The Crewther relation, schemes, gauges and fixed points

J.A. Gracey & R.H. Mason,
Theoretical Physics Division,

Department of Mathematical Sciences,
University of Liverpool,
P.O. Box
L69 3BX,
United Kingdom.

Abstract. We investigate the Crewther relation at high loop order in a
variety of renormalization schemes and gauges. By examining the properties of
the relation in schemes other than modified minimal subtraction ($\MSbar$) at
the fixed points of Quantum Chromodynamics we propose a generalization of the
Crewther relation that extends the $\MSbar$ construction of Broadhurst and
Kataev. A derivation based on the properties of the renormalization group
equation is provided for the generalization which is tested in various

LTH 1344

In [1] Crewther made an interesting and remarkable connection between the
Adler $D$-function and the Bjorken sum rule in non-abelian gauge theories.
Using the then available explicit perturbative expressions for those seemingly
unrelated quantities Crewther showed that their product was a constant and
there was no $O(a)$ correction where $a$ is related to the strong coupling
constant. When the higher order corrections subsequently became available in
the modified minimal subtraction ($\MSbar$) scheme [2, 3, 4] it was apparent
that Crewther's observation of constancy of the product was not retained at
these orders. It was demonstrated in [5] however that the non-zero $a$
dependence could be written as the product of two functions of $a$. One of the
functions was the $\beta$-function of Quantum Chromodynamics (QCD). While the
presence of a non-zero term appeared to undermine the ethos that Crewther's
observation was an exact result, it was in fact in keeping with a more general
property which is that of conformal symmetry, [6, 7, 8]. Indeed this
non-zero $O(a^2)$ term is referred to as the conformal symmetry breaking term,
[5], since the $\beta$-function vanishes when conformal symmetry is
present. Later the improvement in loop technology meant the $O(a^4)$ terms of
both the Adler $D$-function and Bjorken sum rules became available in [9].
It was shown in [9] that the resulting $O(a^4)$ correction to the
conformal symmetry breaking term could also be accommodated within the product
of the two loop $\beta$-function and a correction to the other function in the
breaking term as a function of $a$. Thus the conformal aspect of the Crewther
relation was reinforced.

In briefly summarizing the background to the development of Crewther's
observation it is important to note that the so-called conformal symmetry
breaking term was always determined in the $\MSbar$ scheme in the first
instance, [5, 9]. More recently there have been several detailed studies of
the Crewther construction in other renormalization schemes in [10, 11, 12].
One of the motivations in carrying out such investigations is to ascertain
whether or not the conformal symmetry breaking term is always the product of
the $\beta$-function and another function of $a$ in schemes other than
$\MSbar$. Amongst the schemes examined in [10, 11, 12] were the $V$ scheme
and the minimal momentum ($\mMOM$) scheme. The former is based on the static
quark potential, [13, 14], while the latter is defined with respect to the
ghost-gluon vertex of QCD, [15]. In particular it preserves the
non-renormalization property of that vertex in the Landau gauge, discovered by
Taylor in [16], but extended to an arbitrary linear covariant gauge. The
renormalization group functions of the $\mMOM$ scheme are known to five loops,
[15, 17, 18, 19]. Although the Crewther relation $\mMOM$ scheme study only
required four loop information one of its main conclusions was that at leading
order the factorization of the conformal symmetry breaking term only occurred
for specific values of the covariant gauge fixing parameter $\alpha$,
[12]. These were $\alpha$ $=$ $0$, $-$ $1$ and $-$ $3$. It was also
recognized that the special cases of $\alpha$ $=$ $0$ and $-$ $3$ arose in
other schemes, [12]. At the subsequent order the factorization only
occurred in the Landau gauge where $\alpha$ vanishes. While disappointing,
detailed investigations of the structure of where the factorization, [12],
failed did provide significant insight into and key clues as to how to
reconcile the hope that the Crewther construction could accommodate other
schemes. Indeed this has to be the case if the Crewther relation is to be
regarded as a fundamental property of a nonabelian gauge theory.

One interesting aspect of the emergence of the particular case of
$\alpha$ $=$ $-$ $3$ is that this value has already been noted as having
special significance in situations primarily associated with phenomenology or
infrared dynamics in QCD. For instance, in [20] Schwinger-Dyson methods
were used to examine models of chiral symmetry breaking and this choice of the
gauge parameter was shown to be necessary for a scale invariant solution. In
other words when the ansätzen for the form factors have a power law
behaviour then the underlying equations can be solved to determine when chiral
symmetry breaking occurs. In a different context, [21, 22], where the
worldline formalism was used to construct gauge invariant variables to study
parton distribution functions the choice of $\alpha$ $=$ $-$ $3$ was required
to ensure there were no divergences associated with rapidity. The same choice
of $\alpha$ was also needed in the related area of deep inelastic scattering in [23] where renormalon chains of bubble graphs were examined in the large
$\Nf$ expansion. At leading order the approximation was relatively accurate for
$\alpha$ $=$ $-$ $3$. This value also appeared in an exploration of Yang-Mills
theory at low energy. In particular in [24, 25, 26] an effective Lagrangian
was constructed in terms of dimension six gluonic operators that was a solution
to the infrared Schwinger-Dyson equations. Indeed it was argued in
[24, 25, 26] that to have a scaling solution for the gluon propagator which
would lead to a linearly rising static quark potential associated with
confinement required $\alpha$ $=$ $-$ $3$. In a similar context the role this
particular value plays in the behaviour of the effective coupling constant
derived in quark-quark scattering was discussed in [27]. Specifically at
$\alpha$ $=$ $-$ $3$ the effective coupling decreases as a function of the
exchanged momentum consistent with asymptotic freedom. A more phenomenological
origin for this gauge parameter choice was noted in [28]. Moreover the
appearance of $\alpha$ $=$ $-$ $3$ was not restricted to the $SU(3)$ group. It
was shown in [29] that the $Z$ boson is multiplicatively renormalizable
for this gauge parameter choice in the electroweak sector of the Standard
model. In support of the infrared connection the critical properties of the QCD
renormalization group equations in [30] indicated that there was a fixed
point similar to that of Banks and Zaks, [31], but with a value of
$\alpha$ close to $-$ $3$. That it was not exactly a negative integer was
primarily because that critical point analysis was carried out at four loops
which was an order much higher than the previous work on this special value.
This would suggest that while the infrared dynamics observations are pointing
to the same underlying property quantum corrections will necessarily have to be
taken into account too. Our Crewther study will also lend weight to that point
of view. Indeed the connection provided in [31] also directs our attention
to perhaps the most significant clue to accommodate schemes for the Crewther
relation. This is the fact that in examining the critical properties of the
renormalization group functions of QCD treated in the $(a,\alpha)$ plane there
is a fixed point in the neighbourhood of $\alpha$ $=$ $-$ $3$ in the conformal
window. While it is not precisely this value it is however suggestive of a
connection to the Crewther relation with the discrepancy somehow being
accounted for by higher order corrections lurking within a reconciliation of
the factorization issue with the conformal symmetry property.

It is therefore the aim of this article to extend the Crewther relation in some
way that it is inclusive of schemes beyond the $\MSbar$ one yet at the same
time obeying the conformal aspects on top of respecting the renormalization
group. In our investigation in order to gain as wide a picture as possible we
will consider not only the mMOM scheme but a similar one used in lattice gauge
theories, termed the modified Regularization Invariant ($\RI$) scheme
[32, 33], as well as the kinematic momentum subtraction (MOM) schemes
developed by Celmaster and Gonsalves, [34, 35]. In addition we will examine
the relation in two other gauges which are nonlinear covariant gauge fixings.
The reason for this is that while the Lorenz suite of covariant gauges are
canonically used in computations, with $\alpha$ arbitrary or zero, gauge
independent results should be blind to the choice of the particular gauge
fixing functional. Our aim is to construct a generalization of the conformal
symmetry breaking term that is consistent with the properties of the
renormalization group equation and preserves the conformal symmetry property as
well as putting the origin of the special case of $\alpha$ $\approx$ $-$ $3$ of
the linear covariant gauge fixing on a solid footing. To achieve all these aims
required both numerical examination of the perturbative expansions of the
product of the Adler $D$-function and Bjorken sum rule as well as the analytic
expressions. These will be studied at the highest available loop order in each
of the different gauges and schemes. Indeed the way the article is structured
it will follow the same development ethos of [5] where numerical and
analytic evidence was presented that led to and justified the factorization
property of the original conformal symmetry breaking term of [5]. By
assemblying this evidence a clear picture emerges which allows us to formulate
the generalization before providing an analytic justification of our
observations. This is then robustly and satisfactorily tested across the
various schemes and gauges. It would be remiss of us at this point not to
mention the related work in this area provided in [36]. There the
principle of maximal conformality was used to provide a way of setting the
renormalization scale for phenomenological applications from the Crewther

The article is organized as follows. We recall the core observations
surrounding the Crewther relation in Section $2$ as well as details of the
factorization property in the $\mMOM$ scheme. The focus of Section $3$ is to
present numerical evidence that the conformal symmetry breaking term vanishes
at a large number of fixed points of the QCD renomalization group equations for
nonzero $\alpha$. Based on those observations the consequences for the Crewther
relation in the $\mMOM$ scheme are studied analytically in Section $4$ where a
generalization of [5] is introduced. The consistency of this
generalization is then tested in other schemes and gauges in Section $5$ as
well as several other schemes in the linear covariant gauge in Section $6$.
The formalization of the generalized Crewther relation within the
renormalization group is presented in Section $7$. Section $8$ by contrast is
devoted to explaining the relation of the gauge parameter critical point values
to the special cases of the gauge parameter that were indicated in the $\mMOM$
scheme in [12]. An overview of the field theory origin of certain
properties of the Crewther relation that we observe is provided in Section $9$.
Conclusions are provided in Section $10$ ahead of two appendices. Expressions
illustrating the structure of key functions in the Crewther relation in a MOM
scheme for a general Lie group are recorded in Appendix A. Finally Appendix B
presents the renormalization group functions for a new renormalization scheme
that is introduced in Section $6$.

It is instructive to first review the construction given in [5] which
revealed Crewther's observation that the product of the Bjorken sum rule and
Adler $D$-function is not a constant. Instead there is a discrepancy which
depends on the $\beta$-function in the $\MSbar$ scheme. We recall the sum rule
and $D$-function to $O(a^3)$ are, [2, 3, 4],

\begin{eqnarray}
C_{{\mbox{\footnotesize{Bjr}}}}^{\MSbars}(a) &=&
1 - 3 C_F a
+ \left[
\frac{21}{2} C_F^2
- 23 C_F C_A
+ 8 \Nf T_F C_F
\right] a^2
\nonumber \\
&& + \left[
\frac{440}{3} \zeta_5 C_F C_A^2
+ \frac{1241}{9} C_F^2 C_A
+ \frac{7070}{27} \Nf T_F C_F C_A
+ 48 \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{10874}{27} C_F C_A^2
- \frac{920}{27} \Nf^2 T_F^2 C_F
- \frac{176}{3} \zeta_3 C_F^2 C_A
- \frac{160}{3} \zeta_5 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{133}{9} \Nf T_F C_F^2
- \frac{80}{3} \zeta_3 \Nf T_F C_F^2
- \frac{3}{2} C_F^3
\right] a^3 ~+~ O(a^4)
\label{bjms}
\end{eqnarray}

\begin{eqnarray}
C_{{\mbox{\footnotesize{Adl}}}}^{\MSbars}(a) &=&
\left[
1 + 3 C_F a
+ \left[
\frac{123}{2} C_F C_A
- \frac{3}{2} C_F^2
- 44 \zeta_3 C_F C_A
- 22 \Nf T_F C_F
+ 16 \zeta_3 \Nf T_F C_F
\right] a^2
\right. \nonumber \\
\left. + \left[
\frac{160}{3} \zeta_5 \Nf T_F C_F C_A
+ \frac{4832}{27} \Nf^2 T_F^2 C_F
+ \frac{7168}{9} \zeta_3 \Nf T_F C_F C_A
+ \frac{90445}{54} C_F C_A^2
\right. \right. \nonumber \\
&& \left. \left. ~~~
- \frac{31040}{27} \Nf T_F C_F C_A
- \frac{10948}{9} \zeta_3 C_F C_A^2
- \frac{1216}{9} \zeta_3 \Nf^2 T_F^2 C_F
- \frac{440}{3} \zeta_5 C_F C_A^2
\right. \right. \nonumber \\
&& \left. \left. ~~~
- \frac{69}{2} C_F^3
- 572 \zeta_3 C_F^2 C_A
- 320 \zeta_5 \Nf T_F C_F^2
- 127 C_F^2 C_A
- 29 \Nf T_F C_F^2
\right. \right. \nonumber \\
&& \left. \left. ~~~
+ 304 \zeta_3 \Nf T_F C_F^2
+ 880 \zeta_5 C_F^2 C_A
\right] a^3 ~+~ O(a^4)
\right] d_R
\label{adms}
\end{eqnarray}

in the $\MSbar$ scheme where $C_F$, $C_A$ and $T_F$ are the usual colour group
Casimirs, $\Nf$ is the number of (massless) quark flavours, $d_R$ is the
dimension of the fundamental representation and $\zeta_n$ is the Riemann zeta
function. Although the $O(a^4)$ terms of both are available [9], and
involve higher rank $4$ colour Casimirs, the $O(a^3)$ expressions are
sufficient for the moment to illustrate the properties of the discrepancy. We
note that our focus will be on the flavour non-singlet construction throughout
the article. For clarity we define the product in the same way as [5]

\begin{equation}
C_{{\mbox{\footnotesize{Adl}}}}^{\MSbars}(a) ~=~ d_R \left[ 1 ~+~
\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a) \right]
\label{crewms}
\end{equation}

where ${\mbox{csb}}$ on $\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a)$
labels the conformal symmetry breaking term, [5]. The scheme that the
variables are in will be indicated by the superscript label throughout. It is
clear from the product of (<ref>) and (<ref>) that
$\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a)$ is $O(a^2)$. The subsequent
$O(a^2)$ term is partly comprised of the sum of the two $O(a^2)$ terms of
(<ref>) and (<ref>) which results in five terms where the coefficient
of the $C_F^2$ term is $9$. The remaining $O(a^2)$ part arises from the product
of the one loop terms of each of (<ref>) and (<ref>) that precisely
cancels the other $C_F^2$ contribution. In total four terms remain. One pair
has rational coefficients and the other pair involves $\zeta_3$. While this
summarizes the underlying algebra the major result of [5] was to observe
that the remaining terms factorize with one of the factors equivalent to the
one loop $\beta$-function coefficient of [37, 38]. Consequently it was
proposed that $\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a)$ took the all
orders form

\begin{equation}
\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a) ~=~
\frac{\beta^{\MSbars}(a)}{a} K_a^{\MSbars}(a) ~.
\label{deltms}
\end{equation}

We recall explicit computation produced, [5],

\begin{eqnarray}
K_a^{\MSbars}(a) &=&
\left[
12 \zeta_3 C_F
- \frac{21}{2} C_F
\right] a
\nonumber \\
+ \left[
\frac{326}{3} \Nf T_F C_F
+ \frac{397}{6} C_F^2
- 240 \zeta_5 C_F^2
+ 136 \zeta_3 C_F^2
+ \frac{884}{3} \zeta_3 C_F C_A
- \frac{629}{2} C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{304}{3} \zeta_3 \Nf T_F C_F
\right] a^2
\nonumber \\
+ \left[
\frac{6496}{9} \zeta_3 \Nf^2 T_F^2 C_F
+ \frac{11900}{3} \zeta_5 C_F C_A^2
- \frac{406043}{36} C_F C_A^2
- \frac{40336}{9} \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{24880}{3} \zeta_5 C_F^2 C_A
- \frac{9824}{9} \Nf^2 T_F^2 C_F
- \frac{8000}{3} \zeta_5 \Nf T_F C_F C_A
- \frac{7729}{18} \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{2471}{12} C_F^3
+ \frac{16570}{3} \zeta_3 C_F^2 C_A
+ \frac{67520}{9} \Nf T_F C_F C_A
+ \frac{72028}{9} \zeta_3 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{99757}{36} C_F^2 C_A
- 5720 \zeta_5 C_F^3
- 3668 \zeta_3 \Nf T_F C_F^2
- 1232 \zeta_3^2 C_F C_A^2
- 840 \zeta_7 C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- 128 \zeta_3^2 \Nf T_F C_F C_A
+ 320 \zeta_5 \Nf^2 T_F^2 C_F
+ 488 \zeta_3 C_F^3
+ 576 \zeta_3^2 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
+ 4000 \zeta_5 \Nf T_F C_F^2
+ 5040 \zeta_7 C_F^3
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

which we include as a reference point for later.

