part of (<ref>). The field strength term
is transverse while the other is purely longitudinal. Therefore the running of
the associated couplings will be different with that of the latter operator
being governed by (<ref>). At a more formal level the treatment
of the transverse and longitudinal components of the gauge field as separate
entities within loop computations has been recognized earlier. See, for
instance, [16, 62, 63, 64, 65, 66]. Indeed in [65, 66] the concept of
invariant charges of the two operators was coined and formalized within the
Hopf algebra construct of renormalization theory. Therefore from the point of
view of (<ref>) a more appropriate formulation would be to treat
$\alpha a$ as a second coupling constant in the renormalization group running
in a gauge theory. Moreover while we concentrated on the linear covariant gauge
repeating the rescaling argument for the Curci-Ferrari gauge and MAG will
produce the same observation with regard to the definition of a second coupling
from the two independent operators of the quadratic sector of
$L^{\mbox{\footnotesize{gluonic}}}$. For completeness we note the first few
terms of (<ref>) for the three gauges are

\begin{eqnarray}
\hat{\gamma}_\alpha^{\lins , \MSbars}(a,\alpha) &=&
-~ \left[ \alpha + 3 \right] \frac{C_A a}{2} ~+~
\left[ 40 \Nf T_F - 6 \alpha^2 C_A - 33 \alpha C_A - 95 C_A \right]
\frac{C_A a^2}{24} ~+~ O(a^3) \nonumber \\
\hat{\gamma}_\alpha^{\CFs , \MSbars}(a,\alpha) &=&
-~ \left[ \alpha + 6 \right] \frac{C_A a}{4} ~+~
\left[ 80 \Nf T_F - 3 \alpha^2 C_A - 51 \alpha C_A - 190 C_A \right]
\frac{C_A a^2}{48} ~+~ O(a^3) \nonumber \\
\hat{\gamma}_\alpha^{\MAGs , \MSbars}(a,\alpha) &=&
-~ \left[ 2 \alpha^2 \Nda + \alpha^2 \Noda + 12 \alpha \Nda + 6 \alpha \Noda
+ 12 \Nda \right] \frac{C_A a}{4 \alpha \Noda} \nonumber \\
&& +~ \left[ 512 \Nda \Noda \Nf T_F + 80 \alpha \Noda^2 \Nf T_F
- 30 \alpha^3 C_A \Nda^2 - 27 \alpha^3 C_A \Nda \Noda
\right. \nonumber \\
&& \left. ~~~~
- 3 \alpha^3 C_A \Noda^2 - 366 \alpha^2 C_A \Nda^2
- 339 \alpha^2 C_A \Nda \Noda - 51 \alpha^2 C_A \Noda^2
\right. \nonumber \\
&& \left. ~~~~
+ 294 \alpha C_A \Nda^2 - 647 \alpha C_A \Nda \Noda - 190 \alpha C_A \Noda^2
+ 160 \alpha \Nda \Noda \Nf T_F
\right. \nonumber \\
&& \left. ~~~~
+ 2016 C_A \Nda^2 - 928 C_A \Nda \Noda \right]
\frac{C_A a^2}{48 \alpha \Noda^2} ~+~ O(a^3) ~.
\end{eqnarray}

Clearly the leading terms of the first two gauges indicate the special gauge
choices that emerged in the Crewther analysis. For the MAG the leading term
vanishes at two values of $\alpha$ which are

\begin{equation}
\alpha_{\MAGs} ~=~ -~ 3 ~\pm~ \sqrt{\frac{3[2\Nda+3\Noda]}{[2\Nda+\Noda]}}
\end{equation}

and reproduces the Curci-Ferrari values in the $\Nda$ $\to$ $0$ limit. Also
for example

\begin{equation}
\left. \alpha_{\MAGs} \right|^{SU(\Nc)} ~=~
-~ 3 ~\pm~ \sqrt{\frac{3[3\Nc+2]}{[\Nc+2]}} ~.
\end{equation}

For $SU(3)$ the two values are $-$ $3$ $\pm$ $\sqrt{\frac{33}{5}}$ which equate
to $-$ $0.43095348$ and $-$ $5.56904651$ respectively. It is important to note
that these special gauge parameter values are constructed from the scheme
independent terms of $\hat{\gamma}(a,\alpha)$ and therefore will be the
foundation for the infrared stable fixed points in all schemes at higher

It is clear the Crewther relation has provided another example where a special
value of the gauge parameter plays a crucial role in a deeper property of QCD
as observed in [12] in the $\mMOM$ scheme. The other situations where this
occurred were in a variety of different problems and it was worth pausing to
understand if there is any underlying connections as to when this arises.
Moreover by doing so would provide a signpost where to expect the special case
to occur in other computations. Examining the various articles where
$\alpha$ $=$ $-$ $3$ has been singled out, [21, 22, 23, 24, 25, 26, 27, 28],
several common themes emerge. For instance, in [21, 22, 23] the quark
$2$-point function was the main topic. In [21] this involved solving the
Schwinger-Dyson equation defining the quark $2$-point function to examine the
quark mass gap to ascertain the condition when chiral symmetry was broken.
Although [22, 23] dealt with mesonic quark bound states in a gauge
invariant setup the path integral and worldline approach effectively trade off
gauge invariance for path dependence. This resulted in what was termed the
connector having a renormalization that at one loop depended on $(\alpha+3)$.
The special choice appeared to remove residual dependence on path and gauge at
one loop. For the approaches in [23, 27] the common thread was the
resummation of a class of Feynman diagrams which invariably were one loop
bubble graphs. In [23] this naive nonabelianization was used to study the
evolution of quark bilinear operators that underpin deep inelastic scattering
processes. As the treatment of these operators was by insertion in a quark
$2$-point function the Green's function is $\alpha$ dependent. What was
interesting is that the choice of $\alpha$ $=$ $-$ $3$ provided a reliable
estimate of the one loop $\Nf$ independent part of the operator anomalous
dimension for a range of moments. This accuracy was not as reliable for the
two loop case. Similarly in [27] a quark based Green's function was
studied from which an effective coupling was constructed. Its partial
resummation indicated that asymptotic freedom was conditional on the special
gauge value. In [28] the same gauge parameter choice showed a remarkable
connection between a set of gluonic bubble insertions in a quark current
correlation function and the one loop gluonic contribution to the $V$ scheme
coupling constant derived from the static quark potential. By contrast the
emergence of $\alpha$ $=$ $-$ $3$ was not restricted to quark based quantities.
In [24, 25, 26] a dimension six operator correction to the Yang-Mills
action was used to probe infrared gluon dynamics to determine what conditions
if any led to the gluon propagator having a double pole in the $p^2$ where
$p$ is the momentum. Such a behaviour would lead to a linear confining
potential. Solving the Schwinger-Dyson equation for the ghost propagator with
a massless double pole gluon propagator, that derived from the dimension six
operator, the solution required $\alpha$ $=$ $-$ $3$.

From an overall point of view the studies that have identified
$\alpha$ $=$ $-$ $3$ as having particular significance share several similar
features. Aside from all using the linear covariant gauge the calculations were
all carried out using the $\MSbar$ scheme. In addition most involved
calculations of Green's functions which would introduce gauge parameter
dependence. For instance, although the twist-$2$ operators used in the operator
product expansion for deep inelastic scattering that were the focus of
[23] are gauge invariant, the quark $2$-point function where they were
inserted is gauge variant. Therefore the expression will be $\alpha$ dependent
but the operator renormalization constant extracted from this Green's function
will be gauge parameter independent in the $\MSbar$ scheme. By contrast if the
$2$-point correlation function for the twist-$2$ operator was evaluated then it
would be gauge parameter independent in the $\MSbar$ scheme since the operator
is gauge invariant. In other words the same result would ensue for the operator
correlation function if it was evaluated in the Curci-Ferrari gauge or MAG in
the $\MSbar$ scheme. However if the operator correlation function was evaluated
in a scheme such as $\mMOM$ or the MOM schemes of [34, 35] then it would
depend on the gauge parameter. This is similar to the situation of the
anomalous dimension of a gauge invariant operator which depends on the gauge
parameter in general but not in a scheme such as $\MSbar$. In other words the
correlation function of gauge invariant operators is in general gauge parameter
dependent but independent of it in a subset of renormalization schemes.