One subsequent question that was studied after the establishment of
(<ref>) and (<ref>) was whether or not this relation took the
same form in renormalization schemes other than $\MSbar$. This was examined at
length in the $\mMOM$ scheme in several articles, [10, 11, 12]. The scheme
is based on preserving the non-renormalization property of the ghost-gluon
vertex in the Landau gauge, that Taylor observed in [16], in an arbitrary
linear covariant gauge. QCD has been renormalized to high loop order in $\mMOM$
in [15, 17, 18, 19]. In order to carry out a Crewther relation study the
expressions for the Adler $D$-function and Bjorken sum rule had first to be
established in the $\mMOM$ scheme, [12]. This was achieved by mapping the
$\MSbar$ coupling constant dependence in
$C_{{\mbox{\footnotesize{Bjr}}}}^{\MSbars}(a)$ and
$C_{{\mbox{\footnotesize{Adl}}}}^{\MSbars}(a)$ to the $\mMOM$ coupling constant
using the explicit relations between the variables in both schemes that are
available in [15, 17] for instance. As this mapping is gauge parameter
dependent it therefore produces expressions that are $\alpha$ dependent,
[12]. In order to facilitate our subsequent analysis we note, [12],

\begin{eqnarray}
C_{{\mbox{\footnotesize{Bjr}}}}^{\mMOMs}(a,\alpha) &=&
1 - 3 C_F a
+ \left[
\frac{3}{2} \alpha C_F C_A
- \frac{107}{12} C_F C_A
+ \frac{3}{4} \alpha^2 C_F C_A
+ \frac{4}{3} \Nf T_F C_F
+ \frac{21}{2} C_F^2
\right] a^2
\nonumber \\
+ \left[
24 \zeta_3 \Nf T_F C_F C_A
- \frac{20585}{144} C_F C_A^2
- \frac{208}{9} \Nf T_F C_F^2
- \frac{176}{3} \zeta_3 C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{160}{3} \zeta_5 \Nf T_F C_F C_A
- \frac{117}{8} \zeta_3 C_F C_A^2
- \frac{40}{3} \Nf^2 T_F^2 C_F
- \frac{21}{2} \alpha C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{21}{4} \alpha^2 C_F^2 C_A
- \frac{9}{8} \alpha^2 \zeta_3 C_F C_A^2
- \frac{4}{3} \alpha \Nf T_F C_F C_A
- \frac{3}{2} C_F^3
\right. \nonumber \\
&& \left. ~~~
- \frac{2}{3} \alpha^2 \Nf T_F C_F C_A
+ \frac{9}{16} \alpha^3 C_F C_A^2
+ \frac{33}{4} \alpha \zeta_3 C_F C_A^2
+ \frac{64}{3} \zeta_3 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{215}{48} \alpha C_F C_A^2
+ \frac{349}{48} \alpha^2 C_F C_A^2
+ \frac{440}{3} \zeta_5 C_F C_A^2
+ \frac{832}{9} \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{1415}{36} C_F^2 C_A
\right] a^3 ~+~ O(a^4)
\label{mmombjr}
\end{eqnarray}

\begin{eqnarray}
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha) &=&
\left[
1 + 3 C_F a
\right. \nonumber \\
&& \left.
+ \left[
\frac{569}{12} C_F C_A
- \frac{46}{3} \Nf T_F C_F
- \frac{3}{2} C_F^2
- \frac{3}{2} \alpha C_F C_A
- \frac{3}{4} \alpha^2 C_F C_A
- 44 \zeta_3 C_F C_A
\right. \right. \nonumber \\
&& \left. \left. ~~~
+ 16 \zeta_3 \Nf T_F C_F
\right] a^2
\right. \nonumber \\
&& \left.
+ \left[
\frac{50575}{48} C_F C_A^2
- \frac{18929}{24} \zeta_3 C_F C_A^2
- \frac{2063}{48} \alpha C_F C_A^2
- \frac{2033}{3} \Nf T_F C_F C_A
\right. \right. \nonumber \\
&& \left. \left. ~~~
- \frac{1355}{12} C_F^2 C_A
- \frac{1273}{48} \alpha^2 C_F C_A^2
- \frac{440}{3} \zeta_5 C_F C_A^2
- \frac{69}{2} C_F^3
- \frac{9}{16} \alpha^3 C_F C_A^2
\right. \right. \nonumber \\
&& \left. \left. ~~~
+ \frac{3}{2} \alpha C_F^2 C_A
+ \frac{3}{4} \alpha^2 C_F^2 C_A
+ \frac{23}{3} \alpha^2 \Nf T_F C_F C_A
+ \frac{46}{3} \alpha \Nf T_F C_F C_A
\right. \right. \nonumber \\
&& \left. \left. ~~~
+ \frac{58}{3} \Nf T_F C_F^2
+ \frac{143}{4} \alpha \zeta_3 C_F C_A^2
+ \frac{160}{3} \zeta_5 \Nf T_F C_F C_A
+ \frac{185}{8} \alpha^2 \zeta_3 C_F C_A^2
\right. \right. \nonumber \\
&& \left. \left. ~~~
+ \frac{1424}{3} \zeta_3 \Nf T_F C_F C_A
- 572 \zeta_3 C_F^2 C_A
- 320 \zeta_5 \Nf T_F C_F^2
- 64 \zeta_3 \Nf^2 T_F^2 C_F
\right. \right. \nonumber \\
&& \left. \left. ~~~
- 16 \alpha \zeta_3 \Nf T_F C_F C_A
- 8 \alpha^2 \zeta_3 \Nf T_F C_F C_A
+ 96 \Nf^2 T_F^2 C_F
+ 256 \zeta_3 \Nf T_F C_F^2
\right. \right. \nonumber \\
&& \left. \left. ~~~
+ 880 \zeta_5 C_F^2 C_A
\right] a^3 ~+~ O(a^4)
\right]
\label{mmomadl}
\end{eqnarray}

where we include the explicit gauge parameter dependence in the $\mMOM$ scheme
in the arguments of the functions. A general observation is that (<ref>)
has to be replaced by the more accommodating formal relation

\begin{equation}
C_{{\mbox{\footnotesize{Bjr}}}}^{\cal S}(a,\alpha)
C_{{\mbox{\footnotesize{Adl}}}}^{\cal S}(a,\alpha) ~=~ d_R \left[ 1 ~+~
\Delta_{\mbox{\footnotesize{csb}}}^{\cal S}(a,\alpha) \right]
\end{equation}

where ${\cal S}$ labels the scheme which for the moment is $\mMOM$. In
[12] the objective was to write
$a \Delta_{\mbox{\footnotesize{csb}}}^{\mMOMs}(a,\alpha)$ in the form
$\beta^{\mMOMs}(a,\alpha) K_a^{\mMOMs}(a,\alpha)$ which was partially
successful. By this we mean that while the coefficient of the leading term of
$K_a^{\mMOMs}(a,\alpha)$ had the same value as the $\MSbar$ scheme the next
order term, being $\alpha$ dependent, could not be accommodated within
(<ref>) except for several specific values of $\alpha$. These were
$\alpha$ $=$ $0$, $-$ $1$ and $-$ $3$, [12]. At the subsequent loop order
neither of the last two values preserved the structure of (<ref>).
Indeed the situation for the failure of $\alpha$ $=$ $-$ $3$ was examined in
depth. In particular where the breakdown arose was distilled to a set of terms
with identifiable colour factors. In summarizing the state of play for the
$\mMOM$ scheme, [12], it might appear that the relation (<ref>)
limits Crewther's original relation, [1], and its extension to include a
term proportional to the $\beta$-function to a subset of schemes. However we do
not believe this to be a satisfactory situation.

Instead while the establishment of (<ref>) and (<ref>) is an
important observation, the properties of the $\MSbar$ scheme perhaps overlook
aspects of the renormalization group functions in other schemes that actually
gave a direction as to how to resolve the difficulty with the $\mMOM$ scheme.
There will be several threads to our argument. One of the $\beta$-function
properties has already been incorporated into (<ref>) and
(<ref>) which is that in general the $\beta$-function is gauge
dependent. In the $\MSbar$ scheme the $\beta$-function is independent of the
gauge parameter, [39]. In addition the gauge parameter is in essence a
second coupling constant in a gauge theory even though its origin is in the
quadratic part of the Lagrangian rather than in a higher order interaction. The
renormalization group function associated with the renormalization of $\alpha$,
that is not usually referred to as a $\beta$-function as such, is the anomalous
dimension of the gauge parameter which we will denote by
$\gamma_\alpha(a,\alpha)$ here. Indeed the relevance of this anomalous
dimension can be seen in the renormalization group equation, derived from
ensuring that a finite renormalized $n$-point Green's function
$\Gamma_{(n)}(\mu,a,\alpha)$ is independent of the renormalization scale $\mu$
associated with a regularization, since

\begin{equation}
\left[ \mu \frac{\partial ~}{\partial \mu} ~+~
\beta(a,\alpha) \frac{\partial ~}{\partial a} ~+~
\alpha \gamma_\alpha(a,\alpha) \frac{\partial ~}{\partial \alpha} ~-~
n \gamma_\phi(a,\alpha) \right] \Gamma_{(n)}(\mu,a,\alpha) ~=~ 0 ~.
\label{rgeeqn}
\end{equation}

We have appended an $\alpha$ dependence in the $\beta$-function since in
general it will depend on the gauge parameter. It is only in a subset of
schemes such as $\MSbar$ and $\RI$ that the $\beta$-function is independent of
$\alpha$. Here $\gamma_\phi(a,\alpha)$ represents the anomalous dimensions of
all the fields of the $n$-function and we have simplified to the massless case.
While this is a standard expression for a gauge theory if one interprets
$\alpha$ as a second coupling constant then the second and third terms of
(<ref>) could be rewritten as the differential operator

\begin{equation}
\beta_1(g_i) \frac{\partial ~}{\partial g_1} ~+~
\beta_2(g_i) \frac{\partial ~}{\partial g_2}
\label{betacomb}
\end{equation}

where $g_1$ $=$ $a$, $g_2$ $=$ $\alpha$ and

\begin{equation}
\beta_1(a,\alpha) ~\equiv~ \beta(a,\alpha) ~~~,~~~
\beta_2(a,\alpha) ~\equiv~ \alpha \gamma_\alpha(a,\alpha) ~.
\label{betai}
\end{equation}

The final thread of our argument rests in the observation of [5] that
$\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a)$ vanishes at a fixed point of
the underlying theory. This is clearly the case in $\MSbar$ as the
$\beta$-function zeros determine the critical points of the renormalization
group flow. In other schemes where the $\beta$-function is gauge parameter
dependent the situation is more involved. For instance, in [40] the fixed
point properties of QCD were studied to several loop orders in schemes other
than $\MSbar$ such as the MOM schemes of Celmaster and Gonsalves, [34, 35].
In the analysis of [40] the gauge parameter was treated as a second
coupling constant and the critical point values of $a$ and $\alpha$, where the
functions of (<ref>) are zero, were determined. On top of the Gaussian
and Banks-Zaks critical points, [31, 41], that were derived from the
vanishing of $\beta^{\MSbars}(a)$, there are fixed points for both $a$ and
$\alpha$ non-zero simultaneously. More recently the analysis of [40] has
been extended to several further loop orders, [42], for the $\MSbar$,
$\mMOM$ and the MOM schemes of [34, 35] in the linear covariant gauge as
well as the nonlinear gauges considered here. Not only is the non-trivial
infrared stable fixed point observed in [40] preserved to higher order it
has analogues in the nonlinear gauges as well. Therefore considering all these
ingredients together it should be the case that the product of
$C_{{\mbox{\footnotesize{Bjr}}}}^{\mMOMs}(a,\alpha)$ and
$C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ gives $d_R$ not only at
the Gaussian and Banks-Zaks fixed points but also at the other non-trivial
critical points.

4|c|$\mMOM$ $2$ loop

$a_{\infty}$   $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0033112583$    $0.0000000000$     $2.9999991596$     $3.0000039877$
$0.0032001941$    $-3.0301823312$     $2.9999982468$     $3.0000012469$
$9.1803474173$    $2.4636080795$      $1271156.8083213258$     $17202735.3015072510$
4|c|$\mMOM$ $3$ loop

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031177883$     $0.0000000000$     $2.9999963264$     $3.0000001212$
$0.0031380724$     $-3.0274210489$     $2.9999973439$     $3.0000001217$
$0.1279084604$     $1.9051106246$     $6.2952539870$     $10.1893903424$
4|c|$\mMOM$ $4$ loop

$a_{\infty}$   $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031213518$     $0.0000000000$    $2.9999963720$     $3.0000001843$
$0.0031430130$      $-3.0273541344$    $2.9999974127$     $3.0000002080$
$0.1162651496$     $0.5286066929$      $5.3930704057$     $11.8942763573$
$0.1902883419$     $0.0000000000$      $13.5399867931$     $66.1969134786$
4|c|$\mMOM$ $5$ loop

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031220809$    $0.0000000000$     $2.9999963814$     $3.0000001972$
$0.0031434144$     $-3.0273765993$     $2.9999974183$     $3.0000002151$
$0.0502252330$     $-3.8653031470$      $3.1912609578$     $3.2787374506$
$0.0577103776$     $0.0000000000$      $3.2818695828$     $3.7273436677$

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

4|c|0pt12pt $\MSbar$ 2 loop

$a_{\infty}$    $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0033112583$     $0.0000000000$     $3.0000005633$     $3.0000037708$
$0.0033112583$     $-2.9529847269$      $3.0000005633$     $3.0000037708$
$0.0033112583$     $-203.8803486064$     $3.0000005633$     $3.0000037708$
4|c|0pt12pt $\MSbar$ 3 loop

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031618421$     $0.0000000000$     $2.9999981298$     $3.0000007963$
$0.0031618421$     $-2.9458392416$      $2.9999981298$     $3.0000007963$
4|c|0pt12pt $\MSbar$ 4 loop

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031699203$     $0.0000000000$     $2.9999982498$     $3.0000009437$
$0.0031699203$     $-2.9459051005$     $2.9999982498$     $3.0000009437$
$0.0031699203$     $-126.8989470199$      $2.9999982498$     $3.0000009437$
$0.0917000502$     $0.0000000000$      $4.1865556391$     $6.0730678402$
$0.0917000502$     $2.6061754737$      $4.1865556391$     $6.0730678402$
$0.0917000502$     $-4.8367657559$      $4.1865556391$     $6.0730678402$
4|c|0pt12pt $\MSbar$ 5 loop

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031708402$     $0.0000000000$     $2.9999982636$     $3.0000009605$
$0.0031708402$     $-2.9459382941$     $2.9999982636$     $3.0000009605$
$0.0468980276$     $0.0000000000$      $3.1531010299$     $3.2821634001$
$0.0468980276$     $-3.7013593081$     $3.1531010299$     $3.2821634001$

Values of $C_{{\mbox{\footnotesize{Bjr}}}}^{\MSbars}(a)
C_{{\mbox{\footnotesize{Adl}}}}^{\MSbars}(a)$ at critical points at successive
loop orders for $\Nf$ $=$ $16$.

Numerical evidence.

In order to test the idea given in [5] that
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ reflects some conformal
property of the underlying field theory at a fixed point, we have carried out a
numerical investigation in the first instance using the data given in
[42]. In particular the critical point values of the coupling constant and
gauge parameter are known to various loop orders in different renormalization
schemes and gauges. The numerical analysis of this section will be for the
$SU(3)$ colour group. Also as the Crewther relation is a purely four
dimensional one the fixed points we consider include all real solutions with
positive coupling constant where the functions of (<ref>) vanish. To
remain within the perturbative domain of applicability our analysis was
restricted to values of $\Nf$ close to the top of the conformal window which is
at $\Nf$ $=$ $16$ for $SU(3)$, [31, 41]. As a guide to this reasoning we
have recorded the successive loop order evaluation of
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ at criticality in Table
<ref> for $\Nf$ $=$ $16$. In the table the various three or four
critical values of $a$ and $\alpha$ are given at two to five loops respectively
in four sub-sectors as indicated by the sub-heading. The critical values are
recorded to higher precision than [42]. In each sub-sector the first fixed
point is the Banks-Zaks solution, [31, 41], which is a saddle point in the
$(a,\alpha)$ plane. The second fixed point is the infrared stable one whose
critical coupling value tends to that of the Banks-Zaks one as the loop order
increases. The remaining one or two fixed point solutions are artefacts of
solving polynomial equations at higher loop order. While their critical
coupling values are outside the range of perturbative validity they are
included as they are relevant to subsequent arguments. The data in the final
two columns are the $O(a^3)$ and $O(a^4)$ values of
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ at criticality.

4|c|$\MOMg$ 3 loops

$a_{\infty}$   $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0029141531$     $0.0000000000$     $2.9999877567$     $2.9999976812$
$0.0029387804$     $-3.0259847376$      $2.9999876182$     $2.9999981783$
4|c|$\MOMc$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031361417$     $0.0000000000$     $2.9999980055$     $2.9999998120$
$0.0031893967$     $-3.0263835218$      $2.9999980117$     $3.0000010487$
$0.0108889865$     $-9.6139587420$     $3.0017410493$     $3.0006478718$
4|c|$\MOMq$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$    $O(a^3)$   $O(a^4)$
$0.0031361417$    $0.0000000000$     $2.9999974139$     $2.9999999355$
$0.0031893967$    $-3.0263835218$     $3.0000000826$     $2.9999999820$
$0.0108889865$    $-9.6139587420$     $3.0015252800$     $3.0013631672$

Values of $C_{{\mbox{\footnotesize{Bjr}}}}^{\MOMis}(a)
C_{{\mbox{\footnotesize{Adl}}}}^{\MOMis}(a)$ at critical points at successive
loop orders for $\Nf$ $=$ $16$.

4|c|Curci-Ferrari $\MOMg$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0029141531$     $0.0000000000$     $2.9999877567$     $2.9999976812$
$0.0031077787$     $-5.8860065418$     $2.9999745145$     $2.9999979271$
$0.0069944637$     $-6.2196239630$     $2.9999956606$     $3.0006212984$
4|c|Curci-Ferrari $\MOMc$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0031822955$     $0.0000000000$     $2.9999996960$     $3.0000000816$
$0.0033905650$     $-5.8484574473$     $3.0000096730$     $2.9999996739$
$0.0256527015$     $-17.6663992530$     $3.0598055895$     $2.4987327333$
$0.0776442642$     $2.1961076699$     $3.7711417659$     $2.5429487508$
4|c|Curci-Ferrari $\MOMq$ 3 loops

$a_{\infty}$   $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0031361417$     $0.0000000000$     $2.9999974139$     $2.9999999354$
$0.0032120881$     $-5.8504247987$     $3.0000013385$     $2.9999998776$
$0.0068658178$     $-12.1450763308$     $3.0002807117$     $3.0002920650$

Values of $C_{{\mbox{\footnotesize{Bjr}}}}^{\MOMis}(a)
C_{{\mbox{\footnotesize{Adl}}}}^{\MOMis}(a)$ at critical points at successive
loop orders for $\Nf$ $=$ $16$ in the Curci-Ferrari gauge.

4|c|MAG $\MOMg$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0028654266$     $-0.4168022707$      $2.9999860489$     $2.9999968722$
$0.0029254124$     $-5.5625872573$     $2.9999889816$     $2.9999980333$
4|c|MAG $\MOMc$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0031683606$     $-0.4162766338$     $2.9999989914$     $3.0000000637$
$0.0031577946$     $-5.5686496127$     $2.9999979720$     $3.0000000132$
$0.0014328961$     $-60.7998590768$     $3.0000187570$     $2.9999749282$
4|c|MAG $\MOMq$ 3 loops

$a_{\infty}$    $\alpha_{\infty}$     $O(a^3)$   $O(a^4)$
$0.0031256674$     $-0.4162158544$     $2.9999968662$     $2.9999999505$
$0.0031921140$     $-5.5677230790$     $3.0000001817$     $3.0000000315$

Values of $C_{{\mbox{\footnotesize{Bjr}}}}^{\MOMis}(a)
C_{{\mbox{\footnotesize{Adl}}}}^{\MOMis}(a)$ at critical points at successive
loop orders for $\Nf$ $=$ $16$ in the MAG.

Several features emerge from the table. The first is that for the first two
critical points the combination of the Adler function and Bjorken sum rule
produces a value very close to the expected value of $3$. Moreover there is a
clear but slow convergence to $3$ as the loop order increases. We qualify the
lack of precise agreement by noting that in computing the product there will be
errors from the truncation of the series. With the available $O(a^4)$
expressions for both series the product will have terms up to $O(a^8)$. However
this would not be the true $O(a^8)$ expression for
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ since the as yet
undetermined $O(a^5)$ term of one factor would lead to an $O(a^8)$ contribution
from the known $O(a^3)$ term of the other factor for instance. For the purposes
of the tables we do not include such incomplete higher order contributions.
Therefore the most reliable indication of the value of
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ is that from the five loop
critical points at both $O(a^3)$ and $O(a^4)$. A guide to this is that the
final two fixed points produce a value significantly closer to $3$ than might
have seemed possible from the corresponding lower loop data even at four loops.
That the value for these two five loop fixed points is not as accurate as the
Banks-Zaks or the infrared stable ones resides in the fact that one is still
outside the domain of perturbative reliability. More importantly for the two
that are clearly within the domain the critical value of
C_{{\mbox{\footnotesize{Adl}}}}^{\mMOMs}(a,\alpha)$ remains close to $3$. We
have analysed the small deviation from $3$ for these two fixed points in a
variety of ways and concluded that the discrepancy is due to the terms beyond
$O(a^4)$ that result from the product of two perturbative functions of $a$.
Perhaps one clear indication of this is to repeat the same exercise that
produced Table <ref> but for the $\MSbar$ scheme as that case led to
the expressions for $\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a)$ in
(<ref>). The results are recorded in Table <ref> to the same
loop order. What is apparent is that the same numerical accuracy emerges for
C_{{\mbox{\footnotesize{Adl}}}}^{\MSbars}(a)$ at each loop order for the
Banks-Zaks and infrared stable fixed points as the corresponding cases in the
$\mMOM$ scheme. So the deviation from the value of $3$ in the $\mMOM$ scheme is
on a similar footing to $\MSbar$. While this reinforces the notion that the
Crewther relation has a connection to conformal properties it does not resolve
how the observations of [12] can be accommodated.

We conclude this section by recording the situation in other schemes as well as
gauges. In [42] the fixed point properties of QCD in the MOM schemes of
Celmaster and Gonsalves were studied in the linear covariant gauge as well as
two nonlinear gauges. These were the Curci-Ferrari gauge and the Maximal
Abelian Gauge (MAG) whose more technical properties we discuss later but note
that for both gauges the perturbative renormalization group functions are only
available to three loops. Our results for the linear gauge are given in Table
<ref> while those for the other gauges are noted in Tables
<ref> and <ref> respectively where $\MOMg$, $\MOMc$
and $\MOMq$ denote the three distinct MOM schemes based on the triple gluon,
ghost-gluon and quark-gluon $3$-point vertices respectively. For brevity we
have solely recorded the situation at three loops. Again a very similar state
to the five loop analysis arises. In each case and particularly at three loops
the product of the Adler function and the Bjorken sum rule in effect evaluate
to $3$. Moreover it reinforces the observation that the Crewther relation has
to accommodate properties consistent with the critical points of the underlying
theory for a variety of schemes and gauges.

$\mMOM$ scheme.

In light of these arguments and numerical evidence as well as the association
of $\beta(a,\alpha)$ and $\gamma_\alpha(a,\alpha)$ with the underlying
$\beta$-functions of gauge fixed QCD we propose that the extension of the
Crewther relation of [5] to accommodate renormalization schemes other than
$\MSbar$ is

\begin{equation}
\Delta_{\mbox{\footnotesize{csb}}}^{\cal S}(a_{\cal S},\alpha_{\cal S}) ~=~
\frac{\beta^{\cal S}(a_{\cal S},\alpha_{\cal S})}{a_{\cal S}}
K_a^{\cal S}(a_{\cal S},\alpha_{\cal S}) ~+~
\alpha_{\cal S} \gamma_\alpha^{\cal S}(a_{\cal S},\alpha_{\cal S})
K_\alpha^{\cal S}(a_{\cal S},\alpha_{\cal S})
\label{deltgen}
\end{equation}

where ${\cal S}$ labels the scheme. Again we have included $\alpha$ in the
argument of the $\beta$-function to be as general as possible. While
(<ref>) reflects the combination (<ref>) with the
identification of (<ref>) in (<ref>), unlike its $\MSbar$
counterpart two functions $K_a(a,\alpha)$ and $K_\alpha(a,\alpha)$ are now
required to satisfy the vanishing of
$\Delta_{\mbox{\footnotesize{csb}}}(a,\alpha)$ at all the critical points of
theory in a scheme ${\cal S}$. In other words if the origin of the Crewther
relation rests in the fact that it reflects conformal symmetry, which is
related to vanishing $\beta$-functions, then this property should not be a pure
$\MSbar$ one. The combination of the Bjorken sum rule and Adler $D$-function
that is the Crewther relation should vanish at the fixed points of another
scheme as suggested by our numerical investigation. So this can be resolved for
other schemes provided the relevant $\beta$-functions are present as proposed
in (<ref>).

In keeping with the ethos of [5] where the coefficients of the
renormalization group functions of (<ref>) were established by explicit
calculation we now focus in this section on the $\mMOM$ scheme to study the
$K$-functions partly to be able to compare and contrast with [10, 11, 12].
Therefore we have constructed the following functions in the $\mMOM$ scheme

\begin{eqnarray}
K_a^{\mMOMs}(a,\alpha) &=&
\left[
12 \zeta_3 C_F
- \frac{21}{2} C_F
\right] a
\nonumber \\
+ \left[
\frac{21}{4} C_F C_A \alpha^2
+ \frac{21}{2} C_F C_A \alpha
- \frac{2591}{12} C_F C_A
+ \frac{397}{6} C_F^2
- 240 \zeta_5 C_F^2
\right. \nonumber \\
&& \left. ~~~
- 48 \zeta_3 \Nf T_F C_F
- 12 \zeta_3 C_F C_A \alpha
- 6 \zeta_3 C_F C_A \alpha^2
+ 62 \Nf T_F C_F
+ 136 \zeta_3 C_F^2
\right. \nonumber \\
&& \left. ~~~
+ 182 \zeta_3 C_F C_A
\right] a^2
\nonumber \\
+ \left[
\frac{2471}{12} C_F^3
- \frac{1840145}{288} C_F C_A^2
- \frac{14740}{3} \zeta_5 C_F^2 C_A
- \frac{8000}{3} \zeta_5 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{5092}{9} \Nf T_F C_F^2
- \frac{4792}{3} \zeta_3 \Nf T_F C_F^2
- \frac{4312}{3} \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{2961}{16} \zeta_3 C_F C_A^2 \alpha^2
- \frac{2113}{2} \zeta_3^2 C_F C_A^2
- \frac{1568}{3} \Nf^2 T_F^2 C_F
- \frac{397}{4} C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{397}{8} C_F^2 C_A \alpha^2
- \frac{63}{4} \zeta_3 C_F C_A^2 \alpha^3
- \frac{31}{2} \Nf T_F C_F C_A \alpha^2
+ \frac{27}{2} \zeta_3^2 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{155}{3} \Nf T_F C_F C_A \alpha
+ \frac{391}{8} \zeta_3 C_F C_A^2 \alpha
+ \frac{561}{32} C_F C_A^2 \alpha^3
+ \frac{4051}{96} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{6251}{32} C_F C_A^2 \alpha^2
+ \frac{11900}{3} \zeta_5 C_F C_A^2
+ \frac{71251}{18} \Nf T_F C_F C_A
+ \frac{132421}{72} C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
+ \frac{152329}{48} \zeta_3 C_F C_A^2
- 5720 \zeta_5 C_F^3
- 840 \zeta_7 C_F^2 C_A
- 204 \zeta_3 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- 102 \zeta_3 C_F^2 C_A \alpha^2
- 99 \zeta_3^2 C_F C_A^2 \alpha
- 40 \zeta_3 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ 12 \zeta_3 \Nf T_F C_F C_A \alpha^2
+ 160 \zeta_3^2 \Nf T_F C_F C_A
+ 180 \zeta_5 C_F^2 C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 224 \zeta_3 \Nf^2 T_F^2 C_F
+ 320 \zeta_5 \Nf^2 T_F^2 C_F
+ 360 \zeta_5 C_F^2 C_A \alpha
+ 488 \zeta_3 C_F^3
\right. \nonumber \\
&& \left. ~~~
+ 2400 \zeta_5 \Nf T_F C_F^2
+ 3608 \zeta_3 C_F^2 C_A
+ 5040 \zeta_7 C_F^3
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

\begin{eqnarray}
K_\alpha^{\mMOMs}(a,\alpha) &=&
\left[
\frac{21}{4} C_F C_A
+ \frac{21}{4} C_F C_A \alpha
- 6 \zeta_3 C_F C_A
- 6 \zeta_3 C_F C_A \alpha
\right] a^2
\nonumber \\
+ \left[
\frac{189}{32} \alpha^2 C_F C_A^2
+ \frac{341}{3} \Nf T_F C_F C_A
+ \frac{2035}{48} \alpha C_F C_A^2
+ \frac{2109}{8} \zeta_3 C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
+ 9 \zeta_3^2 C_F C_A^2 \alpha
+ 120 \zeta_5 C_F^2 C_A
+ 120 \zeta_5 C_F^2 C_A \alpha
- \frac{9773}{32} C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{443}{8} \zeta_3 C_F C_A^2 \alpha
- \frac{397}{12} C_F^2 C_A
- \frac{397}{12} C_F^2 C_A \alpha
- \frac{27}{4} \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 88 \zeta_3 \Nf T_F C_F C_A
- 68 \zeta_3 C_F^2 C_A
- 68 \zeta_3 C_F^2 C_A \alpha
- 33 \zeta_3^2 C_F C_A^2
\right] a^3
\nonumber \\
&& +~ O(a^4) ~.
\end{eqnarray}

These expressions were arrived at perturbatively by solving for the two unknown
$K$-functions at each order in the coupling constant. This approach was taken
from the point of view of seeing whether such an ansatz would admit an explicit
solution and is a valid method to apply. What it did reveal was that one
property of the $K$-functions is they are not strictly unique. This is in the
sense that one could in principle arrange terms in one of the $K$-functions to
be absorbed into the other in such a way that the vanishing of
$\Delta_{\mbox{\footnotesize{csb}}}(a,\alpha)$ at all the fixed points is
preserved. One noteworthy property of $K_\alpha^{\mMOMs}(a,\alpha)$ is that the
leading term, which is always $O(a^2)$ for this function, vanishes at
$\alpha$ $=$ $-$ $1$ but not for the $O(a^3)$ term. This particular gauge
parameter value was singled out for specific comment in [12] as being of
special interest. Equally it was noted that its special status was not
preserved at next order. Similar observations were also made for the case of
$\alpha$ $=$ $-$ $3$ and we will discuss both cases in a later section.

MOM schemes.

While the Crewther relation has been studied previously in various schemes but
at length in the $\mMOM$ scheme in [12], it is instructive to study
several specific schemes as well as gauges other than the commonly used linear
covariant gauge. This is the topic for this section and we examine the MOM
schemes of [34, 35]. In particular we will focus on the $\MOMg$ scheme
associated with the triple gluon vertex as the mathematical content of the
$K$-functions of each of the three MOM schemes is similar. The $\MOMg$
renormalization group functions of [34, 35] and the other two schemes were
extended to three loops in [43] numerically as well as analytically for an
arbitrary linear gauge in [44]. More recently the four loop Landau gauge
renormalization group functions were determined in [45]. One reason we
have not used the results of [45] in our calculations is partly as they
are recorded for a specific gauge. However, another is that those higher order
corrections will not contribute to the $K$-functions. This can be seen by
simply examining the orders of $a$ in the two terms of (<ref>) and the
fact that the Adler and Bjorken expressions are only available to $O(a^4)$. As
the mapping between the coupling constants and gauge parameters in the $\MSbar$
and $\MOMg$ schemes are known, [44], we have found
$C_{{\mbox{\footnotesize{Bjr}}}}^{\MOMgs}(a,\alpha)$ and
$C_{{\mbox{\footnotesize{Adl}}}}^{\MOMgs}(a,\alpha)$ to $O(a^4)$. Consequently
we have been able to determine the two $K$-functions parallel to those of the
$\mMOM$ scheme. For instance the $SU(3)$ Yang-Mills expressions are

\begin{eqnarray}
\left. K_a^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
16 \zeta_3
- 14
\right] a
\nonumber \\
+ \left[
\frac{6448}{9} \zeta_3
- \frac{14158}{27}
- \frac{1280}{3} \zeta_5
- \frac{644}{27} \pi^2
- \frac{368}{9} \psi^\prime(\third) \zeta_3
- \frac{56}{9} \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{32}{3} \psi^\prime(\third) \zeta_3 \alpha^2
+ \frac{28}{3} \psi^\prime(\third) \alpha^2
+ \frac{64}{9} \zeta_3 \pi^2 \alpha^2
+ \frac{322}{9} \psi^\prime(\third)
+ \frac{736}{27} \zeta_3 \pi^2
\right. \nonumber \\
&& \left. ~~~
- 72 \zeta_3 \alpha
- 42 \alpha^2
- 42 \psi^\prime(\third) \alpha
- 32 \zeta_3 \pi^2 \alpha
- 8 \zeta_3 \alpha^3
+ 7 \alpha^3
+ 28 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ 48 \zeta_3 \alpha^2
+ 48 \psi^\prime(\third) \zeta_3 \alpha
+ 63 \alpha
\right] a^2
\nonumber \\
+ \left[
\frac{1}{4} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^3
+ \frac{7}{2} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^2
- \frac{127914947}{2592}
- \frac{696830}{243} \pi^2
\right. \nonumber \\
&& \left. ~~~
- \frac{372568}{81} \zeta_3 \pi^2 \alpha
- \frac{307715}{54} \psi^\prime(\third) \alpha
- \frac{210179}{324} \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{177919}{243} \psi^\prime(\third) \pi^2 \alpha
- \frac{147991}{96} \alpha^2
- \frac{144856}{27} \psi^\prime(\third) \zeta_3
- \frac{133352}{729} \zeta_3 \pi^4 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{54362}{243} \zeta_3 \pi^4
- \frac{50834}{81} \psi^\prime(\third)^2 \zeta_3 \alpha
- \frac{29440}{27} \zeta_5 \pi^2
- \frac{25921}{216} \psi^\prime(\third)^2
\right. \nonumber \\
&& \left. ~~~
- \frac{24514}{3} \zeta_3 \alpha
- \frac{21805}{216} \psi^\prime(\third)^2 \alpha^2
- \frac{14812}{81} \psi^\prime(\third) \zeta_3 \pi^2
\right. \nonumber \\
&& \left. ~~~
- \frac{12460}{81} \psi^\prime(\third) \zeta_3 \pi^2 \alpha^2
- \frac{11663}{9} \psi^\prime(\third) \zeta_3 \alpha^2
- \frac{8918}{243} \pi^4 \alpha^2
- \frac{5349}{32} \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{4399}{72} \alpha^3
- \frac{2989}{32} \psi^{\prime\prime\prime}(\third)
- \frac{2560}{9} \zeta_5 \pi^2 \alpha^2
- \frac{1286}{81} \zeta_3 \pi^4 \alpha^3
- \frac{1205}{6} \zeta_3 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{1115}{18} \pi^2 \alpha^3
- \frac{1078}{27} \psi^\prime(\third) \pi^2 \alpha^3
- \frac{763}{12} \pi^2 \alpha^4
- \frac{308}{9} \psi^\prime(\third)^2 \zeta_3 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{250}{3} \psi^\prime(\third) \zeta_3 \alpha^3
- \frac{153}{2} \zeta_3 \alpha^5
- \frac{147}{32} \alpha^6
- \frac{112}{9} \psi^\prime(\third) \zeta_3 \pi^2 \alpha^4
- \frac{98}{27} \pi^4 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{49}{4} \psi^\prime(\third) \alpha^5
- \frac{49}{6} \psi^\prime(\third)^2 \alpha^4
- \frac{49}{16} \psi^{\prime\prime\prime}(\third) \alpha^2
- \frac{28}{3} \zeta_3 \pi^2 \alpha^5
\right. \nonumber \\
&& \left. ~~~
- \frac{7}{32} \psi^{\prime\prime\prime}(\third) \alpha^3
+ \frac{21}{4} \zeta_3 \alpha^6
+ \frac{28}{3} \psi^\prime(\third)^2 \zeta_3 \alpha^4
+ \frac{49}{6} \pi^2 \alpha^5
+ \frac{63}{2} \psi^{\prime\prime\prime}(\third) \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{98}{9} \psi^\prime(\third) \pi^2 \alpha^4
+ \frac{112}{27} \zeta_3 \pi^4 \alpha^4
+ \frac{218}{3} \zeta_3 \pi^2 \alpha^4
+ \frac{427}{4} \psi^{\prime\prime\prime}(\third) \zeta_3
\right. \nonumber \\
&& \left. ~~~
+ \frac{500}{9} \zeta_3 \pi^2 \alpha^3
+ \frac{539}{18} \psi^\prime(\third)^2 \alpha^3
+ \frac{763}{8} \psi^\prime(\third) \alpha^4
+ \frac{867}{4} \zeta_3 \alpha^4
+ \frac{1071}{16} \alpha^5
\right. \nonumber \\
&& \left. ~~~
+ \frac{1115}{12} \psi^\prime(\third) \alpha^3
+ \frac{1232}{27} \psi^\prime(\third) \zeta_3 \pi^2 \alpha^3
+ \frac{1280}{3} \psi^\prime(\third) \zeta_5 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{3115}{27} \psi^\prime(\third)^2 \zeta_3 \alpha^2
+ \frac{3703}{27} \psi^\prime(\third)^2 \zeta_3
+ \frac{4501}{324} \pi^4 \alpha^3
+ \frac{10192}{243} \zeta_3 \pi^4 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{14720}{9} \psi^\prime(\third) \zeta_5
+ \frac{19415}{8} \zeta_3 \alpha^2
+ \frac{21805}{162} \psi^\prime(\third) \pi^2 \alpha^2
+ \frac{22400}{3} \zeta_7
\right. \nonumber \\
&& \left. ~~~
+ \frac{23326}{27} \zeta_3 \pi^2 \alpha^2
+ \frac{25921}{162} \psi^\prime(\third) \pi^2
+ \frac{116683}{729} \pi^4 \alpha
+ \frac{177919}{324} \psi^\prime(\third)^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{186284}{27} \psi^\prime(\third) \zeta_3 \alpha
+ \frac{190267}{972} \pi^4
+ \frac{203336}{243} \psi^\prime(\third) \zeta_3 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{210179}{216} \psi^\prime(\third) \alpha^2
+ \frac{289169}{48} \alpha
+ \frac{289712}{81} \zeta_3 \pi^2
+ \frac{307715}{81} \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{348415}{81} \psi^\prime(\third)
+ \frac{485200}{27} \zeta_5
+ \frac{9840343}{216} \zeta_3
- 23325 \zeta_3^2
- 1920 \zeta_5 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 1920 \psi^\prime(\third) \zeta_5 \alpha
- 109 \psi^\prime(\third) \zeta_3 \alpha^4
- 36 \zeta_3^2 \alpha^3
- 36 \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha
+ 9 \zeta_3^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 14 \psi^\prime(\third) \zeta_3 \alpha^5
+ 320 \zeta_5 \alpha^3
+ 1280 \zeta_5 \pi^2 \alpha
+ 2052 \zeta_3^2 \alpha
+ 2880 \zeta_5 \alpha
\right] a^3 \nonumber \\
&& +~ O(a^4)
\end{eqnarray}