In distilling the essence of previous observations of a special gauge parameter
value it is evident now where the Crewther relation rests within this
framework. The two ingredients of the relation are correlation functions of
gauge invariant operators which are the vector and axial vector quark currents.
Therefore in the $\MSbar$ scheme each correlation function will be gauge
parameter independent meaning the Crewther relation will be too. Equally
applying the formalism to effect a change of scheme to say $\mMOM$ will produce
gauge parameter dependence in the separate correlation functions and hence
produce a similar dependence in $\Delta_{\mbox{\footnotesize{csb}}}(a,\alpha)$.
This is evident in [12]. We have shown how that can be accommodated
consistently using the renormalization group formalism in (<ref>). A
similar mapping that established (<ref>) for the Crewther case could
equally well be applied to the quantities computed in [20, 21, 22, 23, 27, 28].
What is important to remember is that the gauge parameter dependence of any
operator correlation function does not affect any prediction from its
evaluation. The two underlying parameters, $a$ and $\alpha$, both run with
respect to the renormalization mass scale $\mu$ in a way derived from their
definition in the renormalization group equation. The formal dependence of each
on $\mu$ will be different in different schemes but ultimately within the
domain of perturbative applicability the behaviour of the correlation function
with respect to $\mu$ will be very similar since the scheme and gauge
dependence will wash out as the loop order increases.

We have examined the Crewther relation at high loop order from a variety of
different angles to first ascertain whether it is possible to consistently
extend the observation of [5] to schemes other than $\MSbar$ such as
$\mMOM$ and several kinematic schemes [34, 35] in different non-Lorenz
covariant gauge fixings. In other words to see if the multiplicative form that
the relation has in the $\MSbar$ and $V$ schemes is retained universally in
other schemes and gauges. While we have added the $\RI$ scheme to that list,
albeit for trivial reasons, the main conclusion is that the multiplicative form
is not universal. In schemes where the $\beta$-function depends on the gauge
parameter, which is not the case for the $\MSbar$, $V$ and $\RI$ schemes, the
Crewther relation is already known to  depend on the gauge parameter. This is
despite the fact that the two correlation functions that lead to the relation
involve gauge invariant operators. Instead in this situation the relation has
to be extended to accommodate the gauge parameter dependence. Moreover we have
extensively shown that this is fully consistent with quantum field theoretic
considerations and primarily the properties of the renormalization group
equation. Essentially the gauge parameter acts as a second coupling and
therefore its associated $\beta$-function has to be in included as indicated in
(<ref>). While we deduced this initially by probing the product of the
Bjorken sum rule and Adler $D$-function using the fixed point properties of
QCD, as the $\beta$-function will always be zero at such points, we showed how
to effect scheme changes on the coupling and gauge parameters in general to
verify (<ref>) formally. Subsequently the robustness of this structure
was probed from a different angle when we posed the problem of trying to find
what conditions on the gauge parameter would preserve the multiplicative
structure akin to that of (<ref>). In doing so we showed that the
infrared stable critical point value of the gauge parameter resulted, thereby
completing the circle of our analysis. Equally by doing so we provided a
concrete example of the invariant charge concept defined by the couplings
associated with the transverse and longitudinal components of the gluon field.
It would seem that this is not restricted to the Crewther relation since the
gauge variant quantities studied in [20, 21, 22, 23, 27, 28] could equally well
be recast in the two coupling language whence the reasons for the emergence of
a special gauge parameter value underlying key properties of each quantity
would be apparent. If one took a wider viewpoint concerning where this study
appears to point it might be that caution is advised when trying to adduce
general properties of seemingly gauge independent quantities from one specific
renormalization scheme such as $\MSbar$. The minimal way the underlying
divergences are subtracted in divergent Green's function would appear to cloud
more general considerations.

Acknowledgements. This work was carried out with the support of the STFC
Consolidated Grant ST/T000988/1 (JAG) and an EPSRC Studentship EP/R513271/1
(RHM). We thank A.L. Kataev for several discussions. For the purpose of open
access, the authors have applied a Creative Commons Attribution (CC-BY) licence
to any Author Accepted Manuscript version arising. The data representing the
main results here are accessible in electronic form from the arXiv ancillary
directory associated with the article.

Results for a general colour group.

To illustrate aspects of the $K$-functions for an arbitrary colour group we
record them for the $\MOMg$ scheme in a linear covariant by way of an example.
We have

\begin{eqnarray}
K_a^{\MOMgs}(a,\alpha) &=&
\left[
12 \zeta_3 C_F
- \frac{21}{2} C_F
\right] a
\nonumber \\
+ \left[
\frac{7}{3} \psi^{\prime}(\third) C_F C_A \alpha^2
- \frac{512}{27} \zeta_3 \pi^2 \Nf T_F C_F
- \frac{321}{2} C_F C_A
- \frac{224}{9} \psi^{\prime}(\third) \Nf T_F C_F
\right. \nonumber \\
&& \left. ~~~
- \frac{161}{27} \pi^2 C_F C_A
- \frac{112}{3} \zeta_3 \Nf T_F C_F
- \frac{92}{9} \psi^{\prime}(\third) \zeta_3 C_F C_A
- \frac{21}{2} C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{21}{2} \psi^{\prime}(\third) C_F C_A \alpha
- \frac{14}{9} \pi^2 C_F C_A \alpha^2
- \frac{8}{3} \psi^{\prime}(\third) \zeta_3 C_F C_A \alpha^2
+ \frac{7}{4} C_F C_A \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{16}{9} \zeta_3 \pi^2 C_F C_A \alpha^2
+ \frac{63}{4} C_F C_A \alpha
+ \frac{158}{3} \Nf T_F C_F
+ \frac{161}{18} \psi^{\prime}(\third) C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{184}{27} \zeta_3 \pi^2 C_F C_A
+ \frac{256}{9} \psi^{\prime}(\third) \zeta_3 \Nf T_F C_F
+ \frac{356}{3} \zeta_3 C_F C_A
+ \frac{397}{6} C_F^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{448}{27} \pi^2 \Nf T_F C_F
- 240 \zeta_5 C_F^2
- 18 \zeta_3 C_F C_A \alpha
- 8 \zeta_3 \pi^2 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- 2 \zeta_3 C_F C_A \alpha^3
+ 7 \pi^2 C_F C_A \alpha
+ 12 \zeta_3 C_F C_A \alpha^2
+ 12 \psi^{\prime}(\third) \zeta_3 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ 136 \zeta_3 C_F^2
\right] a^2
\nonumber \\
+ \left[
\frac{1}{48} \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2 \alpha^3
- \frac{5457913}{1152} C_F C_A^2
- \frac{336299}{648} \psi^{\prime}(\third) C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{177919}{2916} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha
- \frac{87808}{729} \psi^{\prime}(\third) \zeta_3 \pi^2 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{78529}{1296} \pi^2 C_F C_A^2 \alpha^2
- \frac{78454}{243} \zeta_3 \pi^2 C_F C_A^2 \alpha
- \frac{76832}{2187} \pi^4 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{66269}{384} C_F C_A^2 \alpha^2
- \frac{47950}{243} \pi^2 \Nf T_F C_F C_A \alpha
- \frac{35689}{54} \psi^{\prime}(\third) \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{33338}{2187} \zeta_3 \pi^4 C_F C_A^2 \alpha
- \frac{28672}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \Nf^2 T_F^2 C_F
- \frac{27181}{1458} \zeta_3 \pi^4 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{26776}{81} \zeta_3 \pi^2 \Nf T_F C_F C_A
- \frac{25921}{2592} \psi^{\prime}(\third)^2 C_F C_A^2
- \frac{25088}{729} \pi^4 \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
- \frac{25417}{486} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha
- \frac{24040}{81} \psi^{\prime}(\third) \zeta_3 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{21805}{2592} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha^2
- \frac{19208}{243} \psi^{\prime}(\third)^2 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{18032}{243} \psi^{\prime}(\third) \pi^2 \Nf T_F C_F C_A
- \frac{14257}{54} \pi^2 C_F C_A^2
- \frac{11008}{27} \zeta_3 \pi^2 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
- \frac{10240}{729} \zeta_3 \pi^4 \Nf T_F C_F C_A
- \frac{9986}{27} \psi^{\prime}(\third) \zeta_3 C_F C_A^2
- \frac{9850}{9} \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{9809}{18} \zeta_3 C_F C_A^2 \alpha
- \frac{9487}{108} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^2
- \frac{9131}{108} \psi^{\prime}(\third) C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{9040}{3} \zeta_5 C_F^2 C_A
- \frac{8768}{81} \pi^2 \Nf^2 T_F^2 C_F
- \frac{8000}{3} \zeta_5 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{7775}{4} \zeta_3^2 C_F C_A^2
- \frac{6656}{81} \pi^2 \Nf T_F C_F^2
- \frac{6272}{81} \psi^{\prime}(\third)^2 \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
- \frac{5152}{81} \psi^{\prime}(\third)^2 \zeta_3 \Nf T_F C_F C_A
- \frac{4459}{1458} \pi^4 C_F C_A^2 \alpha^2
- \frac{4058}{9} \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
- \frac{3920}{9} \Nf T_F C_F^2
- \frac{3703}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2
- \frac{3115}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{2989}{384} \psi^{\prime\prime\prime}(\third) C_F C_A^2
- \frac{2816}{27} \psi^{\prime}(\third) \zeta_3 \Nf^2 T_F^2 C_F
- \frac{2560}{3} \psi^{\prime}(\third) \zeta_5 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
- \frac{2197}{9} \Nf T_F C_F C_A \alpha
- \frac{1840}{9} \zeta_5 \pi^2 C_F^2 C_A
- \frac{1792}{729} \zeta_3 \pi^4 \Nf T_F C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{1783}{128} C_F C_A^2 \alpha^4
- \frac{1568}{243} \psi^{\prime}(\third) \pi^2 \Nf T_F C_F C_A \alpha^2
- \frac{1564}{9} \psi^{\prime}(\third) \zeta_3 C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{1328}{81} \zeta_3 \pi^2 \Nf T_F C_F C_A \alpha^2
- \frac{1191}{8} C_F^2 C_A \alpha
- \frac{1115}{216} \pi^2 C_F C_A^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{763}{144} \pi^2 C_F C_A^2 \alpha^4
- \frac{643}{486} \zeta_3 \pi^4 C_F C_A^2 \alpha^3
- \frac{641}{27} \psi^{\prime}(\third) \Nf T_F C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{539}{162} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha^3
- \frac{448}{81} \psi^{\prime}(\third)^2 \zeta_3 \Nf T_F C_F C_A \alpha^2
- \frac{397}{6} \pi^2 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{397}{18} \psi^{\prime}(\third) C_F^2 C_A \alpha^2
- \frac{397}{24} C_F^2 C_A \alpha^3
- \frac{160}{3} \zeta_5 \pi^2 C_F^2 C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{136}{3} \psi^{\prime}(\third) \zeta_3 C_F^2 C_A \alpha^2
- \frac{125}{18} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^3
- \frac{109}{12} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{77}{27} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha^3
- \frac{51}{8} \zeta_3 C_F C_A^2 \alpha^5
- \frac{49}{48} \psi^{\prime}(\third) C_F C_A^2 \alpha^5
\right. \nonumber \\
&& \left. ~~~
- \frac{49}{72} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha^4
- \frac{49}{128} C_F C_A^2 \alpha^6
- \frac{49}{162} \pi^4 C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{49}{192} \psi^{\prime\prime\prime}(\third) C_F C_A^2 \alpha^2
- \frac{38}{3} \zeta_3 \Nf T_F C_F C_A \alpha^2