\begin{eqnarray}
\left. K_\alpha^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
\frac{63}{2}
- \frac{56}{9} \pi^2 \alpha
- \frac{32}{3} \psi^\prime(\third) \zeta_3 \alpha
+ \frac{21}{2} \alpha^2
+ \frac{28}{3} \psi^\prime(\third) \alpha
+ \frac{64}{9} \zeta_3 \pi^2 \alpha
- 42 \alpha
\right. \nonumber \\
&& \left. ~
- 36 \zeta_3
- 21 \psi^\prime(\third)
- 16 \zeta_3 \pi^2
- 12 \zeta_3 \alpha^2
+ 14 \pi^2
+ 24 \psi^\prime(\third) \zeta_3
+ 48 \zeta_3 \alpha
\right] a^2
\nonumber \\
+ \left[
684 \zeta_3^2
+ 960 \zeta_5
- \frac{528991}{108} \psi^\prime(\third)
- \frac{298948}{81} \zeta_3 \pi^2
- \frac{207487}{324} \psi^\prime(\third) \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{179000}{729} \zeta_3 \pi^4
- \frac{177037}{243} \psi^\prime(\third) \pi^2
- \frac{50582}{81} \psi^\prime(\third)^2 \zeta_3
- \frac{26252}{81} \zeta_3 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{12853}{3} \zeta_3
- \frac{5120}{27} \zeta_5 \pi^2 \alpha
- \frac{2464}{27} \zeta_3 \pi^4 \alpha
- \frac{2331}{8} \alpha^3
- \frac{2009}{9} \psi^\prime(\third) \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{1295}{6} \zeta_3 \alpha^2
- \frac{917}{36} \alpha^2
- \frac{574}{3} \psi^\prime(\third)^2 \zeta_3 \alpha
- \frac{448}{27} \psi^\prime(\third) \zeta_3 \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{392}{81} \pi^4 \alpha^3
- \frac{385}{3} \pi^2 \alpha^3
- \frac{255}{2} \zeta_3 \alpha^4
- \frac{245}{12} \psi^\prime(\third) \alpha^4
- \frac{147}{16} \alpha^5
\right. \nonumber \\
&& \left. ~~~
- \frac{140}{9} \zeta_3 \pi^2 \alpha^4
- \frac{98}{9} \psi^\prime(\third)^2 \alpha^3
- \frac{49}{24} \psi^{\prime\prime\prime}(\third) \alpha
- \frac{35}{2} \pi^2 \alpha^2
- \frac{7}{32} \psi^{\prime\prime\prime}(\third) \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{1}{4} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^2
+ \frac{7}{3} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha
+ \frac{21}{2} \zeta_3 \alpha^5
+ \frac{21}{2} \psi^{\prime\prime\prime}(\third)
\right. \nonumber \\
&& \left. ~~~
+ \frac{70}{3} \psi^\prime(\third) \zeta_3 \alpha^4
+ \frac{105}{4} \psi^\prime(\third) \alpha^2
+ \frac{112}{9} \psi^\prime(\third)^2 \zeta_3 \alpha^3
+ \frac{133}{4} \pi^4 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{147}{2} \psi^\prime(\third)^2 \alpha^2
+ \frac{245}{18} \pi^2 \alpha^4
+ \frac{385}{2} \psi^\prime(\third) \alpha^3
+ \frac{392}{27} \psi^\prime(\third) \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{440}{3} \zeta_3 \pi^2 \alpha^3
+ \frac{448}{81} \zeta_3 \pi^4 \alpha^3
+ \frac{1280}{3} \zeta_5 \pi^2
+ \frac{1785}{16} \alpha^4
+ \frac{2009}{12} \psi^\prime(\third)^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{2156}{27} \pi^4 \alpha
+ \frac{2296}{9} \psi^\prime(\third) \zeta_3 \pi^2 \alpha
+ \frac{2560}{9} \psi^\prime(\third) \zeta_5 \alpha
+ \frac{11215}{12} \zeta_3 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{13126}{27} \psi^\prime(\third) \zeta_3 \alpha
+ \frac{20909}{144} \alpha
+ \frac{78711}{16}
+ \frac{149474}{27} \psi^\prime(\third) \zeta_3
\right. \nonumber \\
&& \left. ~~~
+ \frac{156625}{729} \pi^4
+ \frac{177037}{324} \psi^\prime(\third)^2
+ \frac{202328}{243} \psi^\prime(\third) \zeta_3 \pi^2
+ \frac{207487}{486} \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{528991}{162} \pi^2
- 1280 \zeta_5 \alpha
- 640 \psi^\prime(\third) \zeta_5
- 220 \psi^\prime(\third) \zeta_3 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- 98 \psi^\prime(\third) \pi^2 \alpha^2
- 84 \psi^\prime(\third)^2 \zeta_3 \alpha^2
- 38 \zeta_3 \pi^4 \alpha^2
- 36 \zeta_3^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 30 \psi^\prime(\third) \zeta_3 \alpha^2
- 12 \psi^{\prime\prime\prime}(\third) \zeta_3
+ 6 \zeta_3^2 \alpha
+ 20 \zeta_3 \pi^2 \alpha^2
+ 112 \psi^\prime(\third) \zeta_3 \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 320 \zeta_5 \alpha^2
+ 333 \zeta_3 \alpha^3
\right] a^3 ~+~ O(a^4)
\label{momgkfn}
\end{eqnarray}

where $\psi(z)$ is the Euler psi function. Its appearance is directly related
to the symmetric point configuration of the triple gluon vertex and also arises
in the other two MOM schemes based on the ghost-gluon ($\MOMc$) and quark-gluon
($\MOMq$) vertices. In [44] the expressions for the renormalization group
functions additionally involved harmonic polynomials $H^{(2)}_i$ as well as

\begin{equation}
s_n(z) ~=~ \frac{1}{\sqrt{3}} \mbox{Im} \, \left[ \mbox{Li}_n \left(
\frac{e^{iz}}{\sqrt{3}} \right) \right]
\end{equation}

where $\mbox{Li}_n(z)$ is the polylogarithm function. In [45] it was shown
using the PSLQ algorithm that the actual number basis for MOM schemes was
rationals, $\pi$, $\zeta_2$, $\zeta_3$, $\zeta_4$, $\zeta_5$,
$\psi^\prime(\third)$ and $\psi^{\prime\prime\prime}(\third)$. The absence of
harmonic polylogarithms and the polylogarithm function was because they were
not independent of this basis. Indeed we have demonstrated this by verifying
that the relation

\begin{eqnarray}
s_2 (\pisix) &=& \frac{1}{11664}
\left[ 324 \sqrt{3} \ln(3) \pi - 27 \sqrt{3} \ln^2(3) \pi + 29 \sqrt{3} \pi^3
- 1944 \psi^\prime(\third)
\right. \nonumber \\
&& \left. ~~~~~~~~~
+ 23328 s_2 (\pitwo) + 19440 s_3 (\pisix) - 15552 s_3 (\pitwo)
+ 1296 \pi^2 \right]
\label{harmpolrel}
\end{eqnarray}

holds numerically. We have incorporated (<ref>) in our computations
and therefore any MOM scheme expressions will only involve the basis of
[45]. We have also established relations similar to the $\MOMg$ ones of
(<ref>) for the other MOM schemes. Those results as well as the $\MOMg$
ones are available for an arbitrary colour group and $\Nf$ $\neq$ $0$ in the
data file associated with this article. As there was a special value for
$\alpha$ in the $\mMOM$ case that means the $K_\alpha$ function was absent at
leading order we have examined $K_\alpha^{\MOMgs}(a,\alpha)$ for $SU(3)$ to see
if the same or another value of $\alpha$ arises to produce a zero leading order
term of the $\MOMg$ $K_\alpha$ function. For $\Nf$ $\neq$ $0$ this term is
$\Nf$ independent but unlike the $\mMOM$ case it is quadratic in $\alpha$.
Although there are analytic solutions for $\alpha$ when the leading term
vanishes the numerical values are $\alpha$ $=$ $-$ $1.617608$ and $2.492398$.
The fact that neither values are integer is solely due to the presence of
$\psi(\third)$, $\zeta_3$ and $\pi$ contributions which result from the
renormalization prescription defining each MOM scheme and the particular
external momentum configuration of the $3$-point vertices where the
renormalization constants are defined. For the $\MOMc$ and $\MOMq$ schemes the
respective values of $\alpha$ are $-$ $2.358904$ and $-$ $7.145900$ as both
leading order $K_\alpha$ functions are linear in the gauge parameter.

Having concentrated on the linear covariant gauge to this point we turn our
attention to two nonlinear gauges which are the Curci-Ferrari gauge [46]
and the MAG, [47, 48, 49]. However both gauges are not unrelated. The MAG is
a gauge fixing where the gluon fields are partitioned into those whose
associated group generator form an abelian subgroup of the non-abelian gauge
group and the remainder. The former are referred to as the diagonal gluons and
there are $\Nda$ such fields while there are $\Noda$ remaining off-diagonal
fields where $\Nda$ $=$ $\Nc-1$ and $\Noda$ $=$ $\Nc(\Nc-1)$ for the group
$SU(\Nc)$. The gauge fixing functional takes a different form in both sectors
where the diagonal gluons are fixed in the Landau gauge but the $\Noda$ gluons
have a nonlinear functional that involves the ghosts associated with the
off-diagonal gluons. These ghost fields differ from those of the abelian
subgroup gluons. The MAG has several similarities with the background field
gauge in that the diagonal gluons behave like background field gluons. The
Curci-Ferrari gauge is different from the MAG in that the gluons are not
distinguished in the gauge fixing but do share a similar quadratic ghost term
in the gauge fixing functional. Both gauges have quartic ghost interactions
that have no analogue in the linear covariant gauge fixing. What does connect
the MAG to the Curci-Ferrari gauge, however, is the fact that taking the formal
$\Nda$ $\to$ $0$ limit produces the Curci-Ferrari gauge. This is of practical
benefit as a check on any computations involving the MAG since the
Curci-Ferrari gauge expressions have to be reproduced as $\Nda$ $\to$ $0$.

Although both the Bjorken sum rule and the Adler $D$-function were originally
computed perturbatively to high loop order in the linear gauge, since the
operators involved in the underlying field theory formalism are gauge
independent the $\MSbar$ expressions for both quantities will be the same in
either the Curci-Ferrari gauge or MAG. Therefore to determine the extension
to the Crewther relation we have carried out the mapping of the $\MSbar$
coupling constant to the Curci-Ferrari coupling in the $\MOMg$ scheme as well
as also the $\MOMc$ and $\MOMq$ schemes. As a check on this procedure,
however, we carried out the explicit direct computation of the Adler
$D$-function in both the Curci-Ferrari gauge and the MAG to high loop order.
This could only be achieved by using the Forcer package, [50, 51],
written in the symbolic manipulation language Form, [52, 53]. The
Forcer algorithm evaluates massless $2$-point Feynman graphs up to four
loops very efficiently. In carrying out this exercise it is important to
recognize that in the Curci-Ferrari gauge there are several extra Feynman
graphs to be evaluated for the $D$-function than the linear gauge due to the
extra quartic ghost interaction. By contrast in the case of the MAG there are
considerably more diagrams to evaluate compared to the other two gauges not
only due to quartic ghost interactions but because there are twice as many
gluon and ghost fields. These have intricately connected interactions. On top
of this one has to implement a routine to handle the colour group factors
associated with gluon and ghost fields as they lie in two sectors. So the group
generators and structure functions obey a more complicated algebra. Useful in
this context were the identities used in the three loop $\MSbar$
renormalization of the MAG, [54]. The overall outcome was that the
$\MSbar$ Adler $D$-function resulted as expected. Once that was established for
both gauges it was a simple exercise to change the renormalization of the
$D$-function to another scheme such as the $\MOMg$ one for both gauges.
Consequently this checked that the direct application of the mapping from the
linear gauge expression was consistent. The various coupling constant mappings
were orginally determined in [55].

Having summarized the background to both gauges we have managed to construct
the respective $K$-functions in keeping with (<ref>). The expressions
for the Curci-Ferrari (CF) gauge are not dissimilar to those of the linear
covariant gauge since

\begin{eqnarray}
\left. K_{a\,\CFs}^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
16 \zeta_3
- 14
\right] a
\nonumber \\
+ \left[
28 \pi^2 \alpha
+ 48 \zeta_3 \alpha^2
+ 48 \psi^{\prime}(\third) \zeta_3 \alpha
- \frac{14158}{27}
- \frac{1280}{3} \zeta_5
- \frac{644}{27} \pi^2
\right. \nonumber \\
&& \left. ~~~
- \frac{368}{9} \psi^{\prime}(\third) \zeta_3
- \frac{56}{9} \pi^2 \alpha^2
- \frac{32}{3} \psi^{\prime}(\third) \zeta_3 \alpha^2
+ \frac{28}{3} \psi^{\prime}(\third) \alpha^2
+ \frac{64}{9} \zeta_3 \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{322}{9} \psi^{\prime}(\third)
+ \frac{736}{27} \zeta_3 \pi^2
+ \frac{6448}{9} \zeta_3
- 72 \zeta_3 \alpha
- 42 \alpha^2
- 42 \psi^{\prime}(\third) \alpha
\right. \nonumber \\
&& \left. ~~~
- 32 \zeta_3 \pi^2 \alpha
- 8 \zeta_3 \alpha^3
+ 7 \alpha^3
+ 63 \alpha
\right] a^2
\nonumber \\
+ \left[
\frac{22400}{3} \zeta_7
- \frac{127914947}{2592}
- \frac{696830}{243} \pi^2
- \frac{562472}{81} \zeta_3 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{491041}{54} \psi^{\prime}(\third) \alpha
- \frac{402143}{243} \psi^{\prime}(\third) \pi^2 \alpha
- \frac{389608}{729} \zeta_3 \pi^4 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{238097}{324} \pi^2 \alpha^2
- \frac{164461}{96} \alpha^2
- \frac{144856}{27} \psi^{\prime}(\third) \zeta_3
- \frac{114898}{81} \psi^{\prime}(\third)^2 \zeta_3 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{54362}{243} \zeta_3 \pi^4
- \frac{32389}{216} \psi^{\prime}(\third)^2 \alpha^2
- \frac{29948}{3} \zeta_3 \alpha
- \frac{29440}{27} \zeta_5 \pi^2
\right. \nonumber \\
&& \left. ~~~
- \frac{25921}{216} \psi^{\prime}(\third)^2
- \frac{18508}{81} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^2
- \frac{14812}{81} \psi^{\prime}(\third) \zeta_3 \pi^2
\right. \nonumber \\
&& \left. ~~~
- \frac{13643}{243} \pi^4 \alpha^2
- \frac{12761}{9} \psi^{\prime}(\third) \zeta_3 \alpha^2
- \frac{5415}{32} \alpha^4
- \frac{2989}{32} \psi^{\prime\prime\prime}(\third)
\right. \nonumber \\
&& \left. ~~~
- \frac{2560}{9} \zeta_5 \pi^2 \alpha^2
- \frac{2236}{81} \zeta_3 \pi^4 \alpha^3
- \frac{1862}{27} \psi^{\prime}(\third) \pi^2 \alpha^3
- \frac{875}{12} \pi^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{841}{3} \zeta_3 \alpha^3
- \frac{545}{12} \psi^{\prime}(\third) \alpha^3
- \frac{532}{9} \psi^{\prime}(\third)^2 \zeta_3 \alpha^3
- \frac{380}{9} \zeta_3 \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{153}{2} \zeta_3 \alpha^5
- \frac{147}{32} \alpha^6
- \frac{112}{9} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^4
- \frac{98}{27} \pi^4 \alpha^4
- \frac{63}{16} \psi^{\prime\prime\prime}(\third) \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{49}{4} \psi^{\prime}(\third) \alpha^5
- \frac{49}{6} \psi^{\prime}(\third)^2 \alpha^4
- \frac{28}{3} \zeta_3 \pi^2 \alpha^5
- \frac{7}{16} \psi^{\prime\prime\prime}(\third) \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{1}{2} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^3
+ \frac{9}{2} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^2
+ \frac{21}{4} \zeta_3 \alpha^6
+ \frac{28}{3} \psi^{\prime}(\third)^2 \zeta_3 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{49}{6} \pi^2 \alpha^5
+ \frac{63}{2} \psi^{\prime\prime\prime}(\third) \alpha
+ \frac{98}{9} \psi^{\prime}(\third) \pi^2 \alpha^4
+ \frac{112}{27} \zeta_3 \pi^4 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{190}{3} \psi^{\prime}(\third) \zeta_3 \alpha^3
+ \frac{250}{3} \zeta_3 \pi^2 \alpha^4
+ \frac{427}{4} \psi^{\prime\prime\prime}(\third) \zeta_3
+ \frac{545}{18} \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{825}{4} \zeta_3 \alpha^4
+ \frac{875}{8} \psi^{\prime}(\third) \alpha^4
+ \frac{931}{18} \psi^{\prime}(\third)^2 \alpha^3
+ \frac{1071}{16} \alpha^5
\right. \nonumber \\
&& \left. ~~~
+ \frac{1280}{3} \psi^{\prime}(\third) \zeta_5 \alpha^2
+ \frac{2128}{27} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^3
+ \frac{2917}{36} \alpha^3
+ \frac{3703}{27} \psi^{\prime}(\third)^2 \zeta_3
\right. \nonumber \\
&& \left. ~~~
+ \frac{3913}{162} \pi^4 \alpha^3
+ \frac{4627}{27} \psi^{\prime}(\third)^2 \zeta_3 \alpha^2
+ \frac{5279}{2} \zeta_3 \alpha^2
+ \frac{14720}{9} \psi^{\prime}(\third) \zeta_5
\right. \nonumber \\
&& \left. ~~~
+ \frac{15592}{243} \zeta_3 \pi^4 \alpha^2
+ \frac{25522}{27} \zeta_3 \pi^2 \alpha^2
+ \frac{25921}{162} \psi^{\prime}(\third) \pi^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{32389}{162} \psi^{\prime}(\third) \pi^2 \alpha^2
+ \frac{190267}{972} \pi^4
+ \frac{238097}{216} \psi^{\prime}(\third) \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{281236}{27} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{289712}{81} \zeta_3 \pi^2
+ \frac{340907}{729} \pi^4 \alpha
+ \frac{348415}{81} \psi^{\prime}(\third)
\right. \nonumber \\
&& \left. ~~~
+ \frac{402143}{324} \psi^{\prime}(\third)^2 \alpha
+ \frac{433885}{48} \alpha
+ \frac{459592}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha
+ \frac{485200}{27} \zeta_5
\right. \nonumber \\
&& \left. ~~~
+ \frac{491041}{81} \pi^2 \alpha
+ \frac{9840343}{216} \zeta_3
- 23325 \zeta_3^2
- 1920 \zeta_5 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 1920 \psi^{\prime}(\third) \zeta_5 \alpha
- 125 \psi^{\prime}(\third) \zeta_3 \alpha^4
- 72 \zeta_3^2 \alpha^3
- 54 \zeta_3^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 36 \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha
+ 14 \psi^{\prime}(\third) \zeta_3 \alpha^5
+ 320 \zeta_5 \alpha^3
+ 1280 \zeta_5 \pi^2 \alpha
+ 2052 \zeta_3^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ 2880 \zeta_5 \alpha
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