- \frac{28}{27} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{16}{3} \psi^{\prime\prime\prime}(\third) \zeta_3 \Nf T_F C_F C_A
- \frac{13}{8} \zeta_3 C_F C_A^2 \alpha^3
- \frac{7}{9} \zeta_3 \pi^2 C_F C_A^2 \alpha^5
\right. \nonumber \\
&& \left. ~~~
- \frac{7}{384} \psi^{\prime\prime\prime}(\third) C_F C_A^2 \alpha^3
+ \frac{3}{4} \zeta_3^2 C_F C_A^2 \alpha^2
+ \frac{7}{6} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^5
\right. \nonumber \\
&& \left. ~~~
+ \frac{7}{9} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha^4
+ \frac{7}{16} \zeta_3 C_F C_A^2 \alpha^6
+ \frac{7}{24} \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{14}{3} \psi^{\prime\prime\prime}(\third) \Nf T_F C_F C_A
+ \frac{21}{8} \psi^{\prime\prime\prime}(\third) C_F C_A^2 \alpha
+ \frac{28}{81} \zeta_3 \pi^4 C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{49}{54} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha^4
+ \frac{49}{72} \pi^2 C_F C_A^2 \alpha^5
+ \frac{109}{18} \zeta_3 \pi^2 C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{125}{27} \zeta_3 \pi^2 C_F C_A^2 \alpha^3
+ \frac{217}{96} C_F C_A^2 \alpha^3
+ \frac{253}{12} \Nf T_F C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{272}{9} \zeta_3 \pi^2 C_F^2 C_A \alpha^2
+ \frac{289}{16} \zeta_3 C_F C_A^2 \alpha^4
+ \frac{308}{81} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{357}{64} C_F C_A^2 \alpha^5
+ \frac{392}{81} \psi^{\prime}(\third)^2 \Nf T_F C_F C_A \alpha^2
+ \frac{397}{4} C_F^2 C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{397}{4} \psi^{\prime}(\third) C_F^2 C_A \alpha
+ \frac{397}{27} \pi^2 C_F^2 C_A \alpha^2
+ \frac{427}{48} \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{539}{216} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha^3
+ \frac{664}{27} \psi^{\prime}(\third) \zeta_3 \Nf T_F C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{763}{96} \psi^{\prime}(\third) C_F C_A^2 \alpha^4
+ \frac{920}{3} \psi^{\prime}(\third) \zeta_5 C_F^2 C_A
+ \frac{1115}{144} \psi^{\prime}(\third) C_F C_A^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{1282}{81} \pi^2 \Nf T_F C_F C_A \alpha^2
+ \frac{1448}{9} \zeta_3 \Nf T_F C_F C_A \alpha
+ \frac{1552}{9} \zeta_3 \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
+ \frac{1568}{729} \pi^4 \Nf T_F C_F C_A \alpha^2
+ \frac{1792}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \Nf T_F C_F C_A \alpha^2
+ \frac{2471}{12} C_F^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{2548}{729} \zeta_3 \pi^4 C_F C_A^2 \alpha^2
+ \frac{3115}{324} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha^2
+ \frac{3128}{27} \zeta_3 \pi^2 C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{3328}{27} \psi^{\prime}(\third) \Nf T_F C_F^2
+ \frac{3703}{324} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2
+ \frac{4384}{27} \psi^{\prime}(\third) \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
+ \frac{4501}{3888} \pi^4 C_F C_A^2 \alpha^3
+ \frac{4508}{81} \psi^{\prime}(\third)^2 \Nf T_F C_F C_A
+ \frac{5120}{9} \zeta_5 \pi^2 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{5504}{9} \psi^{\prime}(\third) \zeta_3 \Nf T_F C_F^2
+ \frac{5632}{81} \zeta_3 \pi^2 \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
+ \frac{7168}{81} \psi^{\prime}(\third)^2 \zeta_3 \Nf^2 T_F^2 C_F
+ \frac{7594}{3} \zeta_3 C_F^2 C_A
+ \frac{8960}{729} \pi^4 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{9131}{162} \pi^2 C_F^2 C_A
+ \frac{9487}{162} \zeta_3 \pi^2 C_F C_A^2 \alpha^2
+ \frac{10711}{96} \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{11900}{3} \zeta_5 C_F C_A^2
+ \frac{13388}{27} \psi^{\prime}(\third) \zeta_3 \Nf T_F C_F C_A
+ \frac{14257}{36} \psi^{\prime}(\third) C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{19972}{81} \zeta_3 \pi^2 C_F C_A^2
+ \frac{20608}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{21805}{1944} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha^2
+ \frac{21952}{243} \psi^{\prime}(\third)^2 \zeta_3 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{23975}{81} \psi^{\prime}(\third) \Nf T_F C_F C_A \alpha
+ \frac{25088}{243} \psi^{\prime}(\third) \pi^2 \Nf^2 T_F^2 C_F
\right. \nonumber \\
&& \left. ~~~
+ \frac{25921}{1944} \psi^{\prime}(\third) \pi^2 C_F C_A^2
+ \frac{28672}{729} \zeta_3 \pi^4 \Nf^2 T_F^2 C_F
+ \frac{35689}{81} \pi^2 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{39227}{81} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha
+ \frac{47353}{36} C_F^2 C_A
+ \frac{48080}{243} \zeta_3 \pi^2 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{50834}{729} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha
+ \frac{76832}{729} \psi^{\prime}(\third) \pi^2 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{78529}{864} \psi^{\prime}(\third) C_F C_A^2 \alpha^2
+ \frac{87808}{2187} \zeta_3 \pi^4 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{110051}{36} \Nf T_F C_F C_A
+ \frac{116683}{8748} \pi^4 C_F C_A^2 \alpha
+ \frac{177919}{3888} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{190267}{11664} \pi^4 C_F C_A^2
+ \frac{327281}{576} C_F C_A^2 \alpha
+ \frac{336299}{972} \pi^2 C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{741599}{288} \zeta_3 C_F C_A^2
- 5720 \zeta_5 C_F^3
- 1464 \zeta_3 \Nf T_F C_F^2
- 840 \zeta_7 C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- 360 \zeta_5 C_F^2 C_A \alpha^2
- 360 \psi^{\prime}(\third) \zeta_5 C_F^2 C_A \alpha
- 306 \zeta_3 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- 136 \zeta_3 \pi^2 C_F^2 C_A \alpha
- 34 \zeta_3 C_F^2 C_A \alpha^3
- 3 \zeta_3^2 C_F C_A^2 \alpha^3
- 3 \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ 60 \zeta_5 C_F^2 C_A \alpha^3
+ 80 \psi^{\prime}(\third) \zeta_5 C_F^2 C_A \alpha^2
+ 171 \zeta_3^2 C_F C_A^2 \alpha
+ 204 \zeta_3 C_F^2 C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 204 \psi^{\prime}(\third) \zeta_3 C_F^2 C_A \alpha
+ 240 \zeta_5 \pi^2 C_F^2 C_A \alpha
+ 320 \zeta_5 \Nf^2 T_F^2 C_F
+ 488 \zeta_3 C_F^3
\right. \nonumber \\
&& \left. ~~~
+ 540 \zeta_5 C_F^2 C_A \alpha
+ 640 \zeta_3^2 \Nf T_F C_F C_A
+ 2080 \zeta_5 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
+ 5040 \zeta_7 C_F^3
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

\begin{eqnarray}
K_\alpha^{\MOMgs}(a,\alpha) &=&
\left[
12 \zeta_3 C_F C_A \alpha
- \frac{21}{2} C_F C_A \alpha
- \frac{21}{4} \psi^{\prime}(\third) C_F C_A
- \frac{14}{9} \pi^2 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
- \frac{8}{3} \psi^{\prime}(\third) \zeta_3 C_F C_A \alpha
+ \frac{7}{2} \pi^2 C_F C_A
+ \frac{7}{3} \psi^{\prime}(\third) C_F C_A \alpha
+ \frac{16}{9} \zeta_3 \pi^2 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{21}{8} C_F C_A \alpha^2
+ \frac{63}{8} C_F C_A
- 9 \zeta_3 C_F C_A
- 4 \zeta_3 \pi^2 C_F C_A
- 3 \zeta_3 C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~
+ 6 \psi^{\prime}(\third) \zeta_3 C_F C_A
\right] a^2
\nonumber \\
+ \left[
\frac{1}{2} \zeta_3^2 C_F C_A^2 \alpha
- \frac{548047}{1296} \psi^{\prime}(\third) C_F C_A^2
- \frac{177037}{2916} \psi^{\prime}(\third) \pi^2 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{69841}{243} \zeta_3 \pi^2 C_F C_A^2
- \frac{44750}{2187} \zeta_3 \pi^4 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{39424}{729} \psi^{\prime}(\third) \zeta_3 \pi^2 \Nf T_F C_F C_A
- \frac{34496}{2187} \pi^4 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{28204}{243} \pi^2 \Nf T_F C_F C_A
- \frac{25291}{486} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{20231}{432} \psi^{\prime}(\third) C_F C_A^2 \alpha
- \frac{14608}{81} \psi^{\prime}(\third) \zeta_3 \Nf T_F C_F C_A
- \frac{11221}{36} \zeta_3 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{8624}{243} \psi^{\prime}(\third)^2 \Nf T_F C_F C_A
- \frac{3323}{192} C_F C_A^2 \alpha
- \frac{2783}{18} \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{2009}{108} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha
- \frac{971}{27} \zeta_3 \pi^2 C_F C_A^2 \alpha
- \frac{777}{32} C_F C_A^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{616}{81} \zeta_3 \pi^4 C_F C_A^2 \alpha
- \frac{397}{8} C_F^2 C_A
- \frac{397}{18} \pi^2 C_F^2 C_A
- \frac{397}{24} C_F^2 C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{397}{27} \psi^{\prime}(\third) C_F^2 C_A \alpha
- \frac{385}{36} \pi^2 C_F C_A^2 \alpha^3
- \frac{320}{9} \zeta_5 \pi^2 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{287}{18} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha
- \frac{272}{9} \psi^{\prime}(\third) \zeta_3 C_F^2 C_A \alpha
- \frac{245}{144} \psi^{\prime}(\third) C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{136}{3} \zeta_3 \pi^2 C_F^2 C_A
- \frac{112}{81} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha^3
- \frac{98}{243} \pi^4 C_F C_A^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{85}{8} \zeta_3 C_F C_A^2 \alpha^4
- \frac{55}{3} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^3
- \frac{49}{6} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{49}{54} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha^3
- \frac{49}{64} C_F C_A^2 \alpha^5
- \frac{49}{288} \psi^{\prime\prime\prime}(\third) C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{35}{24} \pi^2 C_F C_A^2 \alpha^2
- \frac{35}{27} \zeta_3 \pi^2 C_F C_A^2 \alpha^4
- \frac{23}{8} \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{19}{6} \zeta_3 \pi^4 C_F C_A^2 \alpha^2
- \frac{7}{384} \psi^{\prime\prime\prime}(\third) C_F C_A^2 \alpha^2
- \frac{5}{2} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{1}{48} \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2 \alpha^2
+ \frac{5}{3} \zeta_3 \pi^2 C_F C_A^2 \alpha^2
+ \frac{7}{8} \zeta_3 C_F C_A^2 \alpha^5
\right. \nonumber \\
&& \left. ~~~
+ \frac{7}{8} \psi^{\prime\prime\prime}(\third) C_F C_A^2
+ \frac{7}{36} \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2 \alpha
+ \frac{28}{3} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{28}{27} \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha^3
+ \frac{35}{16} \psi^{\prime}(\third) C_F C_A^2 \alpha^2
+ \frac{35}{18} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{49}{8} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha^2
+ \frac{98}{81} \psi^{\prime}(\third) \pi^2 C_F C_A^2 \alpha^3
+ \frac{110}{9} \zeta_3 \pi^2 C_F C_A^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{111}{4} \zeta_3 C_F C_A^2 \alpha^3
+ \frac{112}{243} \zeta_3 \pi^4 C_F C_A^2 \alpha^3
+ \frac{133}{48} \pi^4 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{160}{3} \psi^{\prime}(\third) \zeta_5 C_F^2 C_A \alpha
+ \frac{245}{216} \pi^2 C_F C_A^2 \alpha^4
+ \frac{251}{48} C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{279}{16} \zeta_3 C_F C_A^2 \alpha
+ \frac{385}{24} \psi^{\prime}(\third) C_F C_A^2 \alpha^3
+ \frac{397}{6} C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{397}{12} \psi^{\prime}(\third) C_F^2 C_A
+ \frac{539}{81} \pi^4 C_F C_A^2 \alpha
+ \frac{544}{27} \zeta_3 \pi^2 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{574}{27} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2 \alpha
+ \frac{595}{64} C_F C_A^2 \alpha^4
+ \frac{794}{81} \pi^2 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{836}{9} \zeta_3 \Nf T_F C_F C_A
+ \frac{971}{18} \psi^{\prime}(\third) \zeta_3 C_F C_A^2 \alpha
+ \frac{2009}{144} \psi^{\prime}(\third)^2 C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{9856}{243} \psi^{\prime}(\third)^2 \zeta_3 \Nf T_F C_F C_A
+ \frac{14102}{81} \psi^{\prime}(\third) \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{20231}{648} \pi^2 C_F C_A^2 \alpha
+ \frac{29216}{243} \zeta_3 \pi^2 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{34496}{729} \psi^{\prime}(\third) \pi^2 \Nf T_F C_F C_A
+ \frac{39424}{2187} \zeta_3 \pi^4 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{50582}{729} \psi^{\prime}(\third) \zeta_3 \pi^2 C_F C_A^2
+ \frac{69841}{162} \psi^{\prime}(\third) \zeta_3 C_F C_A^2
+ \frac{156625}{8748} \pi^4 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{177037}{3888} \psi^{\prime}(\third)^2 C_F C_A^2
+ \frac{248837}{576} C_F C_A^2
+ \frac{548047}{1944} \pi^2 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- 240 \zeta_5 C_F^2 C_A \alpha
- 120 \psi^{\prime}(\third) \zeta_5 C_F^2 C_A
- 102 \zeta_3 C_F^2 C_A
- 34 \zeta_3 C_F^2 C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 7 \psi^{\prime}(\third)^2 \zeta_3 C_F C_A^2 \alpha^2
- 3 \zeta_3^2 C_F C_A^2 \alpha^2
- \psi^{\prime\prime\prime}(\third) \zeta_3 C_F C_A^2
+ 57 \zeta_3^2 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
+ 60 \zeta_5 C_F^2 C_A \alpha^2
+ 68 \psi^{\prime}(\third) \zeta_3 C_F^2 C_A
+ 80 \zeta_5 \pi^2 C_F^2 C_A
+ 136 \zeta_3 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ 180 \zeta_5 C_F^2 C_A
\right] a^3 ~+~ O(a^4) ~.
\end{eqnarray}