\begin{eqnarray}
\left. K_{\alpha\,\CFs}^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
\frac{63}{2}
+ \frac{64}{9} \zeta_3 \pi^2 \alpha
- \frac{56}{9} \pi^2 \alpha
- \frac{32}{3} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{21}{2} \alpha^2
+ \frac{28}{3} \psi^{\prime}(\third) \alpha
- 42 \alpha
\right. \nonumber \\
&& \left. ~
- 36 \zeta_3
- 21 \psi^{\prime}(\third)
- 16 \zeta_3 \pi^2
- 12 \zeta_3 \alpha^2
+ 14 \pi^2
+ 24 \psi^{\prime}(\third) \zeta_3
+ 48 \zeta_3 \alpha
\right] a^2
\nonumber \\
+ \left[
\frac{1}{2} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^2
- \frac{1149479}{108} \psi^{\prime}(\third)
- \frac{620324}{81} \zeta_3 \pi^2
- \frac{612664}{729} \zeta_3 \pi^4
\right. \nonumber \\
&& \left. ~~~
- \frac{556493}{243} \psi^{\prime}(\third) \pi^2
- \frac{158998}{81} \psi^{\prime}(\third)^2 \zeta_3
- \frac{52297}{81} \psi^{\prime}(\third) \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{26576}{81} \zeta_3 \pi^2 \alpha
- \frac{22049}{3} \zeta_3
- \frac{5120}{27} \zeta_5 \pi^2 \alpha
- \frac{2512}{27} \zeta_3 \pi^4 \alpha
- \frac{2079}{8} \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{2009}{9} \psi^{\prime}(\third) \pi^2 \alpha
- \frac{715}{3} \zeta_3 \alpha^2
- \frac{574}{3} \psi^{\prime}(\third)^2 \zeta_3 \alpha
- \frac{448}{27} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{392}{81} \pi^4 \alpha^3
- \frac{255}{2} \zeta_3 \alpha^4
- \frac{245}{12} \psi^{\prime}(\third) \alpha^4
- \frac{147}{16} \alpha^5
- \frac{140}{9} \zeta_3 \pi^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
- \frac{116}{3} \zeta_3 \pi^4 \alpha^2
- \frac{98}{9} \psi^{\prime}(\third)^2 \alpha^3
- \frac{21}{4} \psi^{\prime}(\third) \alpha^2
- \frac{21}{8} \psi^{\prime\prime\prime}(\third) \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{7}{16} \psi^{\prime\prime\prime}(\third) \alpha^2
+ \frac{7}{2} \pi^2 \alpha^2
+ \frac{21}{2} \zeta_3 \alpha^5
+ \frac{21}{2} \psi^{\prime\prime\prime}(\third)
+ \frac{70}{3} \psi^{\prime}(\third) \zeta_3 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{112}{9} \psi^{\prime}(\third)^2 \zeta_3 \alpha^3
+ \frac{147}{2} \psi^{\prime}(\third)^2 \alpha^2
+ \frac{196}{9} \alpha^2
+ \frac{203}{6} \pi^4 \alpha^2
+ \frac{245}{18} \pi^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{357}{2} \psi^{\prime}(\third) \alpha^3
+ \frac{392}{27} \psi^{\prime}(\third) \pi^2 \alpha^3
+ \frac{448}{81} \zeta_3 \pi^4 \alpha^3
+ \frac{1280}{3} \zeta_5 \pi^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{1785}{16} \alpha^4
+ \frac{2009}{12} \psi^{\prime}(\third)^2 \alpha
+ \frac{2198}{27} \pi^4 \alpha
+ \frac{2296}{9} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{2560}{9} \psi^{\prime}(\third) \zeta_5 \alpha
+ \frac{3076}{3} \zeta_3 \alpha
+ \frac{13288}{27} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{14105}{144} \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{104594}{243} \pi^2 \alpha
+ \frac{310162}{27} \psi^{\prime}(\third) \zeta_3
+ \frac{481037}{48}
+ \frac{536081}{729} \pi^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{556493}{324} \psi^{\prime}(\third)^2
+ \frac{635992}{243} \psi^{\prime}(\third) \zeta_3 \pi^2
+ \frac{1149479}{162} \pi^2
- 1280 \zeta_5 \alpha
\right. \nonumber \\
&& \left. ~~~
- 640 \psi^{\prime}(\third) \zeta_5
- 204 \psi^{\prime}(\third) \zeta_3 \alpha^3
- 119 \pi^2 \alpha^3
- 98 \psi^{\prime}(\third) \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- 84 \psi^{\prime}(\third)^2 \zeta_3 \alpha^2
- 72 \zeta_3^2 \alpha^2
- 36 \zeta_3^2 \alpha
- 12 \psi^{\prime\prime\prime}(\third) \zeta_3
- 4 \zeta_3 \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 3 \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha
+ 6 \psi^{\prime}(\third) \zeta_3 \alpha^2
+ 112 \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^2
+ 136 \zeta_3 \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ 297 \zeta_3 \alpha^3
+ 320 \zeta_5 \alpha^2
+ 684 \zeta_3^2
+ 960 \zeta_5
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

for $SU(3)$ Yang-Mills. For the MAG a similar picture emerges since

\begin{eqnarray}
\left. K_{a\,\MAGs}^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
16 \zeta_3
- 14
\right] a
\nonumber \\
+ \left[
\frac{6448}{9} \zeta_3
- \frac{14158}{27}
- \frac{1280}{3} \zeta_5
- \frac{704}{27} \zeta_3 \pi^2 \alpha
- \frac{512}{27} \zeta_3 \pi^2
- \frac{308}{9} \psi^{\prime}(\third) \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{224}{9} \psi^{\prime}(\third)
- \frac{56}{27} \pi^2 \alpha^2
- \frac{32}{9} \psi^{\prime}(\third) \zeta_3 \alpha^2
+ \frac{28}{9} \psi^{\prime}(\third) \alpha^2
+ \frac{64}{27} \zeta_3 \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{256}{9} \psi^{\prime}(\third) \zeta_3
+ \frac{352}{9} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{448}{27} \pi^2
+ \frac{616}{27} \pi^2 \alpha
- 48 \zeta_3 \alpha
- 14 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 16 \zeta_3 \alpha^2
+ 42 \alpha
\right] a^2
\nonumber \\
+ \left[
\frac{25291531}{540} \zeta_3
- \frac{640705981}{12960}
- \frac{9335291}{3240} \psi^{\prime}(\third) \alpha
- \frac{5544823}{4860} \pi^2
\right. \nonumber \\
&& \left. ~~~
- \frac{1402261}{7290} \pi^4 \alpha
- \frac{448672}{1215} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha
- \frac{224123}{135} \psi^{\prime}(\third) \zeta_3
\right. \nonumber \\
&& \left. ~~~
- \frac{169595}{972} \pi^2 \alpha^2
- \frac{151669}{30} \zeta_3 \alpha
- \frac{112718}{45} \zeta_3 \pi^2 \alpha
- \frac{98147}{405} \psi^{\prime}(\third)^2 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{22883}{36} \alpha^2
- \frac{14080}{9} \psi^{\prime}(\third) \zeta_5 \alpha
- \frac{11896}{45} \zeta_3 \pi^4
- \frac{10240}{9} \psi^{\prime}(\third) \zeta_5
\right. \nonumber \\
&& \left. ~~~
- \frac{9799}{27} \psi^{\prime}(\third) \zeta_3 \alpha^2
- \frac{8155}{243} \pi^4 \alpha^2
- \frac{6944}{81} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^2
- \frac{5135}{216} \psi^{\prime}(\third) \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{2560}{27} \zeta_5 \pi^2 \alpha^2
- \frac{1940}{729} \zeta_3 \pi^4 \alpha^3
- \frac{1810}{81} \zeta_3 \pi^2 \alpha^3
- \frac{1792}{243} \psi^{\prime}(\third) \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
- \frac{1519}{27} \psi^{\prime}(\third)^2 \alpha^2
- \frac{987}{32} \alpha^4
- \frac{944}{15} \psi^{\prime}(\third) \zeta_3 \pi^2
- \frac{749}{8} \psi^{\prime\prime\prime}(\third)
\right. \nonumber \\
&& \left. ~~~
- \frac{653}{18} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha
- \frac{512}{81} \psi^{\prime}(\third)^2 \zeta_3 \alpha^3
- \frac{451}{48} \alpha^3
- \frac{413}{10} \psi^{\prime}(\third)^2
\right. \nonumber \\
&& \left. ~~~
- \frac{112}{81} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^4
- \frac{98}{243} \pi^4 \alpha^4
- \frac{49}{54} \psi^{\prime}(\third)^2 \alpha^4
- \frac{11}{3} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{1}{18} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^3
+ \frac{7}{144} \psi^{\prime\prime\prime}(\third) \alpha^3
+ \frac{17}{12} \zeta_3 \alpha^3
+ \frac{21}{2} \psi^{\prime}(\third) \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{28}{27} \psi^{\prime}(\third)^2 \zeta_3 \alpha^4
+ \frac{77}{24} \psi^{\prime\prime\prime}(\third) \alpha^2
+ \frac{98}{81} \psi^{\prime}(\third) \pi^2 \alpha^4
+ \frac{112}{243} \zeta_3 \pi^4 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{183}{4} \zeta_3 \alpha^4
+ \frac{236}{5} \psi^{\prime}(\third)^2 \zeta_3
+ \frac{448}{81} \psi^{\prime}(\third)^2 \alpha^3
+ \frac{826}{15} \psi^{\prime}(\third) \pi^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{905}{27} \psi^{\prime}(\third) \zeta_3 \alpha^3
+ \frac{1280}{9} \psi^{\prime}(\third) \zeta_5 \alpha^2
+ \frac{1736}{27} \psi^{\prime}(\third)^2 \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{2048}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^3
+ \frac{3395}{1458} \pi^4 \alpha^3
+ \frac{4571}{144} \psi^{\prime\prime\prime}(\third) \alpha
+ \frac{5135}{324} \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{5311}{12} \zeta_3 \alpha^2
+ \frac{6076}{81} \psi^{\prime}(\third) \pi^2 \alpha^2
+ \frac{9320}{243} \zeta_3 \pi^4 \alpha^2
+ \frac{10409}{45} \pi^4
\right. \nonumber \\
&& \left. ~~~
+ \frac{19598}{81} \zeta_3 \pi^2 \alpha^2
+ \frac{20480}{27} \zeta_5 \pi^2
+ \frac{22400}{3} \zeta_7
+ \frac{28160}{27} \zeta_5 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{56359}{15} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{112168}{405} \psi^{\prime}(\third)^2 \zeta_3 \alpha
+ \frac{169595}{648} \psi^{\prime}(\third) \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{392588}{1215} \psi^{\prime}(\third) \pi^2 \alpha
+ \frac{448246}{405} \zeta_3 \pi^2
+ \frac{485200}{27} \zeta_5
+ \frac{695693}{240} \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{801292}{3645} \zeta_3 \pi^4 \alpha
+ \frac{5544823}{3240} \psi^{\prime}(\third)
+ \frac{9335291}{4860} \pi^2 \alpha
- 24900 \zeta_3^2
\right. \nonumber \\
&& \left. ~~~
- 640 \zeta_5 \alpha^2
- 22 \zeta_3^2 \alpha^3
- 12 \zeta_3^2 \alpha^4
- 12 \psi^{\prime}(\third) \zeta_3 \alpha^4
- 7 \pi^2 \alpha^4
+ 8 \zeta_3 \pi^2 \alpha^4
\right. \nonumber \\
&& \left. ~~~
+ 107 \psi^{\prime\prime\prime}(\third) \zeta_3
+ 522 \zeta_3^2 \alpha^2
+ 1576 \zeta_3^2 \alpha
+ 1920 \zeta_5 \alpha
\right] a^3
\end{eqnarray}

\begin{eqnarray}
\left. K_{\alpha\,\MAGs}^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
- \frac{352}{27} \zeta_3 \pi^2
- \frac{154}{9} \psi^{\prime}(\third)
- \frac{56}{27} \pi^2 \alpha
- \frac{32}{9} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{28}{9} \psi^{\prime}(\third) \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{64}{27} \zeta_3 \pi^2 \alpha
+ \frac{176}{9} \psi^{\prime}(\third) \zeta_3
+ \frac{308}{27} \pi^2
- 24 \zeta_3
- 14 \alpha
+ 16 \zeta_3 \alpha
\right] a^2
\nonumber \\
+ \left[
640 \zeta_5
- \frac{16773443}{9720} \psi^{\prime}(\third)
- \frac{1763246}{1215} \zeta_3 \pi^2
- \frac{775271}{7290} \pi^4
\right. \nonumber \\
&& \left. ~~~
- \frac{325472}{1215} \psi^{\prime}(\third) \zeta_3 \pi^2
- \frac{289513}{972} \psi^{\prime}(\third) \alpha
- \frac{71197}{405} \psi^{\prime}(\third)^2
- \frac{64139}{30} \zeta_3
\right. \nonumber \\
&& \left. ~~~
- \frac{41876}{243} \zeta_3 \pi^2 \alpha
- \frac{14080}{27} \psi^{\prime}(\third) \zeta_5
- \frac{12152}{243} \psi^{\prime}(\third) \pi^2 \alpha
- \frac{9136}{729} \zeta_3 \pi^4 \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{5120}{81} \zeta_5 \pi^2 \alpha
- \frac{3472}{81} \psi^{\prime}(\third)^2 \zeta_3 \alpha
- \frac{2140}{243} \zeta_3 \pi^4 \alpha^2
- \frac{1904}{81} \psi^{\prime}(\third) \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{1280}{3} \zeta_5 \alpha
- \frac{665}{8} \psi^{\prime}(\third) \alpha^2
- \frac{653}{54} \psi^{\prime\prime\prime}(\third) \zeta_3
- \frac{544}{27} \psi^{\prime}(\third)^2 \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{448}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^3
- \frac{392}{729} \pi^4 \alpha^3
- \frac{329}{8} \alpha^3
- \frac{190}{3} \zeta_3 \pi^2 \alpha^2
- \frac{121}{4} \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{98}{81} \psi^{\prime}(\third)^2 \alpha^3
- \frac{28}{3} \pi^2 \alpha^3
- \frac{22}{9} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha
- \frac{1}{18} \psi^{\prime\prime\prime}(\third) \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{7}{144} \psi^{\prime\prime\prime}(\third) \alpha^2
+ \frac{32}{3} \zeta_3 \pi^2 \alpha^3
+ \frac{77}{36} \psi^{\prime\prime\prime}(\third) \alpha
+ \frac{112}{81} \psi^{\prime}(\third)^2 \zeta_3 \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{392}{243} \psi^{\prime}(\third) \pi^2 \alpha^3
+ \frac{448}{729} \zeta_3 \pi^4 \alpha^3
+ \frac{476}{27} \psi^{\prime}(\third)^2 \alpha^2
+ \frac{665}{12} \pi^2 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{693}{16} \alpha^2
+ \frac{707}{18} \zeta_3 \alpha
+ \frac{1576}{3} \zeta_3^2
+ \frac{2176}{81} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha^2
+ \frac{2345}{108} \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{2560}{27} \psi^{\prime}(\third) \zeta_5 \alpha
+ \frac{3038}{81} \psi^{\prime}(\third)^2 \alpha
+ \frac{3745}{486} \pi^4 \alpha^2
+ \frac{4571}{432} \psi^{\prime\prime\prime}(\third)
\right. \nonumber \\
&& \left. ~~~
+ \frac{7994}{729} \pi^4 \alpha
+ \frac{13888}{243} \psi^{\prime}(\third) \zeta_3 \pi^2 \alpha
+ \frac{20938}{81} \psi^{\prime}(\third) \zeta_3 \alpha
+ \frac{28160}{81} \zeta_5 \pi^2
\right. \nonumber \\
&& \left. ~~~
- 16 \psi^{\prime}(\third) \zeta_3 \alpha^3
+ \frac{81368}{405} \psi^{\prime}(\third)^2 \zeta_3
+ \frac{284788}{1215} \psi^{\prime}(\third) \pi^2
+ \frac{289513}{1458} \pi^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{443012}{3645} \zeta_3 \pi^4
+ \frac{881623}{405} \psi^{\prime}(\third) \zeta_3
+ \frac{1405969}{720}
+ \frac{16773443}{14580} \pi^2
\right. \nonumber \\
&& \left. ~~~
- 22 \zeta_3^2 \alpha^2
- 16 \zeta_3^2 \alpha^3
+ 14 \psi^{\prime}(\third) \alpha^3
+ 61 \zeta_3 \alpha^3
+ 95 \psi^{\prime}(\third) \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ 348 \zeta_3^2 \alpha
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

for the same theory. What is evident in both cases is that the leading term of
the $\MOMg$ $K_a$ in both gauges, as well as the $\MOMc$ and $\MOMq$ schemes,
is the same as that of the linear gauge in all schemes. In other words the
remaining terms of $K_a$ and all in $K_\alpha$  carry both the scheme and gauge
dependence. Moreover by construction the Crewther relation vanishes at the
corresponding fixed points of the Curci-Ferrari and MAG renormalization group
functions. In these two nonlinear gauges $\gamma_\alpha(a,\alpha)$ is linearly
independent of the gluon anomalous dimension $\gamma_A(a,\alpha)$ unlike the
linear covariant gauge. In other words the critical points of both gauges are