The corresponding MAG expression depends on $\Nda$ and $\Noda$ and is available
in the data file.

$\RIc$ scheme $\beta$-function.

We record the $SU(3)$ expressions for the two key renormalization group
equations that are central to the Crewther relation in the $\RIc$ scheme. The
five loop $\beta$-function is

\begin{eqnarray}
\left. \beta^{\RIcs}(a,\alpha) \right|^{SU(3)} &=&
\left[
\frac{2}{3} \Nf
- 11
\right] a^2
+ \left[
\frac{38}{3} \Nf
+ \frac{39}{2} \alpha
- \frac{9}{2} \alpha^2
- 102
- 2 \Nf \alpha
\right] a^3
\nonumber \\
+ \left[
\frac{21}{4} \Nf \alpha^2
+ \frac{27}{8} \alpha^3
- \frac{2857}{2}
- \frac{2655}{16} \alpha^2
- \frac{325}{54} \Nf^2
- \frac{137}{2} \Nf \alpha
- \frac{81}{8} \zeta_3 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{9}{4} \zeta_3 \Nf \alpha^2
+ \frac{351}{16} \zeta_3 \alpha^2
+ \frac{729}{16} \zeta_3 \alpha
+ \frac{4599}{16} \alpha
+ \frac{5033}{18} \Nf
\right] a^4
\nonumber \\
+ \left[
\frac{29}{2} \zeta_3 \Nf^2 \alpha
+ \frac{243}{8} \zeta_3 \Nf \alpha^3
- \frac{1172325}{128} \alpha
- \frac{717363}{128} \alpha^2
- \frac{149753}{6}
\right. \nonumber \\
&& \left. ~~~
- \frac{50065}{162} \Nf^2
- \frac{6472}{81} \zeta_3 \Nf^2
- \frac{6321}{16} \Nf \alpha
- \frac{2025}{16} \zeta_5 \alpha^4
- \frac{1725}{4} \zeta_3 \Nf \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{1323}{2} \zeta_3 \alpha^3
- \frac{1093}{729} \Nf^3
- \frac{621}{16} \zeta_3 \alpha^2
- \frac{567}{8} \alpha^4
- \frac{243}{2} \zeta_3 \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{177}{8} \Nf \alpha^3
- \frac{75}{4} \zeta_5 \Nf \alpha^3
- \frac{45}{2} \zeta_5 \Nf \alpha
- \frac{13}{8} \Nf^2 \alpha
+ \frac{3465}{16} \zeta_5 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{3645}{4} \zeta_5 \alpha^2
+ \frac{4131}{32} \zeta_3 \alpha^4
+ \frac{4185}{8} \zeta_5 \alpha
+ \frac{6508}{27} \zeta_3 \Nf
+ \frac{17169}{32} \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{33705}{64} \alpha^3
+ \frac{197307}{32} \zeta_3 \alpha
+ \frac{1078361}{162} \Nf
- 3564 \zeta_3
\right] a^5
\nonumber \\
+ \left[
\frac{152}{81} \zeta_3 \Nf^4
+ \frac{159}{4} \zeta_3^2 \Nf^2 \alpha
- \frac{15970474577}{9216} \alpha
- \frac{372908785}{3072} \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{207136629}{8192} \zeta_7 \alpha^2
- \frac{32567937}{512} \zeta_3 \alpha^2
- \frac{25960913}{1944} \Nf^2
\right. \nonumber \\
&& \left. ~~~
- \frac{14959621}{144} \zeta_3 \Nf \alpha
- \frac{13960107}{8192} \zeta_7 \alpha^3
- \frac{9452457}{256} \zeta_3^2 \alpha
- \frac{8157455}{16}
\right. \nonumber \\
&& \left. ~~~
- \frac{5668191}{1024} \alpha^4
- \frac{3852207}{512} \zeta_5 \alpha^4
- \frac{3388995}{128} \zeta_3 \alpha^3
- \frac{3383613}{256} \zeta_4 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{2636597}{216} \Nf^2 \alpha
- \frac{1358995}{27} \zeta_5 \Nf
- \frac{1209579}{1024} \zeta_7 \Nf \alpha
- \frac{941891}{256} \zeta_5 \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{886581}{128} \zeta_5 \Nf \alpha
- \frac{793143}{512} \zeta_5 \alpha^3
- \frac{698531}{81} \zeta_3 \Nf^2
- \frac{621885}{2} \zeta_3
\right. \nonumber \\
&& \left. ~~~
- \frac{580203}{256} \zeta_4 \alpha^2
- \frac{573237}{2048} \zeta_7 \alpha^5
- \frac{476263}{1728} \Nf^2 \alpha^2
- \frac{404607}{128} \Nf \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{191727}{256} \zeta_4 \alpha^3
- \frac{151875}{128} \zeta_6 \Nf \alpha
- \frac{92097}{256} \zeta_3 \alpha^5
- \frac{63693}{2048} \zeta_7 \Nf \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{43011}{256} \zeta_3^2 \alpha^4
- \frac{40365}{64} \zeta_3^2 \Nf \alpha^2
- \frac{35559}{256} \zeta_4 \alpha^4
- \frac{34845}{32} \zeta_5 \Nf \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{33935}{6} \zeta_4 \Nf
- \frac{16235}{96} \zeta_3 \Nf^2 \alpha^2
- \frac{14175}{256} \zeta_6 \alpha^5
- \frac{14175}{256} \zeta_6 \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{5895}{32} \zeta_3 \Nf \alpha^4
- \frac{3105}{256} \zeta_5 \alpha^5
- \frac{1809}{16} \zeta_3^2 \Nf \alpha^3
- \frac{1618}{27} \zeta_4 \Nf^3
\right. \nonumber \\
&& \left. ~~~
- \frac{1575}{256} \zeta_6 \Nf \alpha^4
- \frac{1353}{8} \zeta_4 \Nf^2 \alpha
- \frac{1205}{2916} \Nf^4
- \frac{460}{9} \zeta_5 \Nf^3
- \frac{382}{9} \zeta_3 \Nf^3 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{153}{16} \zeta_4 \Nf^2 \alpha^2
- \frac{100}{3} \zeta_5 \Nf^3 \alpha
+ \frac{243}{32} \zeta_4 \Nf \alpha^3
+ \frac{459}{128} \zeta_4 \Nf \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{1701}{64} \zeta_3^2 \Nf \alpha^4
+ \frac{4131}{128} \zeta_4 \alpha^5
+ \frac{5931}{8} \zeta_3^2 \Nf \alpha
+ \frac{6025}{48} \zeta_5 \Nf^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{6093}{64} \Nf \alpha^4
+ \frac{10526}{9} \zeta_4 \Nf^2
+ \frac{10935}{64} \zeta_3^2 \alpha^5
+ \frac{13273}{16} \zeta_5 \Nf^2 \alpha
+ \frac{16605}{128} \alpha^5
\right. \nonumber \\
&& \left. ~~~
+ \frac{19465}{81} \Nf^3 \alpha
+ \frac{27285}{256} \zeta_5 \Nf \alpha^4
+ \frac{46143}{128} \zeta_4 \Nf \alpha^2
+ \frac{48722}{243} \zeta_3 \Nf^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{88209}{2} \zeta_4
+ \frac{126891}{64} \zeta_4 \Nf \alpha
+ \frac{188529}{64} \zeta_3 \Nf \alpha^3
+ \frac{207225}{1024} \zeta_6 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{324577}{128} \zeta_3 \Nf \alpha^2
+ \frac{381760}{81} \zeta_5 \Nf^2
+ \frac{502929}{256} \zeta_3^2 \alpha^3
+ \frac{630559}{5832} \Nf^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{948933}{512} \zeta_5 \alpha
+ \frac{1075275}{1024} \zeta_6 \alpha^3
+ \frac{1816857}{2048} \zeta_7 \Nf \alpha^2
+ \frac{2023353}{256} \zeta_3 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{2116639}{432} \zeta_3 \Nf^2 \alpha
+ \frac{2480247}{256} \zeta_3^2 \alpha^2
+ \frac{3067875}{1024} \zeta_6 \alpha^2
+ \frac{3572667}{8192} \zeta_7 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{4811164}{81} \zeta_3 \Nf
+ \frac{24148125}{1024} \zeta_6 \alpha
+ \frac{24208323}{512} \zeta_5 \alpha^2
+ \frac{33423111}{1024} \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{51960893}{2304} \Nf \alpha^2
+ \frac{192323061}{8192} \zeta_7 \alpha
+ \frac{336460813}{1944} \Nf
+ \frac{435326091}{512} \zeta_3 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{1723650635}{6912} \Nf \alpha
+ 6 \zeta_4 \Nf^3 \alpha
+ 15 \zeta_3^2 \Nf^2 \alpha^2
+ 288090 \zeta_5
\right] a^6
\nonumber \\
&& +~ O(a^7)
\end{eqnarray}

while the gauge parameter anomalous dimension also in the linear covariant
gauge is

\begin{eqnarray}
\left. \gamma_\alpha^{\RIcs}(a,\alpha) \right|^{SU(3)} &=&
\left[
\frac{13}{2}
- \frac{3}{2} \alpha
- \frac{2}{3} \Nf
\right] a
+ \left[
+ \frac{9}{4} \alpha^2
+ \frac{531}{8}
+ 2 \Nf \alpha
- \frac{255}{8} \alpha
- \frac{61}{6} \Nf
\right] a^2
\nonumber \\
+ \left[
\frac{9}{4} \zeta_3 \Nf \alpha^2
+ \frac{81}{16} \zeta_3 \alpha^3
+ \frac{215}{27} \Nf^2
- \frac{27315}{32} \alpha
- \frac{8155}{36} \Nf
- \frac{675}{16} \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{243}{16} \zeta_3
- \frac{135}{32} \alpha^3
- \frac{27}{4} \zeta_3 \Nf \alpha
- \frac{21}{4} \Nf \alpha^2
+ \frac{409}{4} \Nf \alpha
+ \frac{729}{16} \zeta_3 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{5445}{32} \alpha^2
+ \frac{29895}{32}
+ 33 \zeta_3 \Nf
\right] a^3
\nonumber \\
+ \left[
\frac{17}{2} \zeta_3 \Nf^2 \alpha
+ \frac{45}{4} \zeta_4 \Nf \alpha
+ \frac{75}{4} \zeta_5 \Nf \alpha^3
- \frac{23350603}{5184} \Nf
- \frac{21210899}{768} \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{1012023}{256} \zeta_3
- \frac{184473}{64} \zeta_3 \alpha^2
- \frac{177021}{256} \alpha^3
- \frac{46755}{128} \zeta_5 \alpha^3
- \frac{44331}{64} \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{13781}{108} \Nf^2 \alpha
- \frac{13149}{256} \zeta_3 \alpha^4
- \frac{12549}{16} \zeta_3 \Nf \alpha
- \frac{8955}{16} \zeta_4 \Nf
- \frac{3355}{2} \zeta_5 \Nf
\right. \nonumber \\
&& \left. ~~~
- \frac{255}{4} \zeta_5 \Nf \alpha^2
- \frac{243}{8} \zeta_3 \Nf \alpha^3
- \frac{45}{2} \zeta_5 \Nf \alpha
- \frac{27}{16} \zeta_4 \Nf \alpha^2
- \frac{8}{3} \zeta_3 \Nf^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{177}{8} \Nf \alpha^3
+ \frac{243}{64} \zeta_4 \alpha^3
+ \frac{405}{8} \zeta_4 \alpha^2
+ \frac{1809}{64} \alpha^4
+ \frac{2017}{81} \zeta_3 \Nf^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{4017}{16} \zeta_3 \Nf \alpha^2
+ \frac{4427}{1458} \Nf^3
+ \frac{4455}{128} \zeta_5 \alpha^4
+ \frac{8019}{32} \zeta_4
+ \frac{16443}{64} \zeta_4 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{40905}{4} \zeta_5
+ \frac{43033}{162} \Nf^2
+ \frac{68625}{128} \zeta_5 \alpha^2
+ \frac{87831}{128} \zeta_3 \alpha^3
+ \frac{198315}{128} \zeta_5 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{311301}{128} \zeta_3 \alpha
+ \frac{387649}{216} \zeta_3 \Nf
+ \frac{712315}{144} \Nf \alpha
+ \frac{2137221}{256} \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{10596127}{768}
+ 33 \zeta_4 \Nf^2
\right] a^4
\nonumber \\
+ \left[
\frac{3248889939}{4096} \zeta_5
- \frac{40020851929}{36864} \alpha
- \frac{15059970043}{124416} \Nf
\right. \nonumber \\
&& \left. ~~~
- \frac{9151899939}{16384} \zeta_7
- \frac{514366083}{16384} \zeta_7 \alpha
- \frac{322439503}{4608} \zeta_3 \Nf \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{281043751}{4608} \Nf \alpha^2
- \frac{169349259}{1024} \zeta_3 \alpha^2
- \frac{110282119}{864} \zeta_5 \Nf
\right. \nonumber \\
&& \left. ~~~
- \frac{70747047}{8192} \zeta_7 \alpha^2
- \frac{65529567}{2048} \zeta_3^2
- \frac{46637683}{3456} \Nf^2 \alpha
- \frac{45332901}{2048} \zeta_5 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{44784819}{2048} \zeta_3^2 \alpha
- \frac{32357475}{2048} \zeta_6 \alpha
- \frac{31012901}{768} \zeta_4 \Nf
- \frac{17999415}{16384} \zeta_7 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{12215151}{2048} \zeta_3 \alpha^4
- \frac{3696633}{64} \alpha^3
- \frac{3378739}{1296} \zeta_5 \Nf^2
- \frac{2259077}{11664} \Nf^3
\right. \nonumber \\
&& \left. ~~~
- \frac{2249775}{8} \zeta_6
- \frac{1905231}{512} \zeta_3 \Nf \alpha^3
- \frac{1633959}{1024} \zeta_3^2 \alpha^3
- \frac{987899}{256} \zeta_5 \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{726489}{2048} \zeta_4 \alpha^3
- \frac{393579}{4096} \alpha^5
- \frac{370305}{4096} \zeta_5 \alpha^5
- \frac{257775}{512} \zeta_6 \Nf \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{231399}{2048} \zeta_7 \Nf \alpha^3
- \frac{123525}{1024} \zeta_6 \alpha^2
- \frac{105921}{128} \zeta_3^2 \Nf \alpha^2
- \frac{94527}{2048} \zeta_3^2 \alpha^5
\right. \nonumber \\
&& \left. ~~~
- \frac{80771}{243} \zeta_3 \Nf^3
- \frac{45441}{4096} \zeta_4 \alpha^5
- \frac{40139}{96} \zeta_3 \Nf^2 \alpha^2
- \frac{28575}{512} \zeta_6 \Nf \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{27285}{256} \zeta_5 \Nf \alpha^4
- \frac{16775}{6} \zeta_6 \Nf^2
- \frac{6093}{64} \Nf \alpha^4
- \frac{2659}{3} \zeta_3^2 \Nf^2
\right. \nonumber \\
&& \left. ~~~
- \frac{1701}{64} \zeta_3^2 \Nf \alpha^4
- \frac{1317}{4} \zeta_4 \Nf^2 \alpha
- \frac{729}{2} \zeta_4 \Nf \alpha^2
- \frac{675}{8} \zeta_6 \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{459}{128} \zeta_4 \Nf \alpha^4
- \frac{50}{3} \zeta_5 \Nf^3 \alpha
- \frac{32}{81} \zeta_3 \Nf^4
- \frac{8}{3} \zeta_4 \Nf^4
+ \frac{99}{16} \zeta_4 \Nf^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{139}{9} \zeta_3 \Nf^3 \alpha
+ \frac{159}{8} \zeta_3^2 \Nf^2 \alpha
+ \frac{1321}{27} \zeta_4 \Nf^3
+ \frac{1575}{256} \zeta_6 \Nf \alpha^4
+ \frac{2164}{9} \zeta_5 \Nf^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{2299}{48} \zeta_5 \Nf^2 \alpha^2
+ \frac{4715}{2916} \Nf^4
+ \frac{5895}{32} \zeta_3 \Nf \alpha^4
+ \frac{7425}{1024} \zeta_6 \alpha^4
+ \frac{11045}{144} \Nf^3 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{14851}{32} \zeta_5 \Nf^2 \alpha
+ \frac{16119}{256} \zeta_4 \alpha^4
+ \frac{17469}{512} \zeta_4 \Nf \alpha^3
+ \frac{28305}{256} \zeta_3^2 \Nf \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{63693}{2048} \zeta_7 \Nf \alpha^4
+ \frac{65817}{16} \Nf \alpha^3
+ \frac{70875}{2048} \zeta_6 \alpha^5
+ \frac{86103}{128} \zeta_4 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{114075}{128} \zeta_6 \alpha^3
+ \frac{163139}{72} \zeta_4 \Nf^2
+ \frac{187509}{2048} \zeta_7 \Nf \alpha
+ \frac{440809}{64} \zeta_3^2 \Nf
\right. \nonumber \\
&& \left. ~~~
+ \frac{543079}{432} \Nf^2 \alpha^2
+ \frac{638961}{256} \zeta_3^2 \Nf \alpha
+ \frac{820773}{4096} \zeta_3 \alpha^5
+ \frac{863541}{16384} \zeta_7 \alpha^5
\right. \nonumber \\
&& \left. ~~~
+ \frac{996423}{512} \zeta_5 \Nf \alpha^3
+ \frac{1164213}{2048} \zeta_3^2 \alpha^4
+ \frac{1477789}{864} \zeta_3 \Nf^2 \alpha
+ \frac{1816857}{2048} \zeta_7 \Nf \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{2670405}{512} \zeta_4 \Nf \alpha
+ \frac{2988559}{128} \zeta_3 \Nf \alpha^2
+ \frac{3662827}{48} \zeta_7 \Nf
+ \frac{4089225}{512} \zeta_5 \Nf \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{5660469}{1024} \zeta_3^2 \alpha^2
+ \frac{7740549}{2048} \alpha^4
+ \frac{10890829}{2592} \zeta_3 \Nf^2
+ \frac{17186355}{4096} \zeta_5 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{24104115}{8192} \zeta_7 \alpha^3
+ \frac{25440057}{2048} \zeta_3
+ \frac{32297109}{2048} \zeta_5 \alpha^2
+ \frac{39034359}{512} \zeta_4
\right. \nonumber \\
&& \left. ~~~
+ \frac{48001201}{3888} \Nf^2
+ \frac{61806555}{4096} \zeta_5 \alpha
+ \frac{77357835}{4096} \zeta_4 \alpha
+ \frac{97546581}{2048} \zeta_3 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{302449841}{20736} \zeta_3 \Nf
+ \frac{612462769}{2048}
+ \frac{676597463}{1536} \alpha^2
+ \frac{877406273}{3456} \Nf \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{1067166537}{4096} \zeta_3 \alpha
- 3 \zeta_4 \Nf^3 \alpha
+ 15 \zeta_3^2 \Nf^2 \alpha^2
+ 63175 \zeta_6 \Nf
\right] a^5
\nonumber \\
&& +~ O(a^6) ~.
\end{eqnarray}

The five loop expressions for these as well as the other renormalization group
equations in an arbitrary Lie group are provided in the associated data file.