What is not apparent from the $SU(3)$ expressions is the connection between the
Curci-Ferrari gauge and MAG which requires the results for an arbitrary
colour group. To assist with seeing how the $\Nda$ $\to$ $0$ limit takes effect
we record the situation for the $O(a^2)$ $K$-functions in detail as the next
order expressions are large but available in the associated data file. In the
Curci-Ferrari gauge we have

\begin{eqnarray}
K_{a\,\CFs}^{\MOMgs}(a,\alpha) &=&
\left[
12 \zeta_3 C_F
- \frac{21}{2} C_F
\right] a
\nonumber \\
+ \left[
136 \zeta_3 C_F^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
\right. \nonumber \\
&& \left. ~~~
+ \frac{7}{3} \psi^{\prime}(\third) C_F C_A \alpha^2
+ \frac{7}{4} C_F C_A \alpha^3
+ \frac{16}{9} \zeta_3 \pi^2 C_F C_A \alpha^2
+ \frac{63}{4} C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{158}{3} \Nf T_F C_F
+ \frac{161}{18} \psi^{\prime}(\third) C_F C_A
+ \frac{184}{27} \zeta_3 \pi^2 C_F C_A
\right. \nonumber \\
&& \left. ~~~
+ \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
+ \frac{448}{27} \pi^2 \Nf T_F C_F
\right. \nonumber \\
&& \left. ~~~
- 240 \zeta_5 C_F^2
- 18 \zeta_3 C_F C_A \alpha
- 8 \zeta_3 \pi^2 C_F C_A \alpha
- 2 \zeta_3 C_F C_A \alpha^3
\right. \nonumber \\
&& \left. ~~~
+ 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] a^2 ~+~ O(a^3)
\label{kafuncfgen}
\end{eqnarray}

\begin{eqnarray}
K_{\alpha\,\CFs}^{\MOMgs}(a,\alpha) &=&
\left[
6 \psi^{\prime}(\third) \zeta_3 C_F C_A
+ 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
\right. \nonumber \\
&& \left. ~
- \frac{14}{9} \pi^2 C_F C_A \alpha
- \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
\right. \nonumber \\
&& \left. ~
+ \frac{16}{9} \zeta_3 \pi^2 C_F C_A \alpha
+ \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
\right. \nonumber \\
&& \left. ~
- 3 \zeta_3 C_F C_A \alpha^2
\right] a^2 ~+~ O(a^3)
\label{kalfuncfgen}
\end{eqnarray}

while the MAG partners are

\begin{eqnarray}
K_{a\,\MAGs}^{\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{104}{3} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{92}{9} \psi^{\prime}(\third) \zeta_3 C_F C_A
- \frac{91}{2} \psi^{\prime}(\third) \frac{\Nda}{\Noda} C_F C_A
- \frac{63}{4} \frac{\Nda}{\Noda} C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{35}{9} \pi^2 \frac{\Nda}{\Noda} C_F C_A \alpha
- \frac{32}{9} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A \alpha^2
- \frac{21}{2} C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
- \frac{21}{2} \psi^{\prime}(\third) C_F C_A \alpha
- \frac{21}{4} \frac{\Nda}{\Noda} C_F C_A \alpha^3
- \frac{20}{3} \psi^{\prime}(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{14}{3} \psi^{\prime}(\third) \frac{\Nda}{\Noda} C_F C_A \alpha^2
- \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
\right. \nonumber \\
&& \left. ~~~
+ \frac{7}{4} C_F C_A \alpha^3
+ \frac{16}{3} \psi^{\prime}(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha^2
+ \frac{16}{9} \zeta_3 \pi^2 C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~~~
+ \frac{28}{9} \pi^2 \frac{\Nda}{\Noda} C_F C_A \alpha^2
+ \frac{35}{6} \psi^{\prime}(\third) \frac{\Nda}{\Noda} C_F C_A \alpha
+ \frac{40}{9} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{63}{4} C_F C_A \alpha
+ \frac{91}{3} \pi^2 \frac{\Nda}{\Noda} C_F C_A
+ \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
- 24 \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha^2
- 18 \zeta_3 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- 8 \zeta_3 \pi^2 C_F C_A \alpha
- 2 \zeta_3 C_F C_A \alpha^3
+ 6 \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha^3
+ 7 \pi^2 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ 12 \zeta_3 C_F C_A \alpha^2
+ 12 \psi^{\prime}(\third) \zeta_3 C_F C_A \alpha
+ 18 \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
+ 21 \frac{\Nda}{\Noda} C_F C_A \alpha^2
+ 52 \psi^{\prime}(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A
+ 136 \zeta_3 C_F^2
\right] a^2
~+~ O(a^3)
\label{kafunmaggen}
\end{eqnarray}

\begin{eqnarray}
K_{\alpha\,\MAGs}^{\MOMgs}(a,\alpha) &=&
\left[
\frac{7}{2} \pi^2 C_F C_A
- \frac{63}{8} \frac{\Nda}{\Noda} C_F C_A
- \frac{63}{8} \frac{\Nda}{\Noda} C_F C_A \alpha^2
- \frac{35}{18} \pi^2 \frac{\Nda}{\Noda} C_F C_A
\right. \nonumber \\
&& \left. ~
- \frac{32}{9} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A \alpha
- \frac{21}{2} C_F C_A \alpha
- \frac{21}{4} \psi^{\prime}(\third) C_F C_A
\right. \nonumber \\
&& \left. ~
- \frac{14}{3} \psi^{\prime}(\third) \frac{\Nda}{\Noda} C_F C_A \alpha
- \frac{14}{9} \pi^2 C_F C_A \alpha
- \frac{10}{3} \psi^{\prime}(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A
\right. \nonumber \\
&& \left. ~
- \frac{8}{3} \psi^{\prime}(\third) \zeta_3 C_F C_A \alpha
+ \frac{7}{3} \psi^{\prime}(\third) C_F C_A \alpha
+ \frac{16}{3} \psi^{\prime}(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{16}{9} \zeta_3 \pi^2 C_F C_A \alpha
+ \frac{20}{9} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A
+ \frac{21}{8} C_F C_A \alpha^2
\right. \nonumber \\
&& \left. ~
+ \frac{28}{9} \pi^2 \frac{\Nda}{\Noda} C_F C_A \alpha
+ \frac{35}{12} \psi^{\prime}(\third) \frac{\Nda}{\Noda} C_F C_A
+ \frac{63}{8} C_F C_A
\right. \nonumber \\
&& \left. ~
- 24 \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha
- 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
+ 9 \zeta_3 \frac{\Nda}{\Noda} C_F C_A
+ 9 \zeta_3 \frac{\Nda}{\Noda} C_F C_A \alpha^2
+ 12 \zeta_3 C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ 21 \frac{\Nda}{\Noda} C_F C_A \alpha
\right] a^2 ~+~ O(a^3) ~.
\label{kalfunmaggen}
\end{eqnarray}

Deleting the terms with a $\Nda/\Noda$ factor in (<ref>) and
(<ref>) reproduces (<ref>) and (<ref>)
respectively. We have checked that this is also the case for the $O(a^3)$

Other schemes.

One question of interest concerns whether or not there are any schemes other
than the $\MSbar$ one where $\Delta_{\mbox{\footnotesize{csb}}}(a,\alpha)$
involves only the $\beta$-function. It turns out in fact that, aside from the
$V$ scheme of [13, 14] whose $\beta$-function is independent of the gauge
parameter by construction, there is at least one other such scheme which is the
$\RI$ scheme, [32, 33]. It was introduced in relation to lattice
regularized QCD calculations and has a continuum spacetime analogue which has
been used to renormalize the theory to high loop order [56, 57, 58, 59]. The
prescription requires that the wave function renormalization constants of the
gluon, ghost and quark fields are defined so that there are no $O(a)$
corrections to their $2$-point functions at the subtraction point. In this
sense it incorporates the approach of the MOM schemes of [34, 35]. The
$\RI$ scheme differs however in respect of the coupling constant
renormalization. For each vertex function the coupling renormalization constant
is defined according to the $\MSbar$ prescription at the subtraction point. So
the $\RI$ scheme can be regarded as a halfway house between the $\MSbar$ and
the suite of MOM schemes of [34, 35]. In terms of the resulting
renormalization group functions the relevant issue for this article is that the
$\RI$ $\beta$-function is formally equivalent to that of the $\MSbar$
$\beta$-function. By this we mean that although the coupling constant will be
the variable in the respective schemes the actual coefficients of the coupling
constant polynomial of the $\RI$ scheme $\beta$-function are identically the
same as their $\MSbar$ counterparts. We have checked explicitly that the same
reasoning applies to the structure of $K_a^{\RIs}(a)$ whose coefficients are
precisely the same as those of $K_a^{\MSbars}(a)$ with in addition now

\begin{equation}
K_\alpha^{\RIs}(a,\alpha) ~=~ 0 ~.
\end{equation}

In other words the ansatz of (<ref>) holds primarily as a direct result
of the $\RI$ renormalization prescription of the vertex function.

Although this means that there is a scheme other than $\MSbar$ that takes the
same form as (<ref>) it is a relatively trivial observation. However the
nature of the $\RI$ scheme gives a more important insight into some of the
underlying issues. In essence it indicates that the scheme the fields are
renormalized in plays no role in whether a $K_\alpha(a,\alpha)$ function is
needed. Instead it is the prescription for the coupling constant
renormalization that is key. This is understandable since the $\MSbar$ Crewther
construction is $\alpha$ independent. In light of this, one question that is of
interest is what occurs if one considers a scheme that is opposite to the
prescription of the $\RI$ scheme. By this we mean one where the fields, gauge
parameter and quark mass are renormalized in an $\MSbar$ way but the vertex
functions, and thereby the coupling constant, are renormalized in a
non-$\MSbar$ fashion. It is possible to explore this possibility given the four
loop data that is available in [18, 59]. In this new scheme the gluon,
ghost and quark $2$-point functions as well as the quark $2$-point function
with the quark mass operator inserted, [59], are renormalized in the
linear covariant gauge by including only the poles in $\epsilon$ into the
respective renormalization constants. To determine the coupling constant
renormalization we have chosen to use the ghost-gluon vertex in the
configuration where the momentum of one ghost field is zero. This is the setup
that is used to define the $\mMOM$ renormalization. The reason for choosing
this vertex is that in the Landau gauge the vertex is finite according to
Taylor's observation, [16]. Therefore in this new scheme the
renormalization group functions ought to differ from the $\MSbar$ ones in a
minimal way. We have labelled this scheme as $\RIc$ reflecting that it is based
on the ghost-gluon vertex, being a variation of the $\RI$ scheme, and
constructed the renormalization group equations to five loops. Expressions for
them in an arbitrary gauge for a general colour group are given in the
associated data file. To appreciate their structure we have recorded the
$SU(3)$ expressions for the $\beta$-function and gauge parameter anomalous
dimension in Appendix B as these are relevant to the current discussion.
However it is worth noting that our expectation that the $\RIc$ scheme has a
minimal effect on the $\beta$-function structure is met in that the
coefficients of the coupling constant in $\beta^{\RIcs}(a,0)$ are precisely
the same as those of $\beta^{\MSbars}(a)$ to five loops.

Equipped with the $\RIc$ renormalization group functions we have determined the
relation between the $\MSbar$ and $\RIc$ coupling constant and gauge paremeter
that allows us to examine what
$\Delta_{\mbox{\footnotesize{csb}}}^{\RIcs}(a,\alpha)$ is. By following the
procedure used for other schemes it transpires that
$K_\alpha^{\RIcs}(a,\alpha)$ has to be non-zero. More specifically we found

\begin{eqnarray}
K_a^{\RIcs}(a,\alpha) &=&
\left[
12 \zeta_3 C_F
- \frac{21}{2} C_F
\right] a
\nonumber \\
+ \left[
\frac{326}{3} \Nf T_F C_F
+ \frac{397}{6} C_F^2
+ \frac{884}{3} \zeta_3 C_F C_A
- \frac{629}{2} C_F C_A
- \frac{304}{3} \zeta_3 \Nf T_F C_F
\right. \nonumber \\
&& \left. ~~~
- 240 \zeta_5 C_F^2
- 24 \zeta_3 C_F C_A \alpha
+ 21 C_F C_A \alpha
+ 136 \zeta_3 C_F^2
\right] a^2
\nonumber \\
+ \left[
\frac{81}{2} \zeta_3^2 C_F C_A^2 \alpha
+ \frac{85}{16} \zeta_3 C_F C_A^2 \alpha^2
+ \frac{211}{16} C_F C_A^2 \alpha^2
- \frac{406043}{36} C_F C_A^2
\right. \nonumber \\
&& \left. ~~~
- \frac{45517}{48} \zeta_3 C_F C_A^2 \alpha
- \frac{40336}{9} \zeta_3 \Nf T_F C_F C_A
- \frac{24880}{3} \zeta_5 C_F^2 C_A
\right. \nonumber \\
&& \left. ~~~
- \frac{9824}{9} \Nf^2 T_F^2 C_F
- \frac{8000}{3} \zeta_5 \Nf T_F C_F C_A
- \frac{7729}{18} \Nf T_F C_F^2
- \frac{652}{3} \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- \frac{397}{2} C_F^2 C_A \alpha
- \frac{27}{2} \zeta_3^2 C_F C_A^2 \alpha^2
+ \frac{608}{3} \zeta_3 \Nf T_F C_F C_A \alpha
+ \frac{2471}{12} C_F^3
\right. \nonumber \\
&& \left. ~~~
+ \frac{6496}{9} \zeta_3 \Nf^2 T_F^2 C_F
+ \frac{11900}{3} \zeta_5 C_F C_A^2
+ \frac{16570}{3} \zeta_3 C_F^2 C_A
+ \frac{44939}{48} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~~~
+ \frac{67520}{9} \Nf T_F C_F C_A
+ \frac{72028}{9} \zeta_3 C_F C_A^2
+ \frac{99757}{36} C_F^2 C_A
- 5720 \zeta_5 C_F^3
\right. \nonumber \\
&& \left. ~~~
- 3668 \zeta_3 \Nf T_F C_F^2
- 1232 \zeta_3^2 C_F C_A^2
- 840 \zeta_7 C_F^2 C_A
- 408 \zeta_3 C_F^2 C_A \alpha
\right. \nonumber \\
&& \left. ~~~
- 128 \zeta_3^2 \Nf T_F C_F C_A
+ 320 \zeta_5 \Nf^2 T_F^2 C_F
+ 488 \zeta_3 C_F^3
+ 576 \zeta_3^2 \Nf T_F C_F^2
\right. \nonumber \\
&& \left. ~~~
+ 720 \zeta_5 C_F^2 C_A \alpha
+ 4000 \zeta_5 \Nf T_F C_F^2
+ 5040 \zeta_7 C_F^3
\right] a^3 ~+~ O(a^4)
\end{eqnarray}

\begin{eqnarray}
K_\alpha^{\RIcs}(a,\alpha) &=&
\left[
\frac{21}{2} C_F
- 12 \zeta_3 C_F
\right] C_A a^2
\nonumber \\
&& + \left[
\frac{27}{2} \zeta_3^2 C_F C_A
+ \frac{231}{8} \zeta_3 C_F C_A \alpha
+ \frac{3461}{48} C_F C_A
- \frac{1477}{16} \zeta_3 C_F C_A
- \frac{397}{6} C_F^2
\right. \nonumber \\
&& \left. ~~~
- \frac{147}{8} C_F C_A \alpha
- 136 \zeta_3 C_F^2
- 9 \zeta_3^2 C_F C_A \alpha
+ 240 \zeta_5 C_F^2
\right] C_A a^3 \,+\, O(a^4) ~.
\end{eqnarray}

Although this exercise did not produce another scheme whose Crewther structure
was the same as those of $\MSbar$ and $\RI$ $K_\alpha^{\RIcs}(a,\alpha)$ does
have a different structure. For instance unlike the other schemes considered
earlier the leading term is $\alpha$ independent and the linear terms in
$\alpha$ at the next order are proportional to the leading term. Also like
$K_\alpha^{\mMOMs}(a,\alpha)$ we note that $K_\alpha^{\RIcs}(a,\alpha)$ is
proportional to $C_A$. These properties can be traced directly back to those of
the vertex function that this scheme was based on.

Renormalization group connection.

Having accumulated evidence to support the ansatz (<ref>) a natural
question is whether it can be derived directly and in general from some
renormalization group principle. This is important since our analysis so far
indicating an extension to (<ref>) is needed has been concerned with
values of $\Nf$ in the conformal window. Any generalization should not be
restricted to this range. It turns out the key lies in the way one maps the
parameters of a theory in one scheme to the corresponding parameters in another
scheme as well as the relation between the renormalization group functions. For
instance if for the moment we regard $a$ as the $\MSbar$ coupling constant and
$a_{\cal S}$ and $\alpha_{\cal S}$ as the respective coupling constant and
gauge parameter in another scheme ${\cal S}$ then we found the Adler
$D$-function and the Bjorken sum rule using the relations

\begin{equation}
C_{{\mbox{\footnotesize{Bjr}}}}^{\MSbars}(a) ~=~
C_{{\mbox{\footnotesize{Bjr}}}}^{\cal S}(a_{\cal S},\alpha_{\cal S}) ~~~,~~~
C_{{\mbox{\footnotesize{Adl}}}}^{\MSbars}(a) ~=~
C_{{\mbox{\footnotesize{Adl}}}}^{\cal S}(a_{\cal S},\alpha_{\cal S})
\end{equation}

\begin{equation}
a ~\equiv~ a(a_{\cal S},\alpha_{\cal S})
\label{amapmss}
\end{equation}

expresses the $\MSbar$ coupling in terms of the coupling and gauge parameter in
scheme ${\cal S}$ and implies

\begin{equation}
\Delta_{\mbox{\footnotesize{csb}}}^{\MSbars}(a) ~=~
\Delta_{\mbox{\footnotesize{csb}}}^{\cal S}(a_{\cal S},\alpha_{\cal S}) ~.
\end{equation}

Using properties of the renormalization group equation the $\MSbar$
$\beta$-function is related to that in another scheme by

\begin{equation}
\beta^{\MSbars}(a) ~=~ \beta^{\cal S}(a_{\cal S},\alpha_{\cal S})
\frac{\partial}{\partial a_{\cal S}} a(a_{\cal S},\alpha_{\cal S}) ~+~
\alpha_{\cal S} \gamma_\alpha^{\cal S}(a_{\cal S},\alpha_{\cal S})
\frac{\partial}{\partial \alpha_{\cal S}} a(a_{\cal S},\alpha_{\cal S})
\label{betamap}
\end{equation}

which recalling (<ref>) implies
$\Delta_{\mbox{\footnotesize{csb}}}^{\cal S}(a_{\cal S},\alpha_{\cal S})$
can be written as

\begin{equation}
\Delta_{\mbox{\footnotesize{csb}}}^{\cal S}(a_{\cal S},\alpha_{\cal S}) ~=~
\frac{\beta^{\cal S}(a_{\cal S},\alpha_{\cal S})}{a_{\cal S}}
\bar{K}_a^{\cal S}(a_{\cal S},\alpha_{\cal S}) ~+~
\alpha_{\cal S} \gamma_\alpha^{\cal S}(a_{\cal S},\alpha_{\cal S})
\bar{K}_\alpha^{\cal S}(a_{\cal S},\alpha_{\cal S})
\label{deltgenbar}
\end{equation}

\begin{eqnarray}
\bar{K}_a^{\cal S}(a_{\cal S},\alpha_{\cal S}) &=& a_{\cal S}
\left. \left[ \frac{1}{a} K_a^{\MSbars}(a) \right]
\right|_{\MSbars \to {\cal S}} ~~
\frac{\partial}{\partial a_{\cal S}} a(a_{\cal S},\alpha_{\cal S}) \nonumber \\
\bar{K}_\alpha^{\cal S}(a_{\cal S},\alpha_{\cal S}) &=&
\left. \left[ \frac{1}{a} K_a^{\MSbars}(a) \right]
\right|_{\MSbars \to {\cal S}} ~~
\frac{\partial}{\partial \alpha_{\cal S}} a(a_{\cal S},\alpha_{\cal S}) ~.
\label{krgecon}
\end{eqnarray}

The restriction on each appearance of $K_a^{\MSbars}(a)$ indicates that the
$\MSbar$ coupling needs to be mapped to its ${\cal S}$ scheme counterpart by
inverting (<ref>) so that the variables are in the ${\cal S}$ scheme.
By setting ${\cal S}$ in (<ref>) to be the $\MSbar$ scheme and
comparing with (<ref>) clearly implies

\begin{equation}
\bar{K}_\alpha^{\MSbars}(a,\alpha) ~=~ 0 ~.
\end{equation}

However (<ref>) demonstrates the consistency of this observation
trivially. For instance if in the second relation of (<ref>) we choose
${\cal S}$ to be the $\MSbar$ scheme itself then
$\frac{\partial a}{\partial \alpha}$ vanishes as $a$ and $\alpha$ are
independent in the $\MSbar$ scheme.

While the use of (<ref>) formally justifies our ansatz of
(<ref>) it provides another way of constructing explicit expressions
for the $K$-functions. However it turns out that the perturbative construction
of the $K$-functions are not in agreement with those derived from
(<ref>). In other words for each of the schemes and gauges considered
here we define the quantities

\begin{equation}
\Theta_a^{\cal S}(a,\alpha) ~=~ K_a^{\cal S}(a,\alpha) ~-~
\bar{K}_a^{\cal S}(a,\alpha) ~~,~~
\Theta_\alpha^{\cal S}(a,\alpha) ~=~ K_\alpha^{\cal S}(a,\alpha) ~-~
\bar{K}_\alpha^{\cal S}(a,\alpha)
\end{equation}

to illustrate the lack of uniqueness and note they are non-zero. This
non-uniqueness of the perturbative expansion of the $K$-functions does not
invalidate the observation that
$\Delta_{\mbox{\footnotesize{csb}}}^{\cal S}(a_{\cal S},\alpha_{\cal S})$
vanishes at critical points of the renormalization group equation. This still
follows due to the coefficients of the $K$-functions of (<ref>)
being the quantities defining the fixed point locations. By way of example we
record a few expressions for both $\Theta_i^{\cal S}(a,\alpha)$ in various
schemes. First in the linear covariant gauge we have

\begin{eqnarray}
\Theta_a^{\mMOMs}(a,\alpha) &=&
\left[
\frac{434}{3} \alpha \Nf T_F C_F C_A
- \frac{2821}{12} \alpha C_F C_A^2
+ 182 \alpha \zeta_3 C_F C_A^2
- 112 \alpha \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
+ \frac{31}{8} \alpha^2 C_F C_A^2
+ 31 \alpha^2 \Nf T_F C_F C_A
- 3 \alpha^2 \zeta_3 C_F C_A^2
- 24 \alpha^2 \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
+ \frac{93}{8} \alpha^3 C_F C_A^2
- 9 \alpha^3 \zeta_3 C_F C_A^2
\right] a^3 ~+~ O(a^4) \nonumber \\
\Theta_\alpha^{\mMOMs}(a,\alpha) &=&
\left[
\frac{434}{3} \Nf T_F C_F C_A
- \frac{2387}{6} C_F C_A^2
+ 308 \zeta_3 C_F C_A^2
- 112 \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
- \frac{341}{4} \alpha C_F C_A^2
+ 31 \alpha \Nf T_F C_F C_A
+ 66 \alpha \zeta_3 C_F C_A^2
- 24 \alpha \zeta_3 \Nf T_F C_F C_A
\right] a^3 \nonumber \\
&& +~ O(a^4)
\end{eqnarray}

for the $\mMOM$ scheme. By contrast the partially related $\RIc$ scheme
expressions are simpler since

\begin{eqnarray}
\Theta_a^{\RIcs}(a,\alpha) &=&
\left[
\frac{326}{3} \alpha \Nf T_F C_F C_A
- \frac{2119}{12} \alpha C_F C_A^2
+ \frac{494}{3} \alpha \zeta_3 C_F C_A^2
- \frac{304}{3} \alpha \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
+ \frac{163}{4} \alpha^2 C_F C_A^2
- 38 \alpha^2 \zeta_3 C_F C_A^2
\right] a^3 ~+~ O(a^4) \nonumber \\
\Theta_\alpha^{\RIcs}(a,\alpha) &=&
\left[
\frac{326}{3} \Nf T_F C_F C_A
- \frac{1793}{6} C_F C_A^2
+ \frac{836}{3} \zeta_3 C_F C_A^2
- \frac{304}{3} \zeta_3 \Nf T_F C_F C_A
\right] a^3 \nonumber \\
&& +~ O(a^4) ~.
\end{eqnarray}

For balance we record the $\MOMg$ scheme $SU(3)$ Yang-Mills differences are

\begin{eqnarray}
\left. \Theta_a^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
\frac{98}{3} \pi^2 \alpha^4
+ \frac{411}{8} \alpha^4
- \frac{238303}{108} \psi^\prime(\third) \alpha
- \frac{128596}{81} \zeta_3 \pi^2 \alpha
- \frac{99008}{729} \zeta_3 \pi^4 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{86632}{243} \psi^\prime(\third) \pi^2 \alpha
- \frac{24752}{81} \psi^\prime(\third)^2 \zeta_3 \alpha
- \frac{15680}{243} \zeta_3 \pi^4 \alpha^2
- \frac{15001}{36} \psi^\prime(\third) \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{13720}{81} \psi^\prime(\third) \pi^2 \alpha^2
- \frac{7372}{27} \zeta_3 \pi^2 \alpha^2
- \frac{3920}{27} \psi^\prime(\third)^2 \zeta_3 \alpha^2
- \frac{3445}{3} \zeta_3 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{3299}{8} \alpha^3
- \frac{1792}{27} \psi^\prime(\third) \zeta_3 \pi^2 \alpha^3
- \frac{1568}{81} \pi^4 \alpha^3
- \frac{392}{9} \psi^\prime(\third)^2 \alpha^3
\right. \nonumber \\
&& \left. ~
- \frac{112}{3} \zeta_3 \pi^2 \alpha^4
+ \frac{448}{9} \psi^\prime(\third)^2 \zeta_3 \alpha^3
+ \frac{928}{3} \zeta_3 \pi^2 \alpha^3
+ \frac{1568}{27} \psi^\prime(\third) \pi^2 \alpha^3
\right. \nonumber \\
&& \left. ~
+ \frac{1792}{81} \zeta_3 \pi^4 \alpha^3
+ \frac{2623}{8} \alpha^2
+ \frac{3430}{27} \psi^\prime(\third)^2 \alpha^2
+ \frac{3686}{9} \psi^\prime(\third) \zeta_3 \alpha^2
\right. \nonumber \\
&& \left. ~
+ \frac{13720}{243} \pi^4 \alpha^2
+ \frac{15001}{54} \pi^2 \alpha^2
+ \frac{15680}{81} \psi^\prime(\third) \zeta_3 \pi^2 \alpha^2
+ \frac{21658}{81} \psi^\prime(\third)^2 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{51415}{24} \alpha
+ \frac{64298}{27} \psi^\prime(\third) \zeta_3 \alpha
+ \frac{86632}{729} \pi^4 \alpha
+ \frac{99008}{243} \psi^\prime(\third) \zeta_3 \pi^2 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{238303}{162} \pi^2 \alpha
- 464 \psi^\prime(\third) \zeta_3 \alpha^3
- 284 \pi^2 \alpha^3
- 229 \zeta_3 \alpha^2
- 49 \psi^\prime(\third) \alpha^4
\right. \nonumber \\
&& \left. ~
- 33 \zeta_3 \alpha^4
+ 56 \psi^\prime(\third) \zeta_3 \alpha^4
+ 257 \zeta_3 \alpha^3
+ 426 \psi^\prime(\third) \alpha^3
\right] a^3 ~+~ O(a^4) \nonumber \\
\left. \Theta_\alpha^{\MOMgs}(a,\alpha) \right|^{SU(3)}_{\Nf=0} &=&
\left[
\frac{1078}{3} \psi^\prime(\third) \alpha^2
+ \frac{2464}{9} \zeta_3 \pi^2 \alpha^2
- \frac{217624}{81} \zeta_3 \pi^2
- \frac{201641}{54} \psi^\prime(\third)
\right. \nonumber \\
&& \left. ~
- \frac{167552}{729} \zeta_3 \pi^4
- \frac{146608}{243} \psi^\prime(\third) \pi^2
- \frac{41888}{81} \psi^\prime(\third)^2 \zeta_3
- \frac{39424}{243} \zeta_3 \pi^4 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{34496}{81} \psi^\prime(\third) \pi^2 \alpha
- \frac{29216}{27} \zeta_3 \pi^2 \alpha
- \frac{14102}{9} \psi^\prime(\third) \alpha
- \frac{9856}{27} \psi^\prime(\third)^2 \zeta_3 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{5830}{3} \zeta_3
- \frac{2156}{9} \pi^2 \alpha^2
- \frac{1507}{4} \alpha^2
- \frac{1232}{3} \psi^\prime(\third) \zeta_3 \alpha^2
+ \frac{2783}{2} \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{8624}{27} \psi^\prime(\third)^2 \alpha
+ \frac{14608}{9} \psi^\prime(\third) \zeta_3 \alpha
+ \frac{28204}{27} \pi^2 \alpha
+ \frac{34496}{243} \pi^4 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{36652}{81} \psi^\prime(\third)^2
+ \frac{39424}{81} \psi^\prime(\third) \zeta_3 \pi^2 \alpha
+ \frac{43505}{12}
+ \frac{108812}{27} \psi^\prime(\third) \zeta_3
\right. \nonumber \\
&& \left. ~
+ \frac{146608}{729} \pi^4
+ \frac{167552}{243} \psi^\prime(\third) \zeta_3 \pi^2
+ \frac{201641}{81} \pi^2
- 836 \zeta_3 \alpha
+ 242 \zeta_3 \alpha^2
\right] a^3 \nonumber \\
&& +~ O(a^4)
\end{eqnarray}

to illustrate the situation for a kinematic scheme. While the above linear
covariant gauge expressions for $\Theta_a^{\cal S}(a,\alpha)$ each vanish when
$\alpha$ $=$ $0$ this is not the case for the MAG as the $\alpha$ dependence of
the respective $\gamma_\alpha(a,\alpha)$ leading terms differ, [54]. For
instance in the $\MOMc$ scheme in that gauge we find

\begin{eqnarray}
\Theta_{a\,\MAGs}^{\MOMcs}(a,\alpha) &=&
\left[
\frac{2}{3} \pi^2 C_F C_A^2 \alpha^2
+ \frac{2}{3} \psi^\prime(\third) \zeta_3 C_F C_A^2 \alpha^2
+ \frac{8}{9} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{3751}{3} \frac{\Noda}{[\Noda+2\Nda]} C_F C_A^2 \alpha
- \frac{1425}{4} \frac{\Nda}{\Noda} C_F C_A^2
- \frac{1395}{8} \frac{\Nda}{\Noda} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{1323}{8} \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^2
- \frac{1179}{4} \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha
- \frac{1023}{2} \frac{\Noda}{[\Noda+2\Nda]} C_F C_A^2
\right. \nonumber \\
&& \left. ~
- \frac{775}{18} \psi^\prime(\third) \frac{\Nda}{\Noda} C_F C_A^2 \alpha
- \frac{621}{4} \frac{\Nda^2}{\Noda^2} C_F C_A^2
- \frac{560}{27} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{475}{3} \Nf T_F C_F C_A \alpha
- \frac{344}{27} \pi^2 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
- \frac{153}{2} \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{128}{9} \psi^\prime(\third) \zeta_3 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
- \frac{93}{4} \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^3
- \frac{86}{3} \pi^2 \frac{\Nda^2}{\Noda^2} C_F C_A^2
\right. \nonumber \\
&& \left. ~
- \frac{86}{3} \pi^2 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha
- \frac{64}{27} \zeta_3 \pi^2 \Nf T_F C_F C_A \alpha
- \frac{52}{9} \pi^2 C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{52}{9} \psi^\prime(\third) \zeta_3 C_F C_A^2 \alpha
- \frac{43}{9} \pi^2 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^2
- \frac{19}{18} \pi^2 \frac{\Nda}{\Noda} C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{16}{3} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A^2
- \frac{16}{3} \psi^\prime(\third) \Nf T_F C_F C_A \alpha
- \frac{16}{3} \psi^\prime(\third) \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{9}{2} \zeta_3 C_F C_A^2 \alpha^2
- \frac{9}{2} \zeta_3 C_F C_A^2 \alpha^3
- \frac{4}{3} \psi^\prime(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{4}{9} \zeta_3 \pi^2 C_F C_A^2 \alpha^2
+ \frac{19}{12} \psi^\prime(\third) \frac{\Nda}{\Noda} C_F C_A^2 \alpha^2
+ \frac{26}{3} \psi^\prime(\third) C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{32}{9} \pi^2 \Nf T_F C_F C_A \alpha
+ \frac{32}{9} \zeta_3 \pi^2 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^2
+ \frac{32}{9} \psi^\prime(\third) \zeta_3 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{43}{6} \psi^\prime(\third) \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^2
+ \frac{64}{3} \zeta_3 \pi^2 \frac{\Nda^2}{\Noda^2} C_F C_A^2
+ \frac{64}{3} \zeta_3 \pi^2 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{83}{16} C_F C_A^2 \alpha^2
+ \frac{93}{16} C_F C_A^2 \alpha^3
+ \frac{104}{27} \zeta_3 \pi^2 C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{172}{9} \psi^\prime(\third) \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
+ \frac{256}{27} \zeta_3 \pi^2 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{280}{9} \psi^\prime(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2 \alpha
+ \frac{775}{27} \pi^2 \frac{\Nda}{\Noda} C_F C_A^2 \alpha
+ \frac{1023}{2} C_F C_A^2
\right. \nonumber \\
&& \left. ~
+ \frac{1364}{3} \Nf \frac{\Noda}{[\Noda+2\Nda]} T_F C_F C_A \alpha
+ \frac{1571}{16} \frac{\Nda}{\Noda} C_F C_A^2 \alpha^2
+ \frac{18451}{24} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
- 591 \zeta_3 C_F C_A^2 \alpha
- 396 \zeta_3 C_F C_A^2
- 352 \zeta_3 \Nf \frac{\Noda}{[\Noda+2\Nda]} T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
- 69 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
- 62 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha^2
- 32 \psi^\prime(\third) \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2
\right. \nonumber \\
&& \left. ~
- 32 \psi^\prime(\third) \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha
- 24 \zeta_3 \Nf T_F C_F C_A \alpha^2
- 12 \psi^\prime(\third) \frac{\Nda}{\Noda} C_F C_A^2
\right. \nonumber \\
&& \left. ~
- \psi^\prime(\third) C_F C_A^2 \alpha^2
+ 8 \pi^2 \frac{\Nda}{\Noda} C_F C_A^2
+ 8 \psi^\prime(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2
\right. \nonumber \\
&& \left. ~
+ 18 \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^3
+ 31 \Nf T_F C_F C_A \alpha^2
+ 43 \psi^\prime(\third) \frac{\Nda^2}{\Noda^2} C_F C_A^2
\right. \nonumber \\
&& \left. ~
+ 43 \psi^\prime(\third) \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha
+ 48 \zeta_3 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha^2
+ 56 \zeta_3 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ 120 \zeta_3 \Nf T_F C_F C_A \alpha
+ 125 \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2 \alpha
+ 126 \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2
\right. \nonumber \\
&& \left. ~
+ 129 \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha^2
+ 234 \zeta_3 \frac{\Nda^2}{\Noda^2} C_F C_A^2 \alpha
+ 270 \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2
\right. \nonumber \\
&& \left. ~
+ 396 \zeta_3 \frac{\Noda}{[\Noda+2\Nda]} C_F C_A^2
+ 968 \zeta_3 \frac{\Noda}{[\Noda+2\Nda]} C_F C_A^2 \alpha
\right] a^3 ~+~ O(a^4) \nonumber \\
\Theta_{\alpha\,\MAGs}^{\MOMcs}(a,\alpha) &=&
\left[
\frac{32}{9} \pi^2 \Nf T_F C_F C_A
+ \frac{32}{9} \psi^\prime(\third) \zeta_3 \Nf T_F C_F C_A
+ \frac{44}{3} \psi^\prime(\third) C_F C_A^2
\right. \nonumber \\
&& \left. ~
- \frac{3751}{3} \frac{\Noda}{[\Noda+2\Nda]} C_F C_A^2
- \frac{704}{27} \zeta_3 \pi^2 \frac{\Nda}{\Noda} C_F C_A^2
- \frac{475}{3} \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
- \frac{473}{9} \psi^\prime(\third) \frac{\Nda}{\Noda} C_F C_A^2
- \frac{344}{27} \pi^2 \Nf \frac{\Nda}{\Noda} T_F C_F C_A
- \frac{341}{4} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{128}{9} \psi^\prime(\third) \zeta_3 \Nf \frac{\Nda}{\Noda} T_F C_F C_A
- \frac{88}{9} \pi^2 C_F C_A^2
- \frac{88}{9} \psi^\prime(\third) \zeta_3 C_F C_A^2
\right. \nonumber \\
&& \left. ~
- \frac{64}{27} \zeta_3 \pi^2 \Nf T_F C_F C_A
- \frac{16}{3} \psi^\prime(\third) \Nf T_F C_F C_A
+ \frac{172}{9} \psi^\prime(\third) \Nf \frac{\Nda}{\Noda} T_F C_F C_A
\right. \nonumber \\
&& \left. ~
+ \frac{176}{27} \zeta_3 \pi^2 C_F C_A^2
+ \frac{256}{27} \zeta_3 \pi^2 \Nf \frac{\Nda}{\Noda} T_F C_F C_A
+ \frac{341}{2} \frac{\Nda}{\Noda} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{352}{9} \psi^\prime(\third) \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2
+ \frac{759}{4} \frac{\Nda}{\Noda} C_F C_A^2
+ \frac{946}{27} \pi^2 \frac{\Nda}{\Noda} C_F C_A^2
\right. \nonumber \\
&& \left. ~
+ \frac{1364}{3} \Nf \frac{\Noda}{[\Noda+2\Nda]} T_F C_F C_A
+ \frac{5225}{12} C_F C_A^2
\right. \nonumber \\
&& \left. ~
- 352 \zeta_3 \Nf \frac{\Noda}{[\Noda+2\Nda]} T_F C_F C_A
- 330 \zeta_3 C_F C_A^2
- 154 \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2
\right. \nonumber \\
&& \left. ~
- 132 \zeta_3 \frac{\Nda}{\Noda} C_F C_A^2 \alpha
- 69 \Nf \frac{\Nda}{\Noda} T_F C_F C_A
- 62 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
- 24 \zeta_3 \Nf T_F C_F C_A \alpha
+ 31 \Nf T_F C_F C_A \alpha
+ 48 \zeta_3 \Nf \frac{\Nda}{\Noda} T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ 56 \zeta_3 \Nf \frac{\Nda}{\Noda} T_F C_F C_A
+ 66 \zeta_3 C_F C_A^2 \alpha
+ 120 \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
+ 968 \zeta_3 \frac{\Noda}{[\Noda+2\Nda]} C_F C_A^2
\right] a^3 ~+~ O(a^4) ~.
\end{eqnarray}