[1] R.J. Crewther, Phys. Rev. Lett. 28 (1972), 1421.

[2] S.G. Gorishnii, A.L. Kataev & S.A. Larin, Phys. Lett. B259
(1991), 144.

[3] S.G. Gorishnii, A.L. Kataev & S.A. Larin, Pisma Zh. Eksp. Teor.
Fiz. 53 (1991), 121.

[4] S.A. Larin & J.A.M. Vermaseren, Phys. Lett. B259 (1991),

[5] D.J. Broadhurst & A.L. Kataev, Phys. Lett. B315 (1993), 179.

[6] G.T. Gabadadze & A.L. Kataev, JETP Lett. 61 (1995), 448.

[7] R.J. Crewther, Phys. Lett. B397 (1997), 137.

[8] D. Müller, Phys. Rev. D58 (1998), 054005.

[9] P.A. Baikov, K.G. Chetyrkin & J.H. Kühn, Phys. Rev. Lett.
104 (2010), 132004.

[10] A.L. Kataev & V.S. Molokoedov, Phys. Rev. D92 (2015),

[11] A.L. Kataev & V.S. Molokoedov, J. Phys. Conf. Ser. 938
(2017), 012050.

[12] A.V. Garkusha, A.L. Kataev & V.S. Molokoedov, JHEP 02
(2018), 161.

[13] M. Peter, Phys. Rev. Lett. 78 (1997), 602.

[14] Y. Schröder, Phys. Lett. B447 (1999), 321.

[15] L. von Smekal, K. Maltman & A. Sternbeck, Phys. Lett. B681
(2009), 336.

[16] J.C. Taylor, Nucl. Phys. B33 (1971), 436.

[17] J.A. Gracey, J. Phys. A46 (2013), 225403; J. Phys. A48
(2015), 119501(E).

[18] B. Ruijl, T. Ueda, J.A.M. Vermaseren & A. Vogt, JHEP 06
(2017), 040.

[19] J.A. Gracey & R.H. Mason, J. Phys. A56 (2023), 085401.

[20] T. Appelquist, K.D. Lane & U. Mahanta, Phys. Rev. Lett. 61
(1988), 1553.

[21] N.G. Stefanis, Nuovo Cim. A83 (1984), 205.

[22] N.G. Stefanis, Acta Phys. Polon. Supp. 6 (2013), 71.

[23] S.V. Mikhailov, Phys. Lett. B431 (1998), 387.

[24] A.I. Alekseev, B.A. Arbuzov & V.A. Baikov, Theor. Math. Phys.
52 (1982), 739.

[25] B.A. Arbuzov, E.E. Boos & K.Sh. Turashvili, Z. Phys. C30
(1986), 287.

[26] B.A. Arbuzov, E.E. Boos & A.I. Davydychev, Theor. Math. Phys.
74 (1988), 103.

[27] J.H. Field, Phys. Part. Nucl. 40 (2009), 353.

[28] K.G. Chetyrkin, A.H. Hoang, J.H. Kün, M. Steinhauser & T.
Teubner, Phys. Lett. B384 (1996), 233.

[29] L. Baulieu & R. Coquereaux, Ann. Phys. 140 (1982), 163.

[30] T.A. Ryttov, Phys. Rev. D89 (2014), 016013.

[31] T. Banks & A. Zaks, Nucl. Phys. B196 (1982), 189.

[32] G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa & A.
Vladikas, Nucl. Phys. B445 (1995), 81.

[33] E. Franco & V. Lubicz, Nucl. Phys. B531 (1998), 641.

[34] W. Celmaster & R.J. Gonsalves, Phys. Rev. Lett. 42 (1979),

[35] W. Celmaster & R.J. Gonsalves, Phys. Rev. D20 (1979),

[36] J.-M. Shen, X.-G. Wu, Y. Ma & S.J. Brodsky, Phys. Lett.
B770 (2017), 494.

[37] D.J. Gross & F.J. Wilczek, Phys. Rev. Lett. 30 (1973),

[38] H.D. Politzer, Phys. Rev. Lett. 30 (1973), 1346.

[39] G. 't Hooft, Nucl. Phys. B61 (1973), 455.

[40] O.V. Tarasov & D.V. Shirkov, Sov. J. Nucl. Phys. 51 (1990),

[41] W.E. Caswell, Phys. Rev. Lett. 33 (1974), 244.

[42] J.A. Gracey, R.H. Mason, T.A. Ryttov & R.M. Simms,
arXiv:2306.09056 [hep-ph].
[43] K.G. Chetyrkin & T. Seidensticker, Phys. Lett. B495 (2000),

[44] J.A. Gracey, Phys. Rev. D84 (2011), 085011.

[45] A. Bednyakov & A. Pikelener, Phys. Rev. D101 (2020),

[46] G. Curci & R. Ferrari, Nuovo Cim. A32 (1976), 151.

[47] G. 't Hooft, Nucl. Phys. B190 (1981), 455.

[48] A.S. Kronfeld, G. Schierholz & U.J. Wiese, Nucl. Phys. B293
(1987), 461.

[49] A.S. Kronfeld, M.L. Laursen, G. Schierholz & U.J. Wiese, Phys.
Lett. B198 (1987), 516.

[50] T. Ueda, B. Ruijl & J.A.M. Vermaseren, PoS LL2016 (2016), 070.

[51] T. Ueda, B. Ruijl & J.A.M. Vermaseren, Comput. Phys. Commun.
253 (2020), 107198.

[52] J.A.M. Vermaseren, math-ph/0010025.

[53] M. Tentyukov & J.A.M. Vermaseren, Comput. Phys. Commun. 181
(2010), 1419.

[54] J.A. Gracey, JHEP 0504 (2005), 012.

[55] J.M. Bell & J.A. Gracey, Phys. Rev. D92 (2015), 125001.

[56] K.G. Chetyrkin & A. Rétey, Nucl. Phys. B583 (2000), 3.

[57] K.G. Chetyrkin & A. Rétey, hep-ph/0007088.

[58] J.A. Gracey, Nucl. Phys. B662 (2003), 247.

[59] J.A. Gracey, Eur. Phys. J. C83 (2023), 83.

[60] A.L. Kataev & V.S. Molokoedov, arXiv:2302.03443 [hep-ph].

[61] A.L. Kataev & V.S. Molokoedov, Phys. Rev. D92 (2015),

[62] B.E. Lautrup, Of ghoulies and ghosties: an introduction to
QCD, (1977), NBI-HE-76-14,

[63] P. Cvitanovic, Field Theory, NORDITA Lecture Notes (1983),

[64] D. Kreimer, M. Sars & W. van Suijlekom, Annals Phys. 336
(2013), 180.

[65] H. Kißler, Annals Phys. 372 (2016), 159.

[66] H. Kißler, Computational and diagrammatic techniques for
perturbative Quantum Electrodynamics, Ph.D. thesis (2017),