By contrast the parallel expressions in the Curci-Ferrari gauge are

\begin{eqnarray}
\Theta_{a\,\CFs}^{\MOMcs}(a,\alpha) &=&
\left[
31 \Nf T_F C_F C_A \alpha^2
+ 377 \zeta_3 C_F C_A^2 \alpha
- \frac{11557}{24} C_F C_A^2 \alpha
- \frac{64}{27} \zeta_3 \pi^2 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
- \frac{52}{9} \pi^2 C_F C_A^2 \alpha
- \frac{52}{9} \psi^\prime(\third) \zeta_3 C_F C_A^2 \alpha
- \frac{16}{3} \psi^\prime(\third) \Nf T_F C_F C_A \alpha
- \frac{9}{2} \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
- \frac{9}{2} \zeta_3 C_F C_A^2 \alpha^3
- \frac{4}{9} \zeta_3 \pi^2 C_F C_A^2 \alpha^2
+ \frac{2}{3} \pi^2 C_F C_A^2 \alpha^2
+ \frac{2}{3} \psi^\prime(\third) \zeta_3 C_F C_A^2 \alpha^2
\right. \nonumber \\
&& \left. ~
+ \frac{26}{3} \psi^\prime(\third) C_F C_A^2 \alpha
+ \frac{32}{9} \pi^2 \Nf T_F C_F C_A \alpha
+ \frac{32}{9} \psi^\prime(\third) \zeta_3 \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
+ \frac{83}{16} C_F C_A^2 \alpha^2
+ \frac{93}{16} C_F C_A^2 \alpha^3
+ \frac{104}{27} \zeta_3 \pi^2 C_F C_A^2 \alpha
+ \frac{889}{3} \Nf T_F C_F C_A \alpha
\right. \nonumber \\
&& \left. ~
- 232 \zeta_3 \Nf T_F C_F C_A \alpha
- 24 \zeta_3 \Nf T_F C_F C_A \alpha^2
- \psi^\prime(\third) C_F C_A^2 \alpha^2
\right] a^3 ~+~ O(a^4) \nonumber \\
\Theta_{\alpha\,\CFs}^{\MOMcs}(a,\alpha) &=&
\left[
31 \Nf T_F C_F C_A \alpha
+ 66 \zeta_3 C_F C_A^2 \alpha
+ 638 \zeta_3 C_F C_A^2
- \frac{9779}{12} C_F C_A^2
- \frac{341}{4} C_F C_A^2 \alpha
\right. \nonumber \\
&& \left. ~
- \frac{88}{9} \pi^2 C_F C_A^2
- \frac{88}{9} \psi^\prime(\third) \zeta_3 C_F C_A^2
- \frac{64}{27} \zeta_3 \pi^2 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
- \frac{16}{3} \psi^\prime(\third) \Nf T_F C_F C_A
+ \frac{32}{9} \pi^2 \Nf T_F C_F C_A
+ \frac{32}{9} \psi^\prime(\third) \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
+ \frac{44}{3} \psi^\prime(\third) C_F C_A^2
+ \frac{176}{27} \zeta_3 \pi^2 C_F C_A^2
+ \frac{889}{3} \Nf T_F C_F C_A
- 232 \zeta_3 \Nf T_F C_F C_A
\right. \nonumber \\
&& \left. ~
- 24 \zeta_3 \Nf T_F C_F C_A \alpha
\right] a^3 ~+~ O(a^4) ~.
\end{eqnarray}

Again when $\alpha$ $=$ $0$ for this and each of the other MOM schemes then
$\Theta_{a\,\CFs}^{\cal S}(a,\alpha)$ vanishes.

Another angle on Crewther's relation.

While the relation (<ref>) appears to hold in all the schemes and
gauges that we have considered so far it is instructive to return to Crewther's
original relation (<ref>) and consider gauges for which it is valid
especially in light of the earlier analysis. In other words to determine some
gauge $\bar{\alpha}$ for which

\begin{equation}
\Delta_{\mbox{\footnotesize{csb}}}(a,\bar{\alpha}) ~=~
\frac{\beta(a,\bar{\alpha})}{a} \hat{K}_a(a,\bar{\alpha})
\label{deltgauge}
\end{equation}

holds. This was the approach taken in [12] but here rather than assume the
gauge parameter is a pure constant we will incorporate loop corrections. The
motivation to do so is that various observations noted earlier had properties
requiring a fixed value of $\alpha$ $=$ $-$ $3$ at a particular loop order
which turned out to be inconsistent when higher order effects were included. An
approach in a similar vein to explore schemes where (<ref>) holds was
given recently in [60]. In particular the $V$ scheme of [13, 14] was
again studied since the coupling constant in that scheme is by construction
gauge parameter independent being based on the static quark potential. The four
loop renormalization group functions are known [61] and we note that the
coefficients of the $V$ scheme $\beta$-function are independent of the gauge
parameter. The first step to achieve the form (<ref>) is to recognize
that for a gauge dependent scheme we have a two dimensional space parameterized
by $a$ and $\bar{\alpha}$. We can attempt to find a gauge for which
(<ref>) holds by considering curves
$\bar{\alpha}$ $=$ $\bar{\alpha}(a)$ which are defined by some relation
$F(a,\bar{\alpha})$ $=$ $0$. In particular, we are interested in curves that
contain fixed points of the running coupling constant. A natural choice for
this is the set of curves satisfying

\begin{equation}
\gamma_\alpha(a,\bar{\alpha}) ~+~ S\gamma_a(a,\bar{\alpha}) ~=~ 0
\label{alcond}
\end{equation}

\begin{equation}
\gamma_a(a,\alpha) ~\equiv~ \frac{\beta(a,\alpha)}{a}
\end{equation}

since outside the Landau gauge our fixed points are defined by the requirement
that $\gamma_\alpha(a,\alpha)$ $=$ $0$ and $\gamma_a(a,\alpha)$ $=$ $0$. The
parameter $S$ will be tuned later to examine various scenarios. Under these
constraints (<ref>) becomes

\begin{equation}
\Delta_{\mbox{\footnotesize{csb}}}(a,\bar{\alpha}) ~=~
\left[ K_a(a,\bar{\alpha}) ~-~ SK_\alpha(a,\bar{\alpha}) \right]
\gamma_a(a,\bar{\alpha})
\end{equation}

meaning that

\begin{equation}
\hat{K}_a(a,\bar{\alpha}) ~=~ \left[ K_a(a,\bar{\alpha}) ~-~
SK_\alpha(a,\bar{\alpha}) \right]
\end{equation}

when comparing with (<ref>).

In keeping with our perturbative approach we will expand $\bar{\alpha}$ as a
function of $a$ via

\begin{equation}
\bar{\alpha}(a) ~=~ \sum_{n=0}^\infty \alpha^{(n)} a^n
\label{alexp}
\end{equation}

as well as taking a similar expansion for the two renormalization group

\begin{eqnarray}
\gamma_a(a,\alpha) &=& -~ \beta_0 a ~-~
\sum_{n=1}^\infty \left( \sum_{m=0}^{P_m} \beta_n^{(m)} \alpha^m \right)
a^{n+1} \nonumber \\
\gamma_\alpha(a,\alpha) &=& \sum_{n=1}^\infty \left( \sum_{m=0}^{Q_m}
\gamma_n^{(m)} \alpha^m \right) a^n
\end{eqnarray}

where $P_1$ $=$ $4$, $P_2$ $=$ $7$, $P_3$ $=$ $5$, $P_4$ $=$ $6$,
$Q_1$ $=$ $1$, $Q_2$ $=$ $4$, $Q_3$ $=$ $7$, $Q_4$ $=$ $5$ and $Q_5$ $=$ $6$
cover all the schemes considered here. The specific values of $P_m$ and $Q_m$
were deduced from the $\alpha$ dependence in the $\MOMg$ scheme with the four
and five loop ones corresponding to the $\mMOM$ scheme. With these two
expressions and (<ref>) solving (<ref>) produces equations that
determine $\alpha^{(m)}$. We find the formal solution is

\begin{eqnarray}
\alpha^{(0)}\gamma_1^{(1)} &=& S\beta_0-\gamma_1^{(0)} \nonumber \\
\alpha^{(1)}\gamma_1^{(1)} &=&
S \left[ \beta_1^{(4)}{\alpha^{(0)}}^4+\beta_1^{(3)}{\alpha^{(0)}}^3
+\beta_1^{(0)} \right] \nonumber\\
&&- \left[ \gamma_2^{(4)}{\alpha^{(0)}}^4+\gamma_2^{(3)}{\alpha^{(0)}}^3
\right] \nonumber \\
\alpha^{(2)}\gamma_1^{(1)} &=&
S \left[ \beta_2^{(7)}{\alpha^{(0)}}^7+\beta_2^{(6)}{\alpha^{(0)}}^6
\right. \nonumber \\
&& \left. ~~~
+\beta_2^{(1)}{\alpha^{(0)}}+\beta_2^{(0)} \right]
\nonumber \\
&& - \left[ \gamma_3^{(7)}{\alpha^{(0)}}^7+\gamma_3^{(6)}{\alpha^{(0)}}^6
\right. \nonumber \\
&& \left. ~~~
+\gamma_3^{(1)}{\alpha^{(0)}} + \gamma_3^{(0)} \right] \nonumber\\
&&+~ \alpha^{(1)} \left[ S \left[ 4\beta_1^{(4)}{\alpha^{(0)}}^3
\right] \right. \nonumber\\
&& \left. ~~~~~~~~~- \left[ 4\gamma_2^{(4)}{\alpha^{(0)}}^3
+\gamma_2^{(1)} \right] \right] \nonumber \\
\alpha^{(3)}\gamma_1^{(1)} &=&
\sum_{n=0}^{P_4} S\beta_3^{(n)}{\alpha^{(0)}}^n
- \sum_{n=0}^{Q_4} \gamma_4^{(n)}{\alpha^{(0)}}^n
+\alpha^{(1)}\sum_{n=1}^{P_3} nS\beta_2^{(n)}{\alpha^{(0)}}^{n-1}
\nonumber \\
&& -~ \sum_{n=0}^{Q_3} n\gamma_3^{(n)}{\alpha^{(0)}}^{n-1}
+ 2{\alpha^{(0)}}^{2} \left[ 2{\alpha^{(0)}}{\alpha^{(2)}}
+ 3{\alpha^{(1)}}^{2} \right]
\left[ S\beta_1^{(4)}-\gamma_2^{(4)} \right]
\nonumber \\
&& +~ 3{\alpha^{(0)}} \left[ {\alpha^{(0)}}{\alpha^{(2)}}+{\alpha^{(1)}}^2
\right]
\left[ S\beta_1^{(3)}-\gamma_2^{(3)} \right] \nonumber\\
&& +~ \left[ 2{\alpha^{(0)}}{\alpha^{(2)}}+{\alpha^{(1)}}^2 \right]
\left[ S\beta_1^{(2)}-\gamma_2^{(2)} \right]
+{\alpha^{(2)}} \left[ S\beta_1^{(1)}-\gamma_2^{(1)} \right]
\end{eqnarray}

for the first few orders. We recall that although our first interest at the
moment is on the Crewther relation by finding the expression for $\bar{\alpha}$
that solves (<ref>) the second component is to find the value of
$a_\infty$ that is the solution to
$\gamma_a(a_\infty,\bar{\alpha}(a_\infty))$ $=$ $0$. In this way we will
arrive at the fixed points of the running parameters of the theory. Taking this
approach avoids the Banks-Zaks fixed point and will return a value of
$\alpha$ $\approx$ $-$ $3$. This particular integer value was discussed
previously in [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In fact, varying the
defining equation

\begin{equation}
F(a,\alpha,S) ~\equiv~ \gamma_\alpha(a,\alpha) ~+~ S\gamma_a(a,\alpha)
\end{equation}

with respect to $S$ results in

\begin{equation}
\frac{dF(a,\alpha,S)}{dS} ~=~ \gamma_a(a,\alpha)
\end{equation}

which should evaluate to zero at a fixed point. So the positions of the fixed
points determined in this way should be independent of $S$ up to truncation. We
will produce numerical data as evidence of this. In particular choosing
$S$ $=$ $0$ results in a procedure analogous to the typical process for finding
fixed points where the curves are defined such that the gauge parameter is
stationary and the value of the coupling constant can then be chosen so that it
is also stationary.

Before considering the details for a specific scheme it is instructive to
examine the leading order equation for the various gauges. First while
$\beta_0$ will be the same for all gauges since

\begin{equation}
\beta_0 ~=~ -~ \frac{4}{3} \Nf T_F ~+~ \frac{11}{3}C_A
\end{equation}

$\gamma_1$ is different in each of the three gauges. In a linear covariant
gauge fixing we have

\begin{equation}
\gamma_1 ~=~ -~ \frac{1}{2} \alpha C_A ~+~ \frac{13}{6} C_A ~-~
\frac{4}{3} \Nf T_F
\end{equation}

which means we can write

\begin{equation}
\alpha^{(0)} ~=~ \frac{2}{C_A}
\left[ S \left[ \frac{4}{3} \Nf T_F ~-~ \frac{11}{3} C_A \right] ~-~
\frac{4}{3} \Nf T_F ~+~ \frac{13}{6} C_A \right] ~.
\end{equation}

As this would be singular in the abelian limit it is best to redefine the
parameter $S$ as $S$ $=$ $1$ $+$ $3\sigma C_A$ which gives

\begin{equation}
\alpha^{(0)} ~=~ -~ 3 ~+~ \left[ 8 \Nf T_F ~-~ 22 C_A \right] \sigma ~.
\end{equation}

So $S$ $=$ $1$ produces $\alpha^{(0)}$ $=$ $-$ $3$ corresponding to the
solution pointed out in [12]. The higher order corrections will be scheme
dependent. Repeating this leading order exercise produces

\begin{equation}
\alpha^{(0)}_{\CFs} ~=~ -~ 6 ~+~ \left[ 16 T_F N_f - 44 C_a \right] \sigma
\end{equation}

for the Curci-Ferrari gauge. To analyse the MAG the following modification has
to be made to the defining equation

\begin{equation}
F_{\MAGs}(a,\alpha,S) ~=~ \bar{\alpha} \gamma_\alpha(a,\alpha) ~+~
\bar{\alpha} S\gamma_a(a,\alpha)
\end{equation}

because the leading order term in $\gamma_\alpha^{\MAGs}(a,\alpha)$ is
proportional to $\alpha^{-1}$. So the equations for $\alpha^{(i)}$ then need to
be modified in this case. The resulting leading order equation produces two
solutions. For $S$ $=$ $1$ the numerical values are
$\alpha^{(0)}_{\MAGs}$ $=$ $-$ $0.430953$ and $-$ $5.569047$. The first of
these solutions corresponds to that termed as Banks-Zaks one found in
[42]. For each of the three gauges the $S$ $=$ $1$ solutions are
independent of $\Nf$ unlike the $S$ $=$ $0$ case.

This leading order analysis is scheme independent but this ceases to be the
case when corrections are included. For instance, it was shown in [12]
that for the linear covariant gauge the form of Crewther's original relation,
given in (<ref>) but in the $\mMOM$ scheme, was satisfied at next order
for particular values of $\alpha$ which were $0$, $-$ $1$ and $-$ $3$. For
$\alpha$ $=$ $0$ this followed from the $\alpha$ dependence in
$\gamma_\alpha(a,\alpha)$ whereas $\alpha$ $=$ $-$ $1$ emerged as a special
case since $K_\alpha \gamma_\alpha(a,\alpha)$ was proportional to $(\alpha+1)$
at second order. The situation where $\alpha$ $=$ $-$ $3$ is that value that
solves $\gamma_a(a,\alpha)$ $=$ $-$ $\gamma_\alpha(a,\alpha)$ at leading order
which is suggestive of the $\alpha$ $\approx$ $-$ $3$ infrared stable fixed
point discussed earlier. It is worth examining whether this point of view has
any credibility at higher order. Therefore we calculated higher order terms in
the expansion in (<ref>) for a variety of schemes and gauges. The full
set of expressions for $\alpha^{(i)}$ are provided in the associated data file
but we record the $SU(3)$ values for the first few terms are

\begin{eqnarray